On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six

Abstract

An analytic proof is given which shows that it is impossible to extend any triple of mutually unbiased (MU) product bases in dimension six by a single MU vector. Furthermore, the 16 states obtained by removing two orthogonal states from any MU product triple cannot figure in a (hypothetical) complete set of seven MU bases. These results follow from exploiting the structure of MU product bases in a novel fashion, and they are among the strongest ones obtained for MU bases in dimension six without recourse to computer algebra.

arXiv comment12 pages, identical to published version

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• Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six
• Mutually unbiased bases in dimension six containing a product-vector basis
• Mutually Unbiased Product Bases for Multiple Qudits
• Five open problems in quantum information
• Orbits of mutually unbiased bases
• States that "look the same" with respect to every basis in a mutually unbiased set
• Unextendible Mutually Unbiased Bases from Pauli Classes
• On Two Ways to Look for Mutually Unbiased Bases
• Adaptive tomography of qubits: Purity versus statistical fluctuations
• On mutually unbiased bases: Passing from d to d**2
• Real Entries of Complex Hadamard Matrices and Mutually Unbiased Bases in Dimension Six
• An infinite family of strongly unextendible mutually unbiased bases in ${\mathbb{C}}^{2^{2h}}$
• Constructions of mutually unbiased entangled bases
• $H_2$-reducible matrices in six-dimensional mutually unbiased bases
• Some special complex Hadamard matrices of order six