A bipartite pure state has three nonzero Schmidt probabilities

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

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Problem

A bipartite pure state ∣ψ⟩|\psi\rangle has three nonzero Schmidt probabilities (λ12,λ22,λ32)=(0.5,0.3,0.2)(\lambda_1^2,\lambda_2^2,\lambda_3^2)=(0.5,0.3,0.2), and a second state ∣ϕ⟩|\phi\rangle has Schmidt probabilities (μ12,μ22,μ32)=(0.6,0.3,0.1)(\mu_1^2,\mu_2^2,\mu_3^2)=(0.6,0.3,0.1). ((a)) Decide, using the majorization theorem for single-copy LOCC conversion, whether ∣ψ⟩|\psi\rangle can be converted into ∣ϕ⟩|\phi\rangle with certainty by LOCC. ((b)) Decide the same question for the reverse conversion, ∣ϕ⟩|\phi\rangle into ∣ψ⟩|\psi\rangle.

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