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Entropy of a quantum channel

Gilad Gour, Mark M. Wilde

Published May 5, 2021• 45 citations
OpenAlex

Abstract

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.

Related papers

Entropy of a Quantum Channel: Definition, Properties, and Application

Gilad Gour, Mark M. Wilde

The von Neumann entropy 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 establish that this definition satisfies all of the axioms, recently put forward in [Dour, IEEE Trans. Inf. Theory 65, 5880 (2019)], 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. We define Rényi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2020•4 citations

Entropy of a quantum channel

Gilad Gour, Mark M. Wilde

The authors generalize the von Neumann entropy from quantum states to quantum channels, which model evolutions of quantum physical systems, and they provide a physical interpretation of a quantum channel's entropy in terms of an information-processing task called quantum channel merging.

arXiv:1808.06980•Physical Review Research•May 5, 2021•51 citations

Conditional entropy and information of quantum processes

Siddhartha Das, Kaumudibikash Goswami, Vivek Pandey

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 $S[A|B]_{\mathcal{N}}$ of the channel $\mathcal{N}_{A'B'\to AB}$ is based on the quantum relative entropy, with system pairs $A',A$ and $B',B$ 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 $\mathcal{N}_{A'B'\to AB}$ in terms of its von Neumann conditional entropy $S[A|B]_{\mathcal{N}}$, to have no causal influence from $A'$ to $B$. As a consequence, if $S[A|B]_{\mathcal{N}}< -\log|A|$ then the channel necessarily has causal influence (signaling) from $A'$ to $B$. 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.

arXiv:2410.01740•Oct 2, 2024•0 citations

Relating Entropies of Quantum Channels

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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.

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Hypothesis testing and entropies of quantum channels

Xiao Yuan

Hypothesis testing is an important task in mathematics and physics. Hypothesis testing of two random variables is related to the Kullback-Leibler divergence of the two corresponding distributions. Similarly, quantum hypothesis testing of two quantum states is characterized by the quantum relative entropy. While a quantum state can be abstracted as a device that only has outputs but no inputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend hypothesis testing to general quantum channels. In both the one-shot and asymptotic scenario, we study several quantifiers for hypothesis testing under different assumptions of how the channels are used. As the quantifiers are analogs to the quantum relative entropy of states, we call them the quantum relative entropy of channels. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarizing channel. We investigate the properties that the quantum relative entropy of channels should satisfy, and we study its interplay with entanglement. With the broad applications of the quantum relative entropy of states, our results can be useful for understanding general properties of quantum channels.

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Quantum correlation measure based on min relative entropy for two-partition and k-partition

Junqing Li, Li Huang, Shijie Cui +1 more

As a peculiar resource of quantum mechanics, quantum correlation has been applied to many aspects. In quantum information processing and quantum computing, the quantum correlation plays an extremely important role, and it has been a subject of further studies, principally due to the general belief that it is a fundamental resource for different quantum information processing tasks. In addition, correlation measure is a very important physical quantity in studying the quantum correlation. A well-defined correlation measure needs to have some necessary properties. By proving these necessary properties, we can deepen our understanding of correlation measure. As one of the key concepts of quantum information theory, relative entropy is always used to measure the uncertainty contained in the state of physical system. In order to better understand the properties and applications of correlation measure based on relative entropy, in this paper, according to the properties of the min relative entropy, we give the quantum correlation measure based on min relative entropy for two-partition and k-partition. Furthermore, we prove that it satisfies some necessary properties of quantum correlation measures, including the nonnegativity, the invariance under local unitary operators, and the monotonicity under completely positive trace-preserving. By proving these properties, we show that the given correlation measure is well-defined. Security of communication has received much attention since ancient times. In today's society, the internet, instant messaging and e-commerce applications are all related to the information security, and the information security is related to the vital interests of everyone. The information encryption is one of the important methods to ensure information security. As an important way to ensure information security, quantum channel has received more and more attention. At the end of the paper, we introduce the concept of quantum channel, and discuss the influence of quantum channel on the correlation measure based on min relative entropy under k-partition. By proposing a new correlation measure and proving the effect quantum channel on the measure, we can better describe the uncertainty contained in the state of physical system.

Acta Physica Sinica•Oct 19, 2022•0 citations
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.
The von Neumann entropy 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 establish that this definition satisfies all of the axioms, recently put forward in [Dour, IEEE Trans. Inf. Theory 65, 5880 (2019)], 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. We define Rényi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.
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.
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.
Hypothesis testing is an important task in mathematics and physics. Hypothesis testing of two random variables is related to the Kullback-Leibler divergence of the two corresponding distributions. Similarly, quantum hypothesis testing of two quantum states is characterized by the quantum relative entropy. While a quantum state can be abstracted as a device that only has outputs but no inputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend hypothesis testing to general quantum channels. In both the one-shot and asymptotic scenario, we study several quantifiers for hypothesis testing under different assumptions of how the channels are used. As the quantifiers are analogs to the quantum relative entropy of states, we call them the quantum relative entropy of channels. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarizing channel. We investigate the properties that the quantum relative entropy of channels should satisfy, and we study its interplay with entanglement. With the broad applications of the quantum relative entropy of states, our results can be useful for understanding general properties of quantum channels.
As a peculiar resource of quantum mechanics, quantum correlation has been applied to many aspects. In quantum information processing and quantum computing, the quantum correlation plays an extremely important role, and it has been a subject of further studies, principally due to the general belief that it is a fundamental resource for different quantum information processing tasks. In addition, correlation measure is a very important physical quantity in studying the quantum correlation. A well-defined correlation measure needs to have some necessary properties. By proving these necessary properties, we can deepen our understanding of correlation measure. As one of the key concepts of quantum information theory, relative entropy is always used to measure the uncertainty contained in the state of physical system. In order to better understand the properties and applications of correlation measure based on relative entropy, in this paper, according to the properties of the min relative entropy, we give the quantum correlation measure based on min relative entropy for two-partition and k-partition. Furthermore, we prove that it satisfies some necessary properties of quantum correlation measures, including the nonnegativity, the invariance under local unitary operators, and the monotonicity under completely positive trace-preserving. By proving these properties, we show that the given correlation measure is well-defined. Security of communication has received much attention since ancient times. In today's society, the internet, instant messaging and e-commerce applications are all related to the information security, and the information security is related to the vital interests of everyone. The information encryption is one of the important methods to ensure information security. As an important way to ensure information security, quantum channel has received more and more attention. At the end of the paper, we introduce the concept of quantum channel, and discuss the influence of quantum channel on the correlation measure based on min relative entropy under k-partition. By proposing a new correlation measure and proving the effect quantum channel on the measure, we can better describe the uncertainty contained in the state of physical system.