Friday, February 1, 2013

1301.7551 (Lihua Yang et al.)

Quantum measurements and maps preserving strict convex combinations and
pure states
   [PDF]

Lihua Yang, Jinchuan Hou
Let ${\mathcal S}(H)$ be the convex set of all states (i.e., the positive operators with trace one) on a complex Hilbert space $H$. It is shown that a map $\psi:{\mathcal S}(H)\rightarrow {\mathcal S}(K)$ with $2\leq \dim H<\infty$ preserves pure states and strict convex combinations (i.e., for any $\rho_1,\rho_2$ and $0< t< 1$, there exists $0< s< 1$ such that $\psi(t\rho_1+(1-t)\rho_2)=s\psi(\rho_1)+(1-s)\psi(\rho_2)$) if and only if $\psi$ has one of the forms: (1) $\rho\mapsto \sigma_0$ for any $\rho\in {\mathcal S}(H)$; (2) $\psi({\mathcal Pur}(H))=\{Q_1,Q_2\}$; (3) $\rho\mapsto \frac{M\rho M^*}{{\rm Tr}(M\rho M^*)}$, where $\sigma_0$ is a pure state on $K$ and $M: H \rightarrow K$ is an injective linear or conjugate linear operator. For multipartite systems, we also give a structure theorem for maps that preserve separable pure states and strict convex combinations. These results allow us to characterize injective (local) quantum measurements and answer some conjectures proposed in \cite{HL}.
View original: http://arxiv.org/abs/1301.7551

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