A fundamental question in information theory is to quantify the loss of information under a noisy channel. Partial orders and contraction coefficients are typical tools to that end, however, they are often also challenging to evaluate. For the special class of binary input symmetric output (BISO) channels, Geng et al. showed that among channels with the same capacity, the binary symmetric channel (BSC) and binary erasure channel (BEC) are extremal with respect to the more capable order. Here, we show two main results. First, for channels with the same KL contraction coefficient, the same holds with respect to the less noisy order. Second, for channels with the same Dobrushin coefficient, or equiv. maximum leakage or Doeblin coefficient, the same holds with respect to the degradability order. In the process, we provide a closed-form expression for the contraction coefficients of BISO channels. We also discuss the comparability of BISO channels and extensions to binary channels in general.

The trade-offs between error probabilities in quantum hypothesis testing are by now well-understood in the centralized setting, but much less is known for distributed settings. Here, we study a distributed binary hypothesis testing problem to infer a bipartite quantum state shared between two remote parties, where one of these parties communicates to the tester at zero-rate, while the other party communicates to the tester at zero-rate or higher. As our main contribution, we derive an efficiently computable single-letter formula for the Stein's exponent of this problem, when the state under the alternative is product. As a key tool for proving the converse direction of our results, we develop a quantum version of the blowing-up lemma which may be of independent interest.

Quantum state discrimination is an important problem in many information processing tasks. In this work we are concerned with finding the best possible sample complexity when the states are preprocessed by a quantum channel that is required to be locally differentially private. We give achievability and converse bounds that nearly match the best known classical bounds. On the way, we prove several novel inequalities between quantum divergences that should be of independent interest.

Contraction coefficients give a quantitative strengthening of the data processing inequality. As such, they have many natural applications whenever closer analysis of information processing is required. However, it is often challenging to calculate these coefficients. As a remedy we discuss a quantum generalization of Doeblin coefficients. These give an efficiently computable upper bound on many contraction coefficients. We prove several properties and discuss generalizations and applications. In particular, we give additional stronger bounds. One especially for PPT channels and one for general channels based on a constraint relaxation. Additionally, we introduce reverse Doeblin coefficients that bound certain expansion coefficients.

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 Rényi α-entropies. However, due to the lack of a traditional chain rule for Rényi entropies the picture remained incomplete. In this work we establish the missing link by providing Rényi chain rules connecting different definitions of Rényi entropies by Hayashi and Arimoto. This allows us to provide new information combining bounds for the Arimoto Rényi 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.

Quantum capacities are fundamental quantities that are notoriously hard to compute and can exhibit surprising properties such as superadditivity. Thus, a vast amount of literature is devoted to finding tight and computable bounds on these capacities. We add a new viewpoint by giving operationally motivated bounds on several capacities, including the quantum capacity and private capacity of a quantum channel and the one-way distillable entanglement and private key of a quantum state. These bounds are generally phrased in terms of capacity quantities involving the complementary channel or state. As a tool to obtain these bounds, we discuss partial orders on quantum channels and states, such as the less noisy and the more capable order. Our bounds help to further understand the interplay between different capacities, as they give operational limitations on superadditivity and the difference between capacities in terms of the information-theoretic properties of the complementary channel or state. They can also be used as a new approach towards numerically bounding capacities, as discussed with some examples.

Differential privacy has been an exceptionally successful concept when it comes to providing provable security guarantees for classical computations. More recently, the concept was generalized to quantum computations. While classical computations are essentially noiseless and differential privacy is often achieved by artificially adding noise, near-term quantum computers are inherently noisy and it was observed that this leads to natural differential privacy as a feature. In this work we discuss quantum differential privacy in an information theoretic framework by casting it as a quantum divergence. A main advantage of this approach is that differential privacy becomes a property solely based on the output states of the computation, without the need to check it for every measurement. This leads to simpler proofs and generalized statements of its properties as well as several new bounds for both, general and specific, noise models. In particular, these include common representations of quantum circuits and quantum machine learning concepts. Here, we focus on the difference in the amount of noise required to achieve certain levels of differential privacy versus the amount that would make any computation useless. Finally, we also generalize the classical concepts of local differential privacy, Rényi differential privacy and the hypothesis testing interpretation to the quantum setting, providing several new properties and insights.

Quantum capacities are fundamental quantities that are notoriously hard to compute and can exhibit surprising properties such as superadditivity. Thus a vast amount of literature is devoted to finding close and computable bounds on these capacities. We add a new viewpoint by giving operationally motivated bounds on several capacities, including the quantum capacity and private capacity of a channel and the one-way distillable entanglement and private key of a bipartite state. Our bounds themselves are generally given by certain capacities of the complementary channel or state. As a tool to obtain these bounds we discuss partial orders on quantum channels, such as the less noisy and the more capable order. Our bounds help to further understand the interplay between different capacities and give operational limitations on superadditivity properties and the difference between capacities. They can also be used as a new approach towards numerically bounding capacities, as discussed with some examples.

We consider the sequential quantum channel discrimination problem using adaptive and non-adaptive strategies. In this setting the number of uses of the underlying quantum channel is not fixed but a random variable that is either bounded in expectation or with high probability. We show that, by using adaptive strategies for the discrimination problem, both types of error probabilities decrease to zero exponentially fast and the rates are characterized by the measured relative entropy between two quantum channels. Allowing for quantum memory, we see that the optimal rates are given by the regularized channel relative entropy. We also characterize the error exponents in the discrimination problem if non-adaptive strategies are used.

The information bottleneck function gives a measure of optimal preservation of correlation between some random variable X and some side information Y while compressing X into a new random variable W with bounded remaining correlation to X. As such, the information bottleneck has found many natural applications in machine learning, coding and video compression. The main objective in order to calculate the information bottleneck is to find the optimal representation on W. This could in principle be arbitrarily complicated, but fortunately it is known that the cardinality of W can be restricted as |W| ≤ |X |+1 which makes the calculation possible for finite |X|. Now, for many practical applications, e.g. in machine learning, X represents a potentially very large data space, while Y is from a comparably small set of labels. This raises the question whether the known cardinality bound can be improved in such situations. We show that the information bottleneck function can always be approximated up to an error δ(ε, |Y|) with a cardinality |W| ≤ f(ε, |Y|), for explicitly given functions δ and f of an approximation parameter c> 0 and the cardinality of Y. Finally, we generalize the known cardinality bounds to the case were some of the random variables represent quantum information.

Bounds on information combining are entropic inequalities that determine how the information, or entropy, of a set of random variables can change when they are combined in certain prescribed ways. Such bounds play an important role in information theory, particularly in coding and Shannon theory. The arguably most elementary kind of information combining is the addition of two binary random variables, i.e. a CNOT gate, and the resulting quantities are fundamental when investigating belief propagation and polar coding.In this work we will generalize the concept to Rényi entropies. We give optimal bounds on the conditional Rényi entropy after combination, based on a certain convexity or concavity property and discuss when this property indeed holds. Since there is no generally agreed upon definition of the conditional Rényi entropy, we consider four different versions from the literature.Finally, we discuss the application of these bounds to the polarization of Rényi entropies under polar codes.

It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We show that, for the discrimination of classical-quantum channels, adaptive strategies do not lead to an asymptotic advantage. As our main result, this establishes Stein's lemma for classical-quantum channels. Our proofs are based on the concept of amortized distinguishability of channels, which we analyse using entropy inequalities.

In classical information theory, the information bottleneck method (IBM) can be regarded as a method of lossy data compression which focuses on preserving meaningful (or relevant) information. As such it has of late gained a lot of attention, primarily for its applications in machine learning and neural networks. A quantum analogue of the IBM has recently been defined, and an attempt at providing an operational interpretation of the so-called quantum IB function as an optimal rate of an information-theoretic task, has recently been made by Salek et al. The interpretation given by these authors, however, rests on their conjecture that the quantum IB function is convex. Our first contribution is the proof of this conjecture.Secondly, the expression for the rate function involves certain entropic quantities which occur explicitly in the very definition of the underlying information-theoretic task, thus making the latter somewhat contrived. We overcome this drawback by pointing out an alternative operational interpretation of it as the optimal rate of a bona fide information-theoretic task, namely that of quantum source coding with quantum side information at the decoder, which has recently been solved by Hsieh and Watanabe. We show that the quantum IB function characterizes the rate region of this task.We similarly show that the related privacy funnel function is concave (both in the classical and quantum case). However, we comment that it is unlikely that the quantum privacy funnel function can characterize the optimal asymptotic rate of an information theoretic task, since even its classical version lacks a certain essential additivity property.

“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 ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory. 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. 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; (2) the quantitative improvement of strong subadditivity due to Fawzi-Renner; (3) recent duality results due to Renes et al. We furthermore present conjectures on the optimal bounds under quantum side information, supported by analytical observations and strong numerical evidence. We finally apply our bounds to Polar coding for classical-quantum channels, and show that even non-stationary channels polarize, the blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity and under the 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.

Many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialized to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields ln det VAC+ln det VBC-ln det VABC-ln det VC≥ 0 for all 3 × 3 block matrices VABC, where subscripts identify principal submatrices. We shall refer to the above-mentioned inequality as SSA of log-det entropy. In this paper, we develop further insights on the properties of the above-mentioned inequality and its applications to classical and quantum information theory. In the first part of this paper, we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs. Subsequently, we discuss the role of the classical transpose channel (also known as Petz recovery map) in this problem and find its action explicitly. We then prove some extensions of the saturation theorem, by finding faithful lower bounds on a log-det conditional mutual information. In the second part, we focus on quantum Gaussian states, whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation. For Gaussian states, the log-det entropy is equivalent to the Rényi entropy of order 2. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian Rényi-2 entanglement of formation, a well-behaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a log-det entropy equivalent of the squashed entanglement measure, which is remarkably shown to coincide with the Gaussian Rényi-2 entanglement of formation. This allows us to establish useful properties of such measure(s), such as monogamy, faithfulness, and additivity on Gaussian states.

Polar coding is a method for communication over noisy classical channels, which is provably capacity achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In this paper, we apply the polar coding method to network classical-quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a so-called quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder.

We present a new achievable rate region for the two-user binary-input classical-quantum broadcast channel. The result is a generalization of the classical Marton-Gelfand-Pinsker region and is provably larger than the best previously known rate region for classical-quantum broadcast channels. The proof of achievability is based on the recently introduced polar coding scheme and its generalization to quantum network information theory.

Proving achievability of protocols in quantum Shannon theory usually does not consider the efficiency at which the goal of the protocol can be achieved. Nevertheless it is known that protocols such as coherent state merging are efficiently achievable at optimal rate.We aim to investigate this fact further in a general one-shot setting, by considering certain classes of decoupling theorems and give exact rates for these classes. Moreover we compare results of general decoupling theorems using Haar distributed unitaries with those using smaller sets of operators, in particular ε-approximate 2-designs. We also observe the behavior of our rates in special cases such as ε approaching zero and the asymptotic limit.