Entangling power, cartan decomposition, and braiding operators

The main part of this dissertation starts with a generalization of the entangling power, which quantifies the ability of unitary operators to generate entangled states by their action on the computational basis. The entangling power has been defined for an evenly split bipartite system, whose states live in a Hilbert space of the form [special characters omitted] where [special characters omitted]. I generalize this so that we can consider an odd number of qubits. Entangling power is defined in terms of the linear entropy, a linearization of the von Neumann entropy, whose polynomial form allows one to derive simpler expressions for functions of the entropy. The linear entropy measure is lifted from the state space to the operator space to measure the entanglement of operators. In particular, we focus on the three qubit case, where [special characters omitted] = 2 and [special characters omitted] = 4, as a step to understanding entanglement in many qubit systems. This is the content of Chapter 3. The goal of Chapter 4 is to explore the structure of entanglement producing operators in SU(2n), with the focus on SU(8). I first review the magic matrix formalism for two-qubit operators, a well known matrix that captures the entangling capability of operators in SU(4), using only 3 of the 15 parameters of the group. I then use a recursive definition of the Cartan decomposition of SU(2n) to construct a magic matrix for operators in SU(8). Because we view the entangling operation as between the first qubit and the other two as a single subsystem, the entanglement is invariant under the action of SU(2) ⊗ SU(4), but SU(8)/SU(2) ⊗ SU(4) is not a symmetric space, and we must use the decomposition in two stages to capture enough entangling operators to account for all of SU(8). The number of parameters appearing in the entangling power is then reduced from 63 to 8, keeping track of 7 entangling operators, one of which does not commute with the others. The next chapter then presents calculations of the entangling powers of operators that generate GHZ states from the computational basis, many of which are solutions to the Yang-Baxter equation and have strong connections to topological quantum computing. We find that the Kauffman-Lomonaco two-qubit gate that produces the Bell states from the computational basis has a maximal entangling power, while the GHZ gate on three qubits does not, adding another piece to the puzzle of entanglement.

ENTANGLING POWER, CARTAN DECOMPOSITION, AND BRAIDING OPERATORS by Aaron David Ballard A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2010 Copyright c Aaron David Ballard 2010 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of has been approved by the following supervisory committee members: , Chair Date Approved , Member Date Approved , Member Date Approved , Member Date Approved , Member Date Approved and by , Chair of the Department of and by Charles A. Wight, Dean of The Graduate School. ABSTRACT The main part of this dissertation starts with a generalization of the entangling power, which quantifies the ability of unitary operators to generate entangled states by their action on the computational basis. The entangling power has been defined for an evenly split bipartite system, whose states live in a Hilbert space of the form H1 ⊗ H2 where dimH1 = dimH2. I generalize this so that we can consider an odd number of qubits. Entangling power is defined in terms of the linear entropy, a linearization of the von Neumann entropy, whose polynomial form allows one to derive simpler expressions for functions of the entropy. The linear entropy measure is lifted from the state space to the operator space to measure the entanglement of operators. In particular, we focus on the three qubit case, where dimH1 = 2 and dimH2 = 4, as a step to understanding entanglement in many qubit systems. This is the content of Chapter 3. The goal of Chapter 4 is to explore the structure of entanglement producing oper-ators in SU(2n), with the focus on SU(8). I first review the magic matrix formalism for two-qubit operators, a well known matrix that captures the entangling capability of operators in SU(4), using only 3 of the 15 parameters of the group. I then use a recursive definition of the Cartan decomposition of SU(2n) to construct a magic matrix for operators in SU(8). Because we view the entangling operation as between the first qubit and the other two as a single subsystem, the entanglement is invariant under the action of SU(2) ⊗ SU(4), but SU(8)/SU(2) ⊗ SU(4) is not a symmetric space, and we must use the decomposition in two stages to capture enough entangling operators to account for all of SU(8). The number of parameters appearing in the entangling power is then reduced from 63 to 8, keeping track of 7 entangling opera-tors, one of which does not commute with the others. The next chapter then presents calculations of the entangling powers of operators that generate GHZ states from the computational basis, many of which are solutions to the Yang-Baxter equation and have strong connections to topological quantum computing. We find that the Kauffman-Lomonaco two-qubit gate that produces the Bell states from the compu-tational basis has a maximal entangling power, while the GHZ gate on three qubits does not, adding another piece to the puzzle of entanglement. iv Dedicated to my grandmother Annette, my aunt Victoria, my father David, and my mother Susan. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 INTRODUCTION AND SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Structure of this dissertation . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Chapters 3, 4, and 5: Cartan decomposition and entangling power of braiding quantum gates . . . . . . . . . . . . . . 5 2 BACKGROUND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 EPR and Bell's inequality . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Density operator, reduced density operator, and partial trace . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Entropy and entanglement . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Noiseless and noisy channel coding theorems . . . . . . . . . . 20 2.3.2 Shannon's and Schumacher's noiseless channel coding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Linear entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 ENTANGLING POWER, TWO- AND THREE-QUBIT GATES. . 38 3.1 Entanglement and linear entropy . . . . . . . . . . . . . . . . . . . . 39 3.2 Entangling power for two-qubit gates . . . . . . . . . . . . . . . . . . 44 3.3 Entangling power for three-qubit gates . . . . . . . . . . . . . . . . . 46 3.3.1 The swap operator . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 CARTAN DECOMPOSITION AND THE MAGIC MATRIX. . . . . 60 4.1 Magic matrix and Cartan decomposition of SU(4) . . . . . . . . . . . 61 4.2 Entangling power of two-qubit braiding gates . . . . . . . . . . . . . 64 4.2.1 General entangling power formula . . . . . . . . . . . . . . . . 64 4.3 Cartan decomposition for three qubits . . . . . . . . . . . . . . . . . 65 4.3.1 Direct Cartan decomposition of SU(8) . . . . . . . . . . . . . 65 4.3.2 Further decomposition of sul . . . . . . . . . . . . . . . . . . . 67 4.4 Three-qubit entangling power . . . . . . . . . . . . . . . . . . . . . . 70 4.4.1 Three-qubit partial magic matrix Z . . . . . . . . . . . . . . . 71 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5 BRAIDING GATES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1 Braiding quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.1 Extraspecial 2-groups . . . . . . . . . . . . . . . . . . . . . . . 78 5.1.2 Almost complex structures and representations of Em . . . . . . . . . . . . . . . . . . . . . . . 80 5.1.3 GHZ states and generalized Bellmatrices . . . . . . . . . . . . 82 5.2 Entangling power of braiding operators . . . . . . . . . . . . . . . . . 89 5.2.1 Braiding operator, six-vertex . . . . . . . . . . . . . . . . . . . 90 5.2.2 More braiding operators . . . . . . . . . . . . . . . . . . . . . 91 5.3 Entangling power of GHZ generators . . . . . . . . . . . . . . . . . . 93 6 CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 vii ACKNOWLEDGMENTS Large portions of Chapters 3, 4, and 5 have appeared in the following paper: "Cartan Decomposition and Entangling Power of Braiding Quantum Gates", A. D. Ballard and Yong-Shi Wu; Kazem Mahdavi editor, Proceedings of the Conference on Representation Theory, Quantum Field Theory, Cate-gory Theory, Mathematical Physics, and Quantum Information Theory; University of Texas at Tyler, Oct. 2009. Dark Night of the Soul I am reaching the end of a path that I would not normally have chosen, but that instead I was led to by my conscience and circumstance. I state for the record that aside from pure research and interesting puzzles, I have never enjoyed academic life since my first day of kindergarten. I say this because where my family members tend to believe I have enthusiasm for academics, is merely a sense of duty. The gullies of displeasure started forming when I was forced to leave the care and protection of my parents for the company of teachers and other students. Then the death cloud and its torrents descended as I was first assaulted with homework. This eventually developed into a constant feeling of anxiety over unfinished assignments and studying. And now as the story ends, I hear a soundtrack full of deep trombones, emphasizing the gravitas one experiences in his profoundly normal experience. As I emerge from this dark night, finishing the dissertation seems at once both irrelevant and also the crucial exercise of earning a doctorate. I think it must be that the molding of ideas into a coherent contribution to the scientific community is the essential exercise that earns one a Ph.D. Forming even unoriginal ideas into something readable requires something slightly supernatural, or at least inhuman. At its core, the exercise requires one to both fully comprehend his subject and also to simulate in his mind a reasonably uninformed reader. The former is the natural course in inquiry and research, and the latter is a natural function of the brain, though many have difficult issues with it. But one must also have faith that he is saying something worthwhile and has earned the right to stand exposed to the scrutiny of the world, faith built on vast quantities of original sin. One might think, after having thoroughly blackened ones soul, that the act of committing the evidence to paper would be a small step. But in fact, maintaining that faith, and the motivation to apply long-term consistent effort to polish something forged in the chaos of one's subconscious into a coherent illuminated idea, effort composed of inscrutably many vague and easily neglected activities, are the hardest parts. I want to acknowledge the invaluable assistance from my advisor, Yong-Shi Wu, who first suggested to me that I calculate the entangling power of the braiding op-erators. The role and lessons of the doctoral advisor are indelibly imprinted on the psyche and future work of the doctoral student. Yong-Shi Wu has been exceedingly patient and helpful in guiding my progress toward the Ph.D. His strategy has been to allow ample freedom in my choice of problems and the pace of solving them, offering crucial guidance when necessary, so that I could develop my own tools and style of conducting research. ix I also received essential guidance and training from Mikhail Raikh. Working with him was a close collaboration, my first exposure to conducting advanced research, and resulted in my first published paper. His intense focus and clear conceptual understanding in problems of condensed matter physics reaches the sublime. I also thank Spencer Stirling, a post-doc in our group, for his generosity. Many in the group have benefitted from his guidance in studying category theory. I have spent many entertaining hours with Mark Limes, as well as many hours working. He has been my partner-in-crime and partner-in-music since I started grad school. He says he did it for the lulz, but I was the one who was doing it for the lulz. Finally, my family. Growing up, I received nourishment from grandparents on both sides of the family. My personal activities and well being have been made possible through the undying support of my grandmother Annette and my aunt Victoria. I have also received invaluable quality-time and hamburgers from my father, and inherited invaluable attitude from my mother. (I have never formally retracted my assertion that the letter ‘h' is not and should not be included in the alphabet, a natural consequence of discussing school with a five-year-old eating homemade Italian food, or even hamburgers.) x CHAPTER 1 INTRODUCTION AND SUMMARY 1.1 Structure of this dissertation This dissertation is divided into six chapters. The second chapter provides background and context for the main content of the dissertation. It starts by reviewing some basic tools and ideas, and draws a connection between entanglement, information content, and entropy. We say the entropy S(ρ) is the minimum physical resources necessary to store the output from an information source, which is the reduced density matrix of an entangled state. As entropy increases, we require more resources to store the output information. Also as the entropy increases, the more mixed is the reduced density matrix and thus the more nonlocal, entangled, is the composite state. The less we know about the state, then the more valuable is the information it holds. So, here is a connection between entanglement, information content, and entropy. Chapters 3, 4, and 5 form the main body of the dissertation, and much of the material will appear in two papers I coauthored with Yong-Shi Wu, in preparation for publication, and one that is already cited. Chapter 3 is devoted to a construction of an entangling power measure for operators, a concept lifted from and closely related to 2 state entanglement. A purely quantum phenomenon itself, entanglement is integral to a number of quantum operations, such as teleportation. In the context of quantum computing, it is also widely believed to be an integral resource for solving certain classes of otherwise insoluble problems. However, we lack a precise description of its nature and role in computation. Entangling power is another tool to help us along the way, and perhaps an important part of a satisfying understanding of entanglement. The entangling power quantifies the ability of unitary operators to generate en-tangled states by their action on the computational basis. It is defined in terms of the linear entropy, a linearization of the von Neumann entropy, whose polynomial form allows one to derive simpler expressions for functions of the entropy. The linear entropy measure is first lifted from the state space to the operator space to measure the entanglement of operators. Operators acting on a Hilbert space H1 ⊗ H2 live in a Hilbert space of their own, isomorphic to (H1 ⊗ H2)⊗2. Thus, we use an iso-morphic map to this state space, then calculate the entanglement of operators using the linear entropy. The linear entropy is then used to define the entangling power. Originally, the final expression of the entangling power was defined for an evenly split bipartite system, whose states live in a Hilbert space of the form H1 ⊗ H2 where dimH1 = dimH2. I generalize this so that we can consider an odd number of qubits. In particular, we focus on the three qubit case, where dimH1 = 2 and dimH2 = 4, as a step to understanding entanglement in many qubit systems. Chapter 4 expands the power of the entangling power. I explore the structure of entanglement producing operators in SU(2n), with the focus on SU(8). I first review 3 the magic matrix formalism for two-qubit operators. This is a well known matrix that captures the entangling capability of operators in SU(4), reducing the number of parameters from 15 to 3. It does this using the Cartan decomposition, which ex-ploits the structure of the symmetric space SU(4)/SU(2)⊗SU(2), and the fact that entanglement is invariant under local operations. I then use a recursive definition of the Cartan decomposition of SU(2n) to construct a magic matrix for operators in SU(8). Because we view the entangling operation as between the first qubit and the other two as a single subsystem, the entanglement is invariant under the action of SU(2)⊗SU(4), but SU(8)/SU(2)⊗SU(4) is not a symmetric space, and we must use the decomposition in two stages to capture enough entangling operators to account for all of SU(8). The number of parameters appearing in the entangling power is then reduced from 63 to 8, keeping track of 7 entangling operators, one of which does not commute with the others. Then the next chapter has calculations of the entangling powers of operators that generate GHZ states from the computational basis, many of which are solutions to the Yang-Baxter equation and have strong connections to topo-logical quantum computing. We find that the Kauffman-Lomonaco two-qubit gate that produces the Bell states from the computational basis has a maximal entangling power, while the GHZ gate does not, adding another piece to the entanglement puzzle. In Chapter 5, I discuss our primary application of the tools we develop, generators of Bell and GHZ states. I first show how the Bell and GHZ generators fit into the broader theory for braid groups and braid group representations. Then I use the tools developed in Chapters 3 and 4 to quantify the entangling powers of these generators. 4 In the last chapter, I sketch possible applications of entangling power, which can measure the entangling power of bipartite systems split into uneven dimensions, and also the possibility of using the Cartan decomposition technique to explore the opera-tor spaces of higher dimensions. One example is multiqubit teleportation, where two parties share a multiqubit state. The capacity of this channel depends on the amount of entanglement that can be distilled between the two parties, effectively, the number of shared Bell states they can obtain by local operations and classical communication (abbreviated LOCC). Because the entangling power can measure entanglement across uneven splits, it could be a useful indicator of the possible success of teleportation using such a state. The entangling power constructed for a system of a larger number of qubits, could tell us how many qubits-that is, how many Bell states are shared across a bipartite split between parties-a given operator could provide to us for the purpose of performing some operation or transfer of information. More interesting, however, would be to study how entangling operators transform along with the states they generate. Because it is a linearized version of the von Neumann entropy, calcu-lations are easier, so we can more easily lift the idea of entanglement of states to the operator level, to study the operators that generate entanglement. What follows is a more extensive introduction to the main body of the dissertation, Chapters 3, 4, and 5. 5 1.2 Chapters 3, 4, and 5: Cartan decomposition and entangling power of braiding quantum gates Entanglement is a purely quantum and nonlocal correlation, known to be an im-portant resource for quantum information processing and quantum computing. It is crucial for quantum algorithms, such as teleportation and quantum key distribution, and has been shown to be an important resource for solving problems that are ex-ponentially difficult. So, one of the central problems in quantum information is to improve our understanding of the nature of this resource, including how to produce and quantify it. In this dissertation I report in greater detail than previously recent progress in quantifying the entangling power of two-qubit and three-qubit braiding quantum gates that have been proposed recently. A recent approach to understanding quantum entanglement is motivated by an analogy [1] between entangled quantum states and topological entanglement known in knots. In ref. [15], Kauffman and Lomonaco introduced two-qubit braiding quantum gates, which carry out braiding operations of qubits, in a way similar to those in the theory of knots and links. They have shown that the two-qubit braiding gates are universal, when used together with one-qubit gates. In principle, quantum circuits consisting of only braiding gates may be used to implement the so-called topological quantum computation [11, 10, 22], a new approach to implementing fault-tolerant quantum computation. Another advantage of the two-qubit braiding gate proposed by Kauffman and Lomonaco is that it produces the well-known (maximally entangled) 6 Bell states when acting upon computational basis states, which are nonentangled. More recently, a three-qubit braiding quantum gate has been proposed in refs. [29, 21, 2]. It has been shown that this gate, when acting upon the computational basis, produces the famous (entangled three-qubit) GHZ states. Therefore, in consistency with the analogy with topological entanglement in knots and links, braiding gates indeed produce quantum entanglement among the qubits in a quantum circuit. Conceptually, entangling power quantifies the ability of an operator, or gate, to produce entanglement between qubits in product states, such as the computational basis states. Thus, it can answer questions about the efficiency of gates in producing entangled states from product states. But there are different types of entangled states, in the sense that local operations cannot transform them into each other. In the three-qubit case, there are the well known GHZ states and some types of Werner states. So, while we imagine that entanglement is a measurable property of states, and the entangling power is able to distinguish gates that produce these states because the states happen to have different quantities of entanglement, nevertheless, a more precise understanding of this situation is desirable. So, it is also worth strictly saying that entangling power quantifies the disorder produced in a local subsystem, or the loss of local information to a nonlocal state. To quantify entangling power and operator entanglement we employ some of the tools for quantifying state entanglement, using an isomorphic map from the operator space to the state space. Along this line of thought, several authors have developed formal methods to quantify the entangling power of operators acting on two qubits, 7 which form the Lie group SU(4)[25, 20, 13, 16, 4, 18, 27]. In particular, several have shown that one can quantify the entangling power of any SU(4) operator by considering only the magic matrix [18, 27, 16], which contains only 3 parameters rather than the usual 15 for a general SU(4) operator. Physically this is easy to understand, since the entangling power for genuine two-qubit entanglement is known to be unchanged under any one-qubit operation of either qubit, which form an SU(2)⊗ SU(2) subgroup of SU(4). Mathematically, an arbitrary SU(4) operator can be represented by a representative magic matrix through a Cartan decomposition of the symmetric space SU(4)/SU(2) ⊗ SU(2) [12], if we do not care about local one-qubit operations. In the course of studying the entangling power of three-qubit braiding gates, it was natural for us to generalize the above magic-matrix technique to three-qubit operators, which form the Lie group SU(8). The entangling power we want to compute is that for a bipartite three-qubit system. (Right now entangling power is defined only for a bipartite system; see, e.g., refs. [26, 25].) The three-qubit braiding gates proposed in [29] can actually be understood as the braiding between the first qubit and the other two as a whole [21]. The entangling power of such three-qubit braiding gates is invariant under the local operations of SU(2) ⊗ SU(4). But SU(8)/SU(2) ⊗ SU(4) is not a symmetric space as in the two-qubit case. Therefore, we adopt a recursive Cartan decomposition technique [17] to define the three-qubit magic matrix, and construct it for three-qubit braiding gates that we are interested in. In this way, we have been able to compute the (bipartite) entangling power of three-qubit gates, 8 especially those of braiding gates. Since our work is the first step going beyond two-qubit entangling power, we expect that our method might be useful in further study of multiqubit entanglement, such as entanglement combing[23], as well as that of the discord and entanglement for mixed states [5, 6]. In addition to identifying entanglement producing operators, the entangling power could also be useful as a criterion for protocols such as multipartite teleportation, especially in light of entanglement combing[23]. Entanglement combing allows one to express a multipartite entangled state, without changing the entanglement with a preferred qubit, in a combed form in which all parties are bipartite entangled with the preferred qubit. Since combing does not change the entanglement across the bipartite split, the entangling power can be measured across such a split in an uncombed state to determine suitability for protocols such as teleportation. Some have argued [5] that nonclassical correlation, quantified by discord, rather than entanglement, is the important resource in mixed state computation. Though not as powerful as pure-state quantum computers, these systems may lead to a viable quantum information processor [7] 1 or a quantum computer sooner than we realize a pure-state quantum computer, and yet achieve an exponential improvement over known classical algorithms for some computational problems. However, for pure state systems, discord reduces to a measure of entanglement, so the best known algorithms appear to rely on entanglement. In any case, the true nature of the new resource is not well understood, and a scheme for quantifying entangling power and ordering 1These processors fail to meet one or more of DiVincenzo's five criteria for a quantum computer. 9 the operators that produce entanglement would be useful in studying the flow of information in computation. CHAPTER 2 BACKGROUND This section is a survey of background material that is directly relevant to the entangling power. The bulk of the material is from the book by Nielsen and Chuang [19], but some also comes from other reviews and papers found in the literature, including [25]. 2.1 EPR and Bell's inequality Everything in this paper is an effort to understand entanglement. So, we first look at the origins of the idea. One of the earliest and most prominent attempts to refute quantum mechanics is the EPR thought experiment, named after the authors of the paper in which it appeared, Einstein, Podolsky, and Rosen. The conclusion is that quantum theory is incomplete because it does not include a physical mechanism to explain a predictable observation, namely the result of measuring the second qubit of the spin singlet Bell state, (|01 − |10 )/√2, after the first has been measured. Thirty years after the EPR paper, an experiment designed around Bell's inequality invalidated those authors' point of view. The reasoning is as follows. Using common sense notions of how the world works, 11 we derive Bell's inequality. After the common sense analysis, we perform a quan-tum mechanical analysis, which is shown to be inconsistent with the common sense analysis. Finally, experiments verify one or the other. Now, the common sense thought experiment. One party, Charlie, prepares two particles by any repeatable means. He then sends one each to Alice and Bob. Once Alice receives her particle, she performs one of two measurements of different physical properties, which we call PQ and PR. Her choice of which property to measure is random. For simplicity, we suppose that the outcomes can have one of two values, +1 or -1, and we label the outcomes Q and R, with the obvious correspondence to the physical properties being measured. Bob is in a similar situation, randomly choosing to measure one of two physical properties, PS or PT , with the result being +1 or -1. Alice and Bob time their measurements so that they are causally disconnected. Now we reason algebraically to derive Bell's inequality. We start with QS + RS + RT − QT = (Q + R)S + (R − Q)T. Because R,Q = ±1, either Q + R = 0 or R − Q = 0, and the above expression is equal to ±2. Now suppose that before measurements are performed, the state of the system is Q = q, R = r, S = s, and T = t, with probability p(q, r, s, t) that depends on the preparation and experimental noise. Letting E(·) denote the mean value of a 12 quantity, we have E(QS + RS + RT − QT) = qrst p(q, r, s, t)(qs + rs + rt − qt) ≤ qrst p(q, r, s, t) × 2 = 2. Also, E(QS + RS + RT − QT) = qrst p(q, r, s, t)qs + qrst p(q, r, s, t)rs + qrst p(q, r, s, t)rt − qrst p(q, r, s, t)qt = E(QS) + E(RS) + E(RT) − E(QT). Comparing these two expressions, we obtain the Bell inequality, E(QS) + E(RS) + E(RT) − E(QT) ≤ 2. By repeating the experiment enough times, Alice and Bob can estimate to within arbitrary accuracy all the quantities on the left hand side of the Bell inequality, thus verifying whether it is obeyed in real experiments. Now we imagine a similar experiment in the quantum mechanical regime. Charlie prepares a quantum system of two qubits in the spin singlet Bell state, and again passes the first qubit to Alice and the second to Bob. They perform measurements of the following observables: Q = Z1 S = −Z2 − X2 √2 R = X1 T = Z2 − X2 √2 , 13 where X and Z are Pauli matrices. Calculating the average values of the observables in the Bell inequality and writing in the quantum mechanical notation, we get QS = 1 √2 ; RS = 1 √2 ; RT = 1 √2 ; QT = − 1 √2 . So, QS + RS + RT − QT = 2√2. And this clearly contradicts what we previously found. Fortunately, experiments have overwhelmingly confirmed which analysis is correct. Bell's inequality is not obeyed. Our common sense reasoning is wrong. The assumptions that went into the derivation of Bell's inequality have been widely debated, and it is generally accepted that there are two assumptions that could be wrong. • The assumption that the physical properties, PQ, PR, PS, PT have definite values Q, R, S, T that exist independently of observation. This is known as realism. • The assumption that Alice's measurement does not affect Bob's. This is known as locality. Together, these assumptions are known as local realism. No one can say which one is or if both are violated, but clearly there is something peculiar about entangled states such as the spin singlet that violates Bell's inequality. And this peculiarity opens up the potential for phenomena like teleportation and superdense coding that seem impossible in the classical world, begging the question, "What is the nature and potential of this new resource?" 14 2.2 Density operator, reduced density operator, and partial trace Most attempts at quantifying this resource will revolve around information func-tions, and central to these is entropy. In classical information theory, we often use Shannon entropy, and its analogue in the quantum view of the world is the von Neumann entropy. In this paper, the main tool, the entangling power, is defined in terms of a sort of linearized version of the von Neumann entropy (more will be said about this later). These quantum information functions are expressed in terms of the quantum states that potentially contain the information we care about, and the states themselves are expressed in the density operator formalism. I tend to assume that the reader is reasonably familiar with the density operator formalism, but I will review some main points and material on reduced density operators. The reduced density operator is practically necessary for describing subsystems of composite systems. This is especially true when the composite system is entangled, so that the subsystem is incomplete, and therefore mixed. Indeed, mixed states cannot be expressed as state vectors. To start, let us define the density operator associated to some ensemble of states as a positive operator ρ that has trace equal to one. The positivity and trace conditions are both due to the coefficients (or matrix entries) of the operator being probabilities. Then we can state the postulates of quantum mechanics in the density operator picture. 15 • Postulate 1 : The state space of an isolated physical system is a Hilbert space, a complex vector space with inner product. The system is completely described by the density operator, positive operator ρ with trace one, which acts on the state space of the system. For a quantum system in state ρi = j pij ψij ψij with probability pi, the density operator is a mixture of the states ρi, given by i piρi. • Postulate 2 : The evolution of a closed quantum system is given by a unitary transformation, a unitary operator U that depends on the initial and final times, ρ = UρU†. • Postulate 3 : Quantum measurements are described by a collectionMm of mea-surement operators that act on the state space of the system being measured and satisfy the completeness relation. The index m refers the possible measure-ment outcome. If the state space is ρ immediately before a measurement, then the probability that the outcome is m is given by p(m) = tr(M†m Mmρ), and the resulting state is MmρM†m tr(M†m Mmρ) . • Postulate 4 : The state space of a composite physical system is the tensor product of the spaces of the component physical systems. If the component 16 systems are labelled 1 through n, and system i is prepared in state ρi, then the composite state is ρ1 ⊗ ρ2 ⊗· · ·⊗ρn. Finally, there are at least two more points worth mentioning. The first is a criterion to determine whether a state is mixed or pure, tr(ρ2) ≤ 1, with equality if and only if ρ is a pure state. This allows us one interpretation of the linear entropy, S = 1−tr(ρ21 ), as the "impurity" of a component state of a composite system. Second, density matrices do not uniquely represent an ensemble of quantum states. In general, the ensembles {pi, |ψi } and {qi, |φj } generate the same density matrix if and only if ψi = j uij φj , where ψi = √pi |ψi , φj = √qi |φi , uij is a complex valued unitary matrix, and we "pad" the smaller set of vectors with "0" vectors so that the two sets have the same number of elements. Now suppose we have a composite state of systems A and B, described by the density operator ρAB. The reduced density operator provides an indispensable tool for a description of the subsystems. For subsystem A, the reduced density operator is defined as ρA ≡ trB(ρAB), where trB is the partial trace over system B. Like the trace, the partial trace is a map of operators, and it is defined in terms of the trace: trB a1 a2 ⊗ b1 b2 ≡ a1 a2 tr( b1 b2 ) 17 where |ai and |bi are any two vectors in their respective state spaces. In addition, the partial trace is required to be linear in its input. We identify the reduced density operator for system A with the state of system A. The physical justification for this is that the reduced density operator provides the correct measurement statistics for system A. In fact, the partial trace is the unique operation that gives the correct description of the observables of the subsystems. We show this now. Suppose M is an observable of system A, and we can take measurements of it. That is, M is a Hermitian operator on the state space of system A, and has a spectral decomposition, M = m mPm, where Pm is the projector onto the eigenspace of M with eigenvalue m. ˜M is the corresponding observable for the same measurement performed on the composite system AB. We want to show that ˜M = M ⊗ 11B. So, suppose that system AB is prepared in the state |m |ψ , where |m is the m eigenvalue eigenstate of M, and |ψ is any state of B. Then the measurement must give m with probability one. Thus for Pm, the projector onto the m eigenspace of M, the corresponding projector for ˜M is Pm ⊗11B. Thus we have ˜M = m mPm ⊗11B = M ⊗ 11B. Next, we show that the partial trace gives the correct measurement statistics for subsystems. Suppose we perform a measurement on A described by observable M. Then, physical consistency demands that any state ρA that is associated to A must have the same measurement outcomes as those computed from ρAB. This is expressed 18 by tr(MρA) = tr(M˜ρAB) = tr((M ⊗ 11B)ρAB). (2.1) This equation is satisfied if we do indeed choose ρA ≡ trB(ρAB). But in fact, the partial trace is the unique function that satisfies this equation. To show this, let f(·) be any map of density operators on AB to density operators on A, obeying the above relation, tr(Mf(ρAB)) = tr((M ⊗ 11B)ρAB), for all observables M. Let Mi be an orthonormal basis of operators for the space of Hermitian operators with Hilbert- Schmidt inner product, (X, Y ) ≡ tr(X†Y ). Expanding f(ρAB) in this basis gives f(ρAB) = i Mitr(Mif(ρAB)) = i Mitr((Mi ⊗ 11B)ρAB). Since the expansion uniquely determines f(ρAB), and both f(ρAB) and ρA satisfy Eq. (2.1), the partial trace is the unique function that satisfies Eq. (2.1). Finally, we close with two simple examples to illustrate a property of reduced density operators of entangled states. First, suppose the system is in a product state ρAB = ρ⊗σ. Then ρA = ρtr(σ) = ρ and similarly, ρB = σ. Now, suppose the system is in a Bell state (|00 + |11 )/√2. The composite density operator is ρ = |00 √+2|11 00 + 11 √2 = 00 00 + 11 00 + 00 11 + 11 11 √2 . 19 Tracing out the second qubit, we get ρA = 0 0 0 0 + 1 0 0 1 + 0 1 1 0 + 1 1 1 1 2 = I 2 . This is a completely mixed state, resembling a classical ensemble. We can check this with tr((I/2)2) = 1/2 < 1. The lesson is that we can have a pure composite state, such as this Bell state, that we know exactly, but that gives us no knowledge of either subsystem. In maximally entangled states such as this, the subsystems by themselves contain no accessible information. In general, we cannot have maximal knowledge of subsystems of entangled states. They are intrinsically nonlocal. In this paper, we construct a tool to identify operators that create such nonlocal information. Also, it is not a coincidence that the Bell states are mentioned and used in examples often. Further investigation is needed on the subject (possibilities are mentioned toward the end of this paper in conjunction with entanglement combing), but it might be that Bell states are the basic units of entanglement. Then a measure like the one we will construct, the entangling power perhaps constructed for a system of a larger number of qubits, could tell us how many qubits-that is, how many Bell states are shared across a bipartite split between qubits-a given preparation operator could provide to us for the purpose of performing some operation or transfer of information. 20 2.3 Entropy and entanglement 2.3.1 Noiseless and noisy channel coding theorems The entangling power is defined in terms of the linear entropy. So, this section will provide some background that will illuminate the origins of entropy in information theory and answer the question, "Why do we use entropy to quantify entanglement and entangling power?" Entropy is a tool that helps satisfy one of the fundamental goals of information theory, in particular, the goal to quantify the resources necessary to perform elementary processes. This is closely related to other fundamental goals, such as to identify elementary classes of resources and elementary classes of dynamical processes. Classical and quantum information theory share similar goals, but of course, quantum information theory is broader and more general in scope, containing classical information theory as a special case. The two fundamental results of classical information theory are Shannon's noise-less channel coding theorem and Shannon's noisy channel coding theorem. The chan-nel refers to a process that transmits or stores information, and the process can have error characteristics, or noise. The first quantifies how many bits are required to store information being emitted by a source of information. For now, we can think of a clas-sical information source as emitting units of information, such as letters, according to a source dependent probability distribution, and each emission from the source is in-dependent from the others. With respect to the goals mentioned above, the theorem identifies two resources, the bit and the source, two dynamical processes, compression 21 and decompression, and it also quantifies the resources necessary to perform an opti-mal compression scheme. Slightly more precisely, the theorem tells us that a classical information source described by probabilities pj can be compressed so that on average each instance can be represented using H(pj) bits, where H(pj) = − j pj log pj is a function of the source probability distribution called the Shannon entropy. Moreover, the theorem states that to use fewer bits to store source information would result in a high probability of error in decompression. The noisy channel coding theorem, Shannon's second fundamental result, quan-tifies the information that can be transmitted through a noisy channel. Suppose we wish to transmit information being emitted by an information source through a noisy channel to another location or time (storage). In both instances, transmission or storage, we need to encode the information using some error-correction scheme, so that noise, or error, introduced in the information by the channel, can be corrected at the end. The way error-correcting codes do this is by introducing enough redun-dancy into the information so that it can be recovered even after the channel has introduced errors. So, Shannon's noisy channel coding theorem provides a general procedure to calculate the capacity of the channel, or number of bits that can be reliably transmitted through the channel, taking into account the redundancy and noise. This second result also satisfies the three goals of information theory, mentioned above. Again, it identifies two types of static resources, the information source and the bits being sent through the channel. There are three dynamical processes: the noise 22 in the channel, and the processes of encoding and decoding in the error-correction scheme. Finally, it quantifies how much information can be reliably transmitted for given noise and redundancy. The function used is expressed in terms of the Shannon entropy, so again entropy is being used to quantify resources that are necessary for some elementary quantum information processing task. That entropy quantifies re-sources will remain the main theme of this section, and the main point will be that entanglement is a resource that is quantified by entropy. Of course, quantum information has other static resources besides classical infor-mation, and yet it has similar goals. So, we consider analogues to the fundamental results of classical information theory. We will look at quantum states as a static resource, and a quantum analogue of Shannon's noiseless coding theorem. Though considerable progress has been made using quantum error-correction, no completely satisfactory analogue has been found for the noisy channel coding theorem. As for the information source, in the quantum context it will emit quantum states instead of exactly known letters, but it will still be described by the probability distribution over the states. Also, each use of the source is independent from other uses. Now, if the source emits only the states |0 with probability p and |1 with prob-ability 1-p, we can use the Shannon entropy to calculate the minimum number of qubits to store the information, as before we used Shannon entropy to calculate the minimum number of bits. But this is not possible if the states are not orthogonal, such as with |0 and (|0 +|1 )/2, since we cannot always distinguish these states. (In this case, the only instance where we can distinguish them is when a measurement 23 outcome is 1). In this case, we can perform compression, but it is not error free, and we use a fidelity measure to quantify the distortion introduced by the compression scheme. The fidelity should tend toward a no-error limit of one in the limit of large blocks of data. This brings us to the analogue of Shannon's noiseless channel coding theorem, called Schumacher's noiseless channel coding theorem. Schumacher's noiseless channel coding theorem quantifies the resources required for quantum data compression, just as Shannon's did for classical data. Because of the possibility of nonorthogonal states, there is the added restriction that it be possible to recover the source with fidelity close to one. For orthogonal states |ψj with probabilities pj , Schumacher's theorem tells us the same result as Shannon's: that the information source may be compressed to the classical limit, given by the Shannon entropy H(pj). However, in the general case with nonorthogonal states, Schumacher's theorem tells us that the quantum source may be compressed not to the quantity given by Shannon entropy, but to a different entropic quantity, given by the von Neumann entropy. In general, Shannon entropy and von Neumann entropy agree only when the states are orthogonal. Otherwise, the von Neumann entropy is strictly smaller, due to the fact that we can exploit the redundancy in nonorthogonal states. So far we have used entropic quantities, the Shannon entropy and the von Neu-mann entropy, to quantify the resources necessary to store information, either classical information stored in bits or quantum states stored in qubits. We will see that entan-glement is another important elementary static resource that can be quantified by an 24 entropic quantity. Its properties are very different from any classical resources, and in fact general understanding of entanglement is poor and incomplete. Two important problems related to entanglement are entanglement creation and entanglement transformation. The main subjects of this dissertation are related to entangling power, and operators that create entanglement. The Cartan decomposition seeks a subset of operators in SU(8) that can account for the entangling operation of the whole of SU(8) (in general, SU(2n)). Entangling power quantifies the ability of operators to produce entanglement, and of course utilizes an entropic function to do so. Relating to entanglement transformation, entangling power is also invariant under local operations, and the Cartan decomposition takes advantage of this to narrow down the number of elements in the subset mentioned above. This follows from the fact that the entanglement of states does not change under local operations. Entanglement transformation is also relevant for entanglement combing, which we will see in the last chapter. This is a procedure using local operations and classical communication for transforming a multiqubit state into a tensor product of bipartite states, all spread across a bipartite split, without changing the entanglement across this split. 2.3.2 Shannon's and Schumacher's noiseless channel coding theorems In this section, we examine more closely Shannon's and Schumacher's noiseless channel coding theorems to see exactly how entropy shows up as a measure of in- 25 formation content, and ultimately entanglement. Actually, we will see this happen in the preludes to these theorems, where we prove theorems about typical sequences in the classical case and typical subspaces in the quantum case, from an information source. Then after proving theorems concerning typical sequences or subspaces, each in turn, outlines of the respective proofs of the noiseless channel coding theorems will be given (outline of Schumacher's theorem is by analogy only). In classical terms, we achieve the compression by encoding these typical sequences, while discarding the atypical ones. There are more sophisticated compression schemes than the ones we discuss here, which have several weaknesses. For example, it is possible to have zero-error compression, whereas the schemes here have the potential to fail. Another problem is that the compression scheme depends on the source distribution, but in fact, there are universal compression algorithms. But the intent here is to show an example of why we use entropy as a measure. We make no attempt at a general discussion of compression schemes or error-correction. 2.3.2.1 Shannon's noiseless channel coding theorem Shannon's noiseless channel coding theorem quantifies the minimum resources necessary to store information from a classical information source. In other words, it quantifies the extent to which we may compress this information. First, we need to define the information source. There are many models that do this, and we will choose a simple and common one. A source consists of a sequence of random variables X1,X2, . . . that represent the output of the source. We will assume that these random 26 variables take values from a finite alphabet, though this is not necessary in general. We also assume that each use of the source is independent and identically distributed. These latter two assumptions constitute what is known as an i.i.d. information source. Of course real world sources often do not behave this way. For example, English letters have strong correlations with certain other letters. But the assumptions of an i.i.d. source work fairly well in practice, and the ideas we discuss can be generalized to more sophisticated source models. Suppose an i.i.d. source is producing letters ai, each with probability pi, and m is the size of the alphabet. The idea behind Shannon's theorem is to divide the possible sequences of values x1, x2, . . . , xn for the random variables X1, . . . , Xn into two types: highly likely sequences, called typical sequences, and unlikely sequences, called atypical sequences. As n becomes large, we expect with high probability that a fraction pi of the symbols from the source will be equal to ai. The sequences x1, . . . , xn for which this is true are typical sequences. Combining this definition with the assumption of independence for the source gives p(x1, . . . , xn) = p(x1)p(x2) . . . p(xn) ≈ pnp1 1 pnp2 2 . . . pnpm m for typical sequences. Taking the logarithm of both sides, −log p(x1, . . . , xn) ≈ −np1 log p1 − np2 log p2 · · ·−npm log pm = nH(X), where X is a random variable distributed according to the source distribution and H(X) is the entropy of the source distribution, known as the entropy rate of the source. Thus, p(x1, . . . , xn) ≈ 2−nH(X), and since the total probability of typical sequences 27 cannot be greater than one, there can be at most 2nH(X) typical sequences. Already we are seeing entropy quantifying the distribution of information. Now we look at a simple data compression scheme. Suppose the output from the source is x1, . . . , xn. First, we check whether it is a typical sequence. If it is not, we discard it and declare an error. But in the limit of large n, almost all sequences are typical. If it is typical, we record it, and since there are at most 2nH(X) typical sequences, we need only nH(X) bits to uniquely identify a particular sequence. So, we choose some identification scheme, and compress the output to the corresponding string of nH(X) bits, which can later be decompressed. As n becomes large, this scheme succeeds with probability approaching one. For the theorem of typical sequences that follows, we now make a rigorous def-inition of a typical sequence. Given > 0, a string of source symbols x1, . . . , xn is -typical if 2−n(H(X)+ ) ≤ p(x1, . . . , xn) ≤ 2−n(H(X)− ). We denote the set of all such -typical sequences of length n by T(n, ). We can reformulate this definition into another useful form as 1 n log 1 p(x1, . . . , xn) − H(X) ≤ . Using the law of large numbers (statement without proof immediately following) we can prove the theorem of typical sequences, which makes rigorous the idea that most sequences output by an information source are typical in the limit of large n. Suppose we perform an experiment a large number of times, each time measuring 28 the value of some parameter, X, and labeling it for each experiment, X1,X2, . . .. As-suming the experiments are all independent, we expect that the value of the estimator Sn ≡ n i=1 Xi/n of the average E(X) should approach E(X) as n approaches infinity. The law of large numbers makes this rigorous. Theorem (Law of large numbers) Suppose X1,X2, . . . are independent random variables all having the same distribution as a random variable X with finite first and second moments, |E(X)| < ∞and E(X2) < ∞. Then for any > 0, p(|Sn − E(X)| > ) → 0 as n→∞. We do not present the proof of the law of large numbers, but the theorem is used to prove the theorem of typical sequences, which follows. Theorem (Theorem of typical sequences) 1. Fix > 0. Then for any δ > 0, for sufficiently large n, the probability that a sequence is -typical is at least 1 − δ. 2. For any fixed > 0 and δ > 0, for sufficiently large n, the number|T(n, )| of -typical sequences satisfies (1 − δ)2n(H(X)− ) ≤ |T(n, )| ≤ 2n(H(X)+ ). 3. Let S(n) be a collection of size at most 2nR, of length n sequences from the source, where R < H(X) is fixed. Then for any δ > 0 and for sufficiently large n, x∈S(n) p(x) ≤ δ. Proof Part 1: A direct application of the law of large numbers. Notice that −log p(Xi) are identically distributed and independent random variables. By the law 29 of large numbers for any > 0 and δ > 0 for sufficiently large n we have p n i=1 −log p(Xi) n − E(−log p(X)) ≤ ≥ 1 − δ. But E(log p(X)) = −H(X) and n i=1 log p(Xi) = log(p(X1, . . . , Xn)). Thus p −log p(X1, . . . , Xn) n − H(X) ≤ ≥ 1 − δ. This says that the probability that a sequence is -typical is at least 1 − δ. Part 2: This follows from the definition of typicality, and the observation that the sum of the probabilities of typical sequences must be between 1 − δ and 1. Thus 1 ≥ x∈T(n, ) p(x) ≥ x∈T(n, ) 2−n(H(X)+ ) = |T(n, )| 2−n(H(X)+ ), from which we deduce |T(n, )| ≤ 2n(H(X)+ ). Also, 1 − δ ≤ x∈T(n, ) p(x) ≤ x∈T(n, ) 2−n(H(X)− ) = |T(n, )| 2−n(H(X)− ), from which we deduce |T(n, )| ≥ (1 − δ)2n(H(X)− ). Part 3: We split the sequences in S(n) into typical and atypical sequences. The atypical sequences have a small probability in the large n limit. The number of typical sequences in S(n) is no larger than the total number of sequences in S(n), which is at most 2nR, and each typical sequence has probability approximately 2−nH(X). So, the total probability of the typical sequences in S(n) scales roughly as 2n(R−H(X)), which goes to zero when R < H(X). Shannon's noiseless channel coding theorem follows naturally from the theorem of typical sequences. The idea is to suppose that X1,X2, . . . is an i.i.d. classical 30 information source over a finite alphabet. A compression scheme of rate R maps sequences x = (x1, . . . , xn) to a bit string of length nR = nR . We denote the map by Cn(x) = Cn(x1, . . . , xn). The decompression scheme maps the nR compressed bits back to an n length string of letters from the alphabet, Dn(Cn(x)). The compression-decompression scheme is said to be reliable if the probability that Dn(Cn(x)) = x approaches one as n approaches infinity. Shannon's noiseless channel coding theorem tells us for what values of R that a reliable compression scheme exists. So from Shannon's theorem, we have an operational interpretation for the entropy H(X). It is the necessary and sufficient minimal physical resources to reliably store the output from the source. Theorem (Shannon's noiseless channel coding theorem) Suppose {Xi} is an i.i.d. information source with entropy rate H(X). There exists a reliable compression scheme of rate R > H(X) for the source. Conversely, if R < H(X) then there are no reliable compression schemes. Proof Suppose R > H(X). Choose > 0 such that H(X) + < R. Consider the set T(n, ) of -typical sequences. For any δ > 0 and for sufficiently large n, there are at most 2n(H(X)+ ) < 2nR such sequences, and the probability of the source producing such a sequence is at least 1−δ. The method of compression is to determine whether the output of the source is -typical. If not, then compress it to a fixed nR bit string that indicates a failure. The decompression operation then outputs any n length sequence as a guess for the source information. For typical output sequences, we compress by storing an index or label for the particular sequence using nR bits. 31 Suppose R < H(X). At most 2nR of the sequences from the source can be compressed and decompressed without error, since this is the maximum output of the decompression scheme. By the theorem of typical sequences, for sufficiently large n the probability of a sequence output from the source being in a subset of 2nR sequences goes to zero, since R < H(X). Thus any such compression scheme at this rate cannot be reliable. 2.3.2.2 Schumacher's noiseless channel coding theorem To explore the quantum analogue to Shannon's noiseless channel coding theorem, we must define a quantum information source that outputs quantum states. The theorems in the previous section answered information-theoretic questions about the output of an information source. Similarly, in this section, we discuss information-theoretic questions about the quantum states output from a quantum information source. That is, we are to view quantum states as information, and quantify the extent to which they can be compressed. As before, there are many ways to model an information source. We choose a model based on the idea that entanglement is what we are trying to compress and decompress. Formally, an i.i.d. quantum source is described by a Hilbert space H and a density matrix ρ on that Hilbert space. In general, the state ρ is mixed, as it is an ensemble of pure states. So, we can view ρ as part of a larger composite system that is in a pure state. That is, the state ρ of the source is entangled to the rest of the system. 32 A compression-decompression scheme of rate R for the source consists of two fam-ilies of quantum operations Cn and Dn, analogous to the classical case. Cn is the compression operation and takes states from H⊗n to a 2nR dimensional state space, the compressed space. We can think of this as nR qubits. Dn, the decompression operation, takes states from the compressed space back to the original state space. The combined compression-decompression scheme is given by composition, Dn ◦ Cn. Reliability is measured by the entanglement fidelity, F(ρ⊗n,Dn ◦ Cn), which is es-sentially the square of the inner product between the purified states before and after the compression-decompression scheme and tends toward one when entanglement is preserved. Also note, the fidelity depends only on ρ⊗n and Dn ◦ Cn, and not on the choice of purification, which is arbitrary, since the auxiliary systems of any purifica-tion are related by a unitary transformation. The choice of measure here, like the choice of quantum information source model, is another case where the best one to use is an open question, since knowledge of quantum information theory is incomplete. However, quantifying entanglement preservation seems reasonable, since we generally do not expect noise to preserve entanglement. Here is a point where entanglement can be connected to entropy. We say S(ρ) is the minimum physical resources necessary to store the output from the source, and the output of the source is the reduced density matrix of an entangled state. As entropy increases, we require more resources to store the output information. Also as the entropy increases, the more mixed is the reduced density matrix and thus the more nonlocal, entangled, is the composite state. And the less we know about the 33 state, then the more valuable is the information it holds. So, here is a connection between entanglement, information content, and entropy. Analogous to the classical case, Schumacher's noiseless channel coding theorem will be a consequence of a quantum analogue to the theorem of typical sequences and the law of large numbers. Suppose the density operator ρ associated to a quantum source has orthonormal decomposition ρ = x p(x) x x , where |x is an orthonormal set and p(x) are the eigenvalues of ρ. Under these circumstances, H(p(x)) = S(ρ), and it makes sense to talk about an -sequence x1, . . . , xn for which 1 n log 1 p(x1)p(x2) . . . p(xn) − S(ρ) ≤ , similarly to the classical situation. An -typical state is a state |x1 |x2 . . . |xn for which the sequence x1, x2, . . . , xn is -typical. We define the -typical subspace to be the subspace spanned by all -typical states. This subspace is denoted T(n, ), and the projector onto the subspace, P(n, ). The projector can be written as P(n, ) = x −typical x1 x1 ⊗ x2 x2 ⊗ ·· ·⊗ xn xn . Now we can give the quantum analogue of the theorem of typical sequences. Theorem (Typical subspace theorem) 1. Fix > 0. Then for any δ > 0, for sufficiently large n, tr(P(n, )ρ⊗n) ≥ 1 − δ. 34 2. For any fixed > 0 and δ > 0, for sufficiently large n, the dimension |T(n, )| = tr(P(n, )) of T(n, ) satisfies (1 − δ)2n(S(ρ)− ) ≤ |T(n, )| ≤ 2n(S(ρ)+ ). 3. Let S(n) be a projector onto any subspace of H⊗n of dimension at most 2nR, where R < S(ρ) is fixed. Then for any δ > 0 and sufficiently large n, tr(S(n)ρ⊗n) ≤ δ. Proof Part 1: This follows immediately from the theorem of typical sequences, since tr(P(n, )ρ⊗n) = x −typical p(x1)p(x2) . . . p(xn). Part 2: This also follows immediately from the theorem of typical sequences. Part 3: We split the trace over the typical and atypical subspaces, tr(S(n)ρ⊗n) = tr(S(n)ρ⊗nP(n, )) + tr(S(n)ρ⊗n(1 − P(n, ))). Looking at the first term, ρ⊗nP(n, ) = P(n, )ρ⊗nP(n, ), since P(n, ) is a projector that commutes with ρ⊗n. But by the definition of a typical sequence, the eigenvalues of the right hand side operator are bounded above by 2−n(S(ρ)− ). Then tr(S(n)P(n, )ρ⊗nP(n, )) ≤ 2nR2−n(S(ρ)− ), 35 since there are at most 2nR typical states in S(n). As n → ∞, this term tends to zero. And as n → ∞, the probability of an atypical state tends to zero. So, noting that S(n) ≤ 11 and that S(n) and ρ⊗n(1 − P(n, )) are both positive operators, it follows that 0 ≤ tr(S(n)ρ⊗n(1 − P(n, ))) ≤ tr(ρ⊗n(1 − P(n, ))) → 0 as n→∞. So, the second term also tends to zero as n becomes large, establishing the result. With the typical subspace theorem, the main ideas of the proof of Schumacher's noiseless channel coding theorem follow analogously to Shannon's noiseless coding theorem, with some minor technical hurdles. But, since the point of this section is to illustrate a connection between entanglement, entropy, and information content, and that has been done, we close with a statement of the theorem without proof. Theorem (Schumacher's noiseless channel coding theorem) Let {H, ρ} be an i.i.d. quantum source. If R > S(ρ) then there exists a reliable compression scheme of rate R for the source. If R < S(ρ) then any compression scheme of rate R is not reliable. 2.4 Linear entropy This is a short section to introduce the linear entropy. We will review some of its important properties in Chapter 3, after introducing more notation. Here, we will derive it from the von Neumann entropy, as a linear approximation. Because it is polynomial, it simplifies calculations, though it also loses some of the direct quantitative meaning of the von Neumann entropy, which we saw above. In Chapter 3, we will state and prove the properties, following the paper and using the notation of Zanardi [25]. 36 The von Neumann entropy is given by S(ρ) = −tr(ρ log ρ), where ρ is the density operator for some system. While von Neumann entropy is in general defined for any density operator, the intention here is to use it as a measure of entanglement. When it is used this way, ρ is replaced by the reduced density operator of a bipartite composite system, with the conventional labeling, Alice and Bob. For example, ρA or ρB. It is convenient to re-express the entropy in terms of the eigenvalues of ρ, λx, S(ρ) = − x λx log λx, where we define 0 log 0 ≡ 0. We have already seen the von Neumann entropy in previous sections of this in-troduction. It has a range from zero for pure states to a maximum of log d for the completely mixed state 11/d in a d-dimensional Hilbert space. Also, if ρAB is a pure composite state, then S(ρA) = S(ρB), since the von Neumann entropy depends on the eigenvalues of the density operators and the reduced density operators have the same eigenvalues. This does not assume that the systems of Alice and Bob have the same dimension, though they do have the same dimension in the case of two qubits. This contrasts with Chapter 3 in the three-qubit case, where we use the linear entropy as our measure of entanglement in defining the entangling power across a bipartite split, where the dimensions of the two systems are not equal. As just mentioned, in defining the entangling power we use the linear entropy. 37 This allows us to derive useful expressions for the entangling power that are easier to compute. We start by linearizing the logarithm by a Taylor expansion around the identity, log ρA ≈ log 11 + 11−1(ρA − 11) = ρA − 11. Inserting this into the von Neumann entropy, we get the linear entropy, S(ρA) = −tr(ρ2 A − ρA) = 1 − tr(ρ2 A). Zanardi [25] has proven several important properties of the linear entropy, and we will look at these in Chapter 3. But first, we need to write the linear entropy in the operator notation that he uses. This notation will be convenient later when deriving an expression for the entangling power, but at first it might be difficult to see its meaning. So, in Chapter 3, we will formally reintroduce the linear entropy, but for now we will conclude the introductions. CHAPTER 3 ENTANGLING POWER, TWO- AND THREE-QUBIT GATES In Chapter 2, we introduced the linear entropy as a linearized version of the von Neumann entropy. In this chapter we will use the linear entropy as a measure of entanglement in defining the entangling power. One uses the linear entropy in defining entangling power, because it is easy to compute and allows us to derive a simple expression. However, the advantages do come at the cost of some of the direct quantitative meaning of the von Neumann entropy, such as the rate of compression of states or channel capacity in qubits, thus yielding more qualitative relationships between operators. Before we derive expressions, however, we look at some important properties of the linear entropy and introduce some notation. After introducing the linear entropy, we will proceed to develop a simple expression first for the two-qubit entangling power, then for the three-qubit entangling power. Aside from the paper already cited, much of the material in Chapters 3, 4, and 5 will appear in two other papers I coauthored with Yong-Shi Wu, in preparation for publication. 39 3.1 Entanglement and linear entropy The linear entropy quantifies entanglement in a state (with density operator ρ) of the total system, E(ρ) = 1 − TrA(ρ2 A), where ρA = TrB(ρ) is the reduced density operator of subsystem A, and TrA and TrB the partial trace over subsystems A and B, respectively. For a product state, E(ρ) = 0, so the linear entropy measures the impurity of the subsystem, thus indirectly the entanglement of the composite system. The idea for defining the entangling power of an operator (or quantum gate) is to lift the notion of entanglement from the state level to the operator level [25]. For an operator U acting on H, it is known that it belongs to the Hilbert-Schmidt space HHS, which is isomorphic, as a Hilbert space, to H⊗2. The natural isomorphism, Ψ, from the operator to the state space is known to be |Ψ(U) := (U ⊗ 11) Φ+ , (3.1) with the ket, |Φ+ := d−1 α=0 |α ⊗2, and {|α } (α = 0, . . . , d −1) forming an orthonor-mal basis of H. With this map, the entanglement of operator U is defined [25] as the linear entropy of the state |Ψ(U) ; i.e. E(U) := E(Ψ(U)). For a bipartite system with Hilbert space H1⊗H3, an operator U lives in H⊗2 HS ∼= H⊗4, and the map Ψ is given by |Ψ(U) := U13 ⊗ 1124 Φ+ ⊗2 (3.2) with the subscripts of U indicating on which H the operator acts. For two qubits, the single-qubit Hilbert space H is two-dimensional; the subscripts of U indicate on which qubits the operator acts. In general, dim(H1) and dim(H3) are not equal, 40 and the map is given by Eq. (3.11), with a corresponding |Φ+ . With this map, we can measure the entanglement of a unitary operator U with the linear entropy of the mapped state |Ψ(U) . Let T11 be the swap operator between the first and third factor of some Hilbert space H⊗4, and ˆ T11 its adjoint action ˆ T11(X) := T11XT11. We will be using this swap in what follows. Now, for convenience, we want to express the entanglement of a unitary operator in the form that Zanardi [25] uses. We start by writing ρ = |Ψ(U) Ψ(U) (3.3) Noting that Alice owns Hilbert spaces H1, which is a qubit(s), and H2, which is associated to H1 as in |Φ+ := d−1 α=0 |α ⊗2, the reduced density matrix is ρA = TrB |Ψ(U) Ψ(U) , and the entanglement of U is given by E(U) = 1 − Trρ2 A. In the first step we insert the definitions of the partial trace and the operator maps from the operator space to the state space. E(U) = 1 − 1 d21d23 Tr TrB (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) TrB (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) . The factor (d21 d23 )−1 in the end normalizes the trace, and the subscripts of U label the qubits on which U acts (likewise for the identity). These labels apply to the isomorphic operator map, and are intended to be consistent with the subscripts of the swap, T, which is introduced shortly. 41 In the next two steps, we express the traces in Dirac notation to coincide with the Zanardi's notation in [26, 25]. First, the summations are over the computational basis of system B, then the outermost trace is summed over the computational basis of A. The subscripts on the Dirac brackets are also consistent with their respective maps, though they are later dropped to reduce clutter. E(U) = 1 − 1 d21 d23 Tr k,k ,l,l 34 kl (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) |kl 34 34 k l (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) |k l 34 = 1− 1 d21 d23 ijkl (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) |kl 34 34 k l (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) |ijk l . Next, we use the identity, tr[(A ⊗ B)T] = tr(AB) to express the matrix multi-plication of ρ2 A as a tensor product, as in TrA(ρ2 A) = TrA(ρA ⊗ ρAT). Here, we need special care to interpret the notation. The tensor product is inserted between the maps, and thus the summation labels must be doubled. The labels, i and j, for the outermost trace, are doubled and labeled i' and j' for the second factor of ρA. The labels k and l are for the partial trace of the first factor, and the primed labels are for the second. The identities apply to the two qubits that are not summed over in 42 the factor. The following step is to rearrange the terms into a more compact form. E(U) = 1 − 1 d21 d23 ijkli j ⊗ 11 (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124) |kl 34 ⊗ 34 k l (U13 ⊗ 1124)( Φ+ Φ+ )⊗2(U† 13 ⊗ 1124)T |ij ⊗ 11 ⊗ i j k l = 1− 1 d21 d23 ijkli j k l (U13 ⊗ 1124)⊗2T( Φ+ Φ+ )⊗4 (U† 13 ⊗ 1124)⊗2T |ijkli j k l . The second to last step is the sum over the identities, using the completeness relation. These are tensor product factors, so they fall out easily. We also reintroduce the subscripts for the swap T. Finally, the last step is to express the trace in terms of the Hilbert-Schmidt inner product, < A,B >:= tr(A†B). E(U) = 1 − 1 d21 d23 iki k U⊗2T11U†⊗2T11 |iki k = 1− 1 d21 d23 U⊗2, T11U⊗2T11 . (3.4) Note that the subscripts of T and U do not refer to the same spaces. T arises from the identity tr[(A⊗B)T] = tr(AB), and the subscripts are consistent with U12 ⊗U34 acting on (H1 ⊗H2) ⊗ (H3 ⊗H4). The subscripts of U refer to the labeling of the isomorphic map from the operator space to the state space. With this form of the entanglement of a unitary operator, Eq. (3.4), we now derive some properties. (a) First, from the relation [T11, (U1 ⊗ U2)⊗2] = 0, it follows that ∀U1, U2 ∈ U(H), where U(H) is the group of unitaries acting on H, E[(U1 ⊗ U2)U] = E[U(U1 ⊗ U2)] = E(U) (3.5) 43 This is the invariance of entanglement under local unitary transformations. These actions by local unitary operators form a local equivalence class of the operator U. (b) From U 2 HS = tr(U†U) = tr11 = d1d3 and ˆ T11 = 1, and the Cauchy-Schwarz inequality, we get U⊗2, ˆ T11(U⊗2) ≤ U 2 HS ˆ T11(U⊗2) HS ≤ ˆ T11 U 4 HS = d21 d23 . Therefore E(U) ≥ 0. E is also invariant under Hermitian conjugation, E(U) = E(U∗) = E(U†). (c) From (b), it follows that E(U) = 0 ⇔ ˆ T11(U⊗2) = U⊗2 ⇔ T11, U⊗2 = 0. (3.6) Also, invariance of entanglement under local operations as in (a) allows us to consider, without loss of generality, transformations of the form (Schmidt decomposition) U = r k=1 λkek⊗ek, r ≤ d1d3. Inserting this into the expression above, we find k = h ⇒ λkλh = 0. This itself implies that there is only one nonvanishing Schmidt coefficient. Thus U such that E(U) = 0 is a tensor product in the local equivalence class of the identity. (d) The upper bound is met by operators in the local equivalence class of the swap operator in H1 ⊗H2. One has E(S) = 1 − (d1d3)−2tr[(S ⊗ S)T11(S ⊗ S)T11] = 1 − (d1d3)−2tr[T33T11] = 1 − (d1d3)−1. These maximally entangled transforma-tions are the ones having d1d3 nonvanishing Schmidt coefficients of the same amplitude, analogous to the completely mixed states. Later, when calculating the entangling power, this maximum is scaled so that the range is 0 ≤ E(U) ≤ 1. 44 3.2 Entangling power for two-qubit gates We consider the entangling power of an operator of a bipartite system, with Hilbert spaces, H1 and H2, for subsystems labelled 1 and 2, respectively. The quantity we use to quantify entanglement in a state (with density operator ρ) of the total system is the linear entropy, E(ρ) = 1 − tr1(ρ21 ), where ρ1 = tr2(ρ) is the reduced density operator of subsystem 1, and tr1 and tr2 the partial trace over subsystems 1 and 2, respectively. The linear entropy can be viewed as a linearized version of the von Neumann entanglement entropy. One uses the linear entropy in defining entangling power, because it is easy to compute and allows us to derive a simple expression. For a product state, E(ρ) = 0, so the linear entropy measures the impurity of the subsystem, thus indirectly the entanglement of the composite system. The idea for defining the entangling power of an operator (or quantum gate) is to lift the notion of entanglement from the state level to the operator level [25]. For an operator U acting on H, it is known that it belongs to the Hilbert-Schmidt space HHS, which is isomorphic, as a Hilbert space, to H⊗2. The natural isomorphism, Ψ, from the operator to the state space is known to be |Ψ(U) := (U ⊗ 11) Φ+ , (3.7) with the ket, |Φ+ := d−1 α=0 |α ⊗2, and {|α } (α = 0, . . . , d −1) forming an orthonor-mal basis of H. With this map, the entanglement of operator U is defined [25] as the linear entropy of the state |Ψ(U) , i.e., E(U) := E(Ψ(U)). For a bipartite system with Hilbert space H⊗2, an operator U lives in H⊗2 HS ∼= H⊗4, 45 and the map Ψ is given by |Ψ(U) := U13 ⊗ 1124 Φ+ ⊗2 (3.8) with the subscripts of U indicating on which H the operator acts. For two qubits, the single-qubit Hilbert space H is two-dimensional; the subscripts of U indicate on which qubits the operator acts. With this map, we can measure the entanglement of a unitary operator U with the linear entropy of the mapped state |Ψ(U) . On the other hand, we define the entangling power of an operator, U, as the average entanglement produced over all product states |ψ ⊗ |φ distributed according to a probability density p(ψ, φ) over the manifold of product states, ep(U) := E(U |ψ ⊗ |φ )|ψ ,|φ . (3.9) If we restrict ourselves to the uniform distribution, we can evaluate this average in terms of the entanglement of the operator, arriving at Eq. 3.10. The uniform distribu-tion is a U(2)⊗U(2)-invariant probability distribution, i.e., p(ψ1, ψ3) = p(U1ψ1, U3ψ3), where p is the probability density distribution over the manifold of product states. From this definition it was shown [25] that the entangling power of U is related to the entanglement E(U), given by the linear entropy, and the swap between the first and third qubits, by the following relation: ep(U) = E(U) + E(US) − E(S). (3.10) The entangling power quantifies the ability of an operator to produce entangled states from a uniform product state, which can be maximally entangled [24]. It is 46 invariant under local unitary operations, that is, operators that can be written as U1 ⊗ U3, do not change the entangling power [26]. Thus any operator in the local equivalence class of the identity and swap has zero entangling power. In the case of the swap, we have an example of an operator that is nonlocal (E(S) is maximal), but nevertheless has zero entangling power, mapping product states to product states. While this formula is convenient, we can do better. It is useful to introduce the magic matrix, which we do in the next chapter. 3.3 Entangling power for three-qubit gates The explicit formulas for entangling power of two-qubit gates were given by Za-nardi and collaborators in Refs. [26, 25]. To examine the three-qubit braiding gates, we need to generalize the existing formulas. Obviously the bipartite split of three qubits can not be balanced. We assume the bipartition is between the first and the remaining two qubits. For an arbitrary bipartite split with H1⊗H3, operators live in H⊗2 1 ⊗H⊗2 3 . With d1 = dim(H1) and d3 = dim(H3), then |Φ+ := d1 β=1 d3 γ=1 |β ⊗2 |γ ⊗2, and the map (3.8) is of a similar form |Ψ(U) := (U13 ⊗ 1124) Φ+ , (3.11) with U acting on H1 ⊗H3. With this map, we can quantify the entanglement of unitary operators across the bipartite split, by the linear entropy of the mapped state, |Ψ(U) . Then we define 47 the entangling power of an operator, U, as the average entanglement produced over all product states over the manifold of product states, ep(U) := E(U |ψ ⊗ |φ )|ψ ,|φ . (3.12) Again, we impose the uniform distribution so that we can derive an explicit expression for the entangling power in terms of the entanglement of the operator. The uniform distribution is the U(d1) ⊗ U(d3)-invariant probability distribution, i.e., p(ψ1, ψ3) = p(U1ψ1, U3ψ3), where p is the probability density distribution over the manifold of product states. However, a problem arises in the three-qubit case. It is not possible to define a swap between H1 and H3, since d1 = 2 and d3 = 4. Resolving this problem is our next goal. We define the projectors P±11 := 2−1(11±T11)⊗1133 over the eigenspaces of T11. Note that tr(T11) = tr(T11 ⊗ 1133) = d1d22 . Then ρ+ 11 := 2P±11/[d1d22 (d1 + 1)] represents the uniform state over the eigenvalue 1 subspace, and from Eq. (3.4) the entanglement can be written as E(U) = 2N U⊗2ρ+ 11U†⊗2, P−11 , (3.13) where N is a normalization factor. To justify the claim that ρ+ 11 represents the uniform state, we need the following lemma from [26], which relies on Schur's lemma. Lemma Ωp0 = 4Cd1Cd3P+ 11P+ 33, C−1 d = d(d+1), represents the uniform state with respect to which the entangling power over the uniform state is defined. Proof Since the uniform distribution factorizes, we can consider separately the 48 average ω11 with respect to |ψ1 , on the first and third factor of H⊗2, and ω33 with respect to |ψ2 , on the second and fourth factors. Then Ωp0 = ω11ω33. Let us first note that the entangling power is defined as the average entanglement over all product states distributed according to some probability density over the manifold of product states. Thus, the average occurs over Ωp := dμ(ψ1, ψ2) ψ1 ψ1 ⊗ ψ2 ψ2 ⊗2 ∈ S(H⊗2), (3.14) where S(H⊗2) is the space of density matrices over H⊗2. Ωp is symmetric under the exchange of the first and third, and second and fourth, factors. Moreover, since the uniform distribution is U(d1) × U(d3) invariant, one has [U⊗2 1 , ω11] = 0∀U1 ∈ U(d1), and analogously for ω33. Since U⊗2 1 's act on the totally symmetric subspace irreducibly, it follows from the above commutation relation and Schur's lemma that ω11 = 2CP+ 11, a scalar multiple of the projector onto the space. The normalization constant is found by the condition that trω11 = 1. The same result applies to ω33, and we thus have the result. Now, from the definition of the entangling power and Eq. (3.4) the entangling power can be written as ep0 = 2 U⊗2Ωp0U†⊗2, P11 = 1− d21 d22 U⊗2ρ+ 11ρ+ 33U†⊗2, T11 = 1− 1 d1d3(d1 + 1)(d3 + 1) U⊗2(11 + T11)(11 + T33)U†⊗2, T11 . From now on, we will drop the 0 subscript from ep0 . All of our expressions are derived assuming a uniform distribution. 49 So far, the derivation has proceeded identically to the two-qubit case. In the next section, our assumption that the bipartite split is uneven, i.e., d1 = d3, will prevent us from defining a permutation swap that simply exchanges H1 and H2. We will need to generalize this idea of swap. So, we start the next section by considering the term U⊗2T33U†⊗2, T11 . 3.3.1 The swap operator In the two qubit case, we can see that since its square is the identity, a swap of Hilbert spaces 1 and 2 leads to U⊗2T33U†⊗2, T11 = U⊗2S2 13 ⊗ S2 13T33S2 13 ⊗ S2 13U†⊗2, T11 = (US)⊗2T11(US)†⊗2, T11 = d21 d22 [1 − E(US)]. Between the second and third step we used the relation, S ⊗S T33 S ⊗ S = T11, and the last step is Eq. (3.4). Unfortunately, for the three-qubit case, d1 = d3, so we cannot use the same idea 50 of swap. Instead, we consider swapping half of the computational basis states, as in S = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . To avoid confusion, please note that the basis is in ascending numerical order, and that the swap swaps the states that are asymmetric in the first and third qubit. For example, |001 is swapped with |110 , and |011 is swapped with |100 . Then the step proceeds as before, but S ⊗S T33 S ⊗ S = T11, where S ⊗ S = S13 ⊗ S13, (see the appendix to this chapter) might not hold in general. In the context of the operation in Eqs. (3.15), under the trace, it does hold for the GHZ generators we consider in the next chapter, which come from the partial magic matrix, Eq. (4.5), also introduced in the next chapter, but for the full augmented magic matrix, we might need constraints on the parameters or correction terms in the form of a function of the parameters. Until this correction is made, we replace E(US) in Eq. (3.10) by ˜E (US) = 1 − 1 d21 d23 (US)⊗2 S⊗2 T33 S⊗2 (US)†⊗2, T11 , (3.15) which we show in the appendix is also invariant under local operations. With invari-ance, any possible correction terms for three-qubit operators can be determined by 51 the magic matrix and expressed by its parameters. The rest of the derivation proceeds in a straightforward manner. Returning to the expression for the entangling power with a uniform probability distribution, ep = 1− 1 d1d3(d1 + 1)(d3 + 1) U⊗2(11 + T11)(11 + T33)U†⊗2, T11 . There are four terms, and so far we have dealt with only one, where we had an issue with the swap operator that appears in the final expression. The other three are very similar. The first of these is as follows: U⊗2T11U†⊗2, T11 = d21 d22 [1 − E(U)]. We see that this expression already matched the one for the entanglement. The other two do not need to be re-expressed, since they can be evaluated in terms of the dimensions of the Hilbert spaces. The first of these follows, recalling that T11 = T11 ⊗ 1133, d1 is the dimension of H1, and d23 is the trace over the identity on H3 ⊗H3. U⊗21111 ⊗ 1133U†⊗2, T11 = 1111 ⊗ 1133, T11 ⊗ 1133 = d1d23 . In the final term, we have two T swaps on the left side of the inner product, and we can suppose that they are acting toward the right on U†⊗2. Acting on both indices 52 across the tensor product has the effect of interchanging the two factors of U. The effect combines with the swap on the right also acting on U†⊗2, to produce the same effect as T33 . The result is the same as the previous term, but with T33 instead of T11. So, similar to the previous term, we get d21 d3. Putting these four terms together, the entangling power simplifies as follows, where we use N−1 = d1d3(d1 + 1)(d3 + 1). ep = 1− N U⊗2(1 + T11)(1 + T33)U†⊗2, T11 = 1− N d21 d22 [1 − E(U)] + d21 d22 [1 − ˜E (US)] + d1d23 + d21 d3 = d1d3 (d1 + 1)(d3 + 1) E(U) + ˜E (US) − d1d3 − 1 d1d3 = d1d3 (d1 + 1)(d3 + 1) E(U) + ˜E (US) − E(S) , where E(S) = (d1d3 − 1)/d1d3. In summary, Eq. (3.10) has a similar form for the entangling power in the three-qubit case, with a new swap operator and E(US) replaced by ˜E (US). 3.4 Conclusion Thus, to use Zanardi's formula for entangling power in the three-qubit case, we have used a different type of swap. The swap appears in Zanardi's entangling power formula 3.10 because it was introduced to switch 11⊗T33 to T11⊗11, thus giving a final expression in terms of entanglement. Here we could think of it as swapping qubits or Hilbert spaces. To perform the analogous switch in the three-qubit case, where the Hilbert spaces have different dimensions, we instead used a swap to exchange half of the computational basis states. 53 In the end, we obtain the same simple formula for entangling power. From here, we calculate the entangling power (across a bipartite split) of a three-qubit magic matrix that captures all the entangling power of SU(8) in seven operators and eight coefficients-we need eight coefficients since one of the operators does not commute with the others. Though we measure across a bipartite split, rather than directly cap-turing tripartite entangling power, such a measure is nevertheless useful in analyzing three-qubit states and their operators for protocols such as teleportation. Also, in rederiving the final expression of entangling power for three qubits, we have opened the door to the general multiqubit case. 3.5 Appendix In this appendix, we consider the steps in Eqs. (3.15). We have not yet determined constraints or correction terms for the full augmented magic matrix, introduced in Chapter 4, but we will show that the entanglement measure with this new swap is still invariant under local operations, using the Schmidt decomposition for operators. First, recall the isomorphic operator map. For an arbitrary bipartite split with H1 ⊗ H3, operators live in H⊗2 1 ⊗ H⊗2 3 . With d1 = dim(H1) and d3 = dim(H3), then |Φ+ := d1 β=1 d3 γ=1 |β ⊗2 |γ ⊗2, and the map (3.8) is of a similar form |Ψ(U) := (U13 ⊗ 1124) Φ+ , with U acting on H1⊗H3. For this particular case of three qubits, d1 = 2 and d3 = 4. In Eqs. (3.15), we have the operator U⊗2 = U13⊗U13, acting on (H1⊗H3)⊗(H1⊗H3). 54 Then the swap T11 is an exchange of the first and third Hilbert spaces, the two H1. When we say S ⊗S T33 S ⊗S = T11, we must show that it is true on all the states of (H1 ⊗H3) ⊗ (H1 ⊗H3). However, it is not in fact true on every state in the Hilbert space. The choice of swap could also complicate efforts to generalize the entangling power to a partition independent or a truly multipartite measure. Recall that the partial trace is a trace over Bob's Hilbert spaces. In the three-qubit case, Bob holds two qubits, which live in H3. In the states which fail to match the results of the action of T11, the first of Bob's qubits, which is the middle of the three qubits, is flipped. Now, looking at Eqs. (3.15), since S ⊗S T33 S ⊗ S acts to the left on (US)⊗2 and T11 acts to its left on (US)†⊗2, if a middle qubit is flipped by one and not the other, these terms will be zero after the partial trace, which traces separately over each copy of U U† . To show invariance, we consider the Schmidt decomposition, U1 ⊗ U3 = λ ei ⊗ ej , which, due to locality, must be a product and has at most one term. To show invariance, we must show [(U1 ⊗ U3)⊗2, S⊗2 T33 S⊗2] = 0. We check as follows. (U1 ⊗ U3)⊗2 S⊗2 T33 S⊗2 = S⊗2 T33 S⊗2 (U1 ⊗ U3)⊗2 , and substituting, λ2 (ei ⊗ ej)⊗2 S⊗2 T33 S⊗2 = S⊗2 T33 S⊗2 λ2 (ei ⊗ ej)⊗2 . As S acts, it can change the basis states in the decomposition, but T is an exchange 55 between identical terms and has no effect. Thus the result, λ2 (ek ⊗ el)⊗2 T33 S⊗2 = S⊗2 T33 λ2 (ek ⊗ el)⊗2 ⇒ λ2 (ek ⊗ el)⊗2 S⊗2 = S⊗2 λ2 (ek ⊗ el)⊗2 ⇒ λ2 (ei ⊗ ej)⊗2 = λ2 (ei ⊗ ej)⊗2 , where it is possible that i = k or j = l. We use this result to show the invariance, E(U(U1 ⊗ U3)S) = 1 − N U⊗2(U1 ⊗ U3)⊗2 S⊗2 T11 S⊗2 (U1 ⊗ U3)†⊗2U†⊗2 T11 = 1− N U⊗2 S⊗2 T11 S⊗2 U†⊗2 T11 = E(US). Left invariance follows from [(U1 ⊗ U3)⊗2, T11] = 0, which is one of the original prop-erties from the beginning of the chapter. To complete the proof, we must show that 1 − N (US)⊗2 S⊗2 T33 S⊗2 (US)†⊗2 T11 is equivalent to a direct calculation of E(US) = 1 − N (US)⊗2 T11 (US)†⊗2 T11 . This verification is not yet complete, but with invariance it is enough to show the verification for either the Schmidt decomposition or the augmented magic matrix in Chapter 4. If it does not hold in general, then Eq. (3.10) needs correction terms, which are functions of the magic matrix parameters. In Table 3.1, we list states of (H1 ⊗ H3) ⊗ (H1 ⊗ H3), and read from right to left, we see the action of S ⊗S T33 S ⊗ S. Comparing the left-most and right-most 56 columns, we can see that the action on the first qubit is the same as T11, while as we mentioned it fails in a few cases for the second qubit. Not all possible states are listed in the table, but those that are left out can be deduced from the ones given. For example, for the state |111000 , we can look at the case for |000111 in the table. It is instructive to look at a few terms of a direct calculation of E(ZS) with Z, the partial magic matrix constructed in Chapter 4, and shown in Eq. (4.5). Let us look at the terms of the isomorphic map for the operator ZS. ZS = c1 |000000 − is1 |101010 000000 + c−2 |101001 − is−2 |000011 000011 + c4 |001100 + is4 |100110 001100 + c3 |100101 + is3 |001111 001111 + c3 |011010 + is3 |110000 110000 + c4 |110011 + is4 |011001 110011 + c−2 |010110 − is−2 |111100 111100 + c1 |111111 − is1 |010101 111111 . After forming the reduced density matrix ρA, we get some terms as follows: ρA = . . . −ic1s3 − ic4s−2 + ic4s−2 + ic1s3 00 11 . . . , which we can see, sum to zero. These are associated with the types of terms that fail to hold the relation S ⊗S T33 S ⊗ S = T11 as described above and in Table 3.1. The way to check these in the table is illustrated in the following example. After 57 Table 3.1: The operators must survive the partial trace, so only those terms with identical H34 need to be considered. (H24 is not shown in the table, and its subscripts refer to the map labeling.) Reading from right to left, we see that the results on the ket halves of the two operators are the same as for T11 ⊗ 11, except as noted in the text. Similar results apply to the bra halves, which have matching H34 with their corresponding kets, so that the partial trace ends with the same results. Shown below are enough of the states, so that any remaining can be deduced. As explained in the text, there are a few cases where the first qubit of H3 is flipped for the ket, and might result in the need for correction terms. S ⊗ S 11 ⊗ T33 S ⊗ S H13 ⊗H13 000000 000000 000000 000000 010011 010100 000110 000001 010000 010000 000010 000010 000011 000100 000100 000011 100000 011000 000011 000100 110011 001100 000101 000101 110000 001000 000001 000110 100011 011100 000111 000111 001001 110110 110110 001001 S ⊗ S 11 ⊗ T33 S ⊗ S H13 ⊗H13 001010 110010 110010 001010 011001 100110 110100 001011 111010 111010 110011 001100 101001 101110 110101 001101 101010 101010 110001 001110 111001 111110 110111 001111 010010 010010 010010 010010 000001 000110 010100 010011 100010 011010 010011 010100 58 Table 3.1 continued S ⊗ S 11 ⊗ T33 S ⊗ S H13 ⊗H13 110001 001110 010101 010101 110010 001010 010001 010110 100001 011110 010111 010111 111000 111000 100011 011100 101011 101100 100101 011101 101000 101000 100001 011110 111011 111100 100111 011111 110111 001111 011101 100101 110100 001011 011001 100110 S ⊗ S 11 ⊗ T33 S ⊗ S H13 ⊗H13 100111 011111 011111 100111 101110 101001 101001 101110 111101 111101 101111 101111 100101 011101 001111 110111 59 forming the density matrix but before taking the partial trace, we look at terms like −ic1s3 000000 110000 in one copy of ρ and −ic4s−2 111100 001100 in the other. Normally, we would take the partial trace, where these terms sum to zero with their neighboring terms. Then we would find ρ2 A, and these terms would all go to 02. This happens in a direct calculation, even in the three-qubit case. But in the operator notation we use in Eq. (3.15), we have used an identity to write TrB(ρ2 A) = TrB(ρA ⊗ ρAT11). Recall that the terms appearing in the table are states in (H1⊗H3)⊗(H1⊗H3). So, we read these qubits from the kets of the terms we are considering, arriving at |000110 . We see in the table that this is one of the states that fails to hold the relation. As for the bras, they are swapped by T11, which of course performs correctly. Thus there is a mismatch between the kets and bras such that the partial trace on these terms is zero, whereas otherwise they should have survived to later cancel with other terms. CHAPTER 4 CARTAN DECOMPOSITION AND THE MAGIC MATRIX Much of the material in Chapters 3, 4, and 5 will appear in two papers coauthored by me and Yong-ShiWu, in preparation for publication, as well as in the paper already cited. The idea behind the magic matrix proposed in Refs. [17, 20, 18] is the following: Since entangling power is invariant under local unitary rotations of each subsystem, we want to extract the completely nonlocal entangling operation of any operator, apart from local rotations of either subsystem. The magic matrix represents such a completely nonlocal entangling operation hidden in any operator. Mathematically it is required that any unitary operators of the total system are related to a magic matrix by local unitary operations, which act only on one of the subsystems. Thus the computation of the entangling power of any operator of the total system is reduced to the computation of that of the magic matrix associated with it. 61 4.1 Magic matrix and Cartan decomposition of SU(4) For simplicity, let us consider a two-qubit system. There exists a mathemati-cal theorem (see below) stating that all two-qubit unitary operators in SU(4) are related to a magic matrix by local unitary operations in SU(2) ⊗ SU(2). Thus, the computation of the entangling power of any operator in SU(4) is reduced to the computation of that of the magic matrix associated with it in the coset space SU(4)/SU(2) ⊗ SU(2), which is known to be a symmetric space [12]. Actually the magic matrix does not need to run over the whole symmetric space. The magic ma-trix can be generated by exponentiation of a maximal abelian subalgebra (called the Cartan subalgebra) associated with the symmetric space [17]. Mathematically this procedure corresponds to the Cartan decomposition of SU(4) associated with the Lie algebra pair (su(4), su(2) ⊕ su(2)) [17]. More concretely, we have [17, 20]. Theorem G ≡ SU(4) has the following (Cartan) decomposition G = KMK, where K ≡ SU(2) ⊗ SU(2) and M = exp−i(c1σx ⊗ σx + c2σy ⊗ σy + c3σz ⊗ σz). (4.1) In the computational basis, this magic matrix is explicitly given by Ud = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ e−ic3c− 0 0 −ie−ic3s− 0 eic3c+ −ieic3s+ 0 0 −ieic3s+ eic3c+ 0 −ie−ic3s− 0 0 e−ic3c−. ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ with c± = cos(c1 ± c2), s± = sin(c1 ± c2). The matrix Ud has the magic basis as 62 eigenvectors, so is diagonal in that basis. In the two-qubit case, the magic basis vectors are nothing but the Bell states: |Φ1 = 1 √2 (|00 + |11 ), |Φ2 = −i √2 (|00 − |11 ), |Φ3 = 1 √2 (|01 − |10 ), |Φ4 = −i √2 (|01 + |10 ). The global phases on these states are there for the property that one cannot express product states in this basis with purely real coefficients, though in general global phases are not important, since they are physically undetectable. (Recall that a product state is a pure state that can be expressed as a tensor product of its com-posite systems, so that all reduced density states are also pure. An example is any computational basis state. Product states are not entangled.) A proof of the theorem is given in [18] in terms of linear transformations. However, later in this paper when we want to generalize the magic matrix method to the three-qubit case, we need to employ the whole machinery developed in [17]. So here in the following we present a proof of the above theorem in terms of the orthogonal Lie algebra pair and the Cartan decomposition. Definition Let g be a real semisimple Lie algebra and let the decomposition g = m ⊕ l, m = l⊥, satisfy the following commutation relations: [l, l] ⊂ l, [m, l] = m, [m,m] ⊂ l. (4.2) We refer to this decomposition as a Cartan decomposition of g. Definition If h is a subalgebra of g contained in m, then h is abelian because [m,m] ⊂ l. Amaximal abelian subalgebra contained in m is called a Cartan subalgebra 63 of the pair (g, l). The two-qubit magic matrix is a well known result for the case with g = su(4). It has the direct sum decomposition g = m ⊕ l, where l = span i{σx ⊗ 11, σy ⊗ 11, σz ⊗ 11, 11 ⊗ σx, 11 ⊗ σy, 11 ⊗ σz}, m = span i{σx ⊗ σx, σx ⊗ σy, σx ⊗ σz σy ⊗ σx, σy ⊗ σy, σy ⊗ σz σz ⊗ σx, σz ⊗ σy, σz ⊗ σz} are vector spaces spanned with real coefficients and satisfying the commutation re-lations (4.2). The subalgebra l = su(2) ⊕ su(2), generates the subgroup, K = SU(2) ⊗ SU(2), which are the local unitary operations. Furthermore, h = span i{σx ⊗ σx, σy ⊗ σy, σz ⊗ σz} (4.3) is contained in m and is maximal abelian and hence a Cartan subalgebra of the symmetric space exp(m) = SU(4)/SU(2) ⊗ SU(2). We define the magic matrix M = exp(h) using this Cartan subalgebra, which leads to Eq. (4.1). The above theorem in differential geometry [12, 17] asserts that the Lie group G = SU(4) has a Cartan decomposition G = KMK, because the symmetric-space conditions (4.2) are satisfied. Thus, any SU(4) operator can be related to a magic matrix with merely local operations. 64 4.2 Entangling power of two-qubit braiding gates We first present the general formula for entangling power of an arbitrary two-qubit gate, and then in Chapter 5 we apply it to various two-qubit braiding gates. 4.2.1 General entangling power formula One can easily calculate the entangling power of the magic matrix (4.1), which contains three parameters, by starting from Eq. (3.10), and thus obtain [20] ep(M) = 1 9 [3 − (cos4c1cos4c2 + cos4c2cos4c3 + cos4c3cos4c1)]. (4.4) For any two-qubit operator in SU(4), one needs to find its Cartan decomposition, and extract the associated magic matrix, thus determining the value of the constants c1, c2, and c3 in Eq. (4.1). Substituting these constants in Eq. (4.4), one determines the entangling power. Alternatively, because the local unitary transformation does not change the deter-minant (in particular the characteristic equation) we may also determine the constants by directly comparing the eigenvalues of the magic matrix with the eigenvalues of the operator in question, up to a global phase. In Chapter 5, we show this calculation for some braiding operators. It is easy to show that the maximum entangling power for a two-qubit operator corresponds to {c1, c2, c3} = {π/4, φ, 0} (and its permutations, but we may always set their ordering as given above), with 0 ≤ φ ≤ π/4, π/2-periodic around π/4 (see, 65 e.g., [20, 18]). We also see that for this range of constants, ep(M) = 4/9. 4.3 Cartan decomposition for three qubits We would like to use a recursive procedure that extends the Cartan decomposition to n qubits, developed in [17]. Then we will work out the magic matrix (with n=3), to be used to compute the entangling power of three-qubit gates. 4.3.1 Direct Cartan decomposition of SU(8) Consider the following decomposition of the Lie algebra su(2n): su(2n) = span{σx ⊗ A, σy ⊗ B, σz ⊗ C, 11 ⊗ D, iσ1x, iσ1y, iσ1z A,B,C,D ∈ su(2n−1)} sul(2n) = span{σz ⊗ A, 11 ⊗ B, iσ1z A,B ∈ su(2n−1)} sum(2n) = span{σx ⊗ A, σy ⊗ B, iσ1x, iσ1y A,B ∈ su(2n−1)}, where σix, σiy, σiz are the Pauli matrices in the ith qubit. Lemma 1 The vector space sul(2n) is a Lie subalgebra of su(2n), and su(2n) = sul(2n) ⊕ sum(2n) is a Cartan decomposition; namely sul(2n) and sum(2n) satisfy the symmetric space commutation relations (4.2). Lemma 2 We have exp(sul(2n)) = U(1) ⊗ SU(2n−1) ⊗ SU(2n−1), of SU(2n). From these two lemmas, it follows that 66 Theorem 1 Any element U ∈ SU(2n) admits a Cartan decomposition: U = K1AK2, where K1,K2 ∈ U(1)⊗SU(2n−1)⊗SU(2n−1) and A = exp(Y ) ⊂ G for some Y ∈ h, with h a Cartan (maximal and abelian) subalgebra contained in sum(2n). We can choose the Cartan subalgebra for the matrix A for the n = 3 case as follows. Theorem 2 The vector space h = span{σx ⊗ σx ⊗ σx, σy ⊗ σy ⊗ σx, σy ⊗ σx ⊗ σy, σx ⊗ σy ⊗ σy} is a Cartan subalgebra contained in sum(8) (or that of the Lie algebra pair (su(8), sul(8))). Proof The proof consists of two parts. First, we must show that the basis elements shown above commute with each other. Second, we must show that no other members of sum(8) commute with all elements of this set. Hence, it is maximally abelian. The first part is straightforward to check. To prove the second part, we need to first list the elements of the set that generates m, sum(8) = span{σx ⊗ A, σy ⊗ B, iσ1x, iσ1y A,B ∈ su(4)}, and then explicitly check the relevant commutation relations. Since we wish to calculate the entangling power over a bipartite split between the first and the other two qubits, it will be invariant with respect to the local SU(2) ⊗ SU(4) operations. We note that some operators in sul(8) are still nonlocal, 67 i.e., do not belong to SU(2) ⊗ SU(4), so we need to further decompose sul(8) to isolate the nonlocal operators that are beyond SU(2)⊗SU(4). Eventually, we would like to capture all nonlocal entangling operations with an appropriately defined magic matrix. This will be carried out in next section. 4.3.2 Further decomposition of sul There are operators in sul(8) that are capable of changing the entangling power, so that the matrix A in Theorem 2, above, is incomplete as the magic matrix for three qubits. Our strategy is to make a further Cartan decomposition [17] for sul(8), to incorporate more entangling generators into an appropriate magic matrix. First we note that σ1z in sul(2n) generates a U(1) subgroup, so we need to separate it: sul(2n) = u(1) ⊗ sul(2n), and sul(2n) is semisimple. We Cartan-decompose the latter as follows: sul(2n) = span{σz ⊗ A, 11 ⊗ B A,B ∈ su(2n−1)} sul1(2n) = span{σz ⊗ A A ∈ su(2n−1)} sul0(2n) = span{11 ⊗ A A ∈ su(2n−1)}, They satisfy the commutation relations (4.2). In this way we end up with a decomposition of an SU(8) operator U that looks like U = L0 P = L0 L1 Z L2 = K1 A1 K2 Z K3 A2 K4, 68 where Ki ∈ U(1)⊗SU(4), Li are generated from sul(2n), P is generated from sum(2n), Z is generated from the Cartan subalgebra of the pair (su(2n), sul(2n)), and A is generated from the Cartan subalgebra of the pair (sul(2n), sul0(2n)). Now we would like to construct the magic matrix from the nonlocal Ai and Z, and will show in the next section that the entangling power is invariant under the action of Ki ∈ SU(2) ⊗ SU(4). One is tempted to add to Z the generators of a Cartan subalgebra f = span{σz ⊗ 11 ⊗ σz, σz ⊗ σz ⊗ 11, σz ⊗ σz ⊗ σz}. for the pair (sul(2n), sul0(2n)) that generates Ai. However, these generators do not all commute with the generators of Z. Thus, we arrive at an (augmented) magic matrix: M = exp−i(d8σz ⊗ σz ⊗ σz) × exp−i(c1σx ⊗ σx ⊗ σx + c2σy ⊗ σy ⊗ σx + c3σy ⊗ σx ⊗ σy + c4σx ⊗ σy ⊗ σy +c5σz ⊗ σ11 ⊗ σz + c6σz ⊗ σz ⊗ σ11) × exp−i(d7σz ⊗ σz ⊗ σz) 69 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ω1 0 0 0 0 0 0 ω7 0 ω5 0 0 0 0 ω3 0 0 0 ω9 0 0 ω15 0 0 0 0 0 ω13 ω11 0 0 0 0 0 0 ω12 ω14 0 0 0 0 0 ω16 0 0 ω10 0 0 0 ω4 0 0 0 0 ω6 0 ω8 0 0 0 0 0 0 ω2 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ω1 = e−i(5+6+7+8)c1−2−3−4 ω7 = −ie−i(5+6−7+8)s1−2−3−4 ω5 = e−i(−5−6−7−8)c1−2+3+4 ω3 = −ie−i(−5−6+7−8)s1−2+3+4 ω9 = e−i(−5+6−7−8)c−1−2−3+4 ω15 = ie−i(−5+6+7−8)s−1−2−3+4 ω13 = e−i(5−6+7+8)c−1−2+3−4 ω11 = ie−i(5−6−7+8)s−1−2+3−4 ω12 = ie−i(5−6+7−8)s−1−2+3−4 ω14 = e−i(5−6−7−8)c−1−2+3−4 ω16 = ie−i(−5+6−7+8)s−1−2−3+4 ω10 = e−i(−5+6+7+8)c−1−2−3+4 ω4 = −ie−i(−5−6−7+8)s1−2+3+4 ω6 = e−i(−5−6+7+8)c1−2+3+4 ω8 = −ie−i(5+6+7−8)s1−2−3−4 ω2 = e−i(5+6−7−8)c1−2−3−4, where the nonvanishing entries ωα (α = 1, 2, . . . , 16) are expressed in terms of the sine and cosine functions of ci (i = 1, 2, . . . , 8), c1−2−3−4 = cos(c1 − c2 − c3 − c4), c1−2+3+4 = cos(c1 − c2 + c3 + c4), . . . , s−1−2−3+4 = sin(−c1 − c2 − c3 + c4), and so on. 70 4.4 Three-qubit entangling power We would like to calculate the entangling power of this magic matrix, as we did for the bipartite magic matrix. From the cyclical property of the trace and the invariance of the entangling power [26, 25], ep(U1 ⊗ U3U) = ep(U) [Ui ∈ U(di)], we have ep(U) = ep(K1 A1 K2 Z K3 A2 K4) = ep(A1 Z A2), which justifies construction of the magic matrix without the Ki. Using Eq. (3.10), we have obtained the entangling formula for all ci (i = 1, 2, . . . , 8) nonvanishing, and it is presented in the appendix of this chapter, along with remarks on the maximal choices of parameters. Here we present only the result c7 = c8 = 0: ep(U) = 1 2 1 − 1 18 cos(4c1 − 4c4)cos(4c2 + 4c3) +cos(4c1 + 4c4)cos(4c2 + 4c3) +cos(4c2 + 4c3)cos(4c5 + 4c6) +cos(4c1 − 4c4)cos(4c5 + 4c6) +cos(4c2 − 4c3)cos(4c5 − 4c6) +cos(4c1 + 4c4)cos(4c5 − 4c6) − 1 9 cos(4c1)cos(4c3) + cos(4c1)cos(4c6) +cos(4c2)cos(4c4) + cos(4c2)cos(4c5) +cos(4c3)cos(4c6) + cos(4c5)cos(4c4) . 71 This expression is maximal at ep = 2/3 for {c1, c2, c3, c4, c5, c6} = {π/4, π/4, 0, φ, 0, 0}, with 0 ≤ φ ≤ π/4, π/2-periodic around π/4. 1 Because the entangling power in the multiqubit case requires a choice of partition, one might be tempted to suggest taking the average over all choices of partition. Often, a partition is determined by the problem, but in attempts to construct some genuine multipartite measure this might make sense. So, I note that in the case of GHZ generators, the entangling power does not depend on the choice of partition. 4.4.1 Three-qubit partial magic matrix Z With the Cartan subalgebra, h, we can form a three-qubit partial magic matrix. The Cartan subalgebra is not unique, and was chosen in this case to generate the magic matrix whose eigenstates are the GHZ states. Z = exp−i(c1σx ⊗ σx ⊗ σx + c2σy ⊗ σy ⊗ σx + c3σy ⊗ σx ⊗ σy + c4σx ⊗ σy ⊗ σy) 1With a maximal choice of parameters and φ = 0, we get a matrix that generates entangled Werner states from the sum of computational basis states. As such, it is not a generator in the same sense as a GHZ generator, which generates all the GHZ states, one for each basis state. Werner states satisfy (U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) for unitary operators U on H, as defined in [8]. The matrix in this case is W = 000 000 + 001 001 + 110 110 + 111 111 −i 101 010 + 100 011 + 011 100 + 010 101 , and from the sum of computational basis states, it generates the state |W = |000 + |001 + |110 + |111 −i {|101 + |100 + |011 + |010 } . 72 = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ c1 0 0 0 0 0 0 −is1 0 c−2 0 0 0 0 −is−2 0 0 0 c4 0 0 is4 0 0 0 0 0 c3 is3 0 0 0 0 0 0 is3 c3 0 0 0 0 0 is4 0 0 c4 0 0 0 −is−2 0 0 0 0 c−2 0 −is1 0 0 0 0 0 0 c1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (4.5) where c1 = c1−2−3−4 = cos(c1 − c2 − c3 − c4), c−2 = c1−2+3+4 = cos(c1 − c2 + c3 + c4), . . . , s4 = s−1−2−3+4 = sin(−c1 − c2 − c3 + c4), and so on. This has an analogous maximally entangled eigenbasis, i.e., a magic basis with global phases to maintain concurrence properties (see for example [20]). In SU(4), one cannot express product states in the magic basis with purely real coefficients. We introduce these global phases for the same reason, but they are otherwise arbitrary and physically undetectable. The basis states are |Φ1 = e−iπ3 √2 (|000 + |111 ), |Φ8 = e−i π 6 √2 (|000 − |111 ) |Φ2 = e−iπ3 √2 (|001 − |110 ), |Φ7 = e−i π 6 √2 (|001 + |110 ) |Φ3 = eiπ √2 (|010 + |101 ), |Φ6 = ei π 2 √2 (|010 − |101 ) |Φ4 = eiπ √2 (|011 − |100 ), |Φ5 = ei π 2 √2 (|011 + |100 ). 73 4.5 Conclusion Our original motivation has been to calculate the entangling power of some braid-ing operators, including the GHZ generators. To this end, we have extended the magic matrix method to SU(8) for three qubits. To calculate the entangling power of a unitary operator, it is necessary only to know the eigenvalues, up to a global phase. One compares these eigenvalues to the eigenvalues of the magic matrix to determine the constants that appear in the entangling power formula (4.4). We have found that the GHZ generators in which we were originally interested are not maximally entangling. Instead, other choices of parameters yield maximally en-tangling operators. But until there is a general account of multipartite entanglement, one cannot interpret this result in the right context. Finally we note that our formalism of the recursive Cartan decomposition suggests that information of entangling power, as captured by the parameters of the operators in the Cartan subalgebra, is encoded in roughly of order O(2n) constants for oper-ators in SU(2n). Used in conjunction with entanglement combing, this could be a useful criterion of the success of multiparty protocols such as multipartite quantum communication, multipartite teleportation, and distributed compression. 74 4.6 Appendix Here we present the entangling power formula for the augmented magic matrix, including all eight parameters. We have obtained this formula from Eq. (3.10). ep(U) = 1 2 1 − 1 18 cos(4c1 − 4c2)cos(4c3 + 4c4) + cos(4c1 + 4c2)cos(4c3 − 4c4) +cos(4c1 − 4c3)cos(4c5 + 4c6) + cos(4c1 + 4c3)cos(4c5 − 4c6) +cos(4c2 + 4c4)cos(4c5 + 4c6) + cos(4c2 − 4c4)cos(4c5 − 4c6) +cos(4c1)cos(4c5)cos(4c7)cos(4c8) + cos(4c1)cos(4c6)cos(4c7)cos(4c8) +cos(4c2)cos(4c5)cos(4c7)cos(4c8) + cos(4c2)cos(4c6)cos(4c7)cos(4c8) +cos(4c3)cos(4c5)cos(4c7)cos(4c8) + cos(4c3)cos(4c6)cos(4c7)cos(4c8) +cos(4c4)cos(4c5)cos(4c7)cos(4c8) + cos(4c4)cos(4c6)cos(4c7)cos(4c8) +cos(4c1)cos(4c6)cos(4c8) − cos(4c1)cos(4c5)cos(4c8) +cos(4c2)cos(4c5)cos(4c8) − cos(4c2)cos(4c6)cos(4c8) +cos(4c3)cos(4c5)cos(4c8) − cos(4c3)cos(4c6)cos(4c8) +cos(4c4)cos(4c6)cos(4c8) − cos(4c4)cos(4c5)cos(4c8) − 1 9 cos(4c1)cos(4c4) + cos(4c2)cos(4c3) −cos(4c1 − 4c2)cos(4c3 − 4c4)cos(4c5)sin(4c7)sin(4c8) −cos(4c1 + 4c2)cos(4c3 + 4c4)cos(4c5)sin(4c7)sin(4c8) −cos(4c1 − 4c2)cos(4c3 − 4c4)cos(4c6)sin(4c7)sin(4c8) −cos(4c1 + 4c2)cos(4c3 + 4c4)cos(4c6)sin(4c7)sin(4c8) . 75 This expression is still maximal at ep = 2/3 for {c1, c2, c3, c4, c5, c6, c7, c8} = {π/4, π/4, 0, φ, 0, 0, 0, 0}, with 0 ≤ φ ≤ π/4, π/2-periodic around π/4. We can show this is maximal by checking that all partial derivatives are zero at this point. We can also check double derivatives, though we already can check that ep = 0 when all pa-rameters are zero, so that 2/3 is not a minimum. The monotonicity of the entangling power as parameters range 0 ≤ ci ≤ π/4 is inherited from the von Neumann entropy, since the linear entropy comes from the dominant terms in the Taylor expansion of the logarithm. CHAPTER 5 BRAIDING GATES The material in this chapter relating to the calculation of entangling power for the gates we discuss will appear in two papers I coauthored with Yong-Shi Wu, in preparation for publication, as well as the paper already cited. A recent approach to understanding quantum entanglement is motivated by an analogy [1] between entangled quantum states and topological entanglement known in knots. In Ref. [15], Kauffman and Lomonaco introduced two-qubit braiding quantum gates, which carry out braiding operations of qubits, in a way similar to those in the theory of knots and links. They have shown that the two-qubit braiding gates are universal, when used together with one-qubit gates. In principle, quantum circuits consisting of only braiding gates may be used to implement the so-called topological quantum computation [11, 10, 22], a new approach to implementing fault-tolerant quantum computation. Another advantage of the two-qubit braiding gate proposed by Kauffman and Lomonaco is that it produces the well-known (maximally entangled) Bell states when acting upon computational basis states, which are nonentangled. More recently, a three-qubit braiding quantum gate has been proposed in refs. [29, 21, 2]. It has been shown that this gate, when acting upon the computational basis, 77 produces the famous (entangled three-qubit) GHZ states. Therefore, in consistency with the analogy with topological entanglement in knots and links, braiding gates indeed produce quantum entanglement among the qubits in a quantum circuit. The three-qubit braiding gates proposed in [29] can actually be understood as the braiding between the first qubit and the other two as a whole [21]. The entangling power of such three-qubit braiding gates is invariant under the local operations of SU(2) ⊗ SU(4). But SU(8)/SU(2) ⊗ SU(4) is not a symmetric space as in the two-qubit case. Therefore, we adopted in Chapter 4 a recursive Cartan decomposition technique [17] to define the three-qubit magic matrix, and construct it for three-qubit braiding gates that we are interested in. 5.1 Braiding quantum gates This section begins with a brief introduction to braiding quantum gates, and then continues with some elaboration on the initial ideas introduced. But it is not comprehensive, as the theory behind these ideas spans a few well developed and sophisticated areas of mathematics, with which, in a broader perspective, I have barely passing familiarity. The material can be found in papers authored by my advisor, Yong-Shi Wu, and collaborators [29, 21]. A two-qubit braiding quantum gate is an ˇR -matrix (with p = 2) satisfying the braided Yang-Baxter equations (YBE) [15]: (ˇR ⊗ 11p)(11p ⊗ ˇR )(ˇR ⊗ 11p) = (11p ⊗ ˇR )(ˇR ⊗ 11p)(11p ⊗ ˇR ). (5.1) 78 Here the p2 × p2 matrix ˇR : V ⊗ V → V ⊗ V , where V = Cp, is unitary. The relation (5.1) gives rise to a sequence of representations (πn, (Cp)⊗n) of the braid group Bn: πn(bi) = 11⊗i−1 p ⊗ ˇR ⊗ 11⊗n−i−1 p , since clearly πn(bi) satisfy the braid group relations. A three-qubit braiding quantum gate is a special case of the following definition for solutions of the generalized Yang-Baxter equation [21, 29]: Definition Fix p with 2 ≤ p ∈ N and let l = pk. A unitary pN × pN matrix ˇR is a solution to the generalized Yang-Baxter equations, if (ˇR ⊗ 11l)(11l ⊗ ˇR )(ˇR ⊗ 11l) = (11l ⊗ ˇR )(ˇR ⊗ 11l)(11l ⊗ ˇR ), (5.2) as operators on (Cp)⊗(k+N). When N = 2, k = 1, the generalized YBE (5.2) is the same as the conventional YBE (5.1). If k ≥ N/2, the assignment πn(bi) = 11⊗i−1 l ⊗ ˇR ⊗ 11⊗n−i−1 l defines a sequence of representations (πn, (Cp)⊗(N+k(n−2))) of the braid group Bn. The special case with p = 2,N = 3, k = 2 gives rise to a three-qubit braiding gate, with braiding between the first qubit and the other two. For examples of two- and three-qubit braiding gates, see the references [15, 29, 21], and later in this chapter. We now show how these braiding gates can be formed from solutions to the YBE. 5.1.1 Extraspecial 2-groups Extraspecial 2-groups have been seen to play a role in studying images of the braid group under the 4×4 Bell matrix representation [9]. Along these lines, what will follow 79 in the next couple sections is a similar approach to the GHZ states, higher dimensional generalizations of the Bell states. We start with a brief sketch of extraspecial 2-groups. The group Em is the abstract group generated by e1, . . . , em, obeying the rules, e2i = −11, eiej = ejei, |i − j| ≥ 2 ei+1ei = −eiei+1, 1 ≤ i, j ≤ m − 1. Here −11 is an order 2 central element, and we write −11a simply as −a. A group G of order 2m+1 is an extraspecial 2-group if the center Z(G) and the commutator subgroup G (the subgroup generated by [ei, ej] = e−1 i e−1 j eiej) coincide and are isomorphic to Z2 and G/Z(G) ∼= (Z2)m. The commutator subgroup of Em is {±11}, due to the (anti-) commutation relations. Noting the centers listed below, then E2k is an extraspecial 2-group, but when m is odd, the center of Em has order 4, not coinciding with the commutator subgroup. However, since Em−1 ⊂ Em ⊂ Em+1 we can derive an extraspecial 2-group from Em by adding or removing a generator, and we call it a nearly extraspecial 2-group. Every element in Em can be expressed in a unique normal form, ±eα1 1 . . . eαm m , where αi ∈ Z2 and Em/{±11} ∼= (Z)m. Denote by Z(Em) the center of Em, Z(Em) = ⎧⎪⎪⎪⎨ ⎪⎪⎪⎩ {±11} m even {±11,±e1e2 . . . em} m odd 80 5.1.2 Almost complex structures and representations of Em Now we introduce almost complex structures with the goal of constructing repre-sentations of nearly extraspecial 2-groups. The particular almost complex structures we consider are defined in terms of the 2n × 2n anti-Hermitian matrix M2n, M2n = 2n i=1 (i) i ¯i , ¯i = 2n + 1 − i, where the Dirac kets |i form and orthonormal basis, and the generalized step func-tions, (i) satisfy (k) (k) = 1 and (k) (¯k) = −1. The constraints on the step func-tions have solutions (k) = ±1 and (¯k) = ∓1, and either ¯k = −k or ¯k = 2n + 1 − k depending on the range of k, in this case the latter. M2n satisfies (M2n)2 = −112n and (M2n)† = −M2n, and is determined only up to the n choices of signs for (i), 1 ≤ i ≤ n. For any two such choices of signs, the corresponding matrices are related by conjuga-tion by a diagonal matrix. We define two projectors that we need, in terms of the almost complex structure M2n, by P+ = 1 2 (112n + √−1M2n), P− = 1 2 (112n − √−1M2n), that satisfy the basic properties of two mutually orthogonal projectors, P+ + P− = 112n, P2± = P±, P+P− = P−P+ = 0. It is possible to construct two classes of representations for the group Em in terms of the almost complex structure M2n. We will give two classes of almost complex 81 structures that are for this purpose. However, we will skip the constructions of the representations, and use the almost complex structures to express the generalized Bell matrices, which are associated with the GHZ states and are used to construct the representations of the Braid group. Class (1): The almost complex structure is a (2k)2 × (2k)2 matrix MJJ with complex deformation parameters qij ∈ C, MJJ = J i,j=−J (i)qij ij ¯i ¯j , ¯i = −i, ¯j = −j, J = k − 1 2 , k ∈ N, which will be used to express the Bell matrices that generate GHZ states of an even number of objects. Class (2): There are two natural ways to generalize the almost complex structures of Class(1). First, we may consider more general almost complex structures of the form M2k1⊗P2k2, where k1 = k2 (in Class(1), we have something similar, but k1 = k2). Second, we may use the generalized form of the braided YBE, where the almost complex structure is promoted to spaces of different dimensions. For simplicity and with GHZ states in mind, we consider a special case: k1 = 1 and k2 = 2N−2, N ≥ 2. The almost complex structure is chosen to be M2N ≡ M2 ⊗P2N−1 = √−1σy ⊗σ⊗N−1 x . This allows us to interpret the 3-qubit braiding as occurring between the first qubit and the other two as a whole. 82 5.1.3 GHZ states and generalized Bell matrices GHZ states can be obtained from the unitary braid representations, constructed in terms of the representations of extraspecial 2-groups. They can be generated by the action of the Bell matrices, which are used to construct the unitary braid representations, on the computational basis. We begin this section by discussing the form of the states and unitary transformations of them. Then we will express the Bell matrices in terms of the two classes of almost complex structures, corresponding to an even or odd number of qubits. The two-dimensional Hilbert space H2 spanned by eigenvectors |m , m = ±1/2 of the spin-1/2 operators (Pauli matrices) has coordinate vectors over the complex field C2, 1 2 % := 1 0 , − 1 2 % := 0 1 , α 1 2 % + β − 1 2 % := α β , which determines the actions of σx and √−1σy on the basis |m , σx |m = | ¯m , √−1σy |m = (m) | ¯m , where ¯m = −m and the step function (1/2) = − (−1/2) = −1. A state vector in H2 is called a qubit, and H2 ∼= C2. The Hilbert space H2N is isomorphic to (C2)⊗N and describes a system of N qubits, each spanned by the two linearly independent states, up or down. The orthonormal basis is denoted by |Φk , 1 ≤ k ≤ 2N, which are tensor products of the basis states of H2, |Φk ≡ |m1, . . . , mN ≡ |m1 ⊗ ·· · ⊗ |mN . 83 The index k labels the states in increasing binary numerical order, under a relabeling 1/2 → 0 and −1/2 → 1. This orthonormal basis |Φk is partitioned into two sets, |Φl and |Φ¯l , 1 ≤ l ≤ 2N−1 and ¯l = 2N − l + 1, |Φl = |m1, . . . , mN , |Φ¯l = | ¯m1, . . . , ¯mN , ¯mi = −mi, 1 ≤ l ≤ N. In terms of |Φl and |Φ¯l , H2N is spanned by the 2N orthonormal GHZ states |Ψl and |Ψ¯l of N qubits, |Ψl ≡ 1 √2 (|Φl + |Φ¯l ) , |Ψ¯l ≡ 1 √2 (|Φl − |Φ¯l ). These are the usual GHZ states, and they form an orthonormal basis of H2N as do their unitary transformations |Ψ l = U |Ψl , with fixed unitary transformation U. The unitary transformation matrix from the computational basis |Φl to the orthonormal basis |Ψ l has the form |Ψ 1 , |Ψ 2 , . . . , Ψ 2N . We say it generates the GHZ states from the computational basis. These GHZ states can be generated by the action of these generalized Bell matrices (GHZ generators) on the computational basis |Φl , |Φ¯l . Now we will show how to express the Bell matrices in terms of the almost complex structure. 5.1.3.1 Even number of qubits The GHZ states of an even number of qubits, say 2N, are associated to the gen-eralized Bell matrix BJJ in terms of the almost complex structure MJJ in Class(1), BJJ = 11(2k)2 +MJJ, J= k − 1 2 , k ∈ N. 84 BJJ is a (2k)2 ×(2k)2 matrix, while the 2n qubit GHZ states span a 22n dimensional Hilbert space. So for the GHZ states generated by BJJ to span a 22n dimensional Hilbert space, we must have k = 2n−1 so that J = 2n−1 − 1/2. Then, the almost complex structure in Class(1) has the form, MJJ = M2k ⊗ P2k = M2n ⊗ P2n, J= 2n−1 − 1 2 , k = 2n−1, which can be re-expressed as MJJ = J i,j=−J (i)qiqj ij ¯i ¯j = 2n μ,ν=1 (μ)qμqν μν ¯μ¯ν with the generalized step function (μ) (¯μ) = −1 and ¯μ = 2n + 1 − μ, ¯ν = 2n + 1 − ν, qμ = ei ϕμ 2 , q¯μ = ei ϕ¯μ 2 . Let the orthonormal base |μν have a form in terms of the orthonormal basis |Φk for the 2n-dimensional Hilbert space, |α := |μν = Φ(μ−1)2n+ν . When the Bell matrix BJJ of Class(1) acts on the orthonormal basis |α , the result will be the unitary transformation to GHZ states in the 2n-dimensional Hilbert space, 1 √2 |α + (μ)ei ϕμ+ϕν 2 |¯α , consistent with the earlier definition of GHZ states in terms of |Φl and |Φ¯l . As an example for Class(1), set J = 1/2 or n = 1. The Bell matrix B 1 2 1 2 4 has the 85 form B 1 2 1 2 4 ≡ 1 √2 (114 +M2 ⊗ P2) = 1 √2 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 1 0 0 eiϕ 0 1 1 0 0 −1 1 0 −e−iϕ 0 0 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , where the unimodular deformation parameters follow the convention qij = qiqj and q1 2 1 2 = eiϕ, q−1 2−1 2 = e−iϕ, q1 2−1 2 = q−1 2 1 2 = 1, satisfying certain constraint equations that can be found in [21]. Thus, we have a unitary basis transformation matrix from the computational basis |Φk to the or-thonormal basis by unitary transformations of Bell states |Ψk , B 1 2 1 2 4 = ((112 ⊗ u0,ϕ) |Ψ4 , |Ψ3 , |Ψ2 , (112 ⊗ uϕ,0) |Ψ1 ) , where the two unitary matrices u0,ϕ and uϕ,0 are u0,ϕ = ⎛ ⎜⎜⎝ 1 0 0 e−iϕ ⎞ ⎟⎟⎠ , uϕ,0 = ⎛ ⎜⎜⎝ eiϕ 0 0 1 ⎞ ⎟⎟⎠ . We say that this matrix generates the Bell states |Ψk from the computational basis. As another example for Class(1), uniform deformation parameters are chosen to be 1, and the almost complex structures M2n and P2n take the forms M2n = √−1σy ⊗ σ⊗n−1 x , P2n = σ⊗n x , which gives rise to the Bell matrix, BJJ 22n = 1 √2 (1122n +MJJ 22n), MJJ 22n = √−1σy ⊗ σ⊗2n−1 x . 86 The GHZ states of 2n qubits obtained by this Bell matrix on the computational basis |Φk with the step function (m1) are 1 √2 (|m1, . . . , m2n + (m1) | ¯m1, . . . , ¯m2n ), with which we can express the unitary basis transformation matrix as, BJJ 22n = (|Ψ22n , |Ψ22n−1 , . . . , |Ψ2 , |Ψ1 ). 5.1.3.2 Odd number of qubits The Bell matrices from Class(1) cannot generate GHZ states on an odd number of qubits. But Class(2) can, since we use a generalized form of the YBE that allows the almost complex structure to have spaces of different dimensions. The unitary basis transformation matrices that generate the GHZ states from the product basis are B22n+1 = 1 √2 (1122n+1 +M22n+1), M22n+1 = √−1σy ⊗ (σx)⊗2n, without unimodular deformation parameters or generalized step functions. We em-phasize again differences between Class(1) and Class(2). Every strand in the braid group of Class(1) lives in a vector space of the same dimension. In Class(2), this is not necessarily the case. We can regard this as a sort of generalization, where we rel