Acceleration and Entanglement: a Deteriorating Relationship

Transcription

1 Acceleration and Entanglement: a Deteriorating Relationship R.B. Mann Phys. Rev. Lett (2005) Phys. Rev. A (2006) Phys. Rev. A (2009) Phys. Rev. A (2009) D. Ahn P. Alsing I. Fuentes-Schuller F. Haque B.L. Hu SY Lin N. Menicucci D. Ostapchuk G. Salton T. Tessier V. Villalba

2 Qubits The Simplest Quantum Systems ψ = α 0 + β 1 α 2 + β 2 = 1 Photon Polarization Bloch Sphere Linear Circular = α + β

3 Entanglement Alice Bob Resource for information tasks: teleportation,cyptography, etc. Image copyright Anton Zeilinger, Institute of Experimental Physics, University of Vienna.

4 Describing Entanglement Separable state: φ AB = φ 1 A ψ 2 B Local Operations and Classical Communication Entangled state: Defined as non-separable ( ) φ AB = A 0 B A B Cannot be prepared by LOCC Maximal correlations when measured in any basis

5 Density Matrix Quantum State ψ ψ ψ ρ Density Matrix Expectation Value: ψ A ψ = Tr [ ρa] Pure State: α 0 + β 1 ρ = α 2 αβ * α * β β 2 Entangled state: α αβ * α 0 A 1 0 B 2 + β 1 A 1 1 B 2 ρ = α * β 0 0 β 2 Tr ρ 2 = 1 Mixed State: Tr ρ 2 < 1

6 Quantifying Entanglement For pure states: ( ) φ AB = ω ij i A B 1 j φ 2 AB = ω n n A B 1 n 2 ij Schmidt decomposition n ( ) Measure of entanglement: von-neumann entropy Reduce ρ AB = φ φ S ( ρ ) = Tr ρ log 2 ρ S ρ AB ρ A = Tr B Problem: mixed states ρ AB = ( ω ijkl i j k l ) ijkl [ ] [ ρ ] AB S ( ρ ) A = S ( ρ ) B No analogous Schmidt decomposition Entropy no longer quantifies entanglement. ( ) = 0

7 Mixed State Entanglement Separable state: Necessary separability condition: The partial transpose of the density matrix has positive eigenvalues if the state is separable. Mutual information ( ) = S ( ρ ) + S ( ρ ) S ( ρ ) A B AB I ρ AB ρ AB = i α ij ρ A i To determine separability is easy To measure entanglement is not ρ B j Total correlations quantum + classical

8 Measuring Entanglement Logarithmic negativity N ( ρ ) = log 2 ρ T E F = E D E F N Entanglement of Formation # of maximally entangled pairs # of pure state copies created lim copies Upper bound on all entanglement! Entanglement of Distillation # of non-maximally entangled states E D = lim #states # of maximally entangled states

9 Relativistic Entanglement What happens if Bob moves with respect to Alice? Inertial Frames: Entanglement constant between inertially moving parties Some degrees of freedom can be transferred to others Peres,Scudo and Terno Alsing and Milburn Gingrich and Adami Pachos and Solano Non-inertial frames: Alsing and Milburn, PRL , (2003) Teleportation with a uniformly accelerated partner teleportation fidelity decreases as the acceleration grows Indication of entanglement degradation. Our motivation: Study entanglement in this setting

10 Bosonic Entanglement for Uniformly Accelerated Observers Alice Photon 1 Photon 2 Bob φ AB = 1 ( 2 0 A 0 B A B 1 1 ) 2 Rob Photon 1 Photon 2 φ AB =? Fuentes-Schuller/Mann Phys. Rev. Lett. 95: (2005)

11 Minkowski space ds 2 = dt 2 dr 2 Rindler space ds 2 = e 2aξ ( dη 2 dξ 2 ) ( ) ( ) t = a 1 sinh aτ r = a 1 cosh aτ r < t ( ) ( ) t = a 1 e aξ sinh aη r = a 1 e aξ cosh aη r > t ( ) ( ) t = a 1 e aξ sinh aη r = a 1 e aξ cosh aη The quantization of fields in different coordinates is different!

12 Photon 2 cosh r = Ω= ωc a The inertial Bob Rob is causally disconnected from region II B ( ) 1 exp 2πΩ 0 2 B = Unruh effect! 1 cosh r The accelerated Rob tanh n r n I n II n =0

13 0 B 1 2 = tanh n r n cosh r I n II n =0 1 B 1 2 = tanh n r n + 1 n + 1 cosh r I n II n =0 Trace over region II nm = n A m I

14 Entanglement? Check eigenvalues of positive partial transpose λ n ± = tanh2n r n 4 cosh 2 r sinh 2 r + tanh2 r ± 2 n Z n = sinh 2 r + 4 tanh2 r + cosh 2 r Z n λ n < 0 Always entangled for finite acceleration! How much entanglement? N ( ρ ) = log 2 ρ T I AR = S ( ρ A ) + S ( ρ R ) S ( ρ AR ) Logarithmic negativity vs. acceleration parameter r Mutual information vs cosh(r) Distillable entanglement decreases as acceleration increases

15 Where did the entanglement go? Pure state: S ( ρ ) ARI,II = 0 Zero acceleration S ( ρ ) ARI = S ( ρ ) RII Entanglement between Alice+Rob in region I With modes in region II S ( ρ ) ARI = 0 Alice+Bob are maximally entangled and there is no entanglement with region II Finite acceleration the entanglement between Alice+Rob is degraded and entanglement with region II grows Infinite acceleration S ( ρ ) ARI = 1 Alice+Rob are disentangled and entanglement with region II is maximal

16 Fermionic Entanglement for Uniformly Accelerated Observers Alice Electron 1 Electron 2 Bob ψ AB = 1 ( 2 0 A 0 B A B 1 1 ) 2 Rob Electron 1 Electron 2 ψ AB =? Alsing/Mann/Fuentes-Schuller/Tessier Phys. Rev. A (2006)

17 Electron 2 Fermionic vacuum! The accelerated Rob The inertial Bob 0 2 B Entanglement of Formation cosr = E F = 1 + sin r 2 1 sin r B = cosr 0 I 0 II + sin r 1 I 1 II 1 ( ) 1 + exp 2πΩ 1 + sin r log sin r log 2 2 Ω= ωc a Negativity N = log cos 2 r

18 Rob-I anti-rob-ii anti-rob-ii

19 Infinite Acceleration Limit Bosons Entanglement goes to zero Mutual information goes to 1 State maximally entangled to region II The state is only classically correlated Fermions Entanglement goes to 1 2 Mutual information goes to 1 State maximally entangled to region II The state always has some quantum correlation

20 How Robust is this Difference? Does the number of excited modes matter? No for fermions -- entanglement degradation is still limited for fermions Yes for bosons -- entanglement loss depends on the number of modes occupied Leon/Martin-Martinez Phys.Rev. A Leon/Martin-Martinez Phys.Rev. A Does the surviving entanglement depend on the choice of maximally entangled state? No Does spin matter? Yes, but only quantitatively -- entanglement degradation is greater but still does not vanish in the large acceleration limit Leon/Martin-Martinez

21 Non-Uniform Acceleration No Timelike Killing Vector! II I Costa Rev.Bras.Fis. 17 (1987) 585 Percocca/Villalba CQG 9 (1992) 307 at, ( X 0 ) = ae 2aX 0 e 2aT + e 2aX 3/2 0

22 Alice Electron 1 Electron 2 Bob ψ AB = 1 ( 2 0 A 0 B A B 1 1 ) 2 Rob Electron 1 Electron 2 ψ AB =? Bob

23 Possible Scenario Let Vic define +ve/-ve frequency on a constant time-slice KG eqn Mann/Villalba Phys. Rev. A (2009) T=const Separate variables and solve ν = K w

24 Photon 2 Bob --> Rob The inertial Bob ρ = 1 q 2 2 q 2n ρ n n 0 2 B B 0 k 2 = 1 q 2 q n n k n I k II q = e πν * c ν + c ν + T 0 n =0 * ( T 0 ) c ν ( T 0 ) ( ) + e πν c ν ( T 0 ) ρ n = 0n 0n + 1 q 2 n + 1 0n 1 n q 2 n n + 1 0n + ( 1 q 2 )( n + 1) 1 n n + 1

25 I = log ( 2 q 2 ) + 1 q 2 2 ( ) D n = 1 + n 1 q 2 q 2 log n 1 q q 2 ( 1 + ( n + 1) ( 1 q 2 ))log n + 1 n =0 2 ( ) q 2n D n ( ( )( 1 q 2 )) N K = 0.3, w = 1 K = 0.1, w = 1 K = 0.3, w = 5 I ν = K w K = 0.1, w = 5 K = 0.3, w = 1 1 q 2 N = log n =0 q 2n 4 ( 1 q 2 ) Z n K = 0.1, w = 1 K = 0.3, w = 5 K = 0.1, w = 5 Z n = nq 2 + q 2 1 q ( q 2 )

26 Generating Entanglement Measurements of an accelerated observer generate entanglement! Fermions 0 M + = N exp ( I ( tan r )c II k d ) k 0 k I k 0 M = N exp ( I ( tan r )d II k c ) k 0 k I ψ + k + 0 k II + 0 k II Project onto a single particle mode k in region I I ( k ) = P k 0 M + = sin r + cosr a k b k 0 M + Bogoliubov tmf Trace over antiparticle states: S = log ( 2 csc 2 r ) + cos 2 r log 2 tan 2 r ( ) Han/Olson/Dowling Phys. Rev. A (2008) Mann/Ostapchuk Phys. Rev. A (2009) Minkowski particle vacuum Minkowski antiparticle vacuum

27 Detector Responses Can entanglement be used to distinguish curvature effects from Minkowski-space heating? H I ( τ ) = η τ ρ = A A = ( ( )) e iωτ σ + + e iωτ σ ( )φ x τ D + ( x, x ) = φ τ ( ) coupling detector worldline Compare: Minkowski space at finite temperature T ds 2 = dt 2 + dxid x de Sitter space with expansion rate κ=2πt ds 2 = dt 2 + e 2κ t dxid x ( 1 A) 0 0 dτ d τ η( τ )η ( τ )e iω( τ τ ) D + x, x ( )φ τ ( ) = T 2 φ Rφ = 0 ( ) ( ( )) 4sinh 2 πt t t iε A single detector can t distinguish these 2 possibilities Ω Gibbons/Hawking Phys. Rev. D (1977) What about 2 detectors?

28 Negativity N = max ( X A,0) amplitude detector clicks amplitude for virtual particle exchange Menicucci/Ver Steeg Phys. Rev. D (2009) X = 2 dτd τ η( τ )η ( τ )e iω( τ + τ ) D + x, x τ <τ ( ) x = t + t y = t t iε + D Mink + D ds ( x, x ) = T 8π L ( x, x ) = T coth ( πt ( L y )) + coth ( πt ( L y) ) ( ) sinh 2 πty e 2πTx L 2 4π 2 π 2 T 2 1

29 Now compare 2 uniformly accelerated detectors Mann/Menicucci/Salton

30 Same acceleration

31 Opposite acceleration opposite acceleration same acceleration

32 Hu/Haque/Lin/Mann/Ostapchuk Alice and Rob each have a detector that couples to a scalar field Driven Damped Harmonic Oscillator Frequency Ω Damping Coefficient λ 2 0 Iterate over retarded mutual influences in powers of λ 0 Compute Correlation Matrix V = R μ R ν = AB R μ R ν AB + 0 R μ R ν 0 R μ = { Q A, P A,Q B, P B } Entanglement Σ =det V PT + i ħ 2 M < 0 detectors Λ = vacuum V PT = ΛV Λ M =

33 Σ =det V PT + i ħ 2 M a = 1

34 Σ =det V PT + i ħ 2 M a = 0.1

35 Better Regulator Σ =det V PT + i ħ 2 M

36 Summary Entanglement is an observer/detector dependent phenomenon As acceleration increases Entanglement goes to zero (bosons) finite (fermions) Mutual information goes to unity Entanglement with region II gets larger As observer increases acceleration Entanglement decreases with time Maximum change takes place when change in acceleration is maximal