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Single-Particle Interference
Can Witness Bipartite Entanglement
Torsten Scholak1 3 Florian Mintert2 3 Cord A. Muller1
1 2 3
March 13, 2008
Introduction Proposal Quantum Optics Scenario Motivation Definitions
Motivation! double slit experiment
perfect visibility perfect knowledge*
! atoms provide information about photon path?
❀ photon gathers information about atoms
cf. U. Eichmann, Phys. Rev. Lett. 70, 2359 (1993)
source detector
1
2
A
B
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Motivation Definitions
What Information Can Be Gained by Scattering a Probe?
source detector
1
2
A
B
ϱt
! correlations?
! distinguish between classical and quantum correlations?
❀ witness entanglement
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Motivation Definitions
Entanglement Witnesses
! an observable W is an entanglement witness:⇔
⟨W ⟩ = tr{ϱtW }
{≥ 0, for all separable states ϱt
< 0, for at least one entangled state
⇒ negative outcome: ϱt entangled!❀ look for characteristic sign changes!
separable
entangled
⟨W ⟩ = 0
cf. P. Horodecki, Phys. Lett. A 232, 33 (1997)
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Probe-Target Interactionpr
epar
atio
n
ϱ = ⊗ ϱtϱp
targetprobe
1 2
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Probe-Target Interactionpr
epar
atio
n
ϱ = ⊗ ϱtϱp
targetprobe
1 2
inte
ract
ion
ϱ′ =
T = eiφATA + e
iφBTB φ = φA − φB
T
( )
T †
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Probe-Target Interactionpr
epar
atio
n
ϱ = ⊗ ϱtϱp
targetprobe
1 2
inte
ract
ion
ϱ′ =
T = eiφATA + e
iφBTB φ = φA − φB
T
( )
T †
projector Pprobe
mea
sure
men
t
⟨P⟩′
= ⟨T †APTA⟩ + ⟨T †
BPTB⟩︸ ︷︷ ︸
I0 ≥ 0, no witness
+[eiφ⟨T †
BPTA⟩ + e−iφ⟨T †
APTB⟩]
︸ ︷︷ ︸
|δI (φ)| ≤ I0, witness?
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Central Result: Connection of Pattern and Witness
⇒ designated entanglement witness:
reading offinterference contribution
at φ = 0⇔
measuringexp. value of observable
on the target system
δI (0) = ⟨T †BPTA + T
†APTB⟩ = trt
{ϱt M
}
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Central Result: Connection of Pattern and Witness
⇒ designated entanglement witness:
reading offinterference contribution
at φ = 0⇔
measuringexp. value of observable
on the target system
δI (0) = trp&t
{(ϱp ⊗ ϱt)
(T
†BPTA + T
†APTB
)}= trt
{ϱt M
}
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Central Result: Connection of Pattern and Witness
⇒ designated entanglement witness:
reading offinterference contribution
at φ = 0⇔
measuringexp. value of observable
on the target system
δI (0) = trt
{ϱt trp
[ϱp
(T
†BPTA + T
†APTB
)] }= trt
{ϱt M
}
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Central Result: Connection of Pattern and Witness
⇒ designated entanglement witness:
reading offinterference contribution
at φ = 0⇔
measuringexp. value of observable
on the target system
δI (0) = trt
{ϱt trp
[ϱp
(T
†BPTA + T
†APTB
)] }= trt
{ϱt M
}
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result
Central Result: Connection of Pattern and Witness
⇒ designated entanglement witness:
reading offinterference contribution
at φ = 0⇔
measuringexp. value of observable
on the target system
δI (0) = trt
{ϱt trp
[ϱp
(T
†BPTA + T
†APTB
)] }= trt
{ϱt M
}
⇒ criterion:
δI (0)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
> 0 :
I0+δI
φ0
no detection
< 0 : ϱt entangled!
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario
Quantum Optics Realization of W−
! single probe photon
! elastic single scattering
! 2 atoms carry target qubit pair
! unpolarized incident beam,no polarizer (P = 1)
! for detection at right angles to incidence (k ⊥ k′):
M = 2 (1− 2|Ψ−⟩⟨Ψ−|)
witnesses the singlet state |Ψ−⟩ = 1√2(| , ⟩ − | , ⟩)
! works for all entangled states |Ψ⟩ with |⟨Ψ|Ψ−⟩|2 > 12
1
2
A
k
k′
B
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario
Discussion
! single-particle interference is sensitive for entanglement
! mapping of a two-particle observable to a single-particle observable
! proof-of-concept model found in quantum optics
! arXiv:0710.0825:
1
2
A
B
A
B
AB
AB
1
2 1
2
A A
B B
Φ
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement
Introduction Proposal Quantum Optics Scenario
Thank You for Your Attention!
[email protected] Single-Particle Interference Can Witness Bipartite Entanglement