Maps preserving transition probability from pure product states to pure states

ABSTRACT

Let n>1 be a positive integer, {H₁, …, H_n} be a finite collection of complex Hilbert spaces with $\dim \left(H_k\right)\ge 2$, and P₁(H_k) be the set of all rank-1 self-adjoint projections on H_k, k = 1, …, n. Set $\mathcal{D}\left(P_1\left(\underset{k=1}{\overset{n}{⨂}}{H}_k\right)\right)=\{A_1\otimes \cdots \otimes A_n:A_k\in P_1(H_k),k=1,\dots ,n\}.$We characterize the maps ϕ from $\mathcal{D}\left(P_1\right(\underset{k=1}{\overset{n}{⨂}}{H}_k\left)\right)$ to $P_1\left(\underset{k=1}{\overset{n}{⨂}}{H}_k\right)$ preserving transition probability, i.e. $\mathrm{t}\mathrm{r}\left(AB\right)=\mathrm{t}\mathrm{r}\left(\varphi \right(A\left)\varphi \right(B\left)\right),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}A,B\in D\left(P_1\left(\underset{k=1}{\overset{n}{⨂}}{H}_k\right)\right).$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.

Keywords:

• Wigner's theorem
• preserver
• transition probability
• tensor product

2010 Mathematics Subject Classifications:

• 15A15
• 15A69
• 15A86

Acknowledgments

The authors would like to thank the referee for helpful comments and careful reading of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Jinli Xu is supported by the Fundamental Research Funds for the Central Universities [grant number 2572019BC07], the National Natural Science Foundation of China [grant number 11701075] and the Foundation of Talent Introduction and the Double First-Rate for the Northeast Forestry University [grant number 1020160016].