1. M. Stobińska1, F. Töppel2, P. Sekatski3,
A. Buraczewski4, N. Gisin3
Towards loophole-free Bell test with preselected unsymmetrical singlet states of light
1Institute of Physics, Polish Academy of Sciences
2Max Planck Institute for the Science of Light & Erlangen-Nürnberg University
3GAP-Optique, University of Geneva
4Warsaw University of Technology

2. + _ ? Motivation for Bell Test with Highly Populated States of Light
Bell test is useful for: fundamental interest, CCPs, device-independent assessments, randomness generation,...
Due to macroscopic nature, the quantum systems: _
decohere and lose their quantum features very fast
postselection can lead to false conclusions indicating entanglement in separable states
E. Pomarico, B. Sanguinetti, P. Sekatski, H. Zbinden, N. Gisin, New J. Phys. 13, (2011) +
they are always detectable with classical detectors
preselection strategy with a tap measurement improving distinguishability in detection is possible ?
upper bound for population for which violation can be observed (with increasing size of system, the sensitivity of some measuring devices has to increase as well)
R. Ramanathan, T. Paterek, A. Kay, P. Kurzyński, D. Kaszlikowski, PRL 106, (2011)

3. CHSH-Bell Test with Preselection Strategy & Intensity Measurements
M. Stobińska, P. Horodecki, A. Buraczewski, R. W. Chhajlany, R. Horodecki, G. Leuchs
arXiv:
M. Stobińska, P. Sekatski, A. Buraczewski, N. Gisin, G. Leuchs,
Phys. Rev. A 84, (2011)
M. Stobińska, F. Töppel, P. Sekatski, A. Buraczewski, N. Gisin,
in preparation

4. Unsymmetrical Singlet
Let's consider a general unsymmetrical singlet
Approximation: „qubit” description
„micro”- qubit:
„large”- qubit:
Population of the „large”- qubit is controlled by an external parameter;
it may vary from few photons up to the macroscopic scale
We assume the „large”-qubit suffers from low distinguishability in detection

5. Preselection of Unsymmetrical Singlet
Filtering: a POVM in the basis rotated by with respect
to the source basis
In general, may not posses a rotational symmetry
It transforms the singlet as follows
It may not preserve orthogonality

6. Observables and Correlations
CHSH - Bell parameter
Micro - observable
Macro - observable is a binary threshold operator,
which is adapted to the preselection strategy

7. Correlation Function Rotation on the microscopic part
Correlation function for an unsymmetrical singlet
where
visibility
anti-visibility

8. Bell Parameter
We choose
and obtain

9. Maximal Bell Parameter
The maximal value of Bell parameter with respect to
is set by the condition
It equals
The violation takes place if

10. Non-orthogonal Measurement Directions
The usual (red and blue) and optimal (red and black) angle settings
for CHSH inequality test
After preselection, the unsymmetrical singlet becomes non-maximally
entangled

11. Analogy to Qubit
For a two-qubit state expressed in Hilbert-Schidt basis
where are coefficients of matrix
one obtains
with being mutually orthogonal measurement directions
R. Horodecki, P. Horodecki, M. Horodecki, Phys. Lett. A 200, 340 (1995)

12. Micro-macro Singlet States of Light
They are produced in optimal phase-covariant quantum cloning
Hamiltonian
The „large”-qubit
F. De Martini et al, PRL 100, (2008)

13. Micro-macro Singlet States of Light
In experiment:
amplification gain
average population
but distinuishability is low
preselection is necessary or very good detectors required
special kind of POVM filter with implementation has been proposed:
the modulus of intensity difference filter
M. Stobińska, F. Töppel, P. Sekatski, A. Buraczewski, M. Żukowski, M. V. Chekhova,
G. Leuchs, N. Gisin, arXiv:

14. General Preselection
The MDF filter can be generalized to a device filtering
according to an arbitrary function of the sum
and difference

15. State Analysis
The „large”-qubits are expressed as a convex sum of states
of a fixed photon number distributed over two polarizations

16. Visibility & Antivisibility
Since the states are given by the convex sum of different
photon number sectors
Macro-observable for the modulus of intensity difference filter

17. No Preselection Optimal choice g = 0.8 → B = 2.06 g = 1.1 → B = 2.01
1/√2
g = 0.8 → B = 2.06
g = 1.1 → B = 2.01 2

18. State Analysis after Preselection
The convex sum structure is preserved

19. Example: MDF & Cross Filter
MDF: Bell test does not work
Cross filter:

20. Cross Filter
g = 0.8 → B = 2.44
g = 1.1 → B = 2.28 2

21. Cross Filter
g = 0.8 → B = 2.13
g = 1.1 → B = 2.01 2

22. Summary
The cross filter provides the best filtering for the De Martini “macro”- qubits
It creates superpositions of the N00N states
For these particular states and intensity measurement Bell violation
for N00N states is the best we can get
This is a proof-of-the-concept that the preselection leads to Bell violation
Can we find a different measurement scheme (observable),
which does not depend on the particle number?
We considered truly many photon states, where the operations cannot be
factorized

23. Thank you!