Entanglement: concept, measures and open problems

Transcription

1 Entanglement: concept, measures and open problems Division of Mathematical Physics Lund University June 2013 Project in Quantum information. Supervisor: Peter Samuelsson

2 Outline 1 Motivation for study of entanglement Early attempts to quantify entanglement Quantum teleportation and Quantum cryptography 2 LOCC transformation Kraus operators, measurement and trace preserving operations 3 Maximally entangled states and separable states Manipulation of single bi-partite states Asymptotic manipulation of single bi-partite states 4 Postulates and axioms Survey of entangled measures Representative entangled states 5

3 Early attempts to quantify entanglement Early attempts to quantify entanglement Quantum teleportation and Quantum cryptography Bell state ψ 2 = 0,0 + 1,1 2 EPR paradox Bell s inequality

4 Early attempts to quantify entanglement Quantum teleportation and Quantum cryptography Quantum teleportation and Quantum cryptography Quantum teleportation Quantum cryptography

5 LOCC transformation LOCC transformation Kraus operators, measurement and trace preserving operations Quantum noisy channel Local operations - post process Classical communications

6 LOCC transformation Kraus operators, measurement and trace preserving operations Kraus operators, measurement and trace preserving operations ρ i = A i ρ in A i tr(a i ρ in A i ) σ = i AiρinA i ρ k = A k B k ρ in A k B k tr(a k B k ρ in A k B k )

7 Maximally entangled states and separable states Manipulation of single bi-partite states Asymptotic manipulation of single bi-partite states Maximally entangled states and separable states ψ d = dd d. There are maximally entangled states All non-separable states allow some task to be achieved better than by LOCC alone, so all non-separable states are entangled Entanglement does not increase under LOCC operations The entanglement does not change under local unitary evolutions, without classical communication Separable states contain no entanglement

8 Manipulation of single bi-partite states Maximally entangled states and separable states Manipulation of single bi-partite states Asymptotic manipulation of single bi-partite states ψ 2 = 0,0 + 1,1 2. φ = α 0, 0 + β 1, 1 A 0 = (α β 1 1 ) 1 A 1 = (α β 0 1 ) ( ) 0,0 A 0 B + 0,1 A 1 B 2 00 α00 + β11 and 01 β01 + α10 0 A (α 0,0 AB +β 1,1 AB )+ 1 A (β 0,1 AB +α 1,0 AB ) 2.

9 Maximally entangled states and separable states Manipulation of single bi-partite states Asymptotic manipulation of single bi-partite states Asymptotic manipulation of single bi-partite states ρ σ (partial order) ρ n σ m The supreme of all such achievable rates r = m/n is r exact(ρ ω). ρ n σ m, σ m σ m n with the same constant rate r = m/n, which means m also. If in the asymptotic regime the state ρ m is arbitrarily close to the state ρ m we will call the rate r achievable. The supremal achievable rate r approx is then related to the relative entanglement content of ρ and σ in the asymptotic regime. [ E C = inf (r : lim n infψ D ( ρ n Ψ(Φ(2) rn ) )])

10 Postulates and axioms Postulates and axioms Survey of entangled measures Representative entangled states An entanglement measure E is a map from density matrices to a positive real number. The maximally entangled (Bell state for qubits) are taken as the maximum entanglement, with E( B B ) = log d. If the state is separable, then E(ρ AB ) = 0. LOCC operations cannot increase E, on average. For a pure state, the entanglement measure on ρ AB is given by the von Neumann entropy of one of the subsystems. According to the Schmidt decomposition, the entropy of the subsystems is the same. E(ρ AB ) = S(ρ A ) = S(ρ B ) if ρ AB is pure. Equivalently, E(( ψ ψ ) = S(Tr B { ψ ψ})

11 Survey of entangled measures Postulates and axioms Survey of entangled measures Representative entangled states von Neumann entropy S(ρ) = tr(ρ log(ρ)) = λi log(λi) i entanglement ( cost E C = inf r : lim n [ infψ Tr { ρ n Ψ(Φ(2) rn ) )}]). entanglement ( distillation E D = sup r : lim n [ infψ D { Ψ ( ρ n) Φ(2) rn}]). Entrophy of entanglement E( ψ ψ ) = S(tr B ( ψ ψ )) entanglement of formation E F (ρ) = inf { pie ( ψi ψi )} i logarithmic negativity,...

12 Representative entangled states Postulates and axioms Survey of entangled measures Representative entangled states Bell state B 1 = 1 2 ( 0 A 0 B + 1 A 1 B ) GHZ state GHZ = 1 2 ( ) W state W = 1 3 ( )

13 and Outlook multipartite systems there is a need for further study a good calculatable measure for the entanglement cost, and here the additivity of the entanglement of formation is a very interesting open question different algorithms need different definitions of what is meant by entanglement