Quantum teleportation in a nutshell - Max Planck Society · Quantum teleportation in a nutshell Fabian Kössel Quantum Teleportation Protocol 1. Alice shares with Bob an entangled

Quantum teleportation in a nutshell Fabian Kössel

Quantum teleportation in a nutshell

Fabian Kössel

Technische Universität München and MPQ

June 12th, 2013

Motivation Theory Experiment Summary References


Why Quantum Teleportation?

Transfer of a quantum state or QuBit from one position (Alice) to another

(Bob).

I Useful in Quantum information, Quantum cryptography,. . .

I It’s not about transportation or dis- and reassembling of matter.

Sorry, Trekkies!



Recap: The QuBit

|ψ〉 = cos(θ2

)|0〉+ eiϕ sin

(θ2

)|1〉

QuBit on bloch sphere



How to transport a QuBit?

1.

Classically: Copy bit and transfer it, hence clone it.

I Doesn’t work for QuBit. Can’t retreive complete

information of state.

2.Physically transport QuBit, i.e. carry it from A to B.

I Lossy. Short coherence times of quantum state.

3. Quantum Teleportation




1.










1.









Basic scheme of quantum teleportation

EPR

BSM

2 3

entangled1

|ψin〉1

classical information

Alice

Bob



Recap: Entanglement and Bell basis

EPR

BSM

2 3

entangled1

|ψin〉1


Alice

Bob



Recap: Entanglement and Bell basis

I Strong correlation between quanta that is unique in quantum

mechanics

I A special entangled basis of two-QuBit-system: Bell basis

|ψ±〉 = 1√2(|0〉 |1〉 ± |1〉 |0〉)

|φ±〉 = 1√2(|0〉 |0〉 ± |1〉 |1〉)



Bell State Measurement (BSM)

EPR

BSM

2 3

entangled1

|ψin〉1


Alice

Bob



Bell State Measurement (BSM)

Joined measurement of two incoming particles.

1

2

BSM2 bits

|ψ−〉 , |ψ+〉 , |φ−〉 or |φ+〉

I projection onto one of four Bell states

I thus 2 bits are needed to express outcome

I destroys incoming QuBits



Quantum Teleportation Protocol

EPR

BSM

2 3

entangled1

|ψin〉1


Alice

Bob




1. Alice shares with Bob an entangled pair

of QuBits |ψ−〉2,3.

2. Alice performs a joined BSM on her own

initial state and her own QuBit of the EPR

pair and detects on which of the four Bell

state the incoming QuBits were

projected.

3. Alice sends this information from the

BSM to Bob.

EPR

BSM

23

entangled1

|ψin〉1


Alice

Bob

1.

2.3.










projected.


BSM to Bob.

EPR

BSM

23

entangled1

|ψin〉1


Alice

Bob

1.

2.

3.










projected.


BSM to Bob.EPR

BSM

23

entangled1

|ψin〉1


Alice

Bob

1.

2.3.




Three QuBit-System of incoming state |ψin〉1 and entangled |ψ−〉2,3 EPR pair.

First rewrite it (ignoring normalizations). . .

|ψ〉1,2,3 = |ψin〉1 |ψ−〉2,3

= |ψin〉1 |0〉2 |1〉3 − |ψin〉1 |1〉2 |0〉3

... → change to Bell basis of1

and2

= |ψ−〉1,2 U1 |ψin〉3

+ |ψ+〉1,2 U2 |ψin〉3

+ |φ−〉1,2 U3 |ψin〉3

+ |φ+〉1,2 U4 |ψin〉3

〈ψ+|1,2 U2 |ψin〉3




Three QuBit-System of incoming state |ψin〉1 and entangled |ψ−〉2,3 EPR pair.

First rewrite it (ignoring normalizations). . .

|ψ〉1,2,3 = |ψin〉1 |ψ−〉2,3

= |ψin〉1 |0〉2 |1〉3 − |ψin〉1 |1〉2 |0〉3

... → change to Bell basis of1

and2

= |ψ−〉1,2 U1 |ψin〉3

+ |ψ+〉1,2 U2 |ψin〉3

+ |φ−〉1,2 U3 |ψin〉3

+ |φ+〉1,2 U4 |ψin〉3

〈ψ+|1,2 U2 |ψin〉3










projected.


BSM to Bob.

4. Bob applies one of four unitary

transformations Ui on his now collapsed

QuBit from the EPR pair and receives

Alice’s initial state.

EPR

BSM

23

entangled1

|ψin〉1


Alice

Bob

1.

2.3.

4.










projected.


BSM to Bob.

4. Bob applies one of four unitary

transformations Ui on his now collapsed

QuBit from the EPR pair and receives

Alice’s initial state.

EPR

BSM

23

entangled1

|ψin〉1


Alice

Bob

1.

2.3. 4.



global phase shift

|ψ−〉 :

U1 =

−

1 0

0 1

phase shift

|ψ+〉 :

U2 =−1 0

0 1

spin flip

|φ−〉 :

U3 =0 1

1 0

phase shift + spin flip

|φ+〉 :

U4 =0 −1

1 0



Information

Both information channels are needed to reconstruct state!

unita

rytr

ansf

or

matio

nUi , i = 1, . . . ,4

Alice

BobI Classical information is not enough!

I You need a state on which you can

apply the unitary transformations.

I Otherwise one measurement would

be enough to fully characterize

state.



Information

Both information channels are needed to reconstruct state!

I Quantum channel is not enough!

I Ensemble is in a perfectly mixed

state. Need information about right

transformation.

I Otherwise causality would be

violated and instant transfer of

information would be possible.

∑i Ui |ψin〉



Information

Both information channels are needed to reconstruct state!un

itary

tran

sfor

matio

nUi , i = 1, . . . ,4

Alice

Bob

∑i Ui |ψin〉



First experimental realisation – Setup

EPR

BSM

2 3

entangled1

|ψin〉1


Alice

Bob



First experimental realisation – Setup

EPR

Alice Bob

trigger

f1

f2

d1

d2

UV pulse

beam splitter

polarizing beam splitter

inital state preparation

EPR-Source

1 2 3

entangled

Weinfurter, Zeilinger 1997, Nature



EPR-Source

EPR

trigger

f1

f2

d1

d2

UV pulse

beam splitter



EPR-Source

1 2 3

entangled




EPR-Source

Spontaneous parametric down-conversion, Wikipedia



BSM with a beam splitter

Alice

trigger

f1

f2

d1

d2

UV pulse

beam splitter



EPR-Source

1 2 3

entangled




BSM with a beam splitter

a

b

|Ψ 〉 = |ψ〉inner · |χ〉spatial

With photons (bosons) the state has to be symmetric under

exchange of particles.

|ψ〉inner |χ〉spatial

|ψ−〉 = 1√2(|0〉1 |1〉2 − |1〉1 |0〉2) · 1√

2(|a〉1 |b〉2 − |b〉1 |a〉2)

|ψ+〉 = 1√2(|0〉1 |1〉2 + |1〉1 |0〉2) · 1√

2(|a〉1 |b〉2 + |b〉1 |a〉2)

|φ−〉 = 1√2(|0〉1 |0〉2 − |1〉1 |1〉2) · 1√

2(|a〉1 |b〉2 + |b〉1 |a〉2)

|φ+〉 = 1√2(|0〉1 |0〉2 + |1〉1 |1〉2) · 1√

2(|a〉1 |b〉2 + |b〉1 |a〉2)

spatialantisymmetric

spatial symmetric

Hong-Ou-Mandel



Results

EPR

Alice Bob

trigger

f1

f2

d1

d2

UV pulse

beam splitter



EPR-Source

1 2 3

entangled




Results

I Three-fold coincidence (click) at f1, f2 (→ |ψ−〉1,2) and output port d1 or

d2 corresponding to the input polarisation.

For example:

|ψ〉in = |1〉 Telep.PBS

8|0〉 no click!

3|1〉 click!



Results

Checking teleporation for a complete basis and linear combinations.

Polarization Visibility

+45° 0.63± 0.02

-45° 0.64± 0.02

0° 0.66± 0.02

90° 0.61± 0.02

circular 0.57± 0.02

After substracting background noise a visibility of approximately 70%± 3%

was achieved.



Results

BUT protocoll isn’t fully implemented. Only discriminates |ψ−〉 from the rest at BSM.

→ By now there are more sophisticated experiments.

Experiment What? Remark

Photons 1997 1 photon via photon to photonworks at best in 25% of cases, very

long distances

Cavity 2013 2 atom via photon to atomhigher efficiency due to cavity, long

distance

Ions 2004 3 trapped ions full protocol, short distances

1Weinfurter, Zeilinger, et al. , 1997, Nature2Rempe, Ritter, et al., 2013, APS3Ozeri, Wineland, et al., 2004, Letters to Nature



Summary

EPR

BSM

2 3

entangled1

|ψin〉1


Alice

Bob

Quantum teleportation makes it possible to reliably transfer a QuBit from on place to another.

By taking advantage of entanglement, first only expandable messenger particles have to be

exchanged through a (lossy) channel. And only, when this has worked, the relevant

information is transferred. Thus minimizing transfer losses of QuBit.



Literature

M. D. Barret et al. “Deterministic quantum teleportation of atomic qubits”. In: Letters to

Nature (2004).

Charles H. Bennet et al. “Teleporting an Unkown Quantum State via Dual Classical and

Einstein-Podolsky-Rosen Channels”. In: Physical Review Letters (1993).

Dik Bouwmeester et al. “Experimental quantum teleportation”. In: Nature 390 (1997).

Samuel L. Braunstein and A. Mann. “Measurement of the Bell operator and quantum

teleportation”. In: The American Physical Society (1995).

Richard A. Campos, Bahaa E. A. Saleh, and Malvin C. Teich. “Teleporting an Unkown

Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels”. In: The

American Physical Society (1989).

Christian Noelleke et al. “Efficient Teleportation Between Remote Single-Atom

Quantum Memories”. In: The American Physical Society (2013).


Documents

Quantum teleportation in a nutshell - Max Planck Society · Quantum teleportation in a nutshell Fabian Kössel Quantum Teleportation Protocol 1. Alice shares with Bob an entangled