We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the Rényi parameters tend to one. To this end we present Rényi mutual information decomposition rules, a new approach to the Rényi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of Rényi entropic uncertainty relations. Accordingly, we establish a family of Rényi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the Rényi bipartite uncertainty relations.

arXiv:2106.10415Jun 19, 20210 citations

Variational approach to relative entropies with an application to QFT

Stefan Hollands

Abstract We define a new divergence of von Neumann algebras using a variational expression similar in nature to Kosaki’s formula for Umegaki’s relative entropy. Our divergence satisfies several of the usual desirable properties, upper bounds the sandwiched Renyi entropy and reduces to the fidelity in a limit. As an illustration, we use the formula in quantum field theory to compute our divergence between the vacuum in a bipartite system and an “orbifolded”—in the sense of a conditional expectation—system in terms of the Jones index. We take the opportunity to point out an entropic certainty relation associated with an inclusion of von Neumann factors related to the relative entropy. This certainty relation has an equivalent formulation in terms of error correcting codes.Abstract We define a new divergence of von Neumann algebras using a variational expression similar in nature to Kosaki’s formula for Umegaki’s relative entropy. Our divergence satisfies several of the usual desirable properties, upper bounds the sandwiched Renyi entropy and reduces to the fidelity in a limit. As an illustration, we use the formula in quantum field theory to compute our divergence between the vacuum in a bipartite system and an “orbifolded”—in the sense of a conditional expectation—system in terms of the Jones index. We take the opportunity to point out an entropic certainty relation associated with an inclusion of von Neumann factors related to the relative entropy. This certainty relation has an equivalent formulation in terms of error correcting codes.

arXiv:2009.05024Letters in Mathematical PhysicsOct 30, 202114 citations

On variational expressions for quantum relative entropies

Mario Berta, Omar Fawzi, Marco Tomamichel

arXiv:1512.02615Letters in Mathematical PhysicsSep 7, 201781 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

Quantum Entropies and Their Maximizations

Masanori Ohya

Springer eBooksJan 1, 19930 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.