Download Vacuum-Entanglement

Recent Developments in Quantum Physics
Asher Peres’ 70’th Birthday Honour of In
1-2, 2004February
Vacuum Entanglement
B. Reznik (Tel Aviv Univ.)
Alonso Botero (Los Andes Univ. Columbia)
Alex Retzker (Tel Aviv Univ.)
Jonathan Silman (Tel Aviv Univ.)
Vacuum Entanglement
Motivation:
QI
Fundamentals:
A
B
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,
BH Ent. Entropy.)
Background
Continuum results:
BH Entanglement entropy:
Unruh (76), Bombelli et. Al. (86), Srednicki (93) Callan & Wilczek (94) .
Albebraic Field Theory:
Summers & Werner (85), Halvason & Clifton (2000).
Entanglement probes:
Reznik (2000), Reznik, Retzker & Silman (2003).
Discrete models:
Harmonic chain: Audenaert et. al (2002), Botero & Reznik (2004).
Spin chains: Wootters (2001), Nielsen (2002), Latorre et. al. (2003).
(I) Are A and B entangled?
(II) Are Bells' inequalities violated?
A
B
(III) Where does ent. “come
from”?
(I) Are A and B entangled?
Yes, for arbitrary separation.
("Atom probes”).
A
B
(II) Are Bells' inequalities violated?
Yes, for arbitrary separation.
(Filtration, “hidden” non-locality).
(III) Where does it “come from”?
Localization, shielding.
(Harmonic Chain).
L  cT
A pair of causally disconnected atoms
A
B
Causal Structure
For L>cT, we have [A,B]=0
Therefore UINT=UA UB (LO)
 ETotal =0, but
 EAB >0. (Ent. Swapping)
Vacuum ent ! Atom ent.
Lower bound.
(Why not the use direct approach?
simplicity, 4£ 4, vs. 1£ 1
but wait to the second part.)
Relativistic field + probe
Interaction:
E1
E  
E0
Two-level system
HINT=HA+HB
HA=A(t)(e+i t A+ +e-i tA-) (xA,t)
Window Function
Initial state:
|(0) i =|+Ai |+Bi|VACi
Relativistic field + probe
Interaction:
E1
E  
E0
Two-level system
HINT=HA+HB
HA=A(t)(e+i t A+ +e-i tA-) (xA,t)
Window Function
Initial state:
|(0) i =|+Ai |+Bi|VACi
Do not use the rotating frame approximation!
Probe Entanglement
AB(4£ 4) = TrF (4£1)
?
 i pi A(2£2)B(2£2)
Calculate to the second order (in ) the final state,
and evaluate the reduced density matrix.
Finally, we use Peres’s (96) partial transposition
criteria to check non-separability and use the
Negativity as a measure.
Emission < Exchange
XAB
|++i + h XAB|VACi |**i “+”…
Emission < Exchange
XAB
|++i + h XAB|VACi |**i “+”…


0
d [ (   )]  
2

0
Off resonance
d
Sin( L) A (   ) B (   )
L
Vacuum “window function”
Characteristic Behavior
1) Exponentially decrease: E¼ e-L2.
Super-oscillatory window functions.
(Aharonov(88), Berry(94)).
2) Increasing probe frequency ¼ L2 .
3) Bell inequalities?
Entanglement 9 Bell ineq. Violation.
(Werner(89)).
Bells’ inequalities
N ()
Maximal Ent.
No violation of Bell’s inequalities.
But, by applying local filters
Filtered
|++i + h XAB|VACi |**i “+”…
! 2 |+i|+i + h XAB|VACi|*i|*i “+”…
Negativity

M ()
Maximal violation
CHSH ineq. Violated iff
M ()>1, (Horokecki (95).)

Reznik, Retzker, Silman (2003)
“Hidden” non-locality.
(Popescu(95).)
Comment
{
Comment
Asher Peres Lecture Notes on GR.
(200?).
{
Accelerated probes
Time
QFT
Red Shift !
A&B perceive |VACi
as a thermal state.
(Unruh effect)
|VACi= N (n e- n |ni|ni)
A
B
Space
Final A&B state becomes entangled.
Special case: complementary regions.
Summers & Werner (85).
} Comment
Where does Ent. “come from”?
Where does Ent. “come from”?
1
Hchain! Hscalar field
A
B
Circular chain of coupled
Harmonic oscillators.
-1 q /4
j
chain/ e-qi Q
is a Gaussian state.
! Exact calculation.
“Mode-Wise” structure
A
A
B
Qi
Pi
qi
pi
AB = ci|Aii|Bii
Scmidth decomposition
B
1
AB= 1122…kk ( k+1k+2…)
Mode-Wise decomposition.
kk /  e-k n|ni|ni
Botero, Reznik 2003.
Giedke, Eisert, Cirac, Plenio, 2003.
(are 1£1 Gaussian states.)
Mode Participation
Local
qi , pi
A
qi ! Qi= ui qi
B
pi ! Pi= vi pi
Quantifies the participation of the
local (qi, pi) oscillators in the
collective coordinates (Qi,Pi)
“normal modes” within each block.
Mode Shapes
Botero, Reznik (2004).
Discussion
Atom Probes:
Vacuum Entanglement can be swapped (In theory) to atoms.
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 seem to provide a mechanism for
“shielding” entanglement from exterior regions.
(Therefore in spin or harmonic chains entanglement between single
sites truncates.)