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Manipulating Continuous Variable Photonic Entanglement
Martin Plenio
Imperial College LondonInstitute for Mathematical Sciences
&Department of Physics
Imperial College London Krynica, 15th June 2005
Sponsored by:Royal Society Senior Research Fellowship
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Local preparation
A BEntangled state between distant sites
The vision . . .
Prepare and distribute pure-state entanglement
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. . . and the reality
A BWeakly 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
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The three basic questions of a theory of entanglement
decide which states are entangled and which are disentangled (Characterize)
Provide efficient methods to
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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)
Provide efficient methods to
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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
Krynica, 15th June 2005Imperial College London
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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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Consider n harmonic oscillators
nn PXPXPX , , , 2211
Canonical coordinates ),,...,,(),,...,,( 1121221 nnnn PXPX OOOO
Basics of continuous-variable systems
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Lets go quantum
Harmonic oscillators, light modes or cold atom gases.
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canonical commutation relations
where is a real 2n x 2n matrix is the symplectic matrix
Lets go quantum
Harmonic oscillators, light modes or cold atom gases.
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Characteristic function (Fourier transform of Wigner function)
Characteristic function
Simplest example: Vacuum state = Gaussian function
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A state is called Gaussian, if and only if its characteristic function (or its Wigner function) is a Gaussian
Arbitrary CV states too general: Restrict to Gaussian states
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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
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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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First moments (local displacements in phase space):
First Moments
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Local displacement Local displacement
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The covariance matrix embodies the second moments
Heisenberg uncertainty principle
Uncertainty Relations
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represents a physical Gaussian state iff the uncertainty relations are satisfied.
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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).
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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).
Inconsistent:With general operations one can make any stateImpractical: Experimentally, cannot access all operations
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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 stateImpractical: Experimentally, cannot access all operations
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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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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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Characterization of Gaussian operations
Optical elements and additional field modes
Vacuum detection Homodyne measurement
C1 C3
C3T C2
AAT G
C1 C3(C2 1) 1C3T TCCC 3
121 )(
)0,1,...,0,1(diag
G i iAT A 0
Transformation: Transformation: Transformation:
with where
C1 C3
C3T C2 1
Schur complement of
G
Areal, symmetricreal
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Gaussian manipulation of entanglement
What quantum state transformations can be implemented under Gaussian local operations?
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Gaussian manipulation of entanglement
Apply Gaussian LOCC to the initial state
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Gaussian manipulation of entanglement
Can one reach ’, ie is there a Gaussian LOCC map such that
?
'
E () '
E
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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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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 '
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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 '
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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)
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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 !
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A1 B1
A2 B2
Homodyne measurements
General local unitary Gaussianoperations (any array of beam splitters, phase shifts and squeezers)
SymmetricGaussian two-modestates
Characterised by 20 real numbers When can the degree of entanglement be increased?
Gaussian entanglement distillation on mixed states
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Gaussian entanglement distillation on mixed states
The optimal iterative Gaussian distillation protocol can be identified:
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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)!
J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev. Lett. 89, 137903 (2002)
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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)
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Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
(Gaussian) mixed states
Transmission through noisy channel
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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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Procrustean Approach
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Procrustean Approach
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PD
PD
Yes/No detector
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Procrustean Approach
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• 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 .
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Procrustean Approach
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• Probability of Success depends on q and T:• Example:
– Initial supply with q = 0.01
Entanglement Success Probability
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Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step: non-Gaussian state
Iterative Gaussifier (Gaussian operations)
(Gaussian) mixed states
Transmission through noisy channel
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Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step: non-Gaussian state
Iterative Gaussifier (Gaussian operations)
(Gaussian) mixed states
Transmission through noisy channel
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Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step: non-Gaussian state
Iterative Gaussifier (Gaussian operations)
(Gaussian) mixed states
Transmission through noisy channel
Imperial College London Krynica, 15th June 2005
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Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step: non-Gaussian state
Iterative Gaussifier (Gaussian operations)
(Gaussian) mixed states
Transmission through noisy channel
(Gaussian) two-mode squeezed states
Imperial College London Krynica, 15th June 2005
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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 PlenioPhys. Rev. A 67, 062320 (2003);J Eisert, DE Browne, S Scheel, MB Plenio, Annalsof Physics NY 311, 431 (2004)
Iterative Gaussifier (Gaussian operations)
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Gaussification
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A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
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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 |0| > |1|
The Gaussian state to which it converges is the two-modesqueezed state with q= 1/0.
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)
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Procrustean Approach
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Initial Supply
Procrustean Step
Gaussification
Final State
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Procrustean Approach
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• Example:
Entanglement Fidelity Probability
Initial state 0.0015 0.805
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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
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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
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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
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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
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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
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Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Example:
Probability Fidelity w.r.t. Gaussian target state
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Finite Detector Efficiency
Imperial College London
Entanglement Mixedness
1-Tr[2]
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
Krynica, 15th June 2005
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Improving the Procrustean Step
Imperial College London Krynica, 15th June 2005
Source
T
Fibre-loop detector with loss
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Photon Number Resolving Detectors
Imperial College London Krynica, 15th June 2005
APD
50/50
(2m)LL
2m+1 Light pulses
D. Achilles, Ch. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, Opt. Lett. 28, 2387 (2003).
Fiber based experimental implementation
realization of time-multiplexing with passive linear elements & two APDs
inputpulse
Principle: photons separated into distributed modes
ˆ U •••
inputpulse
APDs
linear network
•••
© W
alm
s ley
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Detector Efficiency
Imperial College London Krynica, 15th June 2005
fi
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Photon Number Resolution
Imperial College London Krynica, 15th June 2005
Enta
ngle
ment
Incr
ease
0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
Number of loops
Conditioned on two photons
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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