Quantum Information Theory is an area of physics which studies both fundamental and applied issues in quantum mechanics from an information-theoretical viewpoint. The underlying techniques are, however, often restricted to the analysis of systems which satisfy a certain independence condition. For example, it is assumed that an experiment can be repeated independently many times or that a large physical system consists of many virtually independent parts. Unfortunately, such assumptions are not always justified. This is particularly the case for practical applications — e.g. in quantum cryptography — where parts of a system might have an arbitrary and unknown behavior. We propose an approach which allows us to study general physical systems for which the above mentioned independence condition does not necessarily hold. It is based on an extension of various information-theoretical notions. For example, we introduce new uncertainty measures, called smooth min- and max-entropy, which are generalizations of the von Neumann entropy. Furthermore, we develop a quantum version of de Finetti's representation theorem, as described below. Consider a physical system consisting of n parts. These might, for instance, be the outcomes of n runs of a physical experiment. Moreover, we assume that the joint state of this n-partite system can be extended to an (n + k)-partite state which is symmetric under permutations of its parts (for some k ≫ 1). The de Finetti representation theorem then says that the original n-partite state is, in a certain sense, close to a mixture of product states. Independence thus follows (approximatively) from a symmetry condition. This symmetry condition can easily be met in many natural situations. For example, it holds for the joint state of n parts, which are chosen at random from an arbitrary (n + k)-partite system. As an application of these techniques, we prove the security of quantum key distribution (QKD), i.e. secret key agreement by communication over a quantum channel. In particular, we show that, in order to analyze QKD protocols, it is generally sufficient to consider so-called collective attacks, where the adversary is restricted to applying the same operation to each particle sent over the quantum channel separately. The proof is generic and thus applies to known protocols such as BB84 and B92 (where better bounds on the secret-key rate and on the the maximum tolerated noise level of the quantum channel are obtained) as well as to continuous variable schemes (where no full security proof has been known). Furthermore, the security holds with respect to a strong so-called universally composable definition. This implies that the keys generated by a QKD protocol can safely be used in any application, e.g. for one-time pad encryption — which, remarkably, is not the case for most standard definitions.

The uncertainty principle, originally formulated by Heisenberg 1 , clearly illustrates the difference between classical and quantum mechanics. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future 2 , it is possible to predict the outcomes for both measurement choices precisely. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. We detail the application of our result to witnessing entanglement and to quantum key distribution.Uncertainty relations constrain the potential knowledge one can have about the physical properties of a system. Although classical theory does not limit the knowledge we can simultaneously have about arbitrary properties of a particle, such a limit does exist in quantum theory. Even with a complete description of its state, it is impossible to predict the outcomes of all possible measurements on the particle. This lack of knowledge, or uncertainty, was quantified by Heisenberg 1 using the standard deviation (which we denote by R for an observable R). If the measurement on a given particle is chosen from a set of two possible observables, R and S, the resulting bound on the uncertainty can be expressed in terms of the commutator 3 :In an information-theoretic context, it is more natural to quantify uncertainty in terms of entropy rather than the standard deviation. Entropic uncertainty relations for position and momentum were derived in ref. 4 and later a relation was developed that holds for any pair of observables 5 . An improvement of this relation was subsequently conjectured 6 and then proved 7 . The improved relation iswhere H (R) denotes the Shannon entropy of the probability distribution of the outcomes when R is measured. The term 1/c quantifies the complementarity of the observables. For non-degenerate observables, c := max j,k | ψ j |φ k | 2 , where |ψ j and |φ k are the eigenvectors of R and S, respectively. One way to think about uncertainty relations is through the following game (the uncertainty game) between two players, Alice and Bob. Before the game commences, Alice and Bob agree on two measurements, R and S. The game proceeds as follows. Bob prepares a particle in a quantum state of his choosing and sends it to Alice. Alice then carries out one of the two measurements and announces her choice to Bob. Bob's task is to minimize his uncertainty about Alice's measurement outcome. This is illustrated in Fig. 1.Equation (1) bounds Bob's uncertainty in the case that he has no quantum memory-all information Bob hold...

Despite enormous theoretical and experimental progress in quantum cryptography, the security of most current implementations of quantum key distribution is still not rigorously established. One significant problem is that the security of the final key strongly depends on the number, M, of signals exchanged between the legitimate parties. Yet, existing security proofs are often only valid asymptotically, for unrealistically large values of M. Another challenge is that most security proofs are very sensitive to small differences between the physical devices used by the protocol and the theoretical model used to describe them. Here we show that these gaps between theory and experiment can be simultaneously overcome by using a recently developed proof technique based on the uncertainty relation for smooth entropies.

Abstract-In this paper, we show that the conditional min-entropy Hmin(AjB) of a bipartite state AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy H max (AjB) to the maximum fidelity of AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B.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 guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B.Index Terms-Entropy measures, max-entropy, min-entropy, operational interpretations, quantum information theory, quantum hypothesis testing, singlet fraction, single-shot information theory.

We present a technique for proving the security of quantum-key-distribution ͑QKD͒ protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel.