Quantum cryptography with polarizing interferometers

In the Ekert cryptographic scheme (E91) [1] pairs of EPR particles were used to generate identical sequences of bits in remote places, while Bell’s theorem certified that the particles were not measured in transit by an eavesdropper. The modification of the Ekert scheme proposed by Bennett, Brassard and Mermin (BBM92) [2] did not use the Bell theorem and was based essentially on the fact that two particles correlated by a singlet state behave similarly to a single particle. A user of the cryptographic channel, traditionally called Alice or Bob, instead of sending a polarized particle through the channel, may make a measurement of polarization of his or her particle and somehow “create” an appropriately polarized object at the other side of the channel. In this way the BBM92 protocol becomes a kind of the original Bennett-Brassard 1984 (BB84) protocol [3] but with “polarization at a distance” [4]. Quantum cryptography based on entangled states is an ingenious practical application of the set of ideas and techniques that were originally developed in order to understand the problem of completeness and limitations of quantum mechanics. Another such application, equally amazing and simultaneously very simple, is the idea of interaction-free measuerements proposed by Elitzur and Vaidman (EV) [5]. Below I will discuss the latter in some detail since the objective of this Letter is to show that combining the ideas of BBM92 and EV one arrives at a particularly suggestive version of a quantum cryptographic scheme. The scheme can be formally shown to be secure by a direct application of the proof given in BBM92. What is interesting, however, its security is so explicit that an explanation of the problem to nonexperts becomes particularly simple. As opposed to the standard schemes one can illustrate the point without referring to technical aspects such as complementarity, non-cloning theorem, or Bell’s theorem with all its loopholes. Let us begin with the EV experiment. It is based on the Mach-Zehnder interferometer consisting of two identical semi-transparent symmetric mirrors (Fig. 1). The action of such mirrors can be described by 2× 2 unitary maps