SHAKING ENTANGLEMENTTELEPORTATION IN MOTION

Nicolai FriisUniversity of Nottingham, School of Mathematical Sciences, Nottingham, UK

RELATIVISTIC QUANTUM INFORMATION (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

MOTIVATION (SEE REFS. [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.

NON-UNIFORM CAVITY MOTION (SEE REFS. [2, 3, 4, 5])

Φn

a bx

ct

Φn

Φm~

Α mn

Β mna b

x

ct

Φn

Η = Η1Φm~

Α mn

Β mn

Α*nm

-Β nm

GnHΗ1L

a bx

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 Gn

Real scalar field: φ =∑n

(φn an + 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 + m2)φ(x) = 0 Rindler quantization: φ =∑n

(φ̃n ãn + φ̃∗n ã†n) Compose transformation A−1(α, β)G(η)A(α, β)

Dirichlet boundaries: φ(t,a) = φ(t,b) = 0 Bogoliubov transformation: φn =∑m

(α∗mn φ̃m − βmn φ̃∗m) , αmn = (φ̃m, φn), βmn = −(φ̃m, φ∗n)

FOCK STATE PROCEDURE(i) Select in-region state: ρ

(ii) Transform to out-region: ρ→ ρ̃

Vacuum: | 0 〉 = exp(

12

∑p,q

Vpq ã†p ã†q

) ∣∣ 0̃ 〉where V = −β∗α−1

Fock states: act on | 0 〉with creation operators

a†n =∑m

(α∗mn ã

†m + βmn ãm

)(iii) Trace out inaccessible modes

(iv) Study entanglement properties

GAUSSIAN STATES — CONTINUOUS VARIABLES (CV) (SEE REF. [6])Covariance matrix: Γij =

〈XiXj + XjXi

〉ρ− 2〈Xi〉ρ

〈Xj〉ρ

Quadratures: X(2n−1) = 1√2 (an + a†n) , X(2n) = −i√2 (an − a

†n)

Bogoliubov transformation→ symplectic transformation S

Γ̃ = S ΓST

Γ =

C11 C12 C13 . . .C21 C22 C23 . . .C31 C32 C33 . . .

......

.... . .

→ Γ̃ =C̃11 C̃12 C̃13 . . .

C̃21 C̃22 C̃23 . . .

C̃31 C̃32 C̃33 . . ....

......

. . .

S =

M11 M12 M13 . . .M21 M22 M23 . . .M31 M32 M33 . . .

......

.... . .

Mmn = Re(

αmn − βmn −i(αmn + βmn)

i(αmn − βmn) αmn + βmn

)

where C̃mn =∑i,jMmiCijMTnj

Trace out/remove inaccessible modes⇒ study entanglement between remaining modes

TELEPORTATION IN MOTION (SEE REF. [7])Alice wants to teleport a coherent state — standard CV protocol:

(i) Resource: two-mode squeezing r: modes k (Alice) & k′ (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: αmn = α(0)mn + α

(1)mn h +O(h

2), βmn = β(1)mn h +O(h

2)

h := δv̇c2

, with v̇ . . . acceleration at center of cavity

Fidelity optimized over local Gaussian operations: F̃optcompensate phases from time evolution ⇒ F̃opt = F̃ (0)opt − F̃

(2)

opt h2 +O(h4)

F̃ (0)opt =(1 + e−2r

)−1, F̃ (2)opt = F̃

(0)

opt12

∑n 6=k′

(|β (1)nk′ |

2 + |α (1)nk′ |

2 Tanh(2r))

SIMULATION (SEE REF. [7, 8])

Leff

L 0 d HtL-d HtL+

1− dim transmission line for microwave radiationterminated by superconducting circuits (SQUIDs)

⇒ simulate boundary conditions for a cavity ofeffective length Leff = L0 + d+(t) + d−(t).

Predicted degradation of optimal teleportationfidelity for realistic parameters: ≈ 4 % of F̃ (0)opt.

REFERENCES[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. QuantumGrav. 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. Huber, I. Fuentes & D. E. Bruschi, Quantum gates and multipartite entanglement resonances realized by non-uniform cavity motion, Phys. Rev. D 86, 105003 (2012).[6] N. Friis & I. Fuentes, Entanglement generation in relativistic quantum fields, J. Mod. Opt. 60, 22 (2013).[7] N. Friis, A. R. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson & I. Fuentes, Relativistic Quantum Teleportation with Superconducting Circuits, Phys. Rev. Lett. 110, 113602 (2013).[8] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori & P. Delsing, Observation of the dynamical Casimir effect in a superconducting circuit,

Nature 479, 376 (2011).