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Physica Scripta

Mutually unbiased coherence of Bell Diagonal States*

An-Ting Zhao, Ming Fang, Yuan-Hong Tao

Published Jun 25, 2024• 1 citation
Publisher

Abstract

Abstract This study explores the coherence of Bell-diagonal states under two complete sets of mutually unbiased bases (MUBs), focusing on four measures: the l 1 norm, l p norm, Rényi relative α-entropy, and relative entropy. We initially construct a complete set of MUBs in C 4 by incorporating two maximally entangled bases unbiased with respect to the autotensor of mutually unbiased bases (AMUBs), subsequently comparing it with the canonical set documented in existing literature. We establish the expressions for Bell-diagonal states under these two complete sets of MUBs and demonstrate that the aggregate of each coherence measure across both sets is equivalent. Additionally, we present the upper bounds for the sums of the l 1 norm, l p norm, and Rényi relative 2-entropy of coherence. Furthermore, we describe the geometric structure associated with the sum of relative entropy coherence, revealing that an increase in the sum of relative entropy of coherence corresponds to an increase in the parameter of the Bell-diagonal states.

DOI
10.1088/1402-4896/ad5b8f

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Abstract This study explores the coherence of Bell-diagonal states under two complete sets of mutually unbiased bases (MUBs), focusing on four measures: the l 1 norm, l p norm, Rényi relative α-entropy, and relative entropy. We initially construct a complete set of MUBs in C 4 by incorporating two maximally entangled bases unbiased with respect to the autotensor of mutually unbiased bases (AMUBs), subsequently comparing it with the canonical set documented in existing literature. We establish the expressions for Bell-diagonal states under these two complete sets of MUBs and demonstrate that the aggregate of each coherence measure across both sets is equivalent. Additionally, we present the upper bounds for the sums of the l 1 norm, l p norm, and Rényi relative 2-entropy of coherence. Furthermore, we describe the geometric structure associated with the sum of relative entropy coherence, revealing that an increase in the sum of relative entropy of coherence corresponds to an increase in the parameter of the Bell-diagonal states.
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Abstract In this paper, we discuss quantum uncertainty relations of Tsallis relative α entropy coherence for a single qubit system based on three mutually unbiased bases. For α ∈ 1 2 , 1 ∪ ( 1 , 2 ] , the upper and lower bounds of sums of coherence are obtained. However, the above results cannot be verified directly for any α ∈ 0 , 1 2 . Hence, we only consider the special case of α = 1 n + 1 , where n is a positive integer, and we obtain the upper and lower bounds. By comparing the upper and lower bounds, we find that the upper bound is equal to the lower bound for the special α = 1 2 , and the differences between the upper and the lower bounds will increase as α increases. Furthermore, we discuss the tendency of the sum of coherence, and find that it has the same tendency with respect to the different θ or φ, which is opposite to the uncertainty relations based on the Rényi entropy and Tsallis entropy.
Abstract We study the skew information-based coherence of quantum states and derive explicit formulas for Werner states and isotropic states in a set of autotensors of mutually unbiased bases (MUBs). We also give surfaces of skew information-based coherence for Bell-diagonal states and a special class of X states in both computational basis and in MUBs. Moreover, we depict the surfaces of the skew information-based coherence for Bell-diagonal states under various types of local nondissipative quantum channels. The results show similar as well as different features compared with relative entropy of coherence and l 1 norm of coherence.
Abstract Quantum coherence stands as a core property of quantum mechanics, its origin rooted in the superposition principle that governs quantum states. Leveraging mutually unbiased bases (MUBs), this paper delves into an in-depth exploration of quantum coherence characteristics through the lens of the l p -norm and s k e w Information. To begin with, we establish rigorous equality relations for the l p -norm coherence of qubit states under Pauli MUBs. Following this, we derive the complementary relations (upper bounds) and uncertainty relations (lower bounds) associated with l p -norm coherence under MUBs within arbitrary d-dimensional systems. Then, by virtue of deducing the connection between the classical purity and quantum purity of the square root of the quantum state, i.e. ρ , we further elaborate on the strict equality relations of skew-information coherence as well as several corresponding entropy-based coherences under MUBs in d-dimensional systems. Additionally, we provide illustrative examples to instantiate each of these relations, enhancing their clarity and practical relevance.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.