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arXiv:quant-ph/0608200 · Physical Review A

Interference in discrete Wigner functions

Cecilia Cormick, Juan Pablo Paz

Published Dec 22, 2006• 7 citations
arXivHTMLPDF

Abstract

We analyze some features of the class of discrete Wigner functions that was recently introduced by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to represent quantum states of systems with power-of-prime dimensional Hilbert spaces. We consider ``cat'' states obtained as coherent superpositions of states with positive Wigner function; for such states we show that the oscillations of the discrete Wigner function typically spread over the entire discrete phase space (including the regions where the two interfering states are localized). This is a generic property, which is in sharp contrast with the usual attributes of Wigner functions that make them useful candidates to display the existence of quantum coherence through oscillations. However, it is possible to find subsets of cat states with a natural phase-space representation, in which the oscillatory regions remain localized. We show that this can be done for interesting families of stabilizer states used in quantum error-correcting codes, and illustrate this by analyzing the phase-space representation of the five-qubit error-correcting code.

DOI
10.1103/physreva.74.062315

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We analyze some features of the class of discrete Wigner functions that was recently introduced by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to represent quantum states of systems with power-of-prime dimensional Hilbert spaces. We consider ``cat'' states obtained as coherent superpositions of states with positive Wigner function; for such states we show that the oscillations of the discrete Wigner function typically spread over the entire discrete phase space (including the regions where the two interfering states are localized). This is a generic property, which is in sharp contrast with the usual attributes of Wigner functions that make them useful candidates to display the existence of quantum coherence through oscillations. However, it is possible to find subsets of cat states with a natural phase-space representation, in which the oscillatory regions remain localized. We show that this can be done for interesting families of stabilizer states used in quantum error-correcting codes, and illustrate this by analyzing the phase-space representation of the five-qubit error-correcting code.
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.
We study the class of discrete Wigner functions proposed by Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] to describe quantum states using a discrete phase space based on finite fields. We find the extrema of such functions for small Hilbert-space dimensions and present a quantum-information application: a construction of quantum random-access codes. These are constructed using the complete set of phase-space point operators to find encoding states and to obtain the codes' average success rates for Hilbert-space dimensions 2, 3, 4, 5, 7, and 8.
Gibbons et al. [Phys. Rev. A 70, 062101 (2004)] have recently defined a class of discrete Wigner functions WWW to represent quantum states in a finite Hilbert space dimension ddd. I characterize the set Cd{C}_{d}Cd​ of states having non-negative WWW simultaneously in all definitions of WWW in this class. For $d\ensuremath{\leqslant}5$ I show Cd{C}_{d}Cd​ is the convex hull of stabilizer states. This supports the conjecture that negativity of WWW is necessary for exponential speedup in pure-state quantum computation.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
We present a phase-space description of the process of quantum teleportation for a system with an N-dimensional space of states. For this purpose we define a discrete Wigner function which is a minor variation of previously existing ones. This function is useful to represent a composite quantum system in phase space and to analyze situations where entanglement between subsystems is relevant (dimensionality of the space of states of each subsystem is arbitrary). We also describe how a direct tomographic measurement of this Wigner function can be performed.
A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.