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Unformatted text preview: Chapter 4 Quantum Entanglement 4.1 Nonseparability of EPR pairs 4.1.1 Hidden quantum information The deep ways that quantum information differs from classical information involve the properties, implications, and uses of quantum entanglement. Re call 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 begin the study of these correlations in this chapter. Recall, for example, the maximally entangled state of two qubits 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 + ) AB AB ( φ + ) = 1 2 1 A , (4.2) (and similarly ρ B = 1 2 1 B ). This means that if we measure spin A along any axis, the result is completely random – we find spin up with probability 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 1 2 CHAPTER 4. QUANTUM ENTANGLEMENT 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 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 . In fact,  φ + ) is one member of a basis of four mutually orthogonal states for the two qubits, all of which are maximally entangled — the basis  φ ± ) = 1 √ 2 (  00 ) ±  11 ) ) ,  ψ ± ) = 1 √ 2 (  01 ) ±  10 ) ) , (4.3) introduced in § 3.4.1. We can choose to prepare one of these four states, thus encoding two bits in the state of the twoqubit system. One bit is the parity bit (  φ ) or  ψ ) ) – are the two spins aligned or antialigned? The other is the phase bit (+ or ) – what superposition was chosen of the two states of like parity. Of course, we can recover the information by performing an orthogonal measurement that projects onto the { φ + ) ,  φ ) ,  ψ + ) ,  ψ )} basis. But if the two qubits are distantly separated, we cannot acquire this information locally; that is, by measuring A or measuring B . What we can do locally is manipulate this information. Suppose that Alice has access to qubit A , but not qubit B . She may apply σ 3 to her qubit, flipping the relative phase of  ) A and  1 ) A . This action flips the phase bit stored in the entangled state:  φ + ) ↔  φ ) ,  ψ + ) ↔  ψ ) . (4.4) On the other hand, she can apply σ 1 , which flips her spin (  ) A ↔  1 ) A ), and also flips the parity bit of the entangled state:  φ + ) ↔  ψ + ) ,  φ...
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 Fall '11
 JohnPreskill
 Quantum, Alice, Quantum entanglement, Bell's theorem

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