Mutually unbiased bases with free parameters

Goyeneche, Dardo; Gómez, Santiago

doi:10.1103/physreva.92.062325

Cited by 7 publications

(5 citation statements)

References 40 publications

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“…M w . However from (11), we have that the positive (resp., negative) entries of M i represent adjacency between vertices in the first and i th fibers of the µ-heavy (resp., ν-heavy) LSSD. Therefore each M i must have k positive entries and v − k negative entries giving that H i must be regular.…”

Section: Hadamard Matrix Equivalencementioning

confidence: 93%

Linked systems of symmetric designs

Kodalen¹

2019

Algebraic Combinatorics

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A linked system of symmetric designs (LSSD) is a w-partite graph (w ≥ 2) where the incidence between any two parts corresponds to a symmetric design and the designs arising from three parts are related. The original construction for LSSDs by Goethals used Kerdock sets, in which v is a power of two. Some four decades later, new examples were given by Davis et. al. and Jedwab et. al. using difference sets, again with v a power of two. In this paper we develop a connection between LSSDs and "linked simplices", full-dimensional regular simplices with two possible inner products between vertices of distinct simplices. We then use this geometric connection to construct sets of equiangular lines and to find an equivalence between regular unbiased Hadamard matrices and certain LSSDs with Menon parameters. We then construct examples of non-trivial LSSDs in which w can be made arbitarily large for fixed even part of v. Finally we survey the known infinite families of symmetric designs and show, using basic number theoretic conditions, that w = 2 in most cases.

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Section: Hadamard Matrix Equivalencementioning

confidence: 93%

Linked systems of symmetric designs

Kodalen¹

2019

Algebraic Combinatorics

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“…[4,12]. Note that subsets of m ≥ 6 MUB are isolated in dimension 8, whereas several families of m = 5 MUB exist [31]. Another observation is that the restricted defect ∆ for maximal sets of d+1 MUB in dimension d coincides with the defect for d MUB.…”

Section: Free Parameters In Mub and Sic-povmmentioning

confidence: 95%

Quantum measurements with prescribed symmetry

Życzkowski

Bruzda²,

Goyeneche³

2017

Phys. Rev. A

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We introduce a method to determine whether a given generalised quantum measurement is isolated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of equations in some relevant cases. As consequence, we provide a simple derivation of the maximal family of Symmetric Informationally Complete measurements (SIC)-POVM in dimension 3. Furthermore, we show that the following remarkable geometrical structures are isolated, so that free parameters cannot be introduced: (a) maximal sets of mutually unbiased bases in prime power dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c) contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18 and 21 vectors, respectively.Comment: 10 pages, 2 figure

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“…One can extract information about the set of vectors by investigating various characteristics of the corresponding Gram matrix, such as eigenvalues, determinant, entropy, trace, etc. Due to its special structure, Gram matrix has found extensive applications in quantum information theory [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. In this work, we generalize Gram matrix of a pure-state ensemble to a mixed-state ensemble via the Hilbert-Schmidt inner product and explore its basic properties.…”

Section: Gram Matrix Of Mixed-state Ensemble Via Hilbert-schmidt Inne...mentioning

confidence: 99%

Post-measurement quantumness

Sun¹,

Luo²

2021

Phys. Scr.

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A quantum measurement (e.g., positive-operator valued measure, or more generally, instrument) performed on a system state yields a post-measurement ensemble (a family of outcome states together with the corresponding outcome probabilities), which may have quantumness due to the impossibility to perfectly discriminate the constituent states. In order to characterize information contents and structure of a measurement, it is desirable to investigate the post-measurement ensemble. In this work, we study post-measurement quantumness (i.e., quantumness of the post-measurement ensemble) by introducing a Gram matrix summarizing the interaction between a measurement and a state, and explore its implications and consequences. Gram matrix of an ensemble can be regarded as a quantum state (density operator) in an auxiliary system with a computational basis labelling the measurement outcomes. Our basic idea is to exploit coherence of the Gram matrix (with respect to the computational basis of the auxiliary system) and employ it to quantify post-measurement quantumness . We propose several quantifiers of quantumness and reveal their basic properties. We further evaluate explicitly quantumness of measurements arising from several important quantum channels, and illustrate a comparison between them. This provides a method for investigating measurements from the perspective of coherence of Gram matrix. When all M i are orthogonal projections (and thus are RECEIVED

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Mutually unbiased bases with free parameters

Cited by 7 publications

References 40 publications

Linked systems of symmetric designs

Linked systems of symmetric designs

Quantum measurements with prescribed symmetry

Post-measurement quantumness

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