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Journal of Mathematics

Symmetry Transformations on the Set of Tensor Products of Rank‐1 Idempotents

Yulong Tian, Jinli Xu

Published Jan 1, 2024• 0 citations
Publisher

Abstract

We obtain a characterization of bijective maps preserving the trace of products on the set of tensor products of rank‐1 idempotents, which may be considered as a generalization of Molnár’s Wigner‐type theorem in multipartite systems. The proof is based on the studies in linear preserver problems on tensor products. Moreover, a corresponding result on the finite‐dimensional case is presented.

DOI
10.1155/2024/2384295

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We obtain a characterization of bijective maps preserving the trace of products on the set of tensor products of rank‐1 idempotents, which may be considered as a generalization of Molnár’s Wigner‐type theorem in multipartite systems. The proof is based on the studies in linear preserver problems on tensor products. Moreover, a corresponding result on the finite‐dimensional case is presented.
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Orthogonality preserving transformations on the set of nnn-dimensional subspaces of a Hilbert space
We characterize bijective transformations on the set of all nnn-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.
Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner's classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of 'uniform' density operators of a suitable fixed rank are symmetry witnesses as well.