skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Three Ways to Look at Mutually Unbiased Bases

Abstract

This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of ''mubness'' is introduced, and applied to some explicit calculations in six dimensions (partly done by Bjoerck and by Grassl). Although this does not yet solve any problem, some appealing structures emerge.

Authors:
 [1]
  1. Fysikum, Stockholm University, 106 91 Stockholm (Sweden)
Publication Date:
OSTI Identifier:
21054919
Resource Type:
Journal Article
Resource Relation:
Journal Name: AIP Conference Proceedings; Journal Volume: 889; Journal Issue: 1; Conference: 4. international conference on foundations of probability and physics, Vaexjoe (Sweden), 4-9 Jun 2006; Other Information: DOI: 10.1063/1.2713445; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GEOMETRY; HILBERT SPACE; MANY-DIMENSIONAL CALCULATIONS; QUANTUM MECHANICS; REVIEWS

Citation Formats

Bengtsson, Ingemar. Three Ways to Look at Mutually Unbiased Bases. United States: N. p., 2007. Web. doi:10.1063/1.2713445.
Bengtsson, Ingemar. Three Ways to Look at Mutually Unbiased Bases. United States. doi:10.1063/1.2713445.
Bengtsson, Ingemar. Wed . "Three Ways to Look at Mutually Unbiased Bases". United States. doi:10.1063/1.2713445.
@article{osti_21054919,
title = {Three Ways to Look at Mutually Unbiased Bases},
author = {Bengtsson, Ingemar},
abstractNote = {This is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of ''mubness'' is introduced, and applied to some explicit calculations in six dimensions (partly done by Bjoerck and by Grassl). Although this does not yet solve any problem, some appealing structures emerge.},
doi = {10.1063/1.2713445},
journal = {AIP Conference Proceedings},
number = 1,
volume = 889,
place = {United States},
year = {Wed Feb 21 00:00:00 EST 2007},
month = {Wed Feb 21 00:00:00 EST 2007}
}
  • We study the robustness of various protocols for quantum key distributions. We first consider the case of qutrits and study quantum protocols that employ two and three mutually unbiased bases. We then derive the optimal eavesdropping strategy for two mutually unbiased bases in dimension 4 and generalize the result to a quantum key distribution protocol that uses two mutually unbiased bases in arbitrary finite dimensions.
  • We consider the average distance between four bases in six dimensions. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in six dimensions. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically found maximum of the average distance, and we conduct a detailedmore » study of the structure of the extremal set of bases.« less
  • A compete orthonormal basis of N-qutrit unitary operators drawn from the Pauli group consists of the identity and 9{sup N}-1 traceless operators. The traceless ones partition into 3{sup N}+1 maximally commuting subsets (MCS's) of 3{sup N}-1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3{sup N} are isomorphic to all MCS's and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). Formore » three qutrits, 728 operators partition into 28 MCS's with less rigid structure, allowing for the coexistence of separable, partially entangled, and totally entangled (GHZ-like) bases. However a minimum of 16 GHZ-like bases must occur. Every basis state is described by an N-digit trinary number consisting of the eigenvalues of N observables constructed from the corresponding MCS.« less
  • The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension d is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6 or d=10 dimensions even if the maximal number of MUB's exists in these dimensions.
  • For a system of N qubits, living in a Hilbert space of dimension d=2{sup N}, it is known that there exists d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method tomore » the general case of N qubits is outlined.« less