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- Quasiprobability representations of qubits

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12110070

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 nonnegative

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.

©2012 Perimeter Institute for Theoretical Physics