TY - JOUR

T1 - Optimal state-determination by mutually unbiased measurements

AU - Wootters, William K.

AU - Fields, Brian D.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1989/5/1

Y1 - 1989/5/1

N2 - For quantum systems having a finite number N of orthogonal states, we investigate a particular relation among different measurements, called "mutual unbiasedness," which we show plays a special role in the problem of state determination. We define two bases {|vi〉} and {|wj〉} to be mutually unbiased if all inner products across their elements have the same magnitude: |〈ν1|wj|= 1 √N for all i, j. Two non-degenerate measurements are defined to be mutually unbiased if the bases comprising their eigenstates are mutually unbiased. We show that if one can find N + 1 mutually unbiased bases for a complex vector space of N dimensions, then the measurements corresponding to these bases provide an optimal means of determining the density matrix of an ensemble of systems having N orthogonal states, in the sense that the effects of statistical error are minimized. We show further that the number of mutually unbiased bases one may find for a given N is at most N + 1. Finally, we show that N + 1 mutually unbiased bases do exist whenever N is a power of a prime, and we construct such bases explicitly.

AB - For quantum systems having a finite number N of orthogonal states, we investigate a particular relation among different measurements, called "mutual unbiasedness," which we show plays a special role in the problem of state determination. We define two bases {|vi〉} and {|wj〉} to be mutually unbiased if all inner products across their elements have the same magnitude: |〈ν1|wj|= 1 √N for all i, j. Two non-degenerate measurements are defined to be mutually unbiased if the bases comprising their eigenstates are mutually unbiased. We show that if one can find N + 1 mutually unbiased bases for a complex vector space of N dimensions, then the measurements corresponding to these bases provide an optimal means of determining the density matrix of an ensemble of systems having N orthogonal states, in the sense that the effects of statistical error are minimized. We show further that the number of mutually unbiased bases one may find for a given N is at most N + 1. Finally, we show that N + 1 mutually unbiased bases do exist whenever N is a power of a prime, and we construct such bases explicitly.

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U2 - 10.1016/0003-4916(89)90322-9

DO - 10.1016/0003-4916(89)90322-9

M3 - Article

AN - SCOPUS:0004586188

VL - 191

SP - 363

EP - 381

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

IS - 2

ER -