There is no generalization of known formulas for mutually unbiased bases

Details There is no generalization of known formulas for mutually unbiased bases There is no generalization of known formulas for mutually unbiased bases

2003-12-26

arxiv.org/abs/quant-ph/0312204v1

Physics

Quantum Physics

17 pages

Scientific paper

In a quantum system having a finite number $N$ of orthogonal states, two orthonormal bases $\{a_i\}$ and $\{b_j\}$ are called mutually unbiased if all inner products $$ have the same modulus $1/\sqrt{N}$. This concept appears in several quantum information problems. The number of pairwise mutually unbiased bases is at most $N+1$ and various constructions of $N+1$ such bases have been found when $N$ is a power of a prime number. We study families of formulas that generalize these constructions to arbitrary dimensions using finite rings.We then prove that there exists a set of $N+1$ mutually unbiased bases described by such formulas, if and only if $N$ is a power of a prime number.

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