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arXiv:quant-ph/0307151 · Physical Review Letters

Entanglement as a Precondition for Secure Quantum Key Distribution

Marcos Curty, Maciej Lewenstein, Norbert Lütkenhaus

Published May 27, 2004• 330 citations
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Abstract

We demonstrate that a necessary precondition for an unconditionally secure quantum key distribution is that both sender and receiver can use the available measurement results to prove the presence of entanglement in a quantum state that is effectively distributed between them. One can thus systematically search for entanglement using the class of entanglement witness operators that can be constructed from the observed data. We apply such analysis to two well-known quantum key distribution protocols, namely, the 4-state protocol and the 6-state protocol. As a special case, we show that, for some asymmetric error patterns, the presence of entanglement can be proven even for error rates above 25% (4-state protocol) and 33% (6-state protocol).

DOI
10.1103/physrevlett.92.217903

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We demonstrate that a necessary precondition for an unconditionally secure quantum key distribution is that both sender and receiver can use the available measurement results to prove the presence of entanglement in a quantum state that is effectively distributed between them. One can thus systematically search for entanglement using the class of entanglement witness operators that can be constructed from the observed data. We apply such analysis to two well-known quantum key distribution protocols, namely, the 4-state protocol and the 6-state protocol. As a special case, we show that, for some asymmetric error patterns, the presence of entanglement can be proven even for error rates above 25% (4-state protocol) and 33% (6-state protocol).
A necessary precondition for secure quantum key distribution (QKD) is that sender and receiver can prove the presence of entanglement in a quantum state that is effectively distributed between them. In order to deliver this entanglement proof one can use the class of entanglement witness operators that can be constructed from the available measurements results. This criterion provides a necessary and sufficient condition for the existence of quantum correlations even when a quantum state cannot be completely reconstructed. We apply such analysis to three well‐known qubit‐based QKD protocols, namely the 6‐state, the 4‐state and the 2‐state protocols.
A necessary precondition for secure quantum key distribution is that sender and receiver can prove the presence of entanglement in a quantum state that is effectively distributed between them. In order to deliver this entanglement proof one can use the class of entanglement witness (EW) operators that can be constructed from the available measurements results. This class of EWs can be used to provide a necessary and sufficient condition for the existence of quantum correlations even when a quantum state cannot be completely reconstructed. The set of optimal EWs for two well-known entanglement-based (EB) schemes, the six-state and the four-state EB protocols, has been obtained recently [M. Curty et al., Phys. Rev. Lett. 92, 217903 (2004).] Here we complete these results, now showing specifically the analysis for the case of prepare and measure (PM) schemes. For this, we investigate the signal states and detection methods of the four-state and the two-state PM schemes. For each of these protocols we obtain a reduced set of EWs. More importantly, each set of EWs can be used to derive a necessary and sufficient condition to prove that quantum correlations are present in these protocols.
Practical quantum key distribution can be understood as a two-step procedure: in a first step two parties exchange quantum mechanical signals and perform measurements on them, in a second step auxiliary classical communication protocols are performed over an authenticated public channel to transform the data of the first step into an information-theoretic secure key. In this article we address the question of necessary conditions on the correlated (classical) data of the first step such that there can be a successful second step at all. As it turns out, a necessary condition is that these data, together with the knowledge about the physical set-up of sender and receiver, allow to establish a proof of effective entanglement between the two parties. We then demonstrate methods to systematically search for such a proof in basic settings, involving the 2-, 4-, and 6-state protocols.
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.
We prove the unconditional security of the six-state protocol with threshold detectors and one-way classical communication. Unlike the four-state protocol (BB84), it has been proven that the squash operator for the six-state does not exist, i.e., the statistics of the measurements cannot be obtained via measurement on qubits. We propose a technique to determine which photon number states are important, and we consider a fictitious measurement on a qubit, which is defined through the squash operator of BB84, for the better estimation of Eve's information. As a result, we prove that the bit error rate threshold for the six-state protocol (12.611%) remains almost the same as the one of the qubit-based six-state protocol (12.619%). This clearly demonstrates the robustness of the six-state protocol against the use of the practical devices.