Abstract:
We investigate the error tolerance of quantum cryptographic protocols using $d$-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper bound tends slowly to zero for sufficiently large dimensions of the information carriers.

Abstract:
The security of a cryptographic key that is generated by communication through a noisy quantum channel relies on the ability to distill a shorter secure key sequence from a longer insecure one. We show that -- for protocols that use quantum channels of any dimension and completely characterize them by state tomography -- the noise threshold for classical advantage distillation is substantially lower than the threshold for quantum entanglement distillation because the eavesdropper can perform powerful coherent attacks. The earlier claims that the two noise thresholds are identical, which were based on analyzing incoherent attacks only, are therefore invalid.

Abstract:
Many different quantum information communication protocols such as teleportation, dense coding and entanglement based quantum key distribution are based on the faithful transmission of entanglement between distant location in an optical network. The distribution of entanglement in such a network is however hampered by loss and noise that is inherent in all practical quantum channels. Thus, to enable faithful transmission one must resort to the protocol of entanglement distillation. In this paper we present a detailed theoretical analysis and an experimental realization of continuous variable entanglement distillation in a channel that is inflicted by different kinds of non-Gaussian noise. The continuous variable entangled states are generated by exploiting the third order non-linearity in optical fibers, and the states are sent through a free-space laboratory channel in which the losses are altered to simulate a free-space atmospheric channel with varying losses. We use linear optical components, homodyne measurements and classical communication to distill the entanglement, and we find that by using this method the entanglement can be probabilistically increased for some specific non-Gaussian noise channels.

Abstract:
The security of a cryptographic key that is generated by communication through a noisy quantum channel relies on the ability to distill a shorter secure key sequence from a longer insecure one. For an important class of protocols, which exploit tomographically complete measurements on entangled pairs of any dimension, we show that the noise threshold for classical advantage distillation is identical with the threshold for quantum entanglement distillation. As a consequence, the two distillation procedures are equivalent: neither offers a security advantage over the other.

Abstract:
The squashed entanglement of a quantum channel is an additive function of quantum channels, which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication. This quantity has led to an upper bound of $\log((1+\eta )/(1-\eta))$ on the capacity of an optical communication channel for such a task, where $\eta$ is the average fraction of photons that make it from the input to the output of the channel. The purpose of the present paper is to extend these results beyond the single-sender single-receiver setting to the more general case of a single sender and multiple receivers (a quantum broadcast channel). We employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel, along the way developing several new properties of these measures. We apply our results to the case of an optical broadcast channel with one sender and two receivers.

Abstract:
We propose an information reconciliation protocol that uses two-way classical communication. In the case of the BB84 protocol and the six-state protocol, the key rates of the quantum key distribution (QKD) protocols that use our proposed information reconciliation protocol are higher than previously known protocols for wide range of error rates. We also clarify the relation between the proposed protocol and known QKD protocols, and the relation between the proposed protocol and entanglement distillation protocols (EDPs).

Abstract:
Assume that two distant parties, Alice and Bob, as well as an adversary, Eve, have access to (quantum) systems prepared jointly according to a tripartite state. In addition, Alice and Bob can use local operations and authenticated public classical communication. Their goal is to establish a key which is unknown to Eve. We initiate the study of this scenario as a unification of two standard scenarios: (i) key distillation (agreement) from classical correlations and (ii) key distillation from pure tripartite quantum states. Firstly, we obtain generalisations of fundamental results related to scenarios (i) and (ii), including upper bounds on the key rate. Moreover, based on an embedding of classical distributions into quantum states, we are able to find new connections between protocols and quantities in the standard scenarios (i) and (ii). Secondly, we study specific properties of key distillation protocols. In particular, we show that every protocol that makes use of pre-shared key can be transformed into an equally efficient protocol which needs no pre-shared key. This result is of practical significance as it applies to quantum key distribution (QKD) protocols, but it also implies that the key rate cannot be locked with information on Eve's side. Finally, we exhibit an arbitrarily large separation between the key rate in the standard setting where Eve is equipped with quantum memory and the key rate in a setting where Eve is only given classical memory. This shows that assumptions on the nature of Eve's memory are important in order to determine the correct security threshold in QKD.

Abstract:
Prepare and measure quantum key distribution protocols can be decomposed into two basic steps: delivery of the signals over a quantum channel and distillation of a secret key from the signal and measurement records by classical processing and public communication. Here we formalize the distillation process for a general protocol in a purely quantum-mechanical framework and demonstrate that it can be viewed as creating an ``effective'' quantum channel between the legitimate users Alice and Bob. The process of secret key generation can then be viewed as entanglement distribution using this channel, which enables application of entanglement-based security proofs to essentially any prepare and measure protocol. To ensure secrecy of the key, Alice and Bob must be able to estimate the channel noise from errors in the key, and we further show how symmetries of the distillation process simplify this task. Applying this method, we prove the security of several key distribution protocols based on equiangular spherical codes.

Abstract:
A continuous-variable quantum key distribution protocol based on squeezed states and heterodyne detection is introduced and shown to attain higher secret key rates over a noisy line than any other one-way Gaussian protocol. This increased resistance to channel noise can be understood as resulting from purposely adding noise to the signal that is converted into the secret key. This notion of noise-enhanced tolerance to noise also provides a better physical insight into the poorly understood discrepancies between the previously defined families of Gaussian protocols.

Abstract:
We present a simple method to obtain an upper bound on the achievable secret key rate in quantum key distribution (QKD) protocols that use only unidirectional classical communication during the public-discussion phase. This method is based on a necessary precondition for one-way secret key distillation; the legitimate users need to prove that there exists no quantum state having a symmetric extension that is compatible with the available measurements results. The main advantage of the obtained upper bound is that it can be formulated as a semidefinite program, which can be efficiently solved. We illustrate our results by analysing two well-known qubit-based QKD protocols: the four-state protocol and the six-state protocol. Recent results by Renner et al., Phys. Rev. A 72, 012332 (2005), also show that the given precondition is only necessary but not sufficient for unidirectional secret key distillation.