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AIP conference proceedings

Quantum key distribution for composite dimensional finite systems

Mohamed Ahmed Shalaby, Yasser Kamal

Published Jan 1, 2017• 1 citation
Publisher

Abstract

The application of quantum mechanics contributes to the field of cryptography with very important advantage as it offers a mechanism for detecting the eavesdropper. The pioneering work of quantum key distribution uses mutually unbiased bases (MUBs) to prepare and measure qubits (or qudits). Weak mutually unbiased bases (WMUBs) have weaker properties than MUBs properties, however, unlike MUBs, a complete set of WMUBs can be constructed for systems with composite dimensions. In this paper, we study the use of weak mutually unbiased bases (WMUBs) in quantum key distribution for composite dimensional finite systems. We prove that the security analysis of using a complete set of WMUBs to prepare and measure the quantum states in the generalized BB84 protocol, gives better results than using the maximum number of MUBs that can be constructed, when they are analyzed against the intercept and resend attack.

DOI
10.1063/1.4981962

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The application of quantum mechanics contributes to the field of cryptography with very important advantage as it offers a mechanism for detecting the eavesdropper. The pioneering work of quantum key distribution uses mutually unbiased bases (MUBs) to prepare and measure qubits (or qudits). Weak mutually unbiased bases (WMUBs) have weaker properties than MUBs properties, however, unlike MUBs, a complete set of WMUBs can be constructed for systems with composite dimensions. In this paper, we study the use of weak mutually unbiased bases (WMUBs) in quantum key distribution for composite dimensional finite systems. We prove that the security analysis of using a complete set of WMUBs to prepare and measure the quantum states in the generalized BB84 protocol, gives better results than using the maximum number of MUBs that can be constructed, when they are analyzed against the intercept and resend attack.
We present a detailed security analysis of a d -dimensional quantum key distribution protocol based on two and three mutually unbiased bases (MUBs) both in an asymptotic and finite-key-length scenario. The finite secret key rates (in bits per detected photon) are calculated as a function of the length of the sifted key by (i) generalizing the uncertainly relation-based insight from BB84 to any d -level 2-MUB QKD protocol and (ii) by adopting recent advances in the second-order asymptotics for finite block length quantum coding (for both d -level 2- and 3-MUB QKD protocols). Since the finite and asymptotic secret key rates increase with d and the number of MUBs (together with the tolerable threshold) such QKD schemes could in principle offer an important advantage over BB84. We discuss the possibility of an experimental realization of the 3-MUB QKD protocol with the orbital angular momentum degrees of freedom of photons.
We investigate the advantages of high-dimensional encoding for a quantum key distribution protocol. In particular, we address a BBM92-like protocol where the dimension of the systems can be larger than two and more than two mutually unbiased bases (MUBs) can be employed. Indeed, it is known that, for a system whose dimension ddd is a prime or the power of a prime, up to d+1d+1d+1 MUBs can be found. We derive an analytic expression for the asymptotic key rate when d+1d+1d+1 MUBs are exploited and show the effects of using different numbers of MUBs on the performance of the protocol. Then, we move to the non-asymptotic case and optimize the finite key rate against collective and coherent attacks for generic dimension of the systems and all possible numbers of MUBs. In the finite-key scenario, we find that, if the number of rounds is small enough, the highest key rate is obtained by exploiting three MUBs, instead of d+1d+1d+1 as one may expect.
Mutually unbiased bases is an important topic in the recent quantum system \nresearches. Although there is much work in this area, many problems \nrelated to mutually unbiased bases are still open. For example, constructing \na complete set of mutually unbiased bases in the Hilbert spaces with composite \ndimensions has not been achieved yet. This thesis defines a weaker \nconcept than mutually unbiased bases in the Hilbert spaces with composite \ndimensions. We call this concept, weak mutually unbiased bases. There is \na duality between such bases and the geometry of the phase space Zd × Zd, \nwhere d is the phase space dimension. To show this duality we study the \nproperties of lines through the origin in Zd × Zd, then we explain the correspondence \nbetween the properties of these lines and the properties of the \nweak mutually unbiased bases. We give an explicit construction of a complete \nset of weak mutually unbiased bases in the Hilbert space Hd, where \nd is odd and d = p1p2; p1, p2 are prime numbers. We apply the concept of \nweak mutually unbiased bases in the context of quantum tomography and \nquantum cryptography.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
Mutually unbiased bases, as a mathematical concept, has important implications in quantum information theory where the information is encoded as linear combinations of vectors in Hilbert spaces instead of as arrays of digits. Offering a designation on the preparation and measurement of the quantum states, mutually unbiased bases provide mathematics based security to many quantum computation protocols including the famous quantum key distribution protocol named BB84. The construction of mutually unbiased bases, however, is not straight forward and it requires an extensive mathematical approach based on the properties of the finite fields.