S HAKING ENTANGLEMENT T ELEPORTATION IN M OTION Nicolai Friis University of Nottingham, School of Mathematical Sciences, Nottingham, UK R ELATIVISTIC Q UANTUM I NFORMATION (RQI) RQI aims to investigate the relationship of quantum information and relativity: Overlap of quantum information, quantum field theory, quantum optics, special- & general relativity Identify physical systems to store, process and transmit quantum information in relativistic settings Effects of gravity and motion on quantum information: Unruh-Hawking- & dynamical Casimir effect M OTIVATION ( SEE R EFS . [1, 8]) Experiments are reaching regimes where relativistic effects are non-negligible, e.g., quantum communication between moving satellites: need toy models & table top setups to inform future space-based experiments. N ON - UNIFORM CAVITY MOTION ( SEE R EFS . [2, 3, 4, 5]) Φ n a b x ct Φ n Φ m ~ Α mn Β mn a b x ct Φ n Η=Η 1 Φ m ~ Α mn Β mn Α * nm -Β nm G n HΗ 1 L a b x ct Fig.1(a): Inertial cavity of width δ = b - a Fig.1(b): Sudden acceleration at t =0, cavity rigid Fig.1(c): Finite duration: modes pick up phases G n Real scalar field: φ = ∑ n (φ n a n + φ * n a † n ) Rindler coordinates: ct = χ sinh η , x = χ cosh η Inverse transformation: A -1 (α, β )= A(α * , -β ) Klein-Gordon Equation: (∂ μ ∂ μ + m 2 ) φ(x)=0 Rindler quantization: φ = ∑ n ( ˜ φ n ˜ a n + ˜ φ * n ˜ a † n ) Compose transformation A -1 (α, β )G(η )A(α, β ) Dirichlet boundaries: φ(t, a)= φ(t, b)=0 Bogoliubov transformation: φ n = ∑ m (α * mn ˜ φ m - β mn ˜ φ * m ) , α mn =( ˜ φ m ,φ n ), β mn = -( ˜ φ m ,φ * n ) F OCK STATE PROCEDURE (i) Select in-region state: ρ (ii) Transform to out-region: ρ → ˜ ρ Vacuum: | 0 i = exp ( 1 2 ∑ p,q V pq ˜ a † p ˜ a † q ) ˜ 0 where V = -β * α -1 Fock states: act on | 0 i with creation operators a † n = ∑ m α * mn ˜ a † m + β mn ˜ a m (iii) Trace out inaccessible modes (iv) Study entanglement properties G AUSSIAN STATES — CONTINUOUS VARIABLES (CV) ( SEE R EF . [6]) Covariance matrix: Γ ij = X i X j + X j X i ρ - 2 X i ρ X j ρ Quadratures: X (2n-1) = 1 √ 2 (a n + a † n ) , X (2n) = -i √ 2 (a n - a † n ) Bogoliubov transformation → symplectic transformation S ˜ Γ= S Γ S T Γ= C 11 C 12 C 13 ... C 21 C 22 C 23 ... C 31 C 32 C 33 ... . . . . . . . . . . . . → ˜ Γ= ˜ C 11 ˜ C 12 ˜ C 13 ... ˜ C 21 ˜ C 22 ˜ C 23 ... ˜ C 31 ˜ C 32 ˜ C 33 ... . . . . . . . . . . . . S = M 11 M 12 M 13 ... M 21 M 22 M 23 ... M 31 M 32 M 33 ... . . . . . . . . . . . . M mn = Re α mn - β mn -i(α mn + β mn ) i(α mn - β mn ) α mn + β mn ! where ˜ C mn = ∑ i,j M mi C ij M T nj Trace out/remove inaccessible modes ⇒ study entanglement between remaining modes T ELEPORTATION IN M OTION ( SEE R EF . [7]) Alice wants to teleport a coherent state — standard CV protocol: (i) Resource: two-mode squeezing r : modes k (Alice) & k 0 (Rob) (ii) Alice’s Bell measurement: double homodyne detection (iii) Results: sent to Rob classically to retrieve teleported state Fidelity of teleportation degraded by Rob’s motion: Coefficients: α mn = α (0) mn + α (1) mn h + O (h 2 ), β mn = β (1) mn h + O (h 2 ) h := δ ˙ v c 2 , with ˙ v ... acceleration at center of cavity Fidelity optimized over local Gaussian operations: ˜ F opt compensate phases from time evolution ⇒ ˜ F opt = ˜ F (0) opt - ˜ F (2) opt h 2 + O (h 4 ) ˜ F (0) opt = ( 1+ e -2r ) -1 , ˜ F (2) opt = ˜ F (0) opt 1 2 ∑ n6=k 0 |β (1) nk 0 | 2 + |α (1) nk 0 | 2 Tanh(2r ) S IMULATION ( SEE R EF . [7, 8]) L eff L 0 d Ht L - d Ht L + 1 - dim transmission line for microwave radiation terminated by superconducting circuits (SQUIDs) ⇒ simulate boundary conditions for a cavity of effective length L eff = L 0 + d + (t)+ d - (t). Predicted degradation of optimal teleportation fidelity for realistic parameters: ≈ 4% of ˜ F (0) opt . R EFERENCES [1] D. Rideout, T. Jennewein, G. Amelino-Camelia, T. F. Demarie, B. L. Higgins, A. Kempf, A. Kent, R. Laflamme, X. Ma, R. B. Mann, E. Martín-Martínez, N. C. Menicucci, J. Moffat, C. Simon, R. Sorkin, L. Smolin, and D. R. Terno, Fundamental quantum optics experiments conceivable with satellites – reaching relativistic distances and velocities, Class. Quantum Grav. 29, 224011 (2012), Focus Issue on ‘Relativistic Quantum Information’. [2] D. E. Bruschi, I. Fuentes & J. Louko, Voyage to Alpha Centauri: Entanglement degradation of cavity modes due to motion, Phys. Rev. D 85, 061701(R) (2012). [3] N. Friis, A. R. Lee, D. E. Bruschi & J. Louko, Kinematic entanglement degradation of fermionic cavity modes, Phys. Rev. D 85, 025012 (2012). [4] N. Friis, D. E. Bruschi, J. Louko & I. Fuentes, Motion generates entanglement, Phys. Rev. D 85, 081701(R) (2012). [5] N. Friis, M. 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