1.
Manipulating Continuous Variable Photonic Entanglement
Martin Plenio
Imperial College London
Institute for Mathematical Sciences
&
Department of Physics
Imperial College London Krynica, 15th June 2005
Sponsored by:
Royal Society Senior Research Fellowship
2.
Local preparation
A BEntangled state between distant sites
The vision . . .
Prepare and distribute pure-state entanglement
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3.
. . . and the reality
A B
Weakly entangled state
Noisy channel
Local preparation
Decoherence will degrade entanglement
Can Alice and Bob ‘repair’ the damaged entanglement?
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They are restricted to Local Operations and Classical Communication
4.
The three basic questions of a theory of entanglement
decide which states are entangled and which are disentangled
(Characterize)
decide which LOCC entanglement manipulations are possible
and provide the protocols to implement them
(Manipulate)
decide how much entanglement is in a state and how efficient
entanglement manipulations can be
(Quantify)
Provide efficient methods to
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5.
Practically motivated entanglement theory
Theory of entanglement is usually purely abstract
For example: accessibility of all QM allowed operations
Doesn’t match experimental reality very well!
All results assume availability of unlimited experimental resources
Develop theory of entanglement under
experimentally accessible operations
BUT
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6.
Consider n harmonic oscillators
nn PXPXPX ,,, 2211 •••
Canonical coordinates ),,...,,(),,...,,( 1121221 nnnn PXPX=− OOOO
Basics of continuous-variable systems
•••
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7.
canonical commutation relations
where is a real 2n x 2n matrix is the symplectic matrixσ
Lets go quantum
Harmonic oscillators, light modes or cold atom gases.
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8.
Characteristic function (Fourier transform of Wigner function)
Characteristic function
Simplest example: Vacuum state = Gaussian function
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9.
A state is called Gaussian, if and
only if its characteristic function (or its
Wigner function) is a Gaussian
Gaussian states are completely
determined by their first and second
moments
Are the states that can be made
experimentally with current technology
(see in a moment)
Arbitrary CV states too general: Restrict to Gaussian states
coherent states
squeezed states
(one and two modes)
thermal states
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10.
First moments (local displacements in phase space):
First Moments
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Local displacement Local displacement
11.
The covariance matrix embodies the second moments
Heisenberg uncertainty principle
Uncertainty Relations
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γ represents a physical Gaussian state
iff the uncertainty relations are satisfied.
12.
CV entanglement of Gaussian states
Separability + Distillability
Necessary and sufficient criterion known for M x N systems
Simon, PRL 84, 2726 (2000); Duan, Giedke, Cirac Zoller, PRL 84, 2722 (2000); Werner and Wolf, PRL 86, 3658 (2001);
G. Giedke, Fortschr. Phys. 49, 973 (2001)
These statements concern Gaussian states, but assume the
availability of all possible operations (even very hard ones).
Develop theory of what you can and cannot do under Gaussian
entanglement under Gaussian operations.
Programme:
Inconsistent:With general operations one can make any state
Impractical: Experimentally, cannot access all operations
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13.
Characterization of Gaussian operations
For all general Gaussian operations, a ‘dictionary’
would be helpful that links the
physical manipulation that can be done in an
experiment to
the mathematical transformation law
J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002)
J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89, 097901 (2002)
J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89, 137902 (2002)
G. Giedke and J.I. Cirac, Phys. Rev. A 66, 032316 (2002)
B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math. Phys. 2, 161 (1977)
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14.
In a quantum optical setting
Application of linear optical elements:
Beam splitters
Phase plates
Squeezers
Gaussian operations can be implemented ‘easily’!
Measurements:
Homodyne measurements
Addition of vacuum modes
Gaussian operations: Map any Gaussian state to a Gaussian state
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15.
Characterization of Gaussian operations
Optical elements
and additional
field modes
Vacuum detection Homodyne
measurement
γ=
C1 C3
C3
T
C2
γa AγA
T
+G γa C1−C3(C2+1)−1
C3
T T
CCC 3
1
21 )( −
− ππγ a
)0,1,...,0,1(diag=π
G+iσ−iA
T
σA≥0
Transformation: Transformation: Transformation:
with where
γ=
C1 C3
C3
T
C2+1
Schur complement of
G
A
real, symmetric
real
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16.
Gaussian manipulation of entanglement
What quantum state transformations can be
implemented under Gaussian local operations?
ρ
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17.
Gaussian manipulation of entanglement
Apply Gaussian LOCC to the initial state ρ
ρ
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18.
Gaussian manipulation of entanglement
Can one reach ρ’, ie is there a Gaussian LOCC map such that
?
ρ'
E(ρ)=ρ'
E
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ρ
19.
Normal form for pure state entanglement
A B A B
r1
r2
rN
Gaussian local
unitary
G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)
A. Botero and B. Reznik, Phys. Rev. A 67, 052311 (2003)
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20.
The general theorem
Necessary and sufficient condition for the transformation of pure
Gaussian states under Gaussian local operations (GLOCC):
under GLOCC
if and only if (componentwise)r≥r'
G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)
A B
r1
r2
rN
A B
1'r
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2'r
Nr'
21.
The general theorem
Necessary and sufficient condition for the transformation of pure
Gaussian states under Gaussian local operations (GLOCC):
under GLOCC
if and only if (componentwise)r≥r'
G. Giedke, J. Eisert, J.I. Cirac, and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)
A B
r1
r2
rN
A B
11 'rr ≥
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22 'rr ≥
NN rr '≥
22.
Comparison
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General LOCC
r1
r2
01 =r
2
'
2 rr >
Gaussian LOCC
r1
r2
01 =r
2
'
2 rr >
G. Giedke, J. Eisert, J.I. Cirac and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)
23.
Comparison
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General LOCC
r1
r2
01 =r
2
'
2 rr >
Gaussian LOCC
r1
r2
01 =r
2
'
2 rr >
G. Giedke, J. Eisert, J.I. Cirac and M.B. Plenio, Quant. Inf. Comp. 3, 211 (2003)
Cannot compress Gaussian pure state entanglement with Gaussian operations !
24.
A1 B1
A2 B2
Homodyne
measurements
General local unitary Gaussian
operations (any array of beam
splitters, phase shifts and squeezers)
Symmetric
Gaussian two-mode
states ρ
Characterised by 20 real numbers
When can the degree of entanglement be increased?
Gaussian entanglement distillation on mixed states
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25.
Gaussian entanglement distillation on mixed states
The optimal iterative Gaussian distillation protocol
can be identified:
Do nothing at all (then at least no entanglement is lost)!
Subsequently it was shown that even for the most general
scheme with N-copy Gaussian inputs the best is to do nothing
Challenge for the preparation of entangled Gaussian
states over large distances as there are no quantum
repeaters based on Gaussian operations (cryptography).
G. Giedke and J.I. Cirac, Phys. Rev. A 66, 032316 (2002)
J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002)
Krynica, 15th June 2005Imperial College London
26.
Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step:
non-Gaussian state
(Gaussian) mixed states
Transmission through noisy channel
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27.
Procrustean Approach
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PD
PD
Yes/No detector
28.
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Simple protocol to generate non-Gaussian states of higher
entanglement from a weakly squeezed 2-mode squeezed state.
If both detector click – keep the state.
If |q| ¿1 the remaining state has essentially the form:
Choose transmittivity T of the beam splitter to get desired λ.
29.
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Probability of Success depends on q and T:
• Example:
– Initial supply with q = 0.01
Entanglement Success Probability
30.
Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step:
non-Gaussian state
(Gaussian) mixed states
Transmission through noisy channel
(Gaussian) two-mode squeezed states
Imperial College London
Theory: DE Browne, J Eisert, S Scheel, MB Plenio
Phys. Rev. A 67, 062320 (2003);
J Eisert, DE Browne, S Scheel, MB Plenio, Annals
of Physics NY 311, 431 (2004)
Iterative Gaussifier (Gaussian operations)
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31.
Gaussification
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A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
32.
Procrustean Approach
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A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
Can prove that this converges to a Gaussian state for |α0| > |α1|
The Gaussian state to which it converges is the two-mode
squeezed state with q= α1/α0.
For rigorous proof see
Browne, Eisert, Scheel, Plenio Phys. Rev. A 67, 062320 (2003);
Eisert, Browne, Scheel, Plenio, Annals of Physics NY 311, 431 (2004)
33.
Procrustean Approach
Imperial College London Krynica, 15th June 2005
Initial Supply
Procrustean Step
Gaussification
Final State
34.
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Example:
Entanglement Fidelity Probability
Initial state 0.0015 0.805
Procrustean
(T=0.017)
0.82 0.932 0.0004
Gaussification 1 0.97 0.933 0.75
2 1.11 0.967 0.74
3 1.24 0.987 0.71
4 1.33 0.996 0.69
35.
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Example:
Probability Fidelity w.r.t. Gaussian target state
36.
Finite Detector Efficiency
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Entanglement Mixedness
1-Tr[ρ2
]
log. neg.
1
2
NG 1
2
Input: Weakly entangled two-mode squeezed state (logneg <0.1)
Non-Gaussian step
Two Gaussification steps
Plot resulting entanglement and mixedness versus detector efficiency
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37.
Improving the Procrustean Step
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Source
T
Fibre-loop detector with loss
39.
Detector Efficiency
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fi
λ
40.
Photon Number Resolution
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EntanglementIncrease
0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
λ
Number
of loops
Conditioned on two photons
41.
Summary
Imperial College London Krynica, 15th June 2005
• Gaussian operations on Gaussian states cannot
distill entanglement
• Single non-Gaussian step allows for subsequent
distillation by Gaussian operations
• Fibre loop detector based schemes robust against
against finite detector efficiencies and low number
resolution.
• Robustness suggests experimental feasibility
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