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

+ _ ? 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, 063031 (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, 180402 (2011)

CHSH-Bell Test with Preselection Strategy & Intensity Measurements M. Stobińska, P. Horodecki, A. Buraczewski, R. W. Chhajlany, R. Horodecki, G. Leuchs arXiv:0909.1545 M. Stobińska, P. Sekatski, A. Buraczewski, N. Gisin, G. Leuchs, Phys. Rev. A 84, 034104 (2011) M. Stobińska, F. Töppel, P. Sekatski, A. Buraczewski, N. Gisin, in preparation

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

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

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

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

Bell Parameter We choose and obtain

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

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

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)

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, 253601 (2008)

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:1108.4906

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

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

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

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

State Analysis after Preselection The convex sum structure is preserved

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

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

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

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

Thank you!