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Unformatted text preview: Lecture Notes for Ph219/CS219: Quantum Information and Computation Chapter 4 John Preskill California Institute of Technology November 2, 2001 Contents 4 Quantum Entanglement 4 4.1 Nonseparability of EPR pairs 4 4.1.1 Hidden quantum information 4 4.1.2 Einstein locality and hidden variables 8 4.2 The Bell inequality 10 4.2.1 Three quantum coins 10 4.2.2 Quantum entanglement vs. Einstein locality 13 4.3 More Bell inequalities 17 4.3.1 CHSH inequality 17 4.3.2 Maximal violation 18 4.3.3 Quantum strategies outperform classical strategies 20 4.3.4 All entangled pure states violate Bell inequalities 22 4.3.5 Photons 24 4.3.6 Experiments and loopholes 26 4.4 Using entanglement 27 4.4.1 Dense coding 28 4.4.2 Quantum teleportation 30 4.4.3 Quantum teleportation and maximal entanglement 32 4.4.4 Quantum software 35 4.5 Quantum cryptography 36 4.5.1 EPR quantum key distribution 36 4.5.2 No cloning 39 4.6 Mixedstate entanglement 41 4.6.1 Positivepartialtranspose criterion for separability 43 4.7 Nonlocality without entanglement 45 4.8 Multipartite entanglement 48 4.8.1 Three quantum boxes 49 4.8.2 Cat states 55 4.8.3 Entanglementenhanced communication 57 2 Contents 3 4.9 Manipulating entanglement 59 4.10 Summary 59 4.11 Bibliographical notes 59 4.12 Exercises 59 4 Quantum Entanglement 4.1 Nonseparability of EPR pairs 4.1.1 Hidden quantum information The deep ways that quantum information differs from classical informa tion involve the properties, implications, and uses of quantum entangle ment. Recall from § 2.4.1 that a bipartite pure state is entangled if its Schmidt number is greater than one. Entangled states are interesting because they exhibit correlations that have no classical analog. We will study these correlations in this chapter. Recall, for example, the maximally entangled state of two qubits (or EPR pair ) defined in § 3.4.1:  φ + ) AB = 1 √ 2 (  00 ) AB +  11 ) AB ) . (4.1) “Maximally entangled” means that when we trace over qubit B to find the density operator ρ A of qubit A , we obtain a multiple of the identity operator ρ A = tr B (  φ + )( φ +  ) = 1 2 I A , (4.2) (and similarly ρ B = 1 2 I B ). This means that if we measure spin A along any axis, the result is completely random — we find spin up with proba bility 1/2 and spin down with probability 1 / 2. Therefore, if we perform any local measurement of A or B , we acquire no information about the preparation of the state, instead we merely generate a random bit. This situation contrasts sharply with case of a single qubit in a pure state; there we can store a bit by preparing, say, either  ↑ ˆ n ) or  ↓ ˆ n ) , and we can recover that bit reliably by measuring along the ˆ naxis. With two 4 4.1 Nonseparability of EPR pairs 5 qubits, we ought to be able to store two bits, but in the state  φ + ) AB this information is hidden ; at least, we can’t acquire it by measuring A or B ....
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 Fall '11
 JohnPreskill
 Alice, Quantum entanglement

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