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International Journal of Quantum Information

Comparing two entropy functions of a quantum channel

Junfang Cheng, Yanjun Chu, Fang Huang

Published Dec 17, 2023• 0 citations
Publisher

Abstract

Inspired by a conjecture related to numerical investigations for a quantum channel in a system of large size, we shall study relations between two different entropies of a quantum channel. One of them is defined by Gour and Wilde as the generalization of quantum relative entropy and maximally mixed state. The second one is defined as the difference between the entropy of the Choi state of the quantum channel and a constant number. At first, we show that the latter entropy is an upper boundary of the former entropy. Moreover, we also give a sufficient and necessary condition that the two entropies are equal. That is, if and only if the relative entropy reaches its supremum on the maximal entangled pure state in the former entropy. At last, to study the difference between the two entropies, we expect to find some quantum channels in which their related entropies do not reach their supremum on the maximal entangled pure states for the former entropy. By calculating the two entropies, we find that amplitude-damping channels are desired examples.

DOI
10.1142/s0219749923500491

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Inspired by a conjecture related to numerical investigations for a quantum channel in a system of large size, we shall study relations between two different entropies of a quantum channel. One of them is defined by Gour and Wilde as the generalization of quantum relative entropy and maximally mixed state. The second one is defined as the difference between the entropy of the Choi state of the quantum channel and a constant number. At first, we show that the latter entropy is an upper boundary of the former entropy. Moreover, we also give a sufficient and necessary condition that the two entropies are equal. That is, if and only if the relative entropy reaches its supremum on the maximal entangled pure state in the former entropy. At last, to study the difference between the two entropies, we expect to find some quantum channels in which their related entropies do not reach their supremum on the maximal entangled pure states for the former entropy. By calculating the two entropies, we find that amplitude-damping channels are desired examples.
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.
The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the "relative entropy distance"between ρ and the maximally mixed state, here we define the entropy of a channel N as the difference of the number of physical qubits of the channel output with the "relative entropy distance"between N and the completely depolarizing channel. We prove that this definition satisfies all of the axioms, recently put forward by Gour [IEEE Trans. Inf. Theory 65, 5880 (2019)IETTAW0018-944810.1109/TIT.2019.2907989], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. The entropy of a channel can be negative for certain channels, but this negativity has an operational interpretation in terms of the channel merging protocol. We define Rényi and min-entropies of a channel and prove that they satisfy the axioms required for a channel entropy function. Among other results, we also prove that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.
Abstract It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.
This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Koßmann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.
What would be a reasonable definition of the conditional entropy of bipartite quantum processes, and what novel insight would it provide? We develop this notion using four information-theoretic axioms and define the corresponding quantitative formulas. Our definitions of the conditional entropies of channels are based on the generalized state and channel divergences, for instance, quantum relative entropy. We find that the conditional entropy of quantum channels has potential to reveal insights for quantum processes that aren't already captured by the existing entropic functions, entropy or conditional entropy, of the states and channels. The von Neumann conditional entropy S[A∣B]NS[A|B]_{\mathcal{N}}S[A∣B]N​ of the channel NA′B′→AB\mathcal{N}_{A'B'\to AB}NA′B′→AB​ is based on the quantum relative entropy, with system pairs A′,AA',AA′,A and B′,BB',BB′,B being nonconditioning and conditioning systems, respectively. We identify a connection between the underlying causal structure of a bipartite channel and its conditional entropy. In particular, we provide a necessary and sufficient condition for a bipartite quantum channel NA′B′→AB\mathcal{N}_{A'B'\to AB}NA′B′→AB​ in terms of its von Neumann conditional entropy S[A∣B]NS[A|B]_{\mathcal{N}}S[A∣B]N​, to have no causal influence from A′A'A′ to BBB. As a consequence, if S[A∣B]N<−log⁡∣A∣S[A|B]_{\mathcal{N}}< -\log|A|S[A∣B]N​<−log∣A∣ then the channel necessarily has causal influence (signaling) from A′A'A′ to BBB. Our definition of the conditional entropy establishes the strong subadditivity of the entropy for quantum channels. We also study the total amount of correlations possible due to quantum processes by defining the multipartite mutual information of quantum channels.