2003/07/25 by K. R. Parthasarathy, Parthasarathy, K. R. · 1 citation
Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0307182
arxiv created 2003/07/25 · arxiv updated 2009/12/01
Let \cal H1, \cal H2 be finite dimensional complex Hilbert spaces describing the states of two finite level quantum systems. Suppose ρi is a state in \cal Hi, i=1,2. Let \cal C (ρ1, ρ2) be the convex set of all states ρ in \cal H = \cal H1 ⊗ \cal H2 whose marginal states in \cal H1 and \cal H2 are ρ1 and ρ2 respectively. Here we present a necessary and sufficient criterion for a ρ in \cal 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 \cal C (ρ1, ρ2) it is necessary that the rank of ρ does not exceed (d12 + d22 - 1)1/2, where di = dim \cal Hi, i=1,2. When \cal H1 and \cal H2 coincide with the 1-qubit Hilbert space ℂ2 with its standard orthonormal basis \|0 >, |1> \ and ρ1 = ρ2 = 1/2 I it turns out that a state ρ∈ \cal 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.