Gaussian states and Gaussian transformations represent an interesting counterpart to two-level photonic systems in the field of quantum information processing. On the theoretical side, Gaussian states are easily described using first and second moments of the quadrature operators; from the experimental point of view, Gaussian operations can be implemented using linear optics and optical parametric amplifiers. The biggest advantage compared to two-level photonic systems, is deterministic generation of entangled states in parametric amplifiers and highly efficient homodyne detection. In this presentation, we propose new protocols for manipulation of entanglement of Gaussian states. Firstly, we study entanglement concentration of split single-mode squeezed vacuum states by photon subtraction enhanced by local coherent displacements. These states can be obtained by mixing a single-mode squeezed vacuum state with vacuum on a beam splitter and are, therefore, generated more easily than two-mode squeezed vacuum states. We show that performing local coherent displacements prior to photon subtraction can lead to an enhancement of the output entanglement. This is seen in weak-squeezing approximation where destructive quantum interference of dominant Fock states occurs, while for arbitrarily squeezed input states, we analyze a realistic scenario, including limited transmittance of tap-off beam splitters and limited efficiency of heralding detectors. Next, motivated by results obtained for bipartite Gaussian states, we study symmetrization of multipartite Gaussian states by local Gaussian operations. Namely, we analyze strategies based on addition of correlated noise and on quantum non-demolition interaction. We use fidelity of assisted quantum teleportation as a figure of merit to characterize entanglement of the state before and after the symmetrization procedure. Analyzing the teleportation protocol and considering more general transformations of multipartite Gaussian states, we show that the fidelity can be improved significantly.

1. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Transformations of Continuous-Variable Entangled
States of Light
Ondˇrej ˇCernot´ık
Department of Optics, Palack´y University Olomouc, Czech Republic
Niels Bohr Institute, July 2013

2. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Outline
1 Motivation
2 Enhancing entanglement concentration by coherent displacements
3 Symmetrization of multipartite states by local Gaussian operations
4 Conclusions

3. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Outline
1 Motivation
2 Enhancing entanglement concentration by coherent displacements
3 Symmetrization of multipartite states by local Gaussian operations
4 Conclusions

4. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Gaussian States
Wigner Function
W (x) =
1
2π
√
det γ
exp −
1
2
(x − ¯x)T
γ−1
(x − ¯x)
Mathematical description
in phase space.
Feasible using linear
optics, squeezers and
homodyne detection.1
1
S. L. Braunstein and P. van Loock, RMP 77, 513, C. Weedbrook et al.,
RMP 84, 621

5. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Quantum Entanglement
Important resource in quantum information processing.
Applications of Entanglement
Quantum teleportation,
Quantum key distribution,
Quantum dense coding,
One-way quantum computing,
Quantum metrology,. . .

6. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Bipartite and Multipartite Entanglement
Bipartite Entanglement
Relatively easy identification and quantification.

7. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Bipartite and Multipartite Entanglement
Bipartite Entanglement
Relatively easy identification and quantification.
Multipartite Entanglement
Complexity grows with number of parties. (Tripartite Gaussian
entanglement – 5 entanglement classesa).
Problematic quantification.
Applications: one-way quantum computingb, quantum
networksc.
a
G. Giedke et al., PRA 64, 052303
b
N. C. Menicucci et al., PRL 97, 110501
c
P. van Loock and S. L. Braunstein, PRL 84, 3482

8. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Outline
1 Motivation
2 Enhancing entanglement concentration by coherent displacements
3 Symmetrization of multipartite states by local Gaussian operations
4 Conclusions

9. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Why Entanglement Concentration?
Distribution of entangled states is subject to losses and
decoherence.
State degradation can be probabilistically eliminated using
local operations and classical communication.

10. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Entanglement Concentration and CV Systems
Non-Gaussianity Required
Gaussian states → non-Gaussian operations.a
Non-Gaussian states → Gaussian operations.b
a
J. Eisert et al., PRL 89, 137903, J. Fiur´aˇsek, PRL 89, 137904
b
R. Dong et al., Nat. Phys. 4, 919, B. Haage et al., Nat. Phys. 4, 915

11. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Photon Subtraction
Unbalanced beam splitter
and single-photon
detection.1
Can be improved by local
Gaussian operations.2
BS
APD
1
H. Takahashi et al., Nat. Photon. 4, 178
2
J. Fiur´aˇsek, PRA 84, 012335, S. L. Zhang and P. van Loock, PRA 84,
062309

12. Motivation Entanglement concentration Gaussian symmetrization Conclusions
The Protocol
ˆD(α) ˆD(−α)
ˆF = ˆa + α

13. Motivation Entanglement concentration Gaussian symmetrization Conclusions
The Protocol
ˆD(α) ˆD(−α)
ˆF = ˆa + α
ˆa + α
ˆF1 = ˆa + α

14. Motivation Entanglement concentration Gaussian symmetrization Conclusions
The Protocol
ˆD(α) ˆD(−α)
ˆF = ˆa + α
ˆa + α
ˆF1 = ˆa + α
ˆa + α ˆb + β
ˆF2 = (ˆa + α) ⊗ (ˆb + β)

15. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Input State
Single-mode squeezed vacuum split on a beam splitter.
|ψin =
4
1 − λ2
∞
n=0
2n
k=0
λn
2nn!
(2n)!t2n−krk
k!(2n − k)!
|2n − k, k

16. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Weak Input Squeezing
Zero- and two-photon contributions,
|ψin ≈ |00 + λrt|11 +
λ
√
2
(t2
|20 + r2
|02 ).
Destructive quantum interference leads to enhancement of
entanglement.

17. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Weak Input Squeezing
Single-Mode Subtraction
|ψ1 = λt(t|10 + r|01 ) + α|ψin
Zero displacement is optimal.
0.0
0.2
0.4
0.6
0.8
1.0
E
-0.05 0.0 0.05

18. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Weak Input Squeezing
Two-Mode Subtraction
|ψ2 = (λrt + αβ)|00 + λ(αr + βt)(t|10 + r|01 ) +
+
λ
√
2
αβ(t2
|20 +
√
2rt|11 + r2
|02 )
-0.15
0.0
0.15
-0.15 0.0 0.15

19. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Weak Input Squeezing
Two-Mode Subtraction
|ψ2 = (λrt + αβ)|00 + λ(αr + βt)(t|10 + r|01 ) +
+
λ
√
2
αβ(t2
|20 +
√
2rt|11 + r2
|02 )
Vacuum term elimination,
αβ = −λrt.
|ψ2 =
√
2rt|11 +t2|20 +r2|02
-0.15
0.0
0.15
-0.15 0.0 0.15

20. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Weak Input Squeezing
Two-Mode Subtraction
|ψ2 = (λrt + αβ)|00 + λ(αr + βt)(t|10 + r|01 ) +
+
λ
√
2
αβ(t2
|20 +
√
2rt|11 + r2
|02 )
Vacuum term elimination,
αβ = −λrt.
Single-photon contributions,
α =
√
λt, β =
√
λr.
|ψ2 =
√
2rt|11 +t2|20 +r2|02 0.8
1.0
1.2
1.4
1.6
E
0.0 0.05 0.1 0.15

21. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Arbitrary Squeezing
A Realistic Scenario
Experimental Realization
Stronger squeezing.
On-off detectors for photon subtraction with limited efficiency.
Finite transmittance of tap-off beam splitters.

22. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Arbitrary Squeezing
A Realistic Scenario
Experimental Realization
Stronger squeezing.
On-off detectors for photon subtraction with limited efficiency.
Finite transmittance of tap-off beam splitters.
Higher photon numbers, mixed output state, more complicated
filtering operation.

23. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Arbitrary Squeezing
Results
0.0
0.2
0.4
0.6
0.8
1.0
1.2
EN
-0.5 0.0 0.5
0.9
1.0
1.1
1.2
EN
0.0 0.1 0.2 0.3 0.4
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
P1
0.0 0.1 0.2 0.3 0.4
-1.0
-0.5
0.0
0.5
1.0
-1.0 -0.5 0.0 0.5 1.0
1.3
1.4
1.5
EN
0.0 0.1 0.2 0.3 0.4
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
P2
0.0 0.1 0.2 0.3 0.4

24. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Arbitrary Squeezing
Results
Conclusions
Single-mode subtraction optimal without displacements.
Two-mode subtraction gives more output entanglement; the
success probability is smaller.
a
O. ˇCernot´ık and J. Fiur´aˇsek, PRA 86, 052339

25. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Arbitrary Squeezing
Results
Conclusions
Single-mode subtraction optimal without displacements.
Two-mode subtraction gives more output entanglement; the
success probability is smaller.
a
O. ˇCernot´ık and J. Fiur´aˇsek, PRA 86, 052339
Extensions
Losses limit usability of the protocol.
No Gaussian entanglement at the output.
b
A. Tipsmark et al., Opt. Exp. 21, 6670

26. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Outline
1 Motivation
2 Enhancing entanglement concentration by coherent displacements
3 Symmetrization of multipartite states by local Gaussian operations
4 Conclusions

27. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Symmetrization of Multipartite Gaussian States
Equalization of quadrature correlations.








n 0 c 0 c 0
0 n 0 −d 0 −d
c 0 n 0 c 0
0 −d 0 n 0 −d
c 0 c 0 n 0
0 −d 0 −d 0 n








→








n′ 0 c′ 0 c′ 0
0 n′ 0 −c′ 0 −c′
c′ 0 n′ 0 c′ 0
0 −c′ 0 n′ 0 −c′
c′ 0 c′ 0 n′ 0
0 −c′ 0 −c′ 0 n′








Generalization of protocols for bipartite Gaussian states.1
1
J. Fiur´aˇsek, PRA 86, 032317

28. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Equivalent State Preparation
N
1 2
. . .
rN , nN
1:12:1(N − 1):1
r1, n1 r1, n1 r1, n1r1, n1
BS1 BSN−2 BSN−1BS2
(N − 2):1
N − 2 N − 1
Simplified analysis – working with two separable modes.
Similarity to experimental realizations of quantum networks.1
1
T. Aoki et al., PRL 91, 080404, H. Yonezawa et al., Nature 431, 430

29. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Equivalent State Preparation
N
1 2
. . .
rN , nN
1:12:1(N − 1):1
r1, n1 r1, n1 r1, n1r1, n1
BS1 BSN−2 BSN−1BS2
(N − 2):1
N − 2 N − 1
n =
1
N
[nNe2rN
+ (N − 1)n1e2r1
]
c =
1
N
(nNe2rn
− n1e2r1
)
d =
1
N
(n1e−2r1
− nNe−2rN
)

30. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Assisted Quantum Teleportation
Entanglement
characterization – assisted
teleportation fidelity.1
F = 1√
(n−c+1)(n−d+1−2d2/n)
More general
transformations
(n, c, d) → (n′, c′, kc′).
A
B
C
in
(qin − qA)/
√
2
(pin + pA)/
√
2
1
P. van Loock and S.L. Braunstein, PRL 84, 3482

31. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Strategies
Correlated Noise Addition
ργNS γN S
γN
S
Adding correlated noise γN.
Squeezing S.

32. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Strategies
Quantum Non-Demolition Interaction
ρgdAS g dB S
g
dC
S
QND interaction g, measurement on ancillas and
displacement d.
Squeezing S.

33. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Results for Tripartite States
Noise Addition
0.45
0.5
0.55
0.6
0.65
F
0.45 0.5 0.55 0.6 0.65 0.7 0.75
k
QND Interaction
0.4
0.45
0.5
0.55
F
0.3 0.4 0.5 0.6 0.7
k
Both strategies work best for noisy states.

34. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Outlook
More general class of transformations,
(n1, n2, c, d) → (n′, k1n′, c′, k2c′).
Formalism of complex symplectic matrices for
purity-preserving Gaussian quantum filters.1
1
J. Fiur´aˇsek, PRA 87, 052301

35. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Outline
1 Motivation
2 Enhancing entanglement concentration by coherent displacements
3 Symmetrization of multipartite states by local Gaussian operations
4 Conclusions

36. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Entanglement Concentration
Value of Squeezing
Weak squeezing: Destructive quantum interference.
Arbitrary squeezing: Realistic experimental scenario.

37. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Entanglement Concentration
Value of Squeezing
Weak squeezing: Destructive quantum interference.
Arbitrary squeezing: Realistic experimental scenario.
Strategies
Single-mode photon subtraction optimal without
displacements.
Local displacements can improve two-mode subtraction.

38. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Entanglement Concentration
Value of Squeezing
Weak squeezing: Destructive quantum interference.
Arbitrary squeezing: Realistic experimental scenario.
Strategies
Single-mode photon subtraction optimal without
displacements.
Local displacements can improve two-mode subtraction.
Structure of the entanglement.

39. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Symmetrization of multipartite Gaussian states
Tools
Equivalent state preparation for analyzing protocols.
Assisted teleportation fidelity for state characterization.

40. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Symmetrization of multipartite Gaussian states
Tools
Equivalent state preparation for analyzing protocols.
Assisted teleportation fidelity for state characterization.
Strategies
Correlated noise addition: More sensitive to imperfections
(narrow peak).
QND interaction: More challenging experimentally (use of
atomic ensemblesa, linear optical emulationb).
Each strategy optimal for different types of states.
a
K. Hammerer et al., RMP 82, 1041
b
R. Filip et al., PRA 71, 042308

41. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Symmetrization of multipartite Gaussian states
Tools
Equivalent state preparation for analyzing protocols.
Assisted teleportation fidelity for state characterization.
Strategies
Correlated noise addition: More sensitive to imperfections
(narrow peak).
QND interaction: More challenging experimentally (use of
atomic ensemblesa, linear optical emulationb).
Each strategy optimal for different types of states.
a
K. Hammerer et al., RMP 82, 1041
b
R. Filip et al., PRA 71, 042308
Possible extensions of the protocol.

42. Motivation Entanglement concentration Gaussian symmetrization Conclusions
Credits
Jarom´ır Fiur´aˇsek
Radim Filip
Financial support:
Thank you for your attention!