Electronic Research Archive

Generalizations of Wigner's theorem from rank-$ 1 $ projections to rank-$ n $ projections

Yulong Tian, Jinli Xu

Published Jan 1, 2025• 0 citations

Abstract

We provide generalizations of the classical Wigner's theorem as well as Uhlhorn's version of Wigner's theorem by considering maps that send rank-$ 1 $ projections to rank-$ n $ projections. Namely, we describe the general form of maps $ \phi:P_{1}\left (H\right) \to P_{n}\left (K\right) $ multiplying $ n $ times the transition probability and maps $ \phi:P_{1}\left (H\right) \to P_{n}\left (K\right) $ sending each complete orthogonal system of rank-$ 1 $ projections to some complete orthogonal system of rank-$ n $ projections.

DOI: 10.3934/era.2025140

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Electronic Research Archive

Generalizations of Wigner's theorem from rank-$ 1 $ projections to rank-$ n $ projections

Yulong Tian, Jinli Xu

Published Jan 1, 2025• 0 citations

Publisher

Abstract

We provide generalizations of the classical Wigner's theorem as well as Uhlhorn's version of Wigner's theorem by considering maps that send rank-

1

projections to rank-

n

projections. Namely, we describe the general form of maps

\phi:P_{1}\left (H\right) \to P_{n}\left (K\right)

multiplying

n

times the transition probability and maps

\phi:P_{1}\left (H\right) \to P_{n}\left (K\right)

sending each complete orthogonal system of rank-

1

projections to some complete orthogonal system of rank-

n

projections.

DOI: 10.3934/era.2025140

Related papers

Wigner’s Theorem and Its Generalizations

Mark Pankov

Cambridge University Press eBooksJan 6, 20200 citations

Maps preserving transition probability from pure product states to pure states

Jinli Xu, Yuan Xue, Ajda Fošner

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Erling Størmer

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arXiv:1604.05717Apr 19, 20160 citations

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Peter Šemrl

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arXiv:2106.06182Journal of Physics A Mathematical and TheoreticalJun 21, 20217 citations

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Jáchym Barvínek, Jan Hamhalter

Mathematica SlovacaApr 1, 20176 citations

We provide generalizations of the classical Wigner's theorem as well as Uhlhorn's version of Wigner's theorem by considering maps that send rank-

1

projections to rank-

n

projections. Namely, we describe the general form of maps

\phi:P_{1}\left (H\right) \to P_{n}\left (K\right)

multiplying

n

times the transition probability and maps

\phi:P_{1}\left (H\right) \to P_{n}\left (K\right)

sending each complete orthogonal system of rank-

1

projections to some complete orthogonal system of rank-

n

projections.

Extending Wigner's theorem we give a characterization of positive maps of

B(H)

into itself which map the set of rank k projections onto itself.