Real Mutually Unbiased Bases

Abstract

We tabulate bounds on the optimal number of mutually unbiased bases in R^d. For most dimensions d, it can be shown with relatively simple methods that either there are no real orthonormal bases that are mutually unbiased or the optimal number is at most either 2 or 3. We discuss the limitations of these methods when applied to all dimensions, shedding some light on the difficulty of obtaining tight bounds for the remaining dimensions that have the form d=16n^2, where n can be any number. We additionally give a simpler, alternative proof that there can be at most d/2+1 real mutually unbiased bases in dimension d instead of invoking the known results on extremal Euclidean line sets by Cameron and Seidel, Delsarte, and Calderbank et al.

13 pages

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• Evidence for and against Zauner's MUB Conjecture in ${\mathbb{C}}^6$
• Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases
• Constructions on Real Approximate Mutually Unbiased Bases
• Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size
• Almost Hadamard matrices: general theory and examples