INTERNATIONAL CONFERENCE ON QUANTUM INFORMATION Detecting “Schrödinger’s Cat” States of Light : Insights from the Retrodictive Approach Taoufik AMRI and Claude FABRE Quantum Optics Group, Laboratoire Kastler Brossel, France INTERNATIONAL CONFERENCE ON QUANTUM INFORMATION OTTAWA, JUNE 2011

Introduction Preparations Measurements Choice “m” ? Result “n” ?

Predictive and Retrodictive Approaches POVM Elements describing any measurement apparatus Quantum state corresponding to the property checked by the measurement Born’s Rule (1926)

Quantum Properties of Measurements T. Amri et al., Phys. Rev. Lett. 106, 020502 (2011).

Properties of a measurement Retrodictive Approach answers to natural questions when we perform a measurement : What kind of preparations could lead to such a result ? The properties of a measurement are those of its retrodicted state !

Properties of a measurement Non-classicality of a measurement It corresponds to the non-classicality of its retrodicted state Gaussian Entanglement Quantum state conditioned on an expected result “n” Necessary condition !

Properties of a measurement Projectivity of a measurement It can be evaluated by the purity of its retrodicted state For a projective measurement The probability of detecting the retrodicted state Projective and Non-Ideal Measurement !

Properties of a measurement Fidelity of a measurement Overlap between the retrodicted state and a target state Meaning in the retrodictive approach For faithful measurements, the most probable preparation is the target state ! Preparation operator

Detector of “Schrödinger’s Cat” States of Light

Detector of “Schrödinger’s Cat” States of Light Scheme of the detector Photon counting Non-classical Measurements Projective but Non-Ideal ! Squeezed Vacuum

Detector of “Schrödinger’s Cat” States of Light Retrodicted States and Quantum Properties : Idealized Case Projective but Non-Ideal !

Applications in Quantum Metrology

Applications in Quantum Metrology General scheme of the Predictive Estimation of a Parameter We must wait the results of measurements !

Applications in Quantum Metrology General scheme of the Retrodictive Estimation of a Parameter

Applications in Quantum Metrology Fisher Information and Cramér-Rao Bound Relative distance Fisher Information

Applications in Quantum Metrology Fisher Information and Cramér-Rao Bound Any estimation is limited by the Cramér-Rao bound Fisher Information is the variation rate of retrodictive probabilities under a variation of the parameter Number of repetitions

Applications in Quantum Metrology Retrodictive Estimation of a Parameter Projective but Non-Ideal ! Predictive Retrodictive The result “n” is uncertain even though we prepare its target state The target state is the most probable preparation leading to the result “n”

Applications in Quantum Metrology Predictive and Retrodictive Estimations of a phase-space displacement The Quantum Cramér-Rao Bound is reached …

Conclusions and Perspectives Quantum Behavior of Measurement Apparatus Some quantum properties of a measurement are only revealed by its retrodicted state. T. Amri et al., Phys. Rev. Lett. 106, 020502 (2011). Exploring the use of non-classical measurements Retrodictive version of a protocol can be more relevant than its predictive version. T. Amri et al., in preparation (2011).

Acknowledgements Many thanks to Stephen M. Barnett and Luiz Davidovich for fruitful discussions !

Detector of “Schrödinger’s Cat” States of Light “We can measure the system with a given property, but we can also prepare the system with this same property !” Main Idea : Predictive Version VS Retrodictive Version

Detector of “Schrödinger’s Cat” States of Light Predictive Version : Conditional Preparation of SCS of light A. Ourjoumtsev et al., Nature 448 (2007)

Applications in Quantum Metrology Illustration : Estimation of a phase-space displacement Optimal Minimum noise influence Fisher Information is optimal only when the measurement is projective and ideal

Applications in Quantum Metrology Retrodictive Estimation of a Parameter No Pain, No Gain ! Maximally mixed ! Von Neumann Entropy Concavity