16_580_hw_12 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 16 16 November 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 12 Prof. P. M. Goldbart University of Illinois 1) Quantifying entanglement : Consider a quantum system involving two parties , tradi- tionally called Alice and Bob. We call such systems bipartite . They may each be, e.g. , a quantum spin (say, one spin-1 and one spin-5/2), but for now let us leave them arbitrary and denote orthonormal sets of vectors spanning Alice’s Hilbert space {| α ⟩} and Bob’s Hilbert space {| β ⟩} . Then arbitrary states | ψ A of Alice can be written | ψ A = n a α =1 A α | α ; arbitrary states | ψ B of Bob can be written | ψ B = n b β =1 B β | β . The generic states of the composite bipartite system | Ψ can be expressed in terms of the amplitude Ψ αβ as | Ψ = ± αβ Ψ αβ | α ⟩ ⊗ | β . The unentangled states of the composite bipartite system are the subset for which the ampli- tude Ψ αβ factorises: Ψ αβ = A α B β . The entangled states of the composite bipartite system are the subset for which the amplitude Ψ αβ does not factorise. a) Show that if the amplitude Ψ αβ factorises then the state of the composite bipartite system | Ψ factorises. Let O A and O B be operators, respectively acting solely on Alice’s and Bob’s Hilbert spaces. (Think, e.g. , of the spin and position operators for a particle with spin.) b) Show that for unentangled states the expectation value Ψ |O A O B | Ψ of the product operator O A O B factorises into a product of expectation values, one involving each party. Brieﬂy explain why we say that such states possess no quantum correlations between the parties. Suppose we are concerned only with properties of one of the parties, say Alice. Rather than retain the full information | Ψ (or, equivalently, the full density matrix | Ψ ⟩⟨ Ψ | ) describing the composite system, we may trace out Bob’s Hilbert space. In this way, we develop a reduced density matrix ρ A Tr B | Ψ ⟩⟨ Ψ | , acting solely on Alice’s Hilbert space. As long as we are only concerned with computing expectation values of operators on Alice, i.e. , operators of the form O A I B , where I B is the identity on Bob’s Hilbert space, this tracing presents no loss of options. c) Show that if

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16_580_hw_12 - Physics 580 Quantum Mechanics I Prof P M...

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