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

McNulty, Daniel; Weigert, Stefan

doi:10.1142/s0219749912500566

Cited by 28 publications

(33 citation statements)

References 22 publications

(49 reference statements)

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“…In a recent paper, it has been analytically proven that given any triplet of MU product bases in dimension six it is not possible to find even a single vector MU to the triplet [25].…”

Section: Complex Hadamard Matrices and Mutually Unbiased Basesmentioning

confidence: 99%

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

Goyeneche¹

2013

J. Phys. A: Math. Theor.

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We study the problem of constructing mutually unbiased bases in dimension six. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family K (2) 6 which we confirm analytically. Also, we obtain strong evidence that no more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.

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“…In a recent paper, it has been analytically proven that given any triplet of MU product bases in dimension six it is not possible to find even a single vector MU to the triplet [25].…”

Section: Complex Hadamard Matrices and Mutually Unbiased Basesmentioning

confidence: 99%

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

Goyeneche¹

2013

J. Phys. A: Math. Theor.

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“…It can be immediately seen that any other basis obtained introducing the fourth one of the space would not be mutually unbiased with the others, since it is missing a different basis in . The set (5) consists of 18 product states and cannot be extended by any other vector in , even if entangled states are considered23; moreover if a complete MUB set in d = 6 existed, then only one among the seven bases therein could be composed of product states, while all others must be entangled24.…”

Section: Resultsmentioning

confidence: 99%

Test of mutually unbiased bases for six-dimensional photonic quantum systems

D’Ambrosio

Cardano

Karimi

et al. 2013

Sci Rep

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In quantum information, complementarity of quantum mechanical observables plays a key role. The eigenstates of two complementary observables form a pair of mutually unbiased bases (MUBs). More generally, a set of MUBs consists of bases that are all pairwise unbiased. Except for specific dimensions of the Hilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, the identification of a maximal set of MUBs remains an open problem, although there is strong numerical evidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developed holographic technique, we implement and test different sets of three MUBs for a single photon six-dimensional quantum state (a “qusix”), encoded exploiting polarization and orbital angular momentum of photons. A close agreement is observed between theory and experiments. Our results can find applications in state tomography, quantitative wave-particle duality, quantum key distribution.

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“…for j = 1, 2, 3, 4 in (12). We shall use these facts in the following arguments of solving (13)- (16).…”

Section: ⊓ ⊔mentioning

confidence: 99%

Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six

Liang

Sun

et al. 2019

Linear and Multilinear Algebra

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We investigate the number of real entries of an n × n complex Hadamard matrix (CHM). We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0 − 22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases (MUBs) in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exists then the real entries in any one of the remaining three matrices does not exceed 22.

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

Cited by 28 publications

References 22 publications

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

Test of mutually unbiased bases for six-dimensional photonic quantum systems

Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six

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