“Bounds on information combining” are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in belief propagation and polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal “quantum Mrs. Gerber's Lemma”. Our proof uses three main ingredients: 1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; 2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; and 3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to polar coding for binary-input classical-quantum channels, and show the following three results: 1) even non-stationary channels polarize under the polar transform; 2) the blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity; and 3) under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.“Bounds on information combining” are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in belief propagation and polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal “quantum Mrs. Gerber's Lemma”. Our proof uses three main ingredients: 1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; 2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; and 3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to polar coding for binary-input classical-quantum channels, and show the following three results: 1) even non-stationary channels polarize under the polar transform; 2) the blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity; and 3) under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.

arXiv:1706.09752IEEE Transactions on Information TheoryMay 31, 201811 citations

Chain Rules for Renyi Information Combining

Christoph Hirche, Guan, Xinyue, Marco Tomamichel

Bounds on information combining are a fundamental tool in coding theory, in particular when analyzing polar codes and belief propagation. They usually bound the evolution of random variables with respect to their Shannon entropy. In recent work this approach was generalized to Renyi $α$ -entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of $α=2$ we give the first optimal information combining bounds with quantum side information.Bounds on information combining are a fundamental tool in coding theory, in particular when analyzing polar codes and belief propagation. They usually bound the evolution of random variables with respect to their Shannon entropy. In recent work this approach was generalized to Renyi $α$-entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of $α=2$ we give the first optimal information combining bounds with quantum side information.

arXiv:2305.02589May 4, 20230 citations

Entropy power inequalities and classical capacities of bosonic noise channels

Robert König, Graeme Smith

Characterizing information-carrying capacities of bosonic communication channels is of significant practical interest. For thermal noise channels, using coherent (product) states yields an achievable rate for classical communication which is conjectured to be optimal. However, it is not known whether coding strategies using entanglement may perform better. Here we discuss upper bounds on classical capacities of thermal noise channels. These imply that coherent-state coding is close to optimal. Our main tool is a quantum analog of the entropy power inequality introduced Shannon. It gives a lower bound on the output von Neumann entropy when two independent signals combine at a beamsplitter.Characterizing information-carrying capacities of bosonic communication channels is of significant practical interest. For thermal noise channels, using coherent (product) states yields an achievable rate for classical communication which is conjectured to be optimal. However, it is not known whether coding strategies using entanglement may perform better. Here we discuss upper bounds on classical capacities of thermal noise channels. These imply that coherent-state coding is close to optimal. Our main tool is a quantum analog of the entropy power inequality introduced Shannon. It gives a lower bound on the output von Neumann entropy when two independent signals combine at a beamsplitter.

IEEE/LEOS Summer Topical MeetingsJul 1, 20130 citations

Quantum Entropy and Correlations in Quantum Information

Reinhold A. Bertlmann, Nicolai Friis

Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.

Sep 7, 20230 citations

From asymptotic hypothesis testing to entropy inequalities

Christoph Hirche

This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi- $2$ entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-$2$ entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.

arXiv:1812.05142Dec 12, 20183 citations

von Neumann capacity of noisy quantum channels

Christoph Adami, Nicolas J. Cerf

We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.

arXiv:quant-ph/9609024Physical Review ANov 1, 1997209 citations

α

-entropies. However, due to the lack of a traditional chain rule for Renyi entropies the picture remained incomplete. In this work we establish the missing link by providing Renyi chain rules connecting different definitions of Renyi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Renyi entropy. In the second part, we generalize the chain rule to the quantum setting and show how they allow us to generalize results and conjectures previously only given for the von Neumann entropy. In the special case of

α=2

we give the first optimal information combining bounds with quantum side information.

2

entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.