Ranks and eigenvalues of states with prescribed reduced states

Abstract

For a quantum state represented as an $n\times n$ density matrix $\sigma \in M_n$, let $\mathcal{S}(\sigma )$ be the compact convex set of quantum states $\rho =({\rho }_{ij})\in M_{m\cdot n}$ with the first partial trace equal to $\sigma$, i.e., ${\mathrm{tr}}_1(\rho )={\rho }_{11}+\cdots +{\rho }_{mm}=\sigma$. It is known that if $m\ge n$ then there is a rank one matrix $\rho \in \mathcal{S}(\sigma )$ satisfying ${\mathrm{tr}}_1(\rho )=\sigma$. If $m<n$, there may not be rank one matrix in $\mathcal{S}(\sigma )$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set $\mathcal{S}$ We also determine ${\rho }^{\ast }\in \mathcal{S}(\sigma )$ with rank bounded by $k$ such that $\Vert {\mathrm{tr}}_1{\rho }^{\ast })-\sigma \Vert$ is minimum for a given unitary similarity invariant norm $\Vert \cdot \Vert$. Furthermore, the relation between the eigenvalues of $\sigma$ and those of $\rho \in \mathcal{S}(\sigma )$ is analyzed. Extension of our results and open problems will be mentioned.

12 pages

References in corpus (1)

• Quantum state transformations and the Schubert calculus