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 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.
Since both coherence and quantum correlations arise from the superposition principle and can be regarded as resources in quantum information tasks, it is of significance to investigate the interplay between them from different perspectives. In this work we focus on the basis-dependent correlations in a bipartite state defined by the coherence difference between global state and local state relative to a local basis and characterize bipartite states with vanishing basis-dependent correlations. Using the relative entropy of coherence, the structure of such states has been determined by Yadin et al. [Phys. Rev. X 6, 041028 (2016)], which we call block-diagonal product states here. The first result of this work is to demonstrate that the set of block-diagonal product states can also be characterized by the property of possessing vanishing basis-dependent correlations using the coherence measure based on skew information. As a by-product of this result, we describe the structure of quantum ensembles saturating the convexity inequality in the resource theory of coherence using the coherence measure based on skew information, which may be of independent interest. Next, we characterize the set of bipartite states with vanishing basis-dependent correlations using the l1 norm of coherence, and show that the set of block-diagonal product states is a subset of it. Furthermore, we provide an operational interpretation of block-diagonal product states in an interference model. Finally, we compare the amount of basis-dependent correlations using the three mentioned coherence measures through several examples such as Werner states, isotropic states, Bell-diagonal states, and a family of classical-quantum states.
We plot the geometry of several distance-based quantifiers of coherence for Bell-diagonal states. We find that along with both l1 norm and relative entropy of coherence changes continuously from zero to one, their surfaces move from the separable regions to the entangled regions. Based on this fact, it is more illuminating to use an intuitive geometry to explain quantum states with nonzero coherence can be used for entanglement creation, rather than the other way around. We find the necessary and sufficient conditions that quantum discord of Bell-diagonal states equal to its relative entropy of coherence and depict the surfaces of the equality. We give surfaces of relative entropy of coherence for X states. We show the surfaces of dynamics of relative entropy of coherence for Bell-diagonal states under local nondissipative channels and find that all coherence under local nondissipative channels decrease.
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 concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.
Quantum coherence is one of the characteristic features of quantum mechanics and underpins many quantum mysteries. To eliminate the influence of the reference basis on the coherence of a quantum state and uncover its intrinsic properties, it is common to study coherence by averaging over different reference bases. Using the metric-adjusted skew information, we explore three natural approaches to average coherence of a state: average over all orthonormal bases, average over all elements of an operator orthonormal base, and average over a complete family of mutually unbiased bases. We establish the equivalence among these three types of average coherence and interpret the unified average coherence as the coherence of a quantum state relative to a depolarizing channel. Additionally, we employ the unified average coherence to introduce a notion of quantum f entropy (where f is an operator monotone function associated with the metric-adjusted skew information) and demonstrate that quantum f entropy possesses properties analogous to the ubiquitous von Neumann entropy. Furthermore, we illuminate some connections between quantum f entropy and quasientropy, and compare f entropy with von Neumann entropy, R\'enyi entropy, and Tsallis entropy for some typical states.