We correct an error in the proof of Theorem 11 in our paper “Approximate Degradable Quantum Channels”, IEEE Trans. Inf. Theory, vol. 63, no. 12, pp. 7832-7844, 2017, concerning an upper bound on the private capacity of an approximate anti-degradable channel. Furthermore, we show how to obtain a tighter bound for the quantum capacity.

Uhlmann’s theorem states that, for any two quantum states ρABand σA, there exists an extension σABof σAsuch that the fidelity between ρABand σABequals the fidelity between their reduced states ρAand σA. In this work, we generalize Uhlmann’s theorem to α-R´enyi relative entropies for α ∈ [1/2, ∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α = 1/2, α = 1, and α = ∞, respectively.

The min-entropy of a quantum system A conditioned on another quantum system E describes how much randomness can be extracted from A with respect to an adversary in possession of E. This quantity plays a crucial role in quantum cryptography: the security proofs of many quantum cryptographic protocols reduce to showing a lower bound on such a min-entropy. Here, we develop a new tool, called generalised entropy accumulation, for computing such bounds. Concretely, we consider a sequential process in which each step outputs a system Aiand updates a side information register E. We prove that if this process satisfies a natural “non-signalling” condition between past outputs and future side information, the min-entropy of the outputs $A_{1},\ldots,\ A_{n}$ conditioned on the side information E at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT) [1], which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition.Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. In particular, it is the first general tool that is applicable to mistrustful device-independent cryptography. To demonstrate this, we give the first security proof for blind randomness expansion [2] against general adversaries. Furthermore, our generalised EAT can be used to give improved security proofs for quantum key distribution [3], and also has applications beyond quantum cryptography.

Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively. The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.

Device-independent security is the gold standard for quantum cryptography: not only is security based entirely on the laws of quantum mechanics, but it holds irrespective of any a priori assumptions on the quantum devices used in a protocol, making it particularly applicable in a quantum-wary environment. While the existence of device-independent protocols for tasks such as randomness expansion and quantum key distribution has recently been established, the underlying proofs of security remain very challenging, yield rather poor key rates, and demand very high quality quantum devices, thus making them all but impossible to implement in practice. We introduce a technique for the analysis of device-independent cryptographic protocols. We provide a flexible protocol and give a security proof that provides quantitative bounds that are asymptotically tight, even in the presence of general quantum adversaries. At a high level our approach amounts to establishing a reduction to the scenario in which the untrusted device operates in an identical and independent way in each round of the protocol. This is achieved by leveraging the sequential nature of the protocol and makes use of a newly developed tool, the “entropy accumulation theorem” of Dupuis, Fawzi, and Renner [ Entropy Accumulation, preprint, 2016]. As concrete applications we give simple and modular security proofs for device-independent quantum key distribution and randomness expansion protocols based on the CHSH inequality. For both tasks, we establish essentially optimal asymptotic key rates and noise tolerance. In view of recent experimental progress, which has culminated in loophole-free Bell tests, it is likely that these protocols can be practically implemented in the near future.

Quantum teleportation uses prior shared entanglement and classical communication to send an unknown quantum state from one party to another. Remote state preparation (RSP) is a similar distributed task in which the sender knows the entire classical description of the state to be sent. (This may also be viewed as the task of nonoblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate RSP (ARSP) in which the goal is to prepare an approximation of the desired quantum state. Jain (Quant. Inf. & Comp., 2006) showed that the worst-case communication complexity of ARSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this paper, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of nonasymptotic information-theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that $n$ bits cannot be communicated with less than $n$ transmitted bits in local operations and classical communication protocols. This strengthens a result due to Nayak and Salzman (J. ACM, 2006) and may be of independent interest.

Complex information-processing systems, for example, quantum circuits, cryptographic protocols, or multi-player games, are naturally described as networks composed of more basic information-processing systems. A modular analysis of such systems requires a mathematical model of systems that is closed under composition, i.e., a network of these objects is again an object of the same type. We propose such a model and call the corresponding systems causal boxes. Causal boxes capture superpositions of causal structures, e.g., messages sent by a causal box A can be in a superposition of different orders or in a superposition of being sent to box B and box C. Furthermore, causal boxes can model systems whose behavior depends on time. By instantiating the abstract cryptography framework with causal boxes, we obtain the first composable security framework that can handle arbitrary quantum protocols and relativistic protocols.

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that a complementary channel can be obtained from the channel by applying a degrading channel. In this paper, we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading channel which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

The quantum relative entropy is well known to obey a monotonicity property (i.e., it does not increase under the action of a quantum channel). Here we present several refinements of this entropy inequality, some of which have a physical interpretation in terms of recovery from the action of the channel. The recovery channel given here is explicit and universal, depending only on the channel and one of the arguments to the relative entropy.

In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1 - α and its repeated version Gn(in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this paper, the case of multiplayer non-signaling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players, we prove a threshold theorem stating that the probability that non-signaling players win more than a fraction 1-α+β of the n games is exponentially small in nβ2for every 0 ≤ β ≤ α. For games with incomplete support, we derive a similar statement for a slightly modified form of repetition. The result is proved using a new technique based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems.

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite-dimensional output under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an additive ε-close estimate to the capacity, the presented algorithm requires O((N ν M)M3log(N)1/2ε-1) steps, where N denotes the input alphabet size and M denotes the output dimension. We then generalize the method to the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite-dimensional output. This, using the idea of a universal encoder, allows us to approximate the Holevo capacity for channels with a finite-dimensional quantum mechanical input and output. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multi-dimensional integration problem. For certain families of quantum channels, we prove that the complexity to derive an additive ε-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that the complementary channel can be obtained from the channel by applying a degrading map. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading map which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

We construct an explicit quantum coding scheme which achieves a communication rate not less than the coherent information when used to transmit the quantum information over a noisy quantum channel. For Pauli and erasure channels, we also present efficient encoding and decoding algorithms for this communication scheme based on polar codes (essentially linear in the blocklength), but which do not require the sender and receiver to share any entanglement before the protocol begins. Due to the existence of degeneracies in the involved error-correcting codes, it is indeed possible that the rate of the scheme exceeds the coherent information. We provide a simple criterion which indicates such performance. Finally, we discuss how the scheme can be used for secret key distillation as well as private channel coding.

We study formal properties of smooth max-information, a generalization of von Neumann mutual information derived from the max-relative entropy. Recent work suggests that it is a useful quantity in one-shot channel coding, quantum rate distortion theory, and the physics of quantum many-body systems. Max-information can be defined in multiple ways. We demonstrate that different smoothed definitions are essentially equivalent (up to logarithmic terms in the smoothing parameters). These equivalence relations allow us to derive new chain rules for the max-information in terms of min- and max-entropies, thus extending the smooth entropy formalism to mutual information.

We analyze a cryptographic protocol for generating a distributed secret key from correlations that violate a Bell inequality by a sufficient amount, and prove its security against eavesdroppers, constrained only by the assumption that any information accessible to them must be compatible with the non-signaling principle. The claim holds with respect to the state-of-the-art security definition used in cryptography, known as universally-composable security. The non-signaling assumption only refers to the statistics of measurement outcomes depending on the choices of measurements; hence security is independent of the internal workings of the devices - they do not even need to follow the laws of quantum theory. This is relevant for practice as a correct and complete modeling of realistic devices is generally impossible. The techniques developed are general and can be applied to other Bell inequality-based protocols. In particular, we provide a scheme for estimating Bell-inequality violations when the samples are not independent and identically distributed.

We construct an explicit entanglement distillation scheme which achieves the coherent information when used to send quantum information over a noisy quantum channel. For Pauli and erasure channels we present efficient encoding and decoding algorithms based on polar codes. Unlike previous constructions, this scheme does not require the sender and receiver to share noiseless entanglement before the protocol begins. It is possible, but still unproven, that the scheme even achieves a rate beyond the coherent information, due to degeneracies of certain error correcting codes. Finally we discuss how the scheme can be used for secret key distillation and private channel coding.

We provide a framework for one-shot quantum rate distortion coding, in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level. We obtain a one-shot characterization of the minimum qubit compression size for an entanglement-assisted quantum rate-distortion code in terms of the smooth max-information, a quantity previously employed in the one-shot quantum reverse Shannon theorem. Next, we show how this characterization converges to the known expression for the entanglement-assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source. Finally, we give a tight, finite blocklength characterization for the entanglement-assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbolwise distortion constraint.

The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here, we consider the chain rule for the more general smooth min- and max-entropies, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality. However, the standard chain rule for the von Neumann entropy is retrieved asymptotically when evaluating the smooth entropies for many identical and independently distributed states.

We construct a channel coding scheme to achieve the capacity of any discrete memoryless channel based solely on the techniques of polar coding. In particular, we show how source polarization and randomness extraction via polarization can be employed to “shape” uniformly-distributed i.i.d. random variables into approximate i.i.d. random variables distributed according to the capacity-achieving distribution. We then combine this shaper with a variant of polar channel coding, constructed by the duality with source coding, to achieve the channel capacity. Our scheme inherits the low complexity encoder and decoder of polar coding. It differs conceptually from Gallager's method for achieving capacity, and we discuss the advantages and disadvantages of the two schemes. An application to the AWGN channel is discussed.

Randomness extraction involves the processing of purely classical information and is therefore usually studied with in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well-known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information. We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g., extract all the conditional (smooth) min-entropy of the source using a seed of length polylogarithmic in the input, or only require the seed to be weakly random.

The task of compressing classical information in the one-shot scenario is studied in the setting where the decompressor additionally has access to some given quantum side information. In this hybrid classical-quantum version of the famous Slepian-Wolf problem, the smooth max entropy is found to govern the number of bits into which classical information can be compressed so that it can be reliably recovered from the compressed version and quantum side information. Combining this result with known results on privacy amplification then yields tight bounds on the amount of common randomness and secret key that can be recovered in one shot from hybrid classical-quantum systems using one-way classical communication.

Smooth entropies characterize basic information-theoretic properties of random variables, such as the number of bits required to store them or the amount of uniform randomness that can be extracted from them (possibly with respect to side information). In this paper, explicit and almost tight bounds on the smooth entropies of n-fold product distributions, Pn, are derived. These bounds are expressed in terms of the Shannon entropy of a single distribution, P . The results can be seen as an extension of the asymptotic equipartition property (AEP).

Let$X_1, \ldots, X_n$be a sequence of$n$classical random variables and consider a sample$X_{s_1}, \ldots, X_{s_r}$of$r \leq n$positions selected at random. Then, except with (exponentially in$r$) small probability, the min-entropy$H_{\min}(X_{s_1} \cdots X_{s_r})$of the sample is not smaller than, roughly, a fraction${r\over n}$of the overall entropy$H_{\min}(X_1 \cdots X_n)$, which is optimal. Here, we show that this statement, originally proved in [S. Vadhan, LNCS 2729, Springer, 2003] for the purely classical case, is still true if the min-entropy$H_{\min}$is measured relative to a quantum system. Because min-entropy quantifies the amount of randomness that can be extracted from a given random variable, our result can be used to prove the soundness of locally computable extractors in a context where side information might be quantum-mechanical. In particular, it implies that key agreement in the bounded-storage model—using a standard sample-and-hash protocol—is fully secure against quantum adversaries, thus solving a long-standing open problem.

We show that optimal protocols for noisy channel coding of public or private information over either classical or quantum channels can be directly constructed from two more primitive information-theoretic protocols: privacy amplification and information reconciliation, also known as data compression with side information. We do this in the one-shot scenario of structureless resources, and formulate our results in terms of the smooth min- and max-entropy. In the context of classical information theory, this shows that essentially all two-terminal protocols can be reduced to these two primitives, which are in turn governed by the smooth min- and max-entropies, respectively. In the context of quantum information theory, the recently-established duality of these two protocols means essentially all two-terminal protocols can be constructed using just a single primitive. As an illustration, we show how optimal noisy channel coding protocols can be constructed solely from privacy amplification.

The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, a strictly more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system is shown. Our result applies to almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing.

The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, we prove a (strictly) more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system. Furthermore, our result applies to arbitrary δ-almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing.

In classical and quantum information theory, operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures, called smooth min-entropy and smooth max-entropy, respectively. While both entropies are equal to the von Neumann entropy in certain special cases (e.g., asymptotically, for many independent repetitions of the given data), their values can differ arbitrarily in the general case. In this paper, a recently discovered duality relation between (nonsmooth) min- and max-entropies is extended to the smooth case. More precisely, it is shown that the smooth min-entropy of a systemAconditioned on a systemBequals the negative of the smooth max-entropy ofAconditioned on a purifying systemC. This result immediately implies that certain operational quantities (such as the amount of compression and the amount of randomness that can be extracted from given data) are related. We explain how such relations have applications in cryptographic security proofs.

New channel coding converse and achievability bounds are derived for a single use of an arbitrary channel. Both bounds are expressed using a quantity called the ldquosmooth 0-divergencerdquo, which is a generalization of Renyi's divergence of order 0. The bounds are also studied in the limit of large block-lengths. In particular, they combine to give a general capacity formula which is equivalent to the one derived by Verdu and Han.

Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the information spectrum method and the smooth entropy framework, have been developed independently. They are based on new entropy measures, called spectral entropy rates andsmoothentropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are closely related. More precisely, the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy. Our results apply to the quantum setting and thus include the classical setting as a special case.

In this paper, we show that the conditional min-entropyHmin(A|B) of a bipartite staterhoABis directly related to the maximum achievable overlap with a maximally entangled state if only local actions on theB-part ofrhoABare allowed. In the special case whereAis classical, this overlap corresponds to the probability of guessingAgivenB. In a similar vein, we connect the conditional max-entropyHmax(A|B) to the maximum fidelity ofrhoABwith a product state that is completely mixed onA. In the case whereAis classical, this corresponds to the security ofAwhen used as a secret key in the presence of an adversary holdingB. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessingAgivenBis a lower bound on the number of uniform secret bits that can be extracted fromArelative to an adversary holdingB.

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, a fully quantum generalization of this property is shown, where both the output of the experiment and side information are quantum. An explicit bound on the convergence is given, which is independent of the dimensionality of the side information. This naturally leads to a family of REacutenyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.

Extractors are functions that transform a weakly random value X into an almost perfectly uniform value Z. Traditionally, extractors have been studied in a context where the side information, relative to which the distributions of X and Z are defined, is purely classical. Only recently, the notion of extractors has been generalized to scenarios where side information might be represented by the state of a quantum-mechanical system (while X and Z are still classical). This generalization is crucial for numerous applications, e.g., in cryptography, where an adversary might hold quantum-mechanical side information. In this article, we review this generalized notion of extractors as well as a construction of extractors based on two-universal hashing.

In this paper we provide the answer to the following question: given a noisy channel PY|Xand epsi > 0, how many bits can be transmitted with an error of at most epsi by a single use of the channel

We address the question whether quantum memory is more powerful than classical memory. In particular, we consider a setting where information about a random n-bit string X is stored in s classical or quantum bits, for s

This paper introduces the public Eve scenario and shows that the secret key rate in this scenario is bounded by the intrinsic information. This elucidates previous results and gives new insights in the gap between formation and extraction of secret information. Intrinsic information, in its function as an upper bound on the secret key rate, is generalized to secret key agreement from arbitrary tripartite quantum states.

We introduce the concept of selectable knowledge, which models the information stored in an arbitrary (e.g., quantum mechanical) device. We then analyze a situation where an entity A holds selectable knowledge about some random variable X and quantify the information A has about the output H(X) of a randomly chosen function H applied to X. This generalizes the setting of privacy amplification by universal hashing. In particular, our result can be used to prove that privacy amplification remains secure even if the enemy possesses quantum instead of classical information.

We introduce a new entropy measure, called smooth Renyi entropy. The measure characterizes fundamental properties of a random variable Z, such as the amount of uniform randomness that can be extracted from Z or the minimum length of an encoding of Z.

There are different approaches to proving the impossibility of classical hidden-variable explanations of quantum-mechanical behavior. Whereas Kochen and Specker proved that a three-or higher-dimensional quantum-mechanical system cannot be "classically" prepared for all possible alternative measurements in a consistent way, Bell showed that the behavior of certain two-partite systems is nonlocal, i.e., inexplicable by shared classical information. We show a close connection between deterministic manifestations of such nonlocality-called "pseudotelepathy" games-and Kochen and Specker's theorem: Every such game leads to a Kochen-Specker contradiction, and vice versa