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 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.

Civil War Book ReviewMay 5, 202145 citations

Quantum Entropy and Information

Nilanjana Datta

Lecture notes in physicsJan 1, 20103 citations

Quantum entropy and quantum measurements

Göran Lindblad

Springer eBooksApr 7, 200834 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.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:2410.01740Oct 2, 20240 citations

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.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, 20204 citations

Quantum Entropy and Its Use

Masanori Ohya, Dėnes Petz

Springer eBooksJan 1, 1993764 citations

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'

B

. As a consequence, if

S[A|B]_{\mathcal{N}}< -\log|A|

then the channel necessarily has causal influence (signaling) from

A'

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.