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CIENCIA ergo sum

One-qubit purity in terms of the discrete Wigner transform

Manuél Ávila

Published Jan 14, 2020• 0 citations
Publisher

Abstract

An explanation and an illustration of the meaning of a discrete phase-space is given. The class of a discrete Wigner transform (DWT) for the specific case of a one-qubit state is introduced. We derive the one-qubit state formalism around its formulation in terms of the DWT in detail. A novel structure of a one-qubit purity in terms of the DWT is introduced. We find a criterion for stating when a one-qubit state is either mixed or pure.

DOI
10.30878/ces.v27n1a9

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An explanation and an illustration of the meaning of a discrete phase-space is given. The class of a discrete Wigner transform (DWT) for the specific case of a one-qubit state is introduced. We derive the one-qubit state formalism around its formulation in terms of the DWT in detail. A novel structure of a one-qubit purity in terms of the DWT is introduced. We find a criterion for stating when a one-qubit state is either mixed or pure.
For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a d×dd \times dd×d phase space when the Hilbert-space dimension ddd is odd. It does not hold for a d×dd \times dd×d phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a [Formula: see text] phase space when the Hilbert-space dimension [Formula: see text] is odd. It does not hold for a [Formula: see text] phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality NNN. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator PPP and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
Abstract. We study the properties of the discrete Wigner distribution for two qubits introduced by Wotters [1]. In particular, we analyze the entanglement properties within the Wigner distribution picture. We formulate the separability criteria coming from the Partial Transposition (PT) criterion, the Local Uncertainty Relations (LUR), and the Generalized Uncertainty Principle (GUP) in terms of discrete Wigner functions. Based on PT criterion, we establish the relation between non-negativity of Wigner function and the separability of the corresponding state. A mathematical object of great importance is the covariance matrix relevant to the axis operators. We show that for a separable state the correlations encoded in this object have an upper limit. This allows us to derive a separability criterion –stronger than the one based on LUR– in terms of the covariance matrix. PACS numbers: 03.67.Mn, 03.65.Wj, 42.50.Dv § To whom correspondence should be addressed riccardo.franco@polito.itDiscrete Wigner distribution for two qubits: a characterization... 2
Gibbons et al., [Phys. Rev. A 70, 062101 (2004)] have recently defined discrete Wigner functions WWW to represent quantum states in a Hilbert space with finite dimension. We show that such a class of Wigner functions WWW can be defined so that the only pure states having non-negative WWW for all such functions are stabilizer states, as conjectured by Galv\~ao, [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of WWW for all definitions of WWW in the class form a subgroup of the Clifford group. This means pure states with non-negative WWW and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.