1. Entanglement-enhanced communication over a correlated-noise channel
Squeezing eigenmodes in parametric down-conversion
Entanglement-enhanced communication over a correlated-noise channel
Andrzej Dragan
Wojciech Wasilewski
Czesław Radzewicz
Warsaw University
Jonathan Ball University of Oxford
Konrad Banaszek
Nicolaus Copernicus University Toruń, Poland
Alex Lvovsky
University of Calgary
National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland

2. All that jazz
Sender
Receiver
Mutual information:
Channel capacity:

3. Depolarization in an optical fibre
Independently of the averaged output state has the form:
Photon in a polarization state
Random polarization transformation V H
Jeżeli każda transformacja unitarna może się zdarzyć z równym prawdopodobieństwem (a matematycy wiedzą co to znaczy, zbiór transformacji tworzy grupę Liego a ta ma kanoniczną miarę/rozkład prawdopodobieństwa), to na wyjściu mamy to co mamy... H V
1/2
Capacity of coding in the polarization state of a single photon:

4. Information coding Sender: V H V
Probabilities of measurement outcomes:
H&H, V&V
H&V, V&H
2/3
1/3 H
Capacity per photon pair:

5. Collective detection Probabilities of measurement outcomes: 2&0, 0&2
1&1 1
1/2
Pierwszy foton musimy troche opoznic
Capacity:

6. Entangled states are useful!
Probabilities of measurement outcomes:
2&0, 0&2
1&1 1
Capacity:

7. Proof-of-principle experiment
2&0, 0&2
1&1 1
1/2
Separable ensemble:
These are optimal ensembles for separable and entangled inputs (assuming collective detection), which follows from optimizing the Holevo bound.
J. Ball, A. Dragan, and K.Banaszek, Phys. Rev. A 69, (2004)
2&0, 0&2
1&1 1
Entangled ensemble:

8. Source of polarization-entangled pairs
P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999)
For a suitable polarization of the pump pulses, the generated two-photon state has the form:
Stare (jeden wiek) odwieczne pytanie: powinnismy dodawac prawdopodobieństwa (procesy rozróżnialne) czy tez amplitudy prawdopodobieństwa (procesy nierozróżnialne). Chcielibyśmy jakoś testować spójność tej superpozycji.
With a half-wave plate in one arm it can be transformed into: or

9. Experimental setup Triplet events: D1 & D2 D3 & D4 Singlet events:
K. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Phys. Rev. Lett. 92, (2004)
Triplet events:
D1 & D D3 & D4
Singlet events:
D1 & D3 D2 & D3 D2 & D3 D2 & D4

10. Experimental results

11. Dealing with collective depolarization
Phase encoding in time bins: fixed input polarization, polarization-independent receiver. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett. 82, 2594 (1999).
Decoherence-free subspaces for a train of single photons. J.-C. Boileau, D. Gottesman, R. Laamme, D. Poulin, and R. W. Spekkens, Phys. Rev. Lett. 92, (2004).

12. General scenario Physical system: arbitrarily many photons
N time bins that encompass two orthogonal polarizations
How many distinguishable states can we send via the channel?
What is the biggest decoherence-free subspace?

13. Mathematical model General transformation: where:
– the entire quantum state of light across N time bins
– element of U(2) describing the transformation of the polarization modes in a single time bin.
– unitary representation of W in a single time bin
We will decompose with
and

14. Schwinger representation
Ordering Fock states in a single time bin according to the combined number of photons l:
...
Representation of W:
...
Here is (2j+1)-dimensional representation of SU(2). Consequently has the explicit decomposition in the form:

15. Decomposition Decomposition into irreducible representations:
Integration over a removes coherence between subspaces with different total photon number L. Also, no coherence is left between subspaces with different j.
tells us:
how many orthogonal states can be sent in the subspace j
dimensionality of the decoherence-free subsystem
Recursion formula for :
J. L. Ball and K. Banaszek, quant-ph/ ; Open Syst. Inf. Dyn. 12, 121 (2005)
Biggest decoherence-free subsystems have usually hybrid character!

16. Questions
Relationship to quantum reference frames for spin systems [S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. Lett. 91, (2003)]
Partial correlations?
Linear optical implementations?
How much entanglement is needed for implementing decoherence-free subsystems?
Shared phase reference?
Self-referencing schemes? [Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 91, (2003)]
Other decoherence mechanisms, e.g. polarization mode dispersion?

17. Multimode squeezing – Single-mode model:
SHG
Single-mode model:
PDC
Brutal reality (still simplified):
[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band, Opt. Comm. 221, 337 (2003)] –

18. Perturbative regime The wave function up to the two-photon term:
W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997); T. E. Keller and M. H. Rubin, Phys. Rev A 56, 1534 (1997)
Schmidt decomposition for a symmetric two-photon wave function:
C. K. Law, I. A. Walmsley, and J. H. Eberly, Phys. Rev. Lett. 84, 5304 (2000)
We can now define eigenmodes which yields:
The spectral amplitudes characterize pure squeezing modes

19. Decomposition for arbitrary pump
As the commutation relations for the output field operators must be preserved, the two integral kernels can be decomposed using the Bloch-Messiah theorem:
S. L. Braunstein, quant-ph/ ; see also R. S. Bennink and R. W. Boyd, Phys. Rev. A 66, (2002) 

20. Squeezing modes
The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields:
which evolve according to
describe modes that are described by pure squeezed states
tell us what modes need to be seeded to retain purity
Some properties:
For low pump powers, usually a large number of modes becomes squeezed with similar squeezing parameters
Any superposition of these modes (with right phases!) will exhibit squeezing
The shape of the modes changes with the increasing pump intensity!
This and much more in a poster by Wojtek Wasilewski

21. The End

22. Theory Everything that emerges are Werner states
One-dimensional optimization problem for the Holevo bound
What about phase encoding?

23. Recursion formula N bins, L photons ... L–L′ photons
Decompostion of the corresponding su(2) algebra:
If we subtract one time bin:
N bins, L photons
N-1 bins, L′ photons
...
L–L′ photons

24. Direct sum
The product of two angular momentum algebras has the standard decomposition as:
Therefore the algebra for L photons in N time bins can be written as a triple direct sum:

25. Decoherence-free subsystems
Rearranging the summation order finally yields:
Underlined entries with correspond to pure phase encoding (with all the input pulses having identical polarizations) – in most cases we can do better than that!