A geometric characterization of invertible quantum measurement maps

Abstract

A geometric characterization is given for invertible quantum measurement maps. Denote by $\mathcal{S}(H)$ the convex set of all states (i.e., trace-1 positive operators) on Hilbert space $H$ with dim$H\le \infty$, and $[{\rho }_1,{\rho }_2]$ the line segment joining two elements ${\rho }_1,{\rho }_2$ in $\mathcal{S}(H)$. It is shown that a bijective map $\varphi :\mathcal{S}(H)\to \mathcal{S}(H)$ satisfies $\varphi ([{\rho }_1,{\rho }_2])\subseteq [\varphi ({\rho }_1),\varphi ({\rho }_2)]$ for any ${\rho }_1,{\rho }_2\in \mathcal{S}$ if and only if $\varphi$ has one of the following forms $$\rho \mapsto \frac{M\rho M^{\ast }}{\mathrm{tr}(M\rho M^{\ast })}or\rho \mapsto \frac{M{\rho }^T{M}^{\ast }}{\mathrm{tr}(M{\rho }^T{M}^{\ast })},$$ where $M$ is an invertible bounded linear operator and ${\rho }^T$ is the transpose of $\rho$ with respect to an arbitrarily fixed orthonormal basis.

14 pages

References in corpus (2)

• A characterization of positive linear maps and criteria of entanglement for quantum states
• Characterization of segment and convexity preserving maps