| Description |
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. |
