Permutation Symmetry Determines the Discrete Wigner Function

Zhu, Huangjun

doi:10.1103/physrevlett.116.040501

Cited by 39 publications

(51 citation statements)

References 43 publications

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“…If Sym (S) acts doubly transitively on S [66, p. 225], i.e., if for every x 1 , x 2 , y 1 , y 2 ∈ S, x 1 = x 2 and y 1 = y 2 , there is g ∈ Sym(S) such that g (x i ) = y i for i = 1, 2, we shall call such a set super-symmetric after [133,135]. It is well known that doubly transitive group action is primitive [66,Lemma 8.16].…”

Section: Proposition 3 If Sym(s) Acts Primitively On S (Ie the Onlmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.

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Section: Proposition 3 If Sym(s) Acts Primitively On S (Ie the Onlmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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“…It cannot be extended to systems of qubits due to the presence of state-independent contextuality [23][24][25]. In addition, no qubit Wigner function that satisfies all the properties of Gross' qudit Wigner function seems to exist [41][42][43].This article is organized as follows. The necessary stabilizer formalism is recalled in section 2.…”

mentioning

confidence: 99%

Equivalence between contextuality and negativity of the Wigner function for qudits

et al. 2017

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Understanding what distinguishes quantum mechanics from classical mechanics is crucial for quantum information processing applications. In this work, we consider two notions of nonclassicality for quantum systems, negativity of the Wigner function and contextuality for Pauli measurements. We prove that these two notions are equivalent for multi-qudit systems with odd local dimension. For a single qudit, the equivalence breaks down. We show that there exist single qudit states that admit a non-contextual hidden variable model description and whose Wigner functions are negative. function in terms of the hidden variable model (HVM) and we observe that if the HVM is non-contextual then the Wigner function is non-negative. Howard et al required a two-qudit experiment to demonstrate contextuality of a single qudit state. We elucidate this by showing explicitly that in a single qudit experiment, the equivalence between contextuality and negativity does not hold. We show this by constructing quantum states that admit a noncontextual HVM (NCHVM) description although their Wigner functions take negative values.Our proof of this equivalence relies on the choice of a simple definition of contextuality based on value assignments introduced in the work of Kochen and Specker [22], whereas the work of Howard et al is based on the graphical formalism of Cabello et al [39]. The relations between these different notions of non-classicality are depicted in figure 1. Following [22][23][24], we consider only contextuality of stabilizer measurements and HVMs are assumed to be deterministic. This last assumption is also present in the graphical formalism of Cabello et al [39] and in the work of Howard et al [18]. Our argument does not apply to the generalized notion of HVMs considered for instance by Spekkens [38,40].Our work clarifies the relationship between discrete Wigner function and stabilizer contextuality for odddimensional qudits. It cannot be extended to systems of qubits due to the presence of state-independent contextuality [23][24][25]. In addition, no qubit Wigner function that satisfies all the properties of Gross' qudit Wigner function seems to exist [41][42][43].This article is organized as follows. The necessary stabilizer formalism is recalled in section 2. Section 3 introduces a notion contextuality based on value assignment and a proof of the equivalence between this notion and the negativity of Gross' discrete Wigner function, i.e. (ii) ⇔ (iv). The purpose of section 4 is to prove the equivalence between the two notions of contextuality (i) and (ii), completing the square in figure 1.

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“…Wootters relations (10) and (11) in [14] follow. Phase point operators have been built to satisfy properties analogous to those of the continuous phase space in the context of the continuous Wigner function W (q, p) = ρ(q + x/2, q − x/2) exp(ipx)dx in which p and q are position and momentum, and ρ(x, x ′ ) = ψ * (x)ψ(x) is a density matrix for a particle of coordinate x in a pure state of wave function ψ(x) [1, p. 477].…”

Section: The Generalized Pauli Group the Discrete Wigner Function Anmentioning

confidence: 92%

“…Wigner function was recognized to contain permutation symmetry in its structure [11]. Interestingly enough, the experimental implementation of a simple quantum algorithm for determining the parity of a permutation was performed [12].…”

Section: Introductionmentioning

confidence: 99%