A gap for the maximum number of mutually unbiased bases

Introduction

for all $k,j=1,\ldots,d$, are said to be mutually unbiased. A famous question regarding mutually unbiased bases (MUB) is the following: in a $d$-dimensional complex space, at most how many orthonormal bases can be given so that any two of them are mutually unbiased?

The motivation of the question is coming from quantum information theory. MUB are useful in quantum state tomography , and the known quantum cryptographic protocols also rely on MUB; see for example .

Simple arguments show that the maximum number $N(d)$ of orthonormal bases in a collection of MUB satisfies the bound $N(d)\leq d+1$ for every $d>1$. A collection of $d+1$ MUB is usually referred as a complete collection. When the dimension $d=p^{\alpha}$ is a power of a prime, such complete collections can be constructed . However, apart from this case, at the moment there is no dimension $d>1$ in which the value of $N(d)$ would be known. So already in dimension six the problem is open. Nevertheless, numerical and other evidences suggests that $N(6)=3$, which is much less than $7$ (that we would need for a complete collection.)

One may have a look at the problem of MUB from several different point of views. It may be considered to regard Lie algebra theory . The original problem, which is formulated in a complex space, may be also turned into a real convex geometrical question and hence may be investigated with tools of convex geometry . Often questions about MUB are rephrased in terms of complex Hadamard matrices; see for example . However, for the author of this work, the most natural point of view is that of operator algebras (or, being in finite dimensions, perhaps better to say: matrix algebras).

There is a natural way to associate a maximal abelian $*$-subalgebra (in short: a MASA) to an orthonormal basis (ONB). In the context of matrix algebras, we consider a system of MASAs instead of a system of bases. Mutual unbiasedness is then expressed as a natural orthogonality relation (sometimes also called “quasi-orthogonality” or “complementarity of subalgebras”). In fact, in the study of matrix algebras one considers systems of orthogonal subalgebras in general (that is, systems consisting of all kind of subalgebras — not only maximal abelian ones). For the topic of orthogonal subalgebras and its relation to mutual unbiasedness see for example and . Note that apart from the finite dimensional case, orthogonal subalgebras were also considered in the context of type II1 von Neumann algebras; see .

Can we find a (closed, “elementary”) expression giving the “missing basis” in terms of the others? It is clear where the “missing” MASA is, but to find the corresponding basis we would need to diagonalize the matrices appearing in our MASA. This might require to find the roots of certain characteristic polynomials. So note that it might well be that in general in dimensions $d\geq 5$ there is no (closed, “elementary”) expression giving the missing basis.

Preliminaries

where $\tau=\frac{1}{d}{\rm Tr}$ is the normalized trace.

pairwise orthogonal, $(d-1)$-dimensional subspaces. So when $d>1$, a collection of orthogonal MASAs can have at most $d+1$ elements; this is one of the ways one can obtain the well-known upper bound on $N(d)$.

showing that the sum appearing in the statement is independent of the chosen ONB. Thus the formula can be verified by an elementary check using the ONB consisting of “matrix units”. ∎

The “missing” basis found

First let us note that $E_{K}$ automatically preserves the trace:

implying that $\sum_{j}(A^{(k)}_{j})^{*}XA^{(k)}_{j}=E_{{\mathcal{A}}_{k}}(X)-\frac{1}{d}X$. On the other hand, by Lemma 2.1,

By our previous lemma $E_{\mathcal{B}}$ is completely positive, so by lemma 3.1 ${\mathcal{B}}$ is an algebra. On the other hand, if $X^{\prime}\in{\mathcal{B}}^{\prime}$ then

Acknowledgements. The author would like to thank prof.​ D.​ Petz who suggested to consider this problem and T.​ Szőnyi for useful information on Latin squares and their letterature.

References