single-particle interference can witness bipartite ... · ⇒ designated entanglement witness:...

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


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



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)


Introduction Proposal Quantum Optics Scenario Implementing the Idea Central Result

Probe-Target Interactionpr

epar

atio

n

ϱ = ⊗ ϱtϱp

targetprobe

1 2




epar

atio

n

ϱ = ⊗ ϱtϱp

targetprobe

1 2

inte

ract

ion

ϱ′ =

T = eiφATA + e

iφBTB φ = φA − φB

T

( )

T †




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?



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

}






at φ = 0⇔



δI (0) = trp&t

{(ϱp ⊗ ϱt)

(T

†BPTA + T

†APTB

)}= trt

{ϱt M

}






at φ = 0⇔



δI (0) = trt

{ϱt trp

[ϱp

(T

†BPTA + T

†APTB

)] }= trt

{ϱt M

}






at φ = 0⇔



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!


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



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



Thank You for Your Attention!


single-particle interference can witness bipartite ... · ⇒ designated entanglement witness:...

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