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 analyzing two well-known qubit-based QKD protocols: the four-state protocol and the six-state protocol.
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 analyzing two well-known qubit-based QKD protocols: the four-state protocol and the six-state protocol.
The use of decoy states in quantum key distribution (QKD) has provided a method for substantially increasing the secret key rate and distance that can be covered by QKD protocols with practical signals. The security analysis of these schemes, however, leaves open the possibility that the development of better proof techniques or better classical postprocessing methods might further improve their performance in realistic scenarios. In this paper, we derive upper bounds on the secure key rate for decoy-state QKD. These bounds are based basically only on the classical correlations established by the legitimate users during the quantum communication phase of the protocol. The only assumption about the possible postprocessing methods is that double click events are randomly assigned to single click events. Further, we consider only secure key rates based on the uncalibrated device scenario which assigns imperfections such as detection inefficiency to the eavesdropper. Our analysis relies on two preconditions for secure two-way and one-way QKD. The legitimate users need to prove that there exists no separable state (in the case of two-way QKD) or that there exists no quantum state having a symmetric extension (one-way QKD) that is compatible with the available measurements results. Both criteria have been previously applied to evaluate single-photon implementations of QKD. Here we use them to investigate a realistic source of weak coherent pulses. The resulting upper bounds can be formulated as a convex optimization problem known as a semidefinite program which can be efficiently solved. For the standard four-state QKD protocol, they are quite close to known lower bounds, thus showing that there are clear limits to the further improvement of classical postprocessing techniques in decoy-state QKD.
We investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret-key rate of those QKD protocols involving only entropies of two-qubit density operators. As an illustration of our results, we determine new bounds for the Bennett-Brassard 1984, the 6-state, and the Bennett 1992 protocols. We show that in all these cases the first classical processing that the legitimate partners should apply consists in adding noise.
Bound secret information is classical information that contains secrecy but from which secrecy cannot be extracted. The existence of bound secrecy has been conjectured but is currently unproven, and in this work we provide analytical and numerical evidence for its existence. Specifically, we consider two-way postprocessing protocols in prepare-and-measure quantum key distribution based on the well-known six-state signal states. In terms of the quantum bit-error rate Q of the classical data, such protocols currently exist for $Q<\frac{5\ensuremath{-}\sqrt{5}}{10}\ensuremath{\approx}27.6%$. On the other hand, for $Q\ensuremath{\ge}\frac{1}{3}$ no such protocol can exist as the observed data are compatible with an intercept-resend attack. This leaves the interesting question of whether successful protocols exist in the interval $\frac{5\ensuremath{-}\sqrt{5}}{10}\ensuremath{\le}Q<\frac{1}{3}$. Previous work has shown that a necessary condition for the existence of two-way postprocessing protocols for distilling secret keys is breaking the symmetric extendability of the underlying quantum state shared by Alice and Bob. Using this result, it has been proven that symmetric extendability can be broken up to the 27.6 lower bound using the advantage distillation protocol. In this work, we first show that to break symmetric extendability it is sufficient to consider a generalized form of advantage distillation consisting of one round of postselection by Bob on a block of his data. We then provide evidence that such generalized protocols cannot break symmetric extendability beyond 27.6. We thus have evidence to believe that 27.6 is an upper bound on two-way postprocessing and that the interval $\frac{5\ensuremath{-}\sqrt{5}}{10}\ensuremath{\le}Q<\frac{1}{3}$ is a domain of bound secrecy.
We investigate the resilience to noise of single-qubit quantum key distribution (QKD) protocols in the scenario of security against independent eavesdropping attacks and key distillation based on one-way classical communication. To this end, we introduce a noise-based metric that quantifies the efficiency of QKD protocols. Within this framework, we analyze the maximal noise levels that allow Alice and Bob to asymptotically establish a secure secret key. Using this assumption, we compare the noise tolerance of general single-qubit QKD protocols, in particular the BB84, B92, E91, and six-state protocols. Our main result determines the noise level threshold for QKD allowing one to distill an asymptotically secure secret key. Additionally, we demonstrate that the six-state protocol achieves the greatest resistance to noise while simultaneously yielding a higher post-selection efficiency than the other analyzed single-qubit protocols, confirming its robustness within the considered security model. Finally, we perform an analysis of the proposed noise-based metric and the conventional quantum bit error rate (QBER) metric.
Recently, Chau [Phys. Rev. A 92, 062324 (2015)] introduced an experimentally feasible qudit-based quantum-key-distribution (QKD) scheme. In that scheme, one bit of information is phase encoded in the prepared state in a 2n-dimensional Hilbert space in the form $(|i\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|j\ensuremath{\rangle})/\sqrt{2}$ with $n\ensuremath{\ge}2$. For each qudit prepared and measured in the same two-dimensional Hilbert subspace, one bit of raw secret key is obtained in the absence of transmission error. Here we show that by modifying the basis announcement procedure, the same experimental setup can generate n bits of raw key for each qudit prepared and measured in the same basis in the noiseless situation. The reason is that in addition to the phase information, each qudit also carries information on the Hilbert subspace used. The additional $(n\ensuremath{-}1)$ bits of raw key comes from a clever utilization of this extra piece of information. We prove the unconditional security of this modified protocol and compare its performance with other existing provably secure qubit- and qudit-based protocols on market in the one-way classical communication setting. Interestingly, we find that for the case of n=2, the secret key rate of this modified protocol using nondegenerate random quantum code to perform one-way entanglement distillation is equal to that of the six-state scheme.
We present a technique for proving the security of quantum-key-distribution (QKD) 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.