A geometric characterization of invertible quantum measurement maps

TL;DR

This paper provides a geometric characterization of invertible quantum measurement maps, showing they are essentially conjugations or transpositions by an invertible operator, preserving the structure of state space line segments.

Contribution

It characterizes all bijective maps on quantum states that preserve line segments as either conjugations or transpositions by invertible operators.

Findings

Maps preserving line segments are characterized as conjugations or transpositions.

Invertible operators define the structure-preserving transformations.

The result applies to all states on Hilbert spaces of any dimension.

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$H\leq \infty$, 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 $\phi:{\mathcal S}(H) \rightarrow {\mathcal S}(H)$ satisfies $\phi([\rho_1, \rho_2]) \subseteq [\phi(\rho_1),\phi(\rho_2)]$ for any $\rho_1, \rho_2 \in {\mathcal S}$ if and only if $\phi$ has one of the following forms $$\rho \mapsto \frac{M\rho M^*}{{\rm tr}(M\rho M^*)}\quad \hbox{or} \quad \rho \mapsto \frac{M\rho^T M^*}{{\rm tr}(M\rho^T M^*)},$$ where $M$ is an invertible bounded linear operator and $\rho^T$ is the transpose of $\rho$ with respect to an arbitrarily fixed orthonormal basis.