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Journal of Experimental and Theoretical Physics Letters

Wavefunctions of a prolate spheroid and multiplexing in relativistic quantum cryptography on orthogonal states

S. N. Molotkov, T. A. Potapova

Published Jan 1, 2015• 1 citation
Publisher

Abstract

Relativistic quantum cryptography involves not only the geometric properties of the state vectors of a quantum system in the Hilbert space but also the properties of carriers of quantum states in the Minkowski spacetime. A physical type of quantum object carrying information with extremely high velocity in spacetime is of fundamental importance. The structure of spacetime, more precisely, irreducible representations of the Poincaré group, in the Hilbert space is responsible for the existence of massless particles, photons. In this sense, the structure of spacetime is in fact used in relativistic quantum cryptography systems for cryptographic key distribution. This makes it possible to guarantee the security of keys even with the use of orthogonal states .

DOI
10.1134/s0021364014210115

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Relativistic quantum cryptography involves not only the geometric properties of the state vectors of a quantum system in the Hilbert space but also the properties of carriers of quantum states in the Minkowski spacetime. A physical type of quantum object carrying information with extremely high velocity in spacetime is of fundamental importance. The structure of spacetime, more precisely, irreducible representations of the Poincaré group, in the Hilbert space is responsible for the existence of massless particles, photons. In this sense, the structure of spacetime is in fact used in relativistic quantum cryptography systems for cryptographic key distribution. This makes it possible to guarantee the security of keys even with the use of orthogonal states .
The security of keys in quantum cryptography is based on fundamental quantum mechanical exclusions (the exclusion of cloning and copying of nonorthogonal quantum states. The physical type of a quantum object that carries information (photon, electron, atom, etc.) is insignificant; only its state vector is important. In relativistic quantum cryptography for open space, both the time of the information carrier (photon that propagates with the extremely allowable velocity in a vacuum) and its quantum state are of fundamental importance. Joint fundamental constraints that are dictated by both special relativity and quantum mechanics on the discrimination of nonorthogonal quantum states allow one to formulate fundamentally new key distribution protocols that are stable against any attacks on a key and guarantee the security of keys for a nonstrictly single-photon source and any losses in the communication channel. Although this protocol is a real-time protocol in the Minkowski space-time, where the attack to the communication channel is detected by the delay of eavesdropper measurement results, the protocol does not require clock synchronization on the transmitter and receiver sides .
A principally new approach ensuring secure key distribution via an open quantum communication channel is proposed. In contrast to the existing schemes, in which the security is based upon special properties of nonorthogonal states in the Hilbert space, the security of the proposed scheme relies on a spacetime structure of states and on certain constraints imposed by special relativity. Using these factors, it is possible to provide for secure key transmission using practically arbitrary quantum states.
Representations of quantum operations – We start by introducing a geometrical representation (real vector space) of quantum states and quantum operations. To do so we exploit an isomorphism from positive matrices to a subcone of the Minkowski future light-cone. Pure states map onto certain light-like vectors, whilst the axis of revolution encodes the overall probability of occurrence for the state. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particular to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real symmetric linear transform. In the case of a qubit the representation becomes all the more interesting since it elegantly associates, to each measurement element of a generalized measurement, a Lorentz transformation in Minkowski space. We formalize explicitly this correspondence between ‘observation of a quantum system’ and ‘special relativistic change of inertial frame’. To end this part we review the state-operator correspondence, which was successfully exploited by Choi to derive the operator-sum representation of quantum operations. We go further and show that all of the important theorems concerning quantum operations can in fact be derived as simple corollaries of those concerning quantum states. Using this methodology we derive novel composition laws upon quantum states and quantum operations, Schmidt-type decompositions for bipartite pure states and some powerful formulae relating to the correspondence. Quantum cryptography – The key principle of quantum cryptography could be summarized as follows. Honest parties communicate using quantum states. To the eavesdropper these states are random and non-orthogonal. In order to gather information she must measure them, but this may cause irreversible damage. Honest parties seek to detect her mischief by checking whether certain quantum states are left intact. Thus tradeoff between the eavesdropper’s information gain, and the disturbance she necessarily induces, can be viewed as the power engine behind quantum cryptographic protocols. We begin by quantifying this tradeoff in the case of a measure distinguishing two non-orthogonal equiprobable pure states. A formula for this tradeoff was first obtained by Fuchs and Peres, but we provide a shorter, geometrical derivation (within the framework of the above mentioned conal representation). Next we proceed to analyze the Information gain versus disturbance tradeoff in a scenario where Alice and Bob interleave, at random, pairwise superpositions of two message words within their otherwise classical communications. This work constitutes one of the few results currently available regarding d-level systems quantum cryptography, and seems to provide a good general primitive for building such protocols. The proof crucially relies on the state-operator correspondence formulae derived in the first part, together with some methods by Banaszek. Finally we make use of this analysis to prove the security of a ‘blind quantum computation’ protocol, whereby Alice gets Bob to perform some quantum algorithm for her, but prevents him from learning her input to this quantum algorithm.
A simple proof of the unconditional security of a relativistic quantum cryptosystem based on orthogonal states is given. Limitations imposed by the special relativity theory allow the proof to be markedly simplified as compared to the case of nonrelativistic cryptosystems based on nonorthogonal states. An important point in the proposed protocol is a space-time structure of the quantum states, which is ignored in the non-relativistic protocols using only the properties of the space of states of the information carriers. As a consequence, the simplification is related to the inefficacy of using the collective measurements against an eavesdropper, the allowance for which is an especially difficult task in the nonrelativistic case.
Quantum Cryptography is a novel approach to make the information and network world more secure. Unlike classical cryptography, quantum cryptography ensures the security of communication because it is based on the laws of Quantum Mechanics; a branch of physics that deals with the fundamental behavior of matter and their energies to the levels of subatomic particles. These particles and their intrinsic properties such as photon polarization and entanglement allow us to develop an unbreakable system of cryptographic information. Conventionally, the security has been primarily concerned with the secured distribution of secret key carried over an insecure channel. However, in Quantum Cryptographic systems the transmission of secret key is done using aforementioned laws of nature through a secured quantum channel highly susceptible to intrusion detection. Consequently, the Quantum Key Distribution (QKD), in conjunction with protocols such as BB84 protocol has emerged as an ultimate modern secured communication system. This article concentrates on briefly reviewing quantum cryptography, some recent developments and reports of its vulnerabilities.