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arXiv:1602.04968 · Journal of Physics A Mathematical and Theoretical

Generalising Wigner's theorem

Gniewomir Sarbicki, Dariusz Chruściński, Marek Mozrzymas

Published Jun 20, 2016• 6 citations
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Abstract

Abstract We analyse linear maps of operator algebras  H (  ) mapping the set of rank- k projectors onto the set of rank- l projectors surjectively. A complete characterisation of such maps for prime n = dim  is provided. A particular case corresponding to k = l = 1 is well known as Wigner’s theorem. Hence our result may be considered as a generalisation of this celebrated Wigner’s result.

DOI
10.1088/1751-8113/49/30/305302

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Abstract We analyse linear maps of operator algebras  H (  ) mapping the set of rank- k projectors onto the set of rank- l projectors surjectively. A complete characterisation of such maps for prime n = dim  is provided. A particular case corresponding to k = l = 1 is well known as Wigner’s theorem. Hence our result may be considered as a generalisation of this celebrated Wigner’s result.
Let n>1 be a positive integer, {H1,…,Hn} be a finite collection of complex Hilbert spaces with dim⁡(Hk)≥2, and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k=1,…,n. Set DP1⨂k=1nHk={A1⊗⋯⊗An:Ak∈P1(Hk),k=1,…,n}.We characterize the maps ϕ from D(P1(⨂k=1nHk)) to P1(⨂k=1nHk) preserving transition probability, i.e. tr(AB)=tr(ϕ(A)ϕ(B)),for all A,B∈DP1⨂k=1nHk.A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.
The relations between several different forms of Wigner′s theorem are investigated,the descriptions of the theorem in terms of physics and geometry are given.By using operator theory and operator algebra,the equivalence between these different forms of propositions is proved.The results show that if the surjection T:R1(H)→R1(K) preserves the inner product between unit rays,the surjection S:R(H)→R(K) preserves the inner product between rays,the surjection Φ:P1(H)→P1(K) preserves the trace of the product of rank-one projections,and the surjection W:H→K preserves the inner product between vectors,then there exists unit or anti-unitary operators U:H→K,such that Uy∈Tx,r(Uy)=Sx,Φ(Px)=UPx U* and W(x)=φ(x)U(x),in which φ:H→C satisfies |φ(x)|=1.
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)B(H)B(H) into itself which map the set of rank k projections onto itself.
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.