Frame Moments and Welch Bound with Erasures

Haikin, Marina; Zamir, Ram; Gavish, Matan

doi:10.48550/arxiv.1801.04548

Cited by 2 publications

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“…Defined by a finite set of vectors in an inner product space [1], the Gram matrix has been extensively applied in many different branches of mathematics and physics. Notable features of the Gram matrix, including the eigenvalues [2], the trace [3], the determinant [4] and the entropy [5], have been investigated. Recently, it has been shown that many important issues in quantum information theory, such as uncertainty relations [6][7][8], state discrimination [9][10][11][12][13][14], transitions between two sets of quantum states [15,16], information-theoretic aspects of superposition [17], quantum information masking [18] and PT-symmetric quantum systems [19], are intimately related to the Gram matrix.…”

Section: Introductionmentioning

confidence: 99%

Quantumness of Pure-State Ensembles via Coherence of Gram Matrix Based on Generalized α-z-Relative Rényi Entropy

Yuan

Fei

2022

Int J Theor Phys

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The Gram matrix of a set of quantum pure states plays key roles in quantum information theory. It has been highlighted that the Gram matrix of a pure-state ensemble can be viewed as a quantum state, and the quantumness of a pure-state ensemble can thus be quantified by the coherence of the Gram matrix [Europhys. Lett. 134 30003]. Instead of the l 1 -norm of coherence and the relative entropy of coherence, we utilize the generalized α-z-relative Rényi entropy of coherence of the Gram matrix to quantify the quantumness of a pure-state ensemble and explore its properties. We show the usefulness of this quantifier by calculating the quantumness of six important pure-state ensembles. Furthermore, we compare our quantumness with other existing ones and show their features as well as orderings.

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Section: Introductionmentioning

confidence: 99%

Quantumness of Pure-State Ensembles via Coherence of Gram Matrix Based on Generalized α-z-Relative Rényi Entropy

Yuan

Fei

2022

Int J Theor Phys

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“…characteristics of the Gram matrix, such as the eigenvalues [24], the entropy [25], and the trace [26], have been studied and used to solve problems related to quantum ensembles.…”

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confidence: 99%

Quantumness of ensemble via coherence of Gram matrix

Sun

Luo

Lei

2021

EPL

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The Gram matrix of a set of vectors, which encapsulates the relations between the constituent vectors, plays an important role in the exploration of both geometric and informationtheoretic aspects of quantum state space. In view of its usefulness and importance, we study the Gram matrix of an ensemble of pure states (a set of pure states with a prior probability distribution) and reveal its fundamental properties. We highlight and exploit the fact that the Gram matrix of an ensemble can be formally regarded as a bona fide quantum state. The key idea is to employ coherence (with respect to the computational basis) of the Gram matrix (regarded as a quantum state) to quantify quantumness of the corresponding ensemble. In particular, we propose to use the l1-norm of coherence and the relative entropy of coherence, of the Gram matrix, as two significant quantifiers of quantumness. We illustrate the effectiveness and power of these quantifiers of quantumness by evaluating them for several important ensembles arising from quantum measurement and quantum cryptography. We further compare the quantumness based on the Gram matrix with various other quantifiers of quantumness in the literature, and show that although they are closely related in general, they have subtle differences and capture different aspects of ensembles, which shed light on the complexity of quantum ensembles.

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Frame Moments and Welch Bound with Erasures

Cited by 2 publications

References 0 publications

Quantumness of Pure-State Ensembles via Coherence of Gram Matrix Based on Generalized α-z-Relative Rényi Entropy

Quantumness of Pure-State Ensembles via Coherence of Gram Matrix Based on Generalized α-z-Relative Rényi Entropy

Quantumness of ensemble via coherence of Gram matrix

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