Joel Wallman: Quasiprobability representations of qubits
Joel Wallman, The University of Sydney
Nov 22, 2012
from 12:00 PM to 01:00 PM
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Negativity in a quasi-probability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude bases that are non-negative from having interesting applications---the single-qubit stabilizer states have non-negative Wigner functions and yet play a fundamental role in many quantum information tasks. We determine what other sets of quantum states and measurements of a qubit can be non-negative in a quasiprobability representation, and identify nontrivial groups of unitary transformations that permute such states. These sets of states and measurements are analogous to the single-qubit stabilizer states. We show that no quasiprobability representation of a qubit can be non-negative for more than two bases in any plane of the Bloch sphere. Furthermore, there is a single family of sets of four bases that can be non-negative in an arbitrary quasiprobability representation of a qubit. We provide an exhaustive list of the sets of single-qubit bases that are non-negative in some quasiprobability representation and are also closed under a group of unitary transformations, revealing two families of such sets of three bases. We also show that not all two-qubit Clifford transformations can preserve non-negativity in any quasiprobability representation that is non-negative for the computational basis. This is in stark contrast to the qutrit case, in which the discrete Wigner function is non-negative for all n-qutrit stabilizer states and Clifford transformations. We also provide some evidence that extending the other sets of non-negative single-qubit states to multiple qubits does not give entangled states.