Constructing Mutually Unbiased Bases from Quantum Latin Squares

Musto, Benjamin

doi:10.4204/eptcs.236.8

Cited by 20 publications

(30 citation statements)

References 15 publications

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“…At QPL 2016 the present authors introduced quantum Latin squares [12,15], as quantum structures generalizing the well-known Latin squares from classical combinatorics [6]. Since then this work has been built on separately by a number of researchers: in particular, by Goyeneche, Raissi, Di Martino andŻyczkowski [7], who propose a notion of orthogonality for quantum Latin squares which allows the construction of quantum codes; and also by Benoist and Nechita [4], who introduce matrices of partial isometries of type (C1,C2,C3,C4), generalizations of quantum Latin squares which characterize system-environment observables preserving a certain set of pointer states.…”

Section: Discussionmentioning

confidence: 99%

“…Since the introduction of quantum Latin squares by the present authors [15], two notions of orthogonality for quantum Latin squares have been introduced, both of which extend the standard notion for classical Latin squares. The first such notion, to which we refer here as left orthogonality, was introduced by the first author [12], who showed it could be used to construct maximally entangled mutually unbiased bases. Given a pair of classical Latin squares which are left orthogonal by this definition, the left conjugates of each square are orthogonal Latin squares in the traditional sense [11].…”

Section: Related Workmentioning

confidence: 99%

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Orthogonality for Quantum Latin Isometry Squares

Musto¹,

Vicary²

2019

Electron. Proc. Theor. Comput. Sci.

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Goyeneche et al recently proposed a notion of orthogonality for quantum Latin squares, and showed that orthogonal quantum Latin squares yield quantum codes. We give a simplified characterization of orthogonality for quantum Latin squares, which we show is equivalent to the existing notion. We use this simplified characterization to give an upper bound for the number of mutually orthogonal quantum Latin squares of a given size, and to give the first examples of orthogonal quantum Latin squares that do not arise from ordinary Latin squares. We then discuss quantum Latin isometry squares, generalizations of quantum Latin squares recently introduced by Benoist and Nechita, and define a new orthogonality property for these objects, showing that it also allows the construction of quantum codes. We give a new characterization of unitary error bases using these structures.

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Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Orthogonality for Quantum Latin Isometry Squares

Musto¹,

Vicary²

2019

Electron. Proc. Theor. Comput. Sci.

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“…Examples of projective permutation matrices are provided by the first author's quantum Latin squares [43,45]; more examples are given in Section 6. We now show that quantum bijections only exist between classical sets of the same cardinality.…”

Section: Quantum Bijections Between Classical Setsmentioning

confidence: 99%

A compositional approach to quantum functions

Musto

Reutter

Verdon

2018

Journal of Mathematical Physics

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We introduce a notion of quantum function, and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication, and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyse dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.

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“…Remark 2: One could find that there is quantum version of Latin square which is known as quantum Latin squares [23,24]. A quantum Latin square of order d is an d × d array of elements of the Hilbert space C d , such that every row and every column is an orthonormal basis.…”

Section: Masking Quantum States Into Tripartite Quantum Systemsmentioning

confidence: 99%

Masking quantum information in multipartite scenario

Wang

2018

Phys. Rev. A

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Recently, Kavan Modi et al. found that masking quantum information is impossible in bipartite scenario. This adds another item of the no-go theorems. In this paper, we present some new schemes different from error correction codes, which show that quantum states can be masked when more participants are allowed in the masking process. Moreover, using a pair of mutually orthogonal Latin squares of dimension d, we show that all the d level quantum states can be masked into tripartite quantum systems whose local dimensions are d or d + 1. This highlight some difference between the no-masking theorem and the classical no-cloning theorem or no-deleting theorem.

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Constructing Mutually Unbiased Bases from Quantum Latin Squares

Cited by 20 publications

References 15 publications

Orthogonality for Quantum Latin Isometry Squares

Orthogonality for Quantum Latin Isometry Squares

A compositional approach to quantum functions

Masking quantum information in multipartite scenario

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