Vacuum Entanglement
B. Reznik (Tel-Aviv Univ.)
A. Botero (Los Andes. Univ. Columbia.)J. I. Cirac (Max Planck Inst., Garching.)A. Retzker (Tel-Aviv Univ.)J. Silman (Tel-Aviv Univ.)
Quantum Information Theory: Present Status and Future DirectionsAug. 23-27 2004, INI, Cambridge
Vacuum Entanglement
Motivation: QI
Fundamentals: SR QM QI: natural set up to study Ent.causal structure ! LO.H1, many body Ent.
.Q. Phys.: Can Ent. shed light on “quantum effects”? (low temp. Q. coherences, Q. phase transitions, DMRG, Entropy Area law.)
A
B
See also Latorre’s, Verstraete’s & Plenio’s talks
Continuum results :BH Entanglement entropy:Unruh (76), Bombelli et. Al. (86), Srednicki (93), Callan & Wilczek (94). Albebraic Field Theory :
Summers & Werner (85), Halvarson & Clifton (00).Entanglement probes:
Reznik (00), Reznik, Retzker & Silman (03).
Discrete models:Harmonic chains: Audenaert et. al (02), Botero & Reznik (04).Spin chains: Wootters (01), Nielsen (02), Latorre et. al. (03).Linear Ion trap: Retzker, Cirac & Reznik (04).
Background
AA
BB
)I (Are A and B entangled?
)II (Are Bell’s inequalities violated?
)III (Where does ent. “come from?”
)IV (Can we detect it?
AA
BB
)I (Are A and B entangled? Yes, for arbitrary separation.
") Atom probes.(”
)II (Are Bell’s inequalities violated? Yes, for arbitrary separation .
) Filtration, “hidden” non-locality .(
)III (Where does it “come from?” Localization, shielding.
(Harmonic Chain).
)IV (Can we detect it? Entanglement Swapping.
)Linear Ion trap.(
Plan:
)1 .(Field entanglement: local probes. Reznik (00), Reznik, Retzker, Silman (03).
)2 .(Harmonic chain: spatial structure of ent. Botero, Reznik (04).
)3 .(Linear Ion trap: detection of ground state ent. Retzker, Cirac, Reznik (04).
A
B
A pair of causally disconnected localized detectorsA pair of causally disconnected localized detectors
Probing Field Entanglement
RFT! Causal structure
QI ! LOCC
L> cT
B. Reznik quant-ph/0008006,0112044B. Reznik, A. Retzker & J. Silman quant-ph/0310058
Causal Structure + LO
For L>cT, we have [A,B]=0Therefore UINT=UA UB
ETotal =0, butEAB >0. (Ent. Swapping)
Vacuum ent ! Detectors’ ent .Lower bound.
LO
0E
1EE
Note: we do not use the rotating wave approximation.
Field – Detectors Interaction
Window Function
Interaction:
HINT=HA+HB
HA=A(t)(e+i t A+ +e-i tA
-) (xA,t)
Initial state:
|(0) i =|+Ai |+Bi|VACi
Two-level system
Unruh (76), B. Dewitt (76), particle-detector models .
Probe Entanglement
Calculate to the second order (in the final state,and evaluate the reduced density matrix.
Finally, we use Peres’s (96) partial transposition criterion to check inseparability and use the Negativity as a measure.
?
AB(4£ 4) = TrF (4£1)
i pi A(2£2)B
(2£2)
21(1 ' )(...)
2Interaction A B A A AU U U i dtH T dtdt H H
( ) 0Interaction A BT U
2
52
2
1 0
0( ) ( )
AB
AB AB
AB
A A B
B A B
X
X XT O
E E E
E E E
0 0 2,
0 1,
AB A B
A A
X or photons
E photon
** ++ +* *+
21(1 ' )(...)
2Interaction A B A A AU U U i dtH T dtdt H H
( ) 0Interaction A BT U
2
52
2
1 0
0( ) ( )
AB
AB AB
AB
A A B
B A B
X
X XT O
E E E
E E E
0 0 2,
0 1,
AB A B
A A
X or photons
E photon
P.T,
P.T.
** ++ +* *+
Emission < Exchange
|++i + h XAB|VACi |**i…”+“ XAB
||EA||||EB||< ||h0|XABi||
2
0 0[ ( )] (( ) ) ( )A BSin L
dd
L
Off resonance Vacuum “window function”
Superocillatory functions (Aharonov (88), Berry(94)) .
We can tailor a superoscillatory window function for every Lto resonate with the vacuum “window function”
Entanglement for every separation
sin( )L
Superocillatory window function
VacuumWindow function
Exchange term / exp(-f(L/T))
Bell’s Inequalities
No violation of Bell’s inequalities .
But, by applying local filters
Maximal EntMaximal Ent..
Maximal violationMaximal violation
NN(())
MM(())
| ++i + h XAB|VACi |**i…”+“ ! |+i|+i + h XAB|VACi|*i|*i…”+“
CHSH ineq. Violated iffM()>1, (Horokecki (95).)
“Hidden” non-locality. Popescu (95). Gisin (96).
NegativityNegativity
FilteredFiltered
Reznik, Retzker, SilmanReznik, Retzker, Silman 0310058
Summary (1)
1 (Vacuum entanglement can be distilled!
2( Lower bound: E ¸ e-(L/T)2
) possibly e-L/T(
3 (High frequency (UV) effect: ¼ L2 .
4 (Bell inequalities violation for arbitrary separation maximal “hidden” non-locality .
Reznik, Retzker, Silman Reznik, Retzker, Silman 0310058
The Harmonic Chain model
Hchain! Hscalar field
=s (x)2+(5)2+m22(x))dx
chain/ e-qi Q-1 qj/4
I) Gaussian state ! Exact
calculation of Ent.
II) Mode-wise decomposition !
Identify spatial Ent. Structure
AA BB
11
Circular Harmonicchain
A. Botero & B. Reznik 0403233
Gaussian (pure) Entanglement
A Covariance Matrix M Second moments h qi qji, h pi pji
SymplecticSpectrum
EntanglementEntropy
i
The reduced density matrix of a Gaussian stateIs a Gaussian density matrix.
AB
E= (+1/2)log(+1/2) – (-1/2)log(-1/2)
Entanglement: block vs. the rest
))Log-2 scaleLog-2 scale((
weak
strong
E' 1/3 lnNb +Ec(,Nb)
c=1, bosonic 1-d FT
Mode-Wise decomposition theorem
Botero, Reznik 02090260209026 (bosonic modes)Botero, Reznik 0404176 0404176 (fermionic modes)
A B A B
AB= 1122…kk 0..,
kk / e-k n|ni|ni
Two modes squeezed state
qi
pi
Qi
Pi
AB = ci|Aii|Bii
Schmidt Mode-Wise decomposition
Local U
Local U
Spatial Entanglement Structure
qi ! Qm= ui qi
pi ! Pm= vi pi
AA BB
qqii, p, pii
quantifies the contribution of local (qi, pi) oscillators
to the collective coordinates (Qi,Pi)
Circular Harmonicchain
local collective
Participation function : Pi=ui vi, Pi=1
Site Participation Function
Weakcoupling
Strongcoupling
N=32+48 osc. Modes are ordered in decreasingEnt. Contribution, from front to back.
Logarithmic dependence in the continuum limit with the overall 1/3 coefficient as predicted by conformal field theory .
Inclusion of an ultraviolet cutoff leads to Localization of the highest frequency inner modes .
Mode shape hierarchy with distinctive layered structure ,with exponential decreasing contribution of the innermost modes.
Summary (2)
Detection of Vacuum Entanglement
in a Linear Ion Trap
Paul Trap
AB
Internal levels
H0=z(zA+z
B)+n any an
Hint=(t)(e-i+(k)+ei-
(k))xk
H=H0+Hint
A. Retzker, J. I. Cirac, B. Reznik 0408059
1/z << T<<1/0
Entanglement in a linear trap
Entanglement between symmetric groups of ions as a function of the total number (left) and separation of finite groups (right).
|vaci=|0ci|0bi/ n e- n|niA|niB
Two trapped ions
Final internal state
Eformation(final) accounts for 97% of the calculatedEntantlement: E(|vaci)=0.136 e-bits.
|vaci|++i !|i(|**i+ e-|++i)
“Swapping” spatial ! internal states
U=(ei x x ei p y) ...
U
Long Ion Chain
A B
But how do we check that ent. is not due to “non-local” interaction?
Htruncated=HA© HB
We compare the cases with a truncated and free Hamiltonians
HAB !
Long Ion Chain
L=6,15, N=20 L=10,11 N=20
denotes the detuning, L the locations of A and B.
exchange/emission >1 , signifies entanglement.
Summary
Atom Probes :Vacuum Entanglement can be “swapped” to detectors.Bell’s inequalities are violated (“hidden” non-locality).Ent. reduces exponentially with the separation .High probe frequencies are needed for large separation.
Harmonic Chain:Persistence of ent. for large separation is linked with localization of the interior modes. This seems to provide a mechanism for
“shielding” entanglement from the exterior regions .
Linear ion trap: -A proof of principle of the general idea is experimentally feasible for
two ions .-One can entangle internal levels of two ions without performing gate
operations.