Extremal Quantum States in Coupled Systems

Let ${\cal H}_1,$ ${\cal H}_2$ be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose $\rho_i$ is a state in ${\cal H}_i, i=1,2.$ Let ${\cal C} (\rho_1, \rho_2)$ be the convex set of all states $\rho$ in ${\cal H} = {\cal H}_1 \otimes {\cal H}_2$ whose marginal states in ${\cal H}_1$ and ${\cal H}_2$ are $\rho_1$ and $\rho_2$ respectively. Here we present a necessary and sufficient criterion for a $\rho$ in ${\cal C} (\rho_1, \rho_2)$ to be an extreme point. Such a condition implies, in particular, that for a state $\rho$ to be an extreme point of ${\cal C} (\rho_1, \rho_2)$ it is necessary that the rank of $\rho$ does not exceed $(d_1^2 + d_2^2 - 1)^{{1/2}},$ where $d_i = \dim {\cal H}_i, i=1,2.$ When ${\cal H}_1$ and ${\cal H}_2$ coincide with the 1-qubit Hilbert space $\mathbb{C}^2$ with its standard orthonormal basis $\{|0 >, |1> \}$ and $\rho_1 = \rho_2 = {1/2} I$ it turns out that a state $\rho \in {\cal C} ({1/2}I, {1/2}I)$ 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.