Optimal state-determination by mutually unbiased measurements

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 {|v_i〉} and {|w_j〉} to be mutually unbiased if all inner products across their elements have the same magnitude: |〈ν₁|w_j|= 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.