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Entropy

Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence

Chengyang Zhang, Zhihua Guo, Huaixin Cao

Published Mar 5, 2020• 6 citations
Publisher

Abstract

Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.

DOI
10.3390/e22030297

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Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure Cr(ρ)C_r(\rho)Cr​(ρ) of coherence for nnn-partite quantum states ρ\rhoρ, which gives lower and upper bounds for Cr(ρ)C_r(\rho)Cr​(ρ). Meanwhile, we discuss the conjecture about the validity of the inequality Cr(ρ)≤Cℓ1(ρ)C_r(\rho)\leq C_{\ell_1}(\rho)Cr​(ρ)≤Cℓ1​​(ρ) for any state ρ\rhoρ. We observe that every mixture η\etaη of a state ρ\rhoρ satisfying Cr(ρ)≤Cℓ1(ρ)C_r(\rho)\leq C_{\ell_1}(\rho)Cr​(ρ)≤Cℓ1​​(ρ) and any incoherent state σ\sigmaσ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm ln⁡\lnln instead of log⁡2\log_2log2​, then the reduced relative entropy measure of coherence Cˉr(ρ)=−ρdiagln⁡ρdiag+ρln⁡ρ\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rhoCˉr​(ρ)=−ρdiag​lnρdiag​+ρlnρ satisfies the inequality Cˉr(ρ)≤Cℓ1(ρ){\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)Cˉr​(ρ)≤Cℓ1​​(ρ) for any mixed state ρ\rhoρ.
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.
Quantum addition channels have been recently introduced in the context of deriving entropic power inequalities for finite dimensional quantum systems. We prove a reverse entropy power equality which can be used to analytically prove an inequality conjectured recently for arbitrary dimension and arbitrary addition weight. We show that the relative entropic difference between the output of such a quantum additon channel and the corresponding classical mixture quantitatively captures the amount of coherence present in a quantum system. This new coherence measure admits an upper bound in terms of the relative entropy of coherence and is utilized to formulate a state-dependent uncertainty relation for two observables. Our results may provide deep insights to the origin of quantum coherence for mixed states that truly come from the discrepancy between quantum addition and the classical mixture.
Quantum coherence is a fundamental feature of quantum physics and plays a significant role in quantum information processing. By generalizing the resource theory of coherence from von Neumann measurements to positive operator-valued measures (POVMs), POVM-based coherence measures have been proposed with respect to the relative entropy of coherence, the l 1 norm of coherence, the robustness of coherence and the Tsallis relative entropy of coherence. We derive analytically the lower and upper bounds on these POVM-based coherence of an arbitrary given superposed pure state in terms of the POVM-based coherence of the states in superposition. Our results can be used to estimate range of quantum coherence of superposed states. Detailed examples are presented to verify our analytical bounds.
Abstract Quantum coherence is the most distinguished signature of quantum mechanics, also recognized to be an essential resource for many promising quantum technologies, playing a central role in phenomena related to quantum information science, quantum thermodynamics and quantum biology, just to mention few examples. However, the resource theory of coherence is in the preliminary stage of its development, being still limited to the study of the manipulation and quantification of coherence, mostly from a theoretical viewpoint. Here, we propose an experimental method to directly measure the relative entropy of coherence, which according to the resource theory is one of the main quantifiers of coherence. To achieve this aim, we discuss a procedure to measure the von Neumann entropy of a generic quantum state directly in terms of the Shannon entropy of the probability distribution of outcomes in a defined measurement basis. In both cases, by direct we mean that tomographic methods are not required. Two quantum-optical applications of our method are discussed in order to give support to our predictions.
Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modification of this coherence has been proposed that does. We introduce three new Tsallis coherence measures coming from a more general definition that also satisfy the strong monotonicity, and compare all five definitions between each other. Using three coherence measures that we discuss, one can also define a discord. Two of these have been used in the literature, and another one is new. We also discuss two correlation measures based on Tsallis relative entropy. We provide explicit expressions for all three discord and two correlation measure on pure states. Lastly, we provide tight upper and lower bounds on two discord and correlations measures on any quantum state, with the condition for equality.
Quantum coherence is the most distinguished feature of quantum mechanics, which characterizes the superposition properties of quantum states. It plays a critical role in various fields, ranging from quantum information technology to quantum biology. Although various coherence quantifiers have been proposed since the resource theory of coherence was established, there are a lack of experimental methods to estimate them efficiently, which restricts the applications of coherence. Relative entropy of coherence is one of the main quantifiers of coherence, and is frequently used in quantum information science. Such nonlinear properties of quantum states are usually calculated from full descriptions of the quantum state, although they are not related to all parameters that specify the state. Here, we experimentally measure the relative entropy of coherence for the arbitrary qubit states directly in the photonic system without using standard state tomography. In the experiment, we directly measure the von Neumann entropy of the quantum states through interference, thus obtaining the relative entropy of coherence, and finding that the experimental results are in good agreement with the theory. Our work provides a nice alternative experimental scheme for measuring the relative entropy of coherence.