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How and why to measure (or quantify) quantum entanglement? • Measuring quantum entanglement is useful: as such measurement may have applications to the design of (still theoretical) quantum computers. It is also a fundamental property of quantum systems. • In pure states, there are various proposals to measure quantum entanglement. Consider the entanglement entropy and let us look at a more complicated system: – With the Hilbert space a tensor product H = s 1 ⊗ s 2 ⊗ · · · ⊗ s N = A ⊗ given state |gs〉 ∈ H, calculate the reduced density matrix: 01 01 01 si x i+1 x ρ A = Tr Ā (|gs〉〈gs|) 01 01 01 01 ... s s ... i−1 x 01 01 01 01 01 01 01 01 01 01 01 01 01 A s s x i+L−1 x i+L – The entanglement entropy is the resulting von Neumann entropy: S A = −Tr A (ρ A log(ρ A )) = − ∑ eigenvalues of ρ A λ≠0 ... λ log(λ) Ā, and a
Interpretation of the entanglement entropy • It is the entropy that is measured in a subsystem A, once the rest of the system Ā – the environment – is forgotten. If we think A is all there is, we will think the system is in a mixed state, with density matrix given by ρ A . The entropy of ρ A measures how mixed ρ A is. This mixing is due to the connections, or entanglement, with the environment. • It was proposed as a way to understand black hole entropy [Bombelli, Koul, Lee, Sorkin 1986]. • Then it was proposed as a measure of entanglement [Bennet, Bernstein, Popescu, Schumacher 1996]. • Examples: – Tensor product state: |gs〉 = |A〉 ⊗ |Ā〉 ⇒ ρ A = |A〉〈A| ⇒ S A = −1 log(1) = 0.
Page 1: Entanglement entropy in 1+1 dimensi
Page 6 and 7: - The maximally entangled state |gs
Page 8 and 9: Short- and large-distance entanglem
Page 10 and 11: Partition functions on multi-sheete
Page 12 and 13: • Another twist field ˜T is asso
Page 14 and 15: Hence we have Short- and large-dist
Page 16 and 17: The entropy of disconnected regions
Page 18 and 19: The entropy of disconnected regions
Page 20 and 21: Some simple limit cases • When r
Page 22 and 23: Non-extensivity • In recent years

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