Abstract The aim of device-independent quantum key distribution (DIQKD) is to study protocols that allow the generation of a secret shared key between two parties under minimal assumptions on the devices that produce the key. These devices are merely modeled as black boxes and mathematically described as conditional probability distributions. A major obstacle in the analysis of DIQKD protocols is the huge space of possible black box behaviors. De Finetti theorems can help to overcome this problem by reducing the analysis to black boxes that have an iid structure. Here we show two new de Finetti theorems that relate conditional probability distributions in the quantum set to de Finetti distributions (convex combinations of iid distributions) that are themselves in the quantum set. We also show how one of these de Finetti theorems can be used to enforce some restrictions onto the attacker of a DIQKD protocol. Finally we observe that some desirable strengthenings of this restriction, for instance to collective attacks only, are not straightforwardly possible.
Abstract The aim of device-independent quantum key distribution (DIQKD) is to study protocols that allow the generation of a secret shared key between two parties under minimal assumptions on the devices that produce the key. These devices are merely modeled as black boxes and mathematically described as conditional probability distributions. A major obstacle in the analysis of DIQKD protocols is the huge space of possible black box behaviors. De Finetti theorems can help to overcome this problem by reducing the analysis to black boxes that have an iid structure. Here we show two new de Finetti theorems that relate conditional probability distributions in the quantum set to de Finetti distributions (convex combinations of iid distributions) that are themselves in the quantum set. We also show how one of these de Finetti theorems can be used to enforce some restrictions onto the attacker of a DIQKD protocol. Finally we observe that some desirable strengthenings of this restriction, for instance to collective attacks only, are not straightforwardly possible.
de Finetti-type theorems enable a substantially simplified analysis of information-processing tasks in various areas, such as quantum cryptography and quantum tomography. The idea is that instead of carrying out the analysis for any possible state it is sufficient to consider one particular de Finetti state, that is, a convex combination of i.i.d. states. It is thus interesting to see whether such de Finetti-type theorems are unique for quantum theory or whether they hold for more general theories. Here we prove a de Finetti-type theorem in the framework of conditional probability distributions. In this framework, a system is described by a conditional probability distribution P_A|X where X denotes the measurement and A the outcome. We show that any permutation invariant system P_A|X can be reduced to a de Finetti system. We then demonstrate how the theorem can be applied to security proofs of device independent cryptographic protocols.
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti theorem to situations where two conjugate measurements obey different statistics, such as the security analysis of quantum key distribution protocols based on squeezed state against coherent attack.
Proving the unconditional security of quantum key distribution (QKD) is a highly challenging task as one needs to determine the most efficient attack compatible with experimental data. This task is even more demanding for continuous-variable QKD as the Hilbert space where the protocol is described is infinite dimensional. A possible strategy to address this problem is to make an extensive use of the symmetries of the protocol. In this paper, we investigate a rotation symmetry in phase space that is particularly relevant to continuous-variable QKD, and explore the way towards a new quantum de Finetti theorem that would exploit this symmetry and provide a powerful tool to assess the security of continuous-variable protocols. As a first step, a single-party asymptotic version of this quantum de Finetti theorem in phase space is derived.
Abstract Device-independent quantum key distribution (DIQKD) generates a secret key among two parties in a provably secure way without making assumptions about the internal working of the devices used in the protocol. The main challenge for a DIQKD physical implementation is that the data observed among the two parties must violate a Bell inequality without fair-sampling, since otherwise the observed correlations can be faked with classical resources and security can no longer be guaranteed. In spite of the advances recently made to achieve higher detection efficiencies in Bell experiments, DIQKD remains experimentally difficult at long distances due to the exponential increase of loss in the channel separating the two parties. Here we describe and analyze plausible solutions to overcome the crucial problem of channel loss in the frame of DIQKD physical implementations.
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.
Quantum de Finetti theorem indicates that a surprising connection exists between the permutation-invariant property of a large system and the independent, identical distribution that describes its subsystems. The theorem is highly nontrivial in that it allows one to infer independence---within a finite error---from symmetry alone, both of which are important and fundamental concepts in physics and mathematics. Here, we present an improved quantum de Finetti theorem that provides a lower error bound, thus establishing an even closer relationship between symmetry and independence than before. As application, we study the security of quantum key distribution under the most general attacks, and show that the key rate is higher than previous ones obtained by existing trace-norm-based de Finetti theorems. Therefore, the application of the quantum de Finetti theorem in quantum information processing, as well as developing the theorem itself, is advanced here.