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Communications in computer and information science

Coherence of Quantum States Based on Mutually Unbiased Bases in $$\mathbb {C}^4$$

Xudong Ma, Yongming Li

Published Jan 1, 2022• 0 citations
Publisher
DOI
10.1007/978-981-19-8152-4_3

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Coherence of Quantum States Based on Mutually Unbiased Bases in C4\mathbb {C}^4C4
Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q+1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].
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.
The Kirkwood-Dirac distribution, serving as an informationally complete representation of a quantum state, has recently garnered increasing attention. We investigate the Kirkwood-Dirac classicality with respect to mutually unbiased bases. For prime dimensional Hilbert spaces, {we demonstrate that a quantum state exhibits Kirkwood-Dirac classicality for two distinct sets of mutually unbiased bases (A,B)(A,B)(A,B) and (A,B′)(A,B')(A,B′) if and only if it is incoherent with respect to AAA}. We subsequently introduce a coherence monotone based on Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases. Additionally, we establish that this coherence monotone can be expressed through weak values, suggesting that quantum coherence can be utilized to detect anomalous weak values.