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arXiv:2506.10810 · Laser Physics Letters

Coherence monotones of quantum channels based on two generalized quantum relative entropies

Jiaorui Fan, Zhaoqi Wu, Shao-Ming Fei

Published Sep 6, 2023• 1 citation
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Abstract

Abstract By using the Choi–Jamiołkowski isomorphism, we propose two classes of coherence monotones of quantum channels based on the unified (r, s)-relative entropy and the sandwiched Rényi relative entropy. Elegant properties of the coherence monotones for quantum channels are explored. Moreover, we present the upper bounds of the coherence monotones and derive the explicit formulas of the coherence monotones for qubit unitary channels.

DOI
10.1088/1612-202x/acf36a

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Abstract By using the Choi–Jamiołkowski isomorphism, we propose two classes of coherence monotones of quantum channels based on the unified (r, s)-relative entropy and the sandwiched Rényi relative entropy. Elegant properties of the coherence monotones for quantum channels are explored. Moreover, we present the upper bounds of the coherence monotones and derive the explicit formulas of the coherence monotones for qubit unitary channels.
Abstract Quantum coherence is an important quantum resource in the field of quantum information. We establish the uncertainty relations for the unified ( α , β )-relative entropy of coherence (UREOC) and the sandwiched Rényi α -relative entropy of coherence (SRREOC) under two and three orthonormal bases in qubit systems. We then prove that UREOC can be upper bounded by the purity in arbitrary finite dimensional spaces, from which a complementarity relation between UREOC and the mixedness is obtained. Finally, we show that the minimum error probability of quantum state discrimination and asymptotic exact coherence distillation rate can be upper bounded by UREOC, and can thus be easily estimated with the purity/mixedness of the state.
The uncertainty relation reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, the quantum uncertainty relations of the sandwiched R\'enyi relative entropy of coherence (SRREOC) and unified $(\ensuremath{\alpha},\ensuremath{\beta})$-relative entropy of coherence (UREOC) are studied. First, the uncertainty relation of the SRREOC for pure states, with extension to mixed states by convex-roof construction, is given in a ddd-dimensional quantum system. Second, the analytical formula of the UREOC is presented; it is shown that the UREOC is a good candidate for a quantum coherence measure. Furthermore, the quantum uncertainty relation of the UREOC for pure states is obtained in a ddd-dimensional quantum system, then it is generalized to mixed states. In addition, our results cover the quantum uncertainty relations of the other three coherence measures. Third, two methods are given to compare the lower bounds of the SRREOC and UREOC for single-qubit pure states. By comparison, we find that the results in this paper not only can be applied to single-qubit states but also can be extended to a ddd-dimensional quantum system.
Abstract The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi α-relative entropy, which present in [Phys. Rev. A 94 (2016) 052336]. We show that the Rényi α-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement, which presents in [Phys. Rev. A 93 (2016) 032136]. Due to the Rényi α-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of Rényi 2-relative entropy of coherence are derived .
In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.
Abstract We establish a rigorous equivalence between quantum coherence measures defined by the sandwiched Rényi relative entropy and generalized conditional entropies associated with auxiliary quantum systems, providing a direct operational interpretation of sandwiched Rényi relative entropy of coherence as the conditional uncertainty of projective measurement outcomes with quantum memory. Motivated by this connection, we develop a coherence-based approach to quantum uncertainty relations. Within this approach, we introduce a coherence measure induced by the two-parameter α - z -Rényi relative entropy and derive corresponding uncertainty relations for both pure and mixed quantum states. The resulting state-independent bounds recover known results in appropriate limits and yield tighter bounds over a broad parameter regime. Our results provide a systematic connection between quantum coherence, generalized conditional entropies, and uncertainty relations, thereby offering a unified perspective on quantum uncertainty in terms of Rényi-type information measures.
In contrast to unitary evolutions, which are reversible, generic quantum processes (operations and quantum channels) are often irreversible. However, the degree of irreversibility is different for different channels, and it is desirable to have a quantitative characterization of irreversibility. In this paper, by exploiting the channel–state duality implemented by the Jamiołkowski–Choi isomorphism, we quantify the irreversibility of channels via entropy of the Jamiołkowski–Choi states of the corresponding channels and compare it with the notions of entanglement fidelity and entropy exchange. General properties of a reasonable measure of irreversibility are discussed from an intuitive perspective, and entropic measures of irreversibility are introduced. Several relations between irreversibility, entanglement fidelity, the degree of nonunitality, and decorrelating power are established. Some measures of irreversibility for a variety of prototypical channels are evaluated explicitly, revealing some information-theoretic aspects of the structure of channels from the perspective of irreversibility.