Extremal Quantum States in Coupled Systems

Abstract

Let ${\mathcal{H}}_1,$ ${\mathcal{H}}_2$ be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose ${\rho }_i$ is a state in ${\mathcal{H}}_i,i=1,2.$ Let $\mathcal{C}({\rho }_1,{\rho }_2)$ be the convex set of all states $\rho$ in $\mathcal{H}={\mathcal{H}}_1\otimes {\mathcal{H}}_2$ whose marginal states in ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are ${\rho }_1$ and ${\rho }_2$ respectively. Here we present a necessary and sufficient criterion for a $\rho$ in $\mathcal{C}({\rho }_1,{\rho }_2)$ to be an extreme point. Such a condition implies, in particular, that for a state $\rho$ to be an extreme point of $\mathcal{C}({\rho }_1,{\rho }_2)$ it is necessary that the rank of $\rho$ does not exceed $(d_1^2+d_2^2-1{)}^{1/2},$ where $d_i=\dim {\mathcal{H}}_i,i=1,2.$ When ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ coincide with the 1-qubit Hilbert space ${\mathbb{C}}^2$ with its standard orthonormal basis $\{|0>,|1>\}$ and ${\rho }_1={\rho }_2=1/2I$ it turns out that a state $\rho \in \mathcal{C}(1/2I,1/2I)$ is extremal if and only if $\rho$ is of the form $|\Omega ><\Omega |$ where $|\Omega >=\frac{1}{\sqrt{2}}(|0>|{\psi }_0>+|1>|{\psi }_1>),$ $\{|{\psi }_0>,|{\psi }_1>\}$ being an arbitrary orthonormal basis of ${\mathbb{C}}^2.$ In particular, the extremal states are the maximally entangled states.