Extremal quantum states in coupled systems

Abstract

Let H₁, H₂ be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose ρ_i is a state in H_i, i=1,2. Let C(ρ₁,ρ₂) be the convex set of all states ρ in H = H₁⊗H₂whose marginal states in H₁ and H₂ 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 (d₁² + d₂²-1)^½, where d_i=dim H_i^,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 =½I it turns out that a state ρ∈C(1/2I,1/2I) 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. Using the Weyl commutation relations in the space L²(A) of a finite Abelian group we exhibit a mixed extremal state in C(1/nI_n,1/n²I_n²).