Extremal Quantum States in Coupled Systems

Let H1,H2 be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose ρi is a state in Hi, i = 1, 2. Let C(ρ1, ρ2) be the convex set of all states ρ inH = H1⊗H2 whose marginal states inH1 andH2 are ρ1 and ρ2 respectively. Here we present a necessary and sufficient criterion for a ρ in C(ρ1, ρ2) to be an extreme point. Such a condition implies, in particular, that for a state ρ to be an extreme point of C(ρ1, ρ2) it is necessary that the rank of ρ does not exceed ( d1 + d 2 2 − 1 ) 1 2 , where di = dim Hi, i = 1, 2. When H1 and H2 coincide with the 1-qubit Hilbert space C2 with its standard orthonormal basis {|0 >, |1 >} and ρ1 = ρ2 = 12I it turns out that a state ρ ∈ C(2I, 12I) is extremal if and only if ρ is of the form |Ω >< Ω| where |Ω >= 1 √ 2 (|0 > |ψ0 > +|1 > |ψ1 >) , {|ψ0 >, |ψ1 >} being an arbitrary orthonormal basis of C2. In particular, the extremal states are the maximally entangled states.