Manipulating Continuous Variable Photonic Entanglement
Martin Plenio
Imperial College London
Institute for Mathematical Sc...
Local preparation
A BEntangled state between distant sites
The vision . . .
Prepare and distribute pure-state entanglement...
. . . and the reality
A B
Weakly entangled state
Noisy channel
Local preparation
Decoherence will degrade entanglement
Can...
The three basic questions of a theory of entanglement
 decide which states are entangled and which are disentangled
(Char...
Practically motivated entanglement theory
Theory of entanglement is usually purely abstract
For example: accessibility of ...
 Consider n harmonic oscillators
nn PXPXPX ,,, 2211 •••
 Canonical coordinates ),,...,,(),,...,,( 1121221 nnnn PXPX=− OO...
 canonical commutation relations
where is a real 2n x 2n matrix is the symplectic matrixσ
Lets go quantum
 Harmonic osci...
 Characteristic function (Fourier transform of Wigner function)
Characteristic function
Simplest example: Vacuum state = ...
 A state is called Gaussian, if and
only if its characteristic function (or its
Wigner function) is a Gaussian
 Gaussian...
 First moments (local displacements in phase space):
First Moments
Krynica, 15th June 2005Imperial College London
Local d...
 The covariance matrix embodies the second moments
 Heisenberg uncertainty principle
Uncertainty Relations
Krynica, 15th...
CV entanglement of Gaussian states
 Separability + Distillability
Necessary and sufficient criterion known for M x N syst...
Characterization of Gaussian operations
For all general Gaussian operations, a ‘dictionary’
would be helpful that links th...
 In a quantum optical setting
Application of linear optical elements:
 Beam splitters
 Phase plates
 Squeezers
Gaussia...
Characterization of Gaussian operations
Optical elements
and additional
field modes
Vacuum detection Homodyne
measurement
...
Gaussian manipulation of entanglement
 What quantum state transformations can be
implemented under Gaussian local operati...
Gaussian manipulation of entanglement
 Apply Gaussian LOCC to the initial state ρ
ρ
Krynica, 15th June 2005Imperial Colle...
Gaussian manipulation of entanglement
 Can one reach ρ’, ie is there a Gaussian LOCC map such that
?
ρ'
E(ρ)=ρ'
E
Krynica...
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....
The general theorem
 Necessary and sufficient condition for the transformation of pure
Gaussian states under Gaussian loc...
The general theorem
 Necessary and sufficient condition for the transformation of pure
Gaussian states under Gaussian loc...
Comparison
Krynica, 15th June 2005Imperial College London
General LOCC
r1
r2
01 =r
2
'
2 rr >
Gaussian LOCC
r1
r2
01 =r
2
...
Comparison
Krynica, 15th June 2005Imperial College London
General LOCC
r1
r2
01 =r
2
'
2 rr >
Gaussian LOCC
r1
r2
01 =r
2
...
A1 B1
A2 B2
Homodyne
measurements
General local unitary Gaussian
operations (any array of beam
splitters, phase shifts and...
Gaussian entanglement distillation on mixed states
 The optimal iterative Gaussian distillation protocol
can be identifie...
Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step:
non-Gaussian state
(Gau...
Procrustean Approach
Imperial College London Krynica, 15th June 2005
PD
PD
Yes/No detector
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Simple protocol to generate non-Gaussian states of ...
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Probability of Success depends on q and T:
• Exampl...
Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step:
non-Gaussian state
(Gau...
Gaussification
Imperial College London Krynica, 15th June 2005
A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
Procrustean Approach
Imperial College London Krynica, 15th June 2005
A1 B1
A2 B2
50/5050/50 50/50
Yes/No Yes/No
A1 B1
A2 B...
Procrustean Approach
Imperial College London Krynica, 15th June 2005
Initial Supply
Procrustean Step
Gaussification
Final ...
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Example:
Entanglement Fidelity Probability
Initial ...
Procrustean Approach
Imperial College London Krynica, 15th June 2005
• Example:
Probability Fidelity w.r.t. Gaussian targe...
Finite Detector Efficiency
Imperial College London
Entanglement Mixedness

1-Tr[ρ2
]

log. neg.
1
2
NG 1
2
 Input: Weak...
Improving the Procrustean Step
Imperial College London Krynica, 15th June 2005
Source
T
Fibre-loop detector with loss
Photon Number Resolving Detectors
Imperial College London Krynica, 15th June 2005
APD
50/50
(2m
)LL
2m+1
Light pulses
D. A...
Detector Efficiency
Imperial College London Krynica, 15th June 2005
fi
λ
Photon Number Resolution
Imperial College London Krynica, 15th June 2005
EntanglementIncrease
0.05 0.06 0.07 0.08 0.09 0.1...
Summary
Imperial College London Krynica, 15th June 2005
• Gaussian operations on Gaussian states cannot
distill entangleme...
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Manipulating continuous variable photonic entanglement

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Manipulating continuous variable photonic entanglement

  1. 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. 2. Local preparation A BEntangled state between distant sites The vision . . . Prepare and distribute pure-state entanglement Krynica, 15th June 2005Imperial College London
  3. 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? Krynica, 15th June 2005Imperial College London They are restricted to Local Operations and Classical Communication
  4. 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 Krynica, 15th June 2005Imperial College London
  5. 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 Krynica, 15th June 2005Imperial College London
  6. 6.  Consider n harmonic oscillators nn PXPXPX ,,, 2211 •••  Canonical coordinates ),,...,,(),,...,,( 1121221 nnnn PXPX=− OOOO Basics of continuous-variable systems ••• Krynica, 15th June 2005Imperial College London
  7. 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. Krynica, 15th June 2005Imperial College London
  8. 8.  Characteristic function (Fourier transform of Wigner function) Characteristic function Simplest example: Vacuum state = Gaussian function Krynica, 15th June 2005Imperial College London
  9. 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 Krynica, 15th June 2005Imperial College London
  10. 10.  First moments (local displacements in phase space): First Moments Krynica, 15th June 2005Imperial College London Local displacement Local displacement
  11. 11.  The covariance matrix embodies the second moments  Heisenberg uncertainty principle Uncertainty Relations Krynica, 15th June 2005Imperial College London γ represents a physical Gaussian state iff the uncertainty relations are satisfied.
  12. 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 Krynica, 15th June 2005Imperial College London
  13. 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) Krynica, 15th June 2005Imperial College London
  14. 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 Krynica, 15th June 2005Imperial College London
  15. 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 Krynica, 15th June 2005Imperial College London
  16. 16. Gaussian manipulation of entanglement  What quantum state transformations can be implemented under Gaussian local operations? ρ Krynica, 15th June 2005Imperial College London
  17. 17. Gaussian manipulation of entanglement  Apply Gaussian LOCC to the initial state ρ ρ Krynica, 15th June 2005Imperial College London
  18. 18. Gaussian manipulation of entanglement  Can one reach ρ’, ie is there a Gaussian LOCC map such that ? ρ' E(ρ)=ρ' E Krynica, 15th June 2005Imperial College London ρ
  19. 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) Krynica, 15th June 2005Imperial College London
  20. 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  Krynica, 15th June 2005Imperial College London 2'r Nr'
  21. 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 ≥  Krynica, 15th June 2005Imperial College London 22 'rr ≥ NN rr '≥
  22. 22. Comparison Krynica, 15th June 2005Imperial College London 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. 23. Comparison Krynica, 15th June 2005Imperial College London 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. 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 Krynica, 15th June 2005Imperial College London
  25. 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. 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 Imperial College London Krynica, 15th June 2005
  27. 27. Procrustean Approach Imperial College London Krynica, 15th June 2005 PD PD Yes/No detector
  28. 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. 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. 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) Krynica, 15th June 2005
  31. 31. Gaussification Imperial College London Krynica, 15th June 2005 A1 B1 A2 B2 50/5050/50 50/50 Yes/No Yes/No
  32. 32. Procrustean Approach Imperial College London Krynica, 15th June 2005 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. 33. Procrustean Approach Imperial College London Krynica, 15th June 2005 Initial Supply Procrustean Step Gaussification Final State
  34. 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. 35. Procrustean Approach Imperial College London Krynica, 15th June 2005 • Example: Probability Fidelity w.r.t. Gaussian target state
  36. 36. Finite Detector Efficiency Imperial College London 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 Krynica, 15th June 2005
  37. 37. Improving the Procrustean Step Imperial College London Krynica, 15th June 2005 Source T Fibre-loop detector with loss
  38. 38. 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 input pulse Principle: photons separated into distributed modes ˆU• • • input pulse APDs linear network • • • ©Walmsley
  39. 39. Detector Efficiency Imperial College London Krynica, 15th June 2005 fi λ
  40. 40. Photon Number Resolution Imperial College London Krynica, 15th June 2005 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. 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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