On the Clifford collineation, transform and similarity groups. II.

Bolt, Beverley; Room, T. G.; Wall, G. E.

doi:10.1017/s1446788700026380

Cited by 48 publications

(90 citation statements)

References 3 publications

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“…Any Clifford unitary U induces a symplectic transformation on the symplectic space that labels displacement operators [25][26][27]. The quotient C/D (G denotes the group G modulo phase factors) can be identified with the symplectic group Sp(2n, p).…”

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confidence: 99%

Permutation Symmetry Determines the Discrete Wigner Function

Zhu

2016

Phys. Rev. Lett.

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The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2-design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension. The Wigner quasiprobability distribution in phase space, originally introduced for studying quantum correction to thermodynamics, has numerous applications in various branches of physics, such as quantum optics, quantum chaos, and quantum computing. It also provides an alternative formulation of quantum mechanics, which is particularly suitable for studying quantumclassical correspondence [1][2][3]. The usefulness of the Wigner function is intimately connected to the high symmetry of the underlying operator basis composed of phase point operators. Reminiscent of the symplectic geometry in classical phase space, this basis is invariant under displacements and linear canonical transformations, which realize translations, rotations, and squeezing in phase space. Therefore, "no point, no direction, no scale is distinguished from any other in phase space" [4]. This means that the symmetry group of the basis acts doubly transitively on phase point operators; that is, any pair of distinct phase point operators can be turned into any other pair by a unitary symmetry transformation.Recently, many discrete analogues of the Wigner function have been introduced and found various applications in quantum information science, such as quantum state tomography and quantum computation [5][6][7][8][9]. In addition, general quasiprobability representations have been found useful for studying quantum foundations [10, 11]. At this point, it is natural to reflect on the following questions: what is so special about the Wigner function? Is there a simple criteria that can single out a particular quasiprobability representation among all potential candidates? Although similar questions have been investigated extensively [1, 5, 7, 12], no satisfactory answer is known, especially in the discrete scenario. In every odd prime dimension, the Wootters discrete Wigner function [5] is uniquely determined by Clifford covariance [12]. However, the situation is not clear for other dimensions, despite the intensive efforts of many researchers over the last three decades.In this paper we show that the ...

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confidence: 99%

Permutation Symmetry Determines the Discrete Wigner Function

Zhu

2016

Phys. Rev. Lett.

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“…Basic theory of extraspecial groups extended upwards by their outer automorphism group has been developed in several places. We shall use [GrEx,GrMont,GrDemp,GrNW,Hup,BRW1,BRW2,B]. Proof.…”

Section: The Nonsplit Defect 1 Casementioning

confidence: 99%

Involutions on the Barnes-Wall Lattices and their Fixed Point Sublattices, I

Griess¹

2005

Pure and Applied Mathematics Quarterly

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“…H i is the Clifford group CT [4,5,14], which in recent years has been used in the classification of finite simple groups (see the references in [9]). H i is relevant for the present work because of its connection with the Barnes-Wall lattices.…”

Section: Firstmentioning

confidence: 99%