Manipulating continuous variable photonic entanglement

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

Local preparation
A BEntangled state between distant sites
The vision . . .
Prepare and distribute pure-state entanglement
Krynica, 15th June 2005Imperial College London

. . . and the reality
A B
Weakly entangled state
Noisy channel
Local preparation
Decoherence will degrade entanglement
Can Alice and Bob ‘repair’ the damaged entanglement?
They are restricted to Local Operations and Classical Communication

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

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

 Consider n harmonic oscillators
nn PXPXPX ,,, 2211 •••
 Canonical coordinates ),,...,,(),,...,,( 1121221 nnnn PXPX=− OOOO
Basics of continuous-variable systems
•••

 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.

 Characteristic function (Fourier transform of Wigner function)
Characteristic function
Simplest example: Vacuum state = Gaussian function

 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

 First moments (local displacements in phase space):
First Moments
Local displacement Local displacement

 The covariance matrix embodies the second moments
 Heisenberg uncertainty principle
Uncertainty Relations
γ represents a physical Gaussian state
iff the uncertainty relations are satisfied.

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

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)

 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

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

Gaussian manipulation of entanglement
 What quantum state transformations can be
implemented under Gaussian local operations?
ρ

 Apply Gaussian LOCC to the initial state ρ
ρ

 Can one reach ρ’, ie is there a Gaussian LOCC map such that
?
ρ'
E(ρ)=ρ'
E
ρ

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)

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'
A B
r1
r2
rN

A B
1'r

2'r
Nr'

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'
A B
r1
r2
rN

A B
11 'rr ≥

22 'rr ≥
NN rr '≥

Comparison
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)

Comparison
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 !

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

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)

Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Initial step:
non-Gaussian state
(Gaussian) mixed states
Transmission through noisy channel

Procrustean Approach
PD
PD
Yes/No detector

• 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.

• Probability of Success depends on q and T:
• Example:
– Initial supply with q = 0.01
Entanglement Success Probability

Distillation by leaving the Gaussian regime once
Initial step:
non-Gaussian state
(Gaussian) mixed states
Transmission through noisy channel
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

Gaussification
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
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)

Initial Supply
Procrustean Step
Gaussification
Final State

• 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

• Example:
Probability Fidelity w.r.t. Gaussian target state

Finite Detector Efficiency
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

Improving the Procrustean Step
Source
T
Fibre-loop detector with loss

Photon Number Resolving Detectors
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

Detector Efficiency
fi
λ

Photon Number Resolution
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

Summary
• 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

Manipulating continuous variable photonic entanglement

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