Preservers of matrix pairs with a fixed inner product value

Li, Chi-Kwong; Plevnik, Lucijan; Šemrl, Peter

doi:10.7153/oam-06-29

Cited by 14 publications

(6 citation statements)

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“…Note that, unlike Wigner's theorem, Uhlhorn's theorem does not generalize for nonbijective maps when the Hilbert space is infinite-dimensional. We also note that the analogous problem for other angles different from π 2 has been only recently resolved in a series of papers by Li, Plevnik, andŠemrl [13], Gehér [7], and Gehér and Mori [9]. c 2021 American Mathematical Society…”

Section: ⇐⇒mentioning

confidence: 89%

Book Review: Wigner-type theorems for Hilbert Grassmannians

Gehér¹

2021

Bull. Amer. Math. Soc.

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Section: ⇐⇒mentioning

confidence: 89%

Book Review: Wigner-type theorems for Hilbert Grassmannians

Gehér¹

2021

Bull. Amer. Math. Soc.

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“…The result was extended to Hilbert modules over matrix algebras, prime C*-algebras, and indefinite inner product spaces; see [21,24]. In [16], the authors extended Uhlhorn's result to Hermitian matrices, symmetric matrices, the set of orthogonal projections, the set of rank one orthogonal projections, and the set of effect algebra, and studied bijective maps on these matrix sets such that tr(AB) = c if and only if tr(φ(A)φ(B)) = c for a given c > 0.…”

Section: Quantum Information Science and Preserversmentioning

confidence: 99%

Linear preservers and quantum information science

Fošner

Huang

et al. 2013

Linear and Multilinear Algebra

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Abstract. In this paper, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn × mn Hermitian matrices such that φ(A ⊗ B) and A ⊗ B have the same spectrum for any m × m Hermitian A and n × n Hermitian B. Such a map has the form A ⊗ B → U (ϕ1(A) ⊗ ϕ2(B))U * for mn × mn Hermitian matrices in tensor form A ⊗ B, where U is a unitary matrix, and for j ∈ {1, 2}, ϕj is the identity map X → X or the transposition map X → X t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A ⊗ B is also obtained. The results are connected bipartite (quantum) systems and are extended to multipartite systems.2010 Math. Subj. Class.: 15A69, 15A86, 15B57, 15A18.

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“…Instead of preserving orthogonality of lines (or equivalently zero transition probability between pure states) in both directions, we will assume that a fixed angle α ∈ 0, π 2 (or equivalently a fixed transition probability cos 2 α) is preserved in both directions. We point out here that this problem has been partially answered in [24] by Li, Plevnik and Šemrl in the special case when 0 < α ≤ π 4 and H is a real Hilbert space with 4 < dim H < ∞. Their method depends heavily on these rather restrictive assumptions.…”

Section: Introductionmentioning

confidence: 99%

Symmetries of Projective Spaces and Spheres

Gehér

2018

International Mathematics Research Notices

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Let H be either a complex inner product space of dimension at least two, or a real inner product space of dimension at least three. Let us fix an α ∈ 0, π 2 . The purpose of this paper is to characterize all bijective transformations on the projective space P (H) obtained from H which preserves the angle α between lines in both directions. (We emphasize that we do not assume anything about other angles). For real inner product spaces and when H = C 2 we do this for every α, and when H is a complex inner product space of dimension at least three we describe the structure of these transformations for α ≤ π 4 . As an application, we give an Uhlhorn-type generalization of a famous theorem of Wigner which is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, we show that under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability cos 2 α in both directions is a Wigner symmetry (i.e. it automatically preserves all transition probability), except for the case when H = C 2 and α = π 4 where an additional possibility occurs. We note that the classical theorem of Uhlhorn is the solution for the α = π 2 case. Usually in the literature, results which are connected to Wigner's theorem are discussed under the assumption of completeness of H, however, here we shall remove this unnecessary hypothesis. Our main tools are a characterization of bijective maps on unit spheres of real inner product spaces which preserve an angle in both directions, and an extension of Uhlhorn's theorem for non-complete inner product spaces.2010 Mathematics Subject Classification. Primary: 47B49, 47N50, 81P05, 51A05.

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Preservers of matrix pairs with a fixed inner product value

Cited by 14 publications

References 13 publications

Book Review: Wigner-type theorems for Hilbert Grassmannians

Book Review: Wigner-type theorems for Hilbert Grassmannians

Linear preservers and quantum information science

Symmetries of Projective Spaces and Spheres

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