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Quantum cryptography with polarizing interferometers

Marek Czachor

Published Jan 1, 1999• 3 citations
OpenAlex

Abstract

Cryptographic scheme proposed by Bennett, Brassard, and Mermin [Phys. Rev. Lett. {\bf 68}, 557 (1992)] is reformulated in a version involving two polarizing Mach-Zehnder interferometers. Such a form, although physically equivalent to the original one, makes its security explicit, suggestive and easy to explain to non-experts.

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As we have said, quantum mechanics has become the standard framework for most of what is done in physics and indeed has played this role for three-quarters of a century. For just as long, physicists and philosophers have, as we have already suggested, raised and discussed questions about the interpretation of quantum mechanics: Why do we single out measurement as a special kind of interaction that evokes a probabilistic, irreversible response from nature, when other kinds of interaction cause deterministic, reversible changes? How should we talk about quantum superpositions? What does entanglement tell us about the nature of reality? These are interesting questions and researchers still argue about the answers. However, in the last couple of decades researchers have also been thinking along the following line: Let us accept that quantum objects act in weird ways, and see if we can use this weirdness technologically. The two best examples of potential uses of quantum weirdness are quantum computation and quantum cryptography. One of the first quantum cryptographic schemes to appear in the literature, and the only one we consider in detail in this book, was introduced by Charles Bennett and Gilles Brassard in 1984. Their idea is not to use quantum signals directly to convey secret information. Rather, they suggest using quantum signals to generate a secret cryptographic key shared between two parties. Thus the Bennett–Brassard scheme is an example of “quantum key distribution.”
We provide a complete proof of the security of quantum cryptography against any eavesdropping attack including coherent measurements even in the presence of noise. Polarization-based cryptographic schemes are shown to be equivalent to EPR-based schemes. We also show that the performance of a noisy channel approaches that of a noiseless one as the error rate tends to zero. (i.e., the secrecy capacity Cs(ε)→Cs(0)C_s (ε) \to C_s (0)Cs​(ε)→Cs​(0) as ε→0ε\to 0ε→0.) One implication of our results is that one can {\it double} the efficiency of a most well-known quantum cryptographic scheme proposed by Bennett and Brassard simply by assigning vastly different probabilities to the two conjugate bases.
Communications in secrecy are often required in many commercial and military applications. Unfortunately, many cryptographic schemes in use today such as public-key cryptography based on the RSA algorithm would be broken with either unanticipated advances in hardware and algorithm or the advent of quantum computers. Quantum cryptography, on the other hand, has been proven secure even against the most general attack allowed by the laws of physics and is a promising technology poised for widespread adoption in realistic cryptographic applications. Quantum cryptography allows two parties to expand on a secret key that they have previously shared. Various quantum cryptographic protocols have been proposed to perform this task. In this paper, we survey some popular quantum cryptographic protocols (including the famous Bennett-Brassard 1984 protocol) and discuss their security. Specifically, we consider their security in two cases: the ideal case where a perfect single-photon source is used and the practical case where a realistic laser source is used. We compare the protocols and find that the efficient six-state protocol outperforms the others both in the tolerable quantum bit error rate and in the key generation rate when a realistic laser source is used.
In this paper, we rigorously prove the intuition that in security proofs for the Bennett-Brassard 1984 (BB84) protocol, one may regard an incoming signal to Bob as a qubit state. From this result, it follows that all security proofs for BB84 protocol based on a virtual qubit entanglement distillation protocol, which was originally proposed by Lo and Chau [Science 283, 2050 (1999)] and by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)], are all valid even if Bob's actual apparatus cannot distill a qubit state explicitly. As a consequence, especially, the well-known result that a higher bit error rate of 20% can be tolerated for BB84 protocol by using two-way classical communications is still valid even when Bob uses threshold detectors. Using the same technique, we also prove the security of Bennett-Brassard-Mermin 1992 (BBM92) protocol where Alice and Bob both use threshold detectors.