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Springer eBooks

A triality between weak mutually unbiased bases, zeros of their analytic representations, and finite geometries

T. Olupitan, Chao Lei, Apostol Vourdas

Published Jan 1, 2017• 0 citations
Publisher
DOI
10.1007/978-3-319-69164-0_54

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An analytic representation of weak mutually unbiased bases

T. Olupitan

Quantum systems in the d-dimensional Hilbert space are considered. The mutually unbiased bases is a deep problem in this area. The problem of finding all mutually unbiased bases for higher (non-prime) dimension is still open. We derive an alternate approach to mutually unbiased bases by studying a weaker concept which we call weak mutually unbiased bases. We then compare three rather different structures. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is 1/√k (where k∣d) or 0. The second is maximal lines through the origin in the Z(d) × Z(d) phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. The analytic representation of the weak mutually unbiased bases is defined with the zeros examined. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. We give an explicit breakdown of this triality.

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Partial ordering of weak mutually unbiased bases in finite quantum systems

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There has being an enormous work on finite quantum systems with variables in Zd, especially on mutually unbiased bases. The reason for this is due to its \nwide areas of application. We focus on partial ordering of weak mutually un-biased bases. In it, we studied a partial ordered relation which exists between a subsystem ^(q) and a larger system ^(d) and also, between a subgeometry Gq and larger geometry Gd. Furthermore, we show an isomorphism between: \n(i) the set {Gd} of subgeometries of a finite geometry Gd and subsets of the set {D(d)} of divisors of d. \n(ii) the set {hd} of subspaces of a finite Hilbert space Hd and subsets of the set {D(d)} of divisors of d. \n(iii) the set {Y(d)} of subsystems of a finite quantum system ^(d) and subsets of the set {D(d)} of divisors of d. \nWe conclude this work by showing a duality between lines in finite geometry Gd and weak mutually unbiased bases in finite dimensional Hilbert space Hd.

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Quantum key distribution for composite dimensional finite systems

Mohamed Ahmed Shalaby, Yasser Kamal

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.

AIP conference proceedings•Jan 1, 2017•1 citation
Quantum systems in the d-dimensional Hilbert space are considered. The mutually unbiased bases is a deep problem in this area. The problem of finding all mutually unbiased bases for higher (non-prime) dimension is still open. We derive an alternate approach to mutually unbiased bases by studying a weaker concept which we call weak mutually unbiased bases. We then compare three rather different structures. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is 1/√k (where k∣d) or 0. The second is maximal lines through the origin in the Z(d) × Z(d) phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. The analytic representation of the weak mutually unbiased bases is defined with the zeros examined. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. We give an explicit breakdown of this triality.
There has being an enormous work on finite quantum systems with variables in Zd, especially on mutually unbiased bases. The reason for this is due to its \nwide areas of application. We focus on partial ordering of weak mutually un-biased bases. In it, we studied a partial ordered relation which exists between a subsystem ^(q) and a larger system ^(d) and also, between a subgeometry Gq and larger geometry Gd. Furthermore, we show an isomorphism between: \n(i) the set {Gd} of subgeometries of a finite geometry Gd and subsets of the set {D(d)} of divisors of d. \n(ii) the set {hd} of subspaces of a finite Hilbert space Hd and subsets of the set {D(d)} of divisors of d. \n(iii) the set {Y(d)} of subsystems of a finite quantum system ^(d) and subsets of the set {D(d)} of divisors of d. \nWe conclude this work by showing a duality between lines in finite geometry Gd and weak mutually unbiased bases in finite dimensional Hilbert space Hd.
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.