Extremal quantum states in coupled systems

Abstract

Let H 1 , H 2 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 ( ρ 1 , ρ 2 ) be the convex set of all states ρ in H = H 1 ⊗ H 2 whose marginal states in H 1 and H 2 are ρ 1 and ρ 2 respectively. Here we present a necessary and sufficient criterion for a ρ in C ( ρ 1 , ρ 2 ) to be an extreme point. Such a condition implies, in particular, that for a state ρ to be an extreme point of C ( ρ 1 , ρ 2 ) it is necessary that the rank of ρ does not exceed ( d 1 2 + d 2 2 − 1 ) 1 / 2 , where d i = dim H i , i = 1 , 2 . When H 1 and H 2 coincide with the 1-qubit Hilbert space C 2 with its standard orthonormal basis { | 0 〉 , | 1 〉 } and ρ 1 = ρ 2 = 1 2 I it turns out that a state ρ ∈ C ( 1 2 I , 1 2 I ) is extremal if and only if ρ is of the form | Ω 〉 〈 Ω | where | Ω 〉 = 1 2 ( | 0 〉 | ψ 0 〉 + | 1 〉 | ψ 1 〉 ) , { | ψ 0 〉 , | ψ 1 〉 } being an arbitrary orthonormal basis of C 2 . In particular, the extremal states are the maximally entangled states. Using the Weyl commutation relations in the space L 2 ( A ) of a finite Abelian group we exhibit a mixed extremal state in C ( 1 n I n , 1 n 2 I n 2 ) .