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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
=ψ
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
+
,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)]