Entanglement Purification - Quantum Optics and Quantum Information

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The maximally entangled state will be in some mixed state with a certain entanglement ... Quantum Entanglement Purification for Quantum Communication.
Lecture Note 4

Entanglement Purification Jian-Wei Pan

Introduction(1)

Both long distance quantum teleportation or global quantum key distribution need to distribute a certain supply of pairs of particles in a maximally entangled state to two distant users.

Introduction(2) However, distributed qubits will interact with the environment and decoherence will happen. (t ) 0 E ⎯U⎯ ⎯ → 0 E0 (t )

(t ) 1 E ⎯U⎯ ⎯ → 1 E1 (t )

Here 0 , 1 represents the qubit state and E represents the environment initial state, U (t ) is the joint unitary time evolution operator. For arbitrary qubit state: (t ) (α 0 0 + α1 1 ) E ⎯U⎯ ⎯ → α 0 0 E0 (t ) + α1 1 E1 (t )

ρ q (t ) = TrE ρ q + E

2 ⎡ α0 =⎢ ⎢⎣α1α 0* E0 E1

α 0α1* E1 E0 ⎤ α1

2

⎥ ⎥⎦

The off-diagonal element of the qubit density matrix will drop down with the − Γt rate E0 (t ) E1 (t ) = e , Γ(t ) depends on the coupling between qubit and environment. The maximally entangled state will be in some mixed state with a certain entanglement fidelity due to the process.

Introduction(3) Solution to the decoherence ¾ Quantum Error Correction for Quantum computation

¾ Quantum Entanglement Purification for Quantum Communication ¾ Quantum Communication based on Decoherence Free Subspace

The basic idea of entanglement purification is to extract from a large ensemble of lowfidelity EPR pairs a small sub-ensemble with sufficiently high fidelity EPR pair. ¾ Entanglement Purification----improve purity of any kind of unknown mixed state ¾ Local filtering----improve entanglement quality of known state ¾ Entanglement Concentration---- improve entanglement quality for pure unknown state

Principle of Entanglement Purification Model: Suppose Alice want to share an ensembles of 2-qubit maximally entangled − states Ψ = ( H V − V H ) / 2 with Bob via a noise channel. After the transmission, the state has been changed into a general mixed state M, the purity of M can be expressed as F = Ψ − M Ψ − .

Several ingredients in the Entanglement Purification: (1) Bell states: Ψ ± =

1 1 ( 0 0 ± 1 1 ); ( 0 1 ± 1 0 ) , Φ± = 2 2

(2) Werner state: WF = F Ψ − Ψ − +

1− F + 1− F + 1− F − Ψ Ψ+ + Φ Φ+ + Φ Φ− 3 3 3

(3) Pauli rotation: σ x , σ y , σ z; (4) CNOT gate:

x y → x x⊕ y

0 0 → 0 0 0 1 → 0 1

,

1 0 →1 1 1 1 →1 0

;

Principle of Entanglement Purification Steps of Entanglement Purification: (1) Random Bilateral Pauli Rotation on each photon in the states. This step can change arbitrary mixed state into Werner state: WF = F Ψ − Ψ − +

1− F + 1− F + 1− F − Ψ Ψ+ + Φ Φ+ + Φ Φ− 3 3 3

(2) A Unilateral σ y Rotations converting the states from mostly Ψ Werner states to the analogous mostly Φ + states, ( σ y maps Ψ ± → Φ µ ) , (3) Bilateral CNOT operations on two photon pairs in the Werner state.

[C. Bennett, et al., PRL 76, 772 (1996)]



Bilateral CNOT operations will convert the Bell states as the form:

For example: F(1-F)/3 Probability, we have CNOT ( Ψ +

S

Φ+

Ψ+

S

Φ+

T

1 ) = CNOT [( 0 SA 1 SB + 1 SA 0 SB )( 0 TA 0 TB + 1 TA 1 TB )] T 2 1 = CNOT ( 0 SA 0 TA 1 SB 0 TB + 0 SA 1 TA 1 SB 1 TB + 1 SA 0 TA 0 SB 0 TB + 1 SA 1 TA 0 SB 1 TB ) 2 1 = ( 0 SA 0 TA 1 SB 1 TB + 0 SA 1 TA 1 SB 0 TB + 1 SA 1 TA 0 SB 0 TB + 1 SA 0 TA 0 SB 1 TB ) 2 = Ψ+ Ψ+ S

T

Principle of Entanglement Purification (4) Measuring the target pair in z basis, if the result is parallel, keep the source

pair, if not, discard the source pairs. After the protocol, the purity of the source pair has been improved:

If F > 1 / 2 , then F ′ > F

Via several this kind processes, we can purify a general mixed state into a highly entangled state. [C. Bennett, et al., PRL 76, 772 (1996)]

Experiment of Entanglement Purification • •

The theoretical proposal needs a C-Not gate which requires high non-linearities. Unluckily, no efficient C-Not gate exists at the moment A better solution for experiments

[J.-W. Pan et al., Nature 410, 1067 (2001)]

Initial State:

ρ ab = F | Φ + 〉 ab 〈Φ + | + (1 − F ) | Ψ + 〉 ab 〈 Ψ + | ′ = F ′ | Φ + 〉 ab 〈Φ + | +(1 − F ′) | Ψ + 〉 ab 〈 Ψ + | ρ ab

Purified State:

F2 F′ = 2 > F (if F > 1 / 2) 2 F + (1 − F ) +

+

+

+

+

+

+

+

F2

| Φ s 〉 ab ⋅ | Φ t 〉 ab

F (1 − F )

| Φ s 〉 ab ⋅ | Ψt 〉 ab

(1 − F ) F

| Ψs 〉 ab ⋅ | Φ t 〉 ab

(1 − F )2

| Ψs 〉 ab ⋅ | Ψt 〉 ab

For the first case,

| Φ + 〉 a1b1⋅ | Φ + 〉 a 2b 2 =

1 2

(| H 〉 a1 | H 〉 b1 + | V 〉 a1 | V 〉 b1 )

⋅ (| H 〉 a 2 | H 〉 b 2 + | V 〉 a 2 | V 〉 b 2 ) H a1 H a 2 H b1H b 2

H a1Va 2 H b1Vb 2

Va1Va 2V b1Vb 2

Va1H a 2V b1H b 2

Four-fold events

No four-fold events

50% 1 (| H 〉 a 3 | H 〉 a 4 | H 〉 b3 | H 〉 b 4 + | V 〉 a 3 | V 〉 a 4 | V 〉 b 3 | V 〉 b 4 ) 2 probability of F 2 / 2

| Φ + > a 3b 3 =

1 (| H > a 3 | H > b3 + | V > a 3 | V > b 3 ) 2

Similarly,

| Ψ + 〉 a1b1⋅ | Ψ + 〉 a 2 b 2

| Φ + 〉 a1b1⋅ | Ψ + 〉 a 2b 2 | Ψ + 〉 a1b1⋅ | Φ + 〉 a 2b 2

50%

No four-fold events

1 (| H 〉 a 3 | H 〉 a 4 | V 〉 b 3 | V 〉 b 4 + | V 〉 a 3 | V 〉 a 4 | H 〉 b3 | H 〉 2

b4

)

probability of (1 − F ) / 2 2

| Ψ + 〉 a 3b 3 =

1 2

(| H 〉 a 3 | V 〉 b3 + | V 〉 a 3 | H 〉 b3 )

In this way…

′ = F ′ | Φ + 〉 ab 〈 Φ + | + (1 − F ′) | Ψ + 〉 ab 〈 Ψ + | ρ ab F2 F′= 2 > F (if F > 1 / 2) 2 F + (1 − F )

Experimental

Realization

[J.-W. Pan et al., Nature 423, 417 (2003)]

Four-fold contribution from double pairs emission

a1b1 contribution

V a 3 H a 4V b 3 H b 4

a2b2 contribution

H a 3Va 4 H b 3Vb 4

| Φ + > a 3b 3 =

1 (| H > a 3 | H > b 3 + | V > a 3 | V > b 3 ) 2

[J.-W. Pan et al., Nature 423, 417 (2003)]

To keep the phase stable, we use the polarization-spatial entanglement

(H

H +V V

)( a

1

7000 6000 5000 4000 3000 2000 1000 0 -150

)( a

3 Two-fold coincidence per 5 seconds

Two-fold coincidence per second

→ ( H H + eiφ V V

b1 + eiφ a2 b2

-100

-50

0

50

Time delay (µm)

100

150

)

b3 + a4 b4

)

35000 30000 25000 20000 15000 10000 5000 0 0

500

1000

1500

Piezo position (nm)

2000

2500

Experimental Result Before purification, F=3/4 0.5

|H>|H>

o

0.4 |+45o>|+45o>

|V>|V> Fraction

Fraction

0.4

0.5

0.3

0.2

o

|-45 >|-45 >

0.3

0.2 o

o

|+45 >|-45 >

|V>|H>

|H>|V> 0.1

0.1

0.0

0.0

o

o

|-45 >|+45 >

After purification, F=13/14 0.5

|H>|H>

0.5

|V>|V>

o

o

o

|-45 >|-45 >

0.4

Fraction

Fraction

0.4

o

|+45 >|+45 >

0.3

0.2

0.3

0.2

0.1

0.1

|H>|V>

o

0.0

[J.-W. Pan et al., Nature 423, 417 (2003)]

o

|+45 >|-45 >

|V>|H> 0.0

o

o

|-45 >|+45 >

Local filtering (Procrustean protocol) •

The purification protocol is such a waste for the photon pair sources.



When we have known some information of the state, there is some more efficient method to improve the purity.



For example, Φ ε = ( 00 + ε 11 ) / 1 + ε 2 ( ε ≠ 1) is non-maximally entangled state, can be converted into the maximally entangled + state Φ = ( 00 + 11 ) / 2 by subjecting one of the qubits to a generalized measurement filtering process: 0 → ε 0 , 1 → 1 . This process is called local filtering. It can only be used in known state and can only improve the entanglement degree.

[P. Kwiat et al., Nature 409, 1014 (2003)]

Local filtering •

Any inseparable two spin-1/2 system Matrices Can Be Distilled to a Singlet Form with local filtering and entanglement purification



A quantum system is called inseparable if its density matrix cannot be written as a mixture of product states:

[M. Horodecki, et al., PRL 78, 574 (1997)]

[P. Kwiat et al., Nature 409, 1014 (2003)]

Local Filtering and Hidden Non-locality •

(1 − λ )

For the state ρε ,λ = λ Φ ε Φ ε + 2 ( HV HV + VH VH ) sometimes it can not violate Bell inequality. After the Local Filtering process, the state 2 2ε λ + ε (1 − λ ) + Φ Φ + ( HV HV + VH VH ) is changed into the state 1 + ε 2 2 which can violate the Bell inequality.

[P. Kwiat et al., Nature 409, 1014 (2003)]

Entanglement Concentration---Scheme ψ

T



12

⊗ψ

34

= (αH1V2 + β V1 H 2 ) ⊗ (αH 3V4 + βV3 H 4 )

R90 (αH1V2 + βV1H 2 ) ⊗ (αV3 H 4 + βH 3V4 ) ⎯⎯→

1 (H1V2 H 3V4 + V1H 2V3 H 4 ) 2 + α 2 H1V2V3 H 4 + β 2V1 H 2 H 3V4 = 2αβ .

2αβ . ⎯PBS ⎯→ ⎯

1 (H1V2 H 3V4 + V1H 2V3 H 4 ) 2

[C. H. Bennett et al., Phys. Rev. A 53, 2046 (1996)] [Z. Zhao et al., Phys. Rev. A64, 014301 (2001)] [T. Yamamoto et al., Phys. Rev. A64, 012304 (2001)]

Experiment Realization

[Z. Zhao et al, Phys. Rev. Lett. 90, 207901(2003).] [T. Yamamoto et al., Nature 421, 343 (2003)]

Difficulties in Long-distance Quantum Communication Due to the noisy quantum channel (1) absorption

photon loss

(2) decoherence

degrading entanglement quality

Solution to problem (1): Entanglement swapping ! [N. Gisin et al., Rev. Mod. Phys. 74, 145 (2002)]

Solution to problem (2): Entanglement purification! [C. H. Bennett et al., Phys. Rev. Lett. 76, 722 (1996)] [D. Deutsch et al., Phys. Rev. Lett. 77, 2818 (1996)]

The Kernel Device for Long Distance Quantum Communication Quantum repeaters:

[H. Briegel et al., Phys. Rev. Lett. 81, 5932(1998)]

Require • entanglement swapping with high precision • entanglement purification with high precision • quantum memory

A proof-in-principle demonstration of a quantum repeater

[Z. Zhao et al, Phys. Rev. Lett. 90, 207901(2003).]

Results for Repeater

V1 = 0.83 ± 0.04

V 2 = 0 .80 ± 0 .05

V 2 = 0 . 80 ± 0 . 04

S = 2.58 ± 0.07

S = 2 .43 ± 0 .08

S = 2 . 42 ± 0 . 08

8.3

5 .4

5 .4

Standard Deviation

Standard Deviation

Standard Deviation

Fidelity − 0.96 ± 0.04 − − − −0.93 ± 0.04 − − − − − −0.93 ± 0.04 [Z. Zhao et al, Phys. Rev. Lett. 90, 207901(2003).]

Drawback in Former Experiments • Absence of quantum memory • Probabilistic entangled photon source • Probabilistic entanglement purification • Huge photon loss in fiber Free-space entanglement distribution - we are working on 20km and 500km scale…

• Synchronization of independent lasers - we are working on entanglement swapping… [T. Yang et al., PRL 96, 110501 (2006)]

Quantum memory

Drawback in Former Experiments

P

P

2 /P 2 • In multi-stage experiments, the cost of resource is proportional to N/P N thus not scalable • If one knows when the photon pair is created and the entangled pair can be stored as demanded, the total cost is then

N/P

Another Solution---Quantum Communication based on Decoherence free Subspace • •

For special noise, we can utilize some entanglement subspace to directly implement quantum communication. For example, the phase flip error channel: iφ 0 → 0 , 1 →e 1



The Bell state ψ

+





are immune to the noise:

( 0 1 + 1 0 ) / 2 → ( 0 e iΦ 1 + eiφ 1 0 ) / 2 = e iΦ ( 0 1 + 1 0 ) / 2 ( 0 1 − 1 0 ) / 2 → ( 0 eiΦ 1 − e iφ 1 0 ) / 2 = e iΦ ( 0 1 − 1 0 ) / 2 •

We can take 0 = Ψ + and 1 = Ψ − , Each state combined by the two basis can be used for decoherence free communication. The subspace is called decoherence free subspace. [Q. Zhang et al., PRA 73. 030201(R) (2006)] [T.-Y. Chen et al., PRL 96, 150504 (2006)]