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Taoufik AMRI

Taoufik AMRI. Overview. Chapter I Quantum Description of Light. Chapter II Quantum Protocols. Chapter V Experimental Illustration. Chapter VI Detector of « Schrödinger’s Cat » States Of Light. Chapter III Quantum States and Propositions. The Wigner’s Friend. Chapter IV

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Taoufik AMRI

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  1. Taoufik AMRI

  2. Overview

  3. Chapter I Quantum Description of Light Chapter II Quantum Protocols Chapter V Experimental Illustration Chapter VI Detector of « Schrödinger’s Cat » States Of Light Chapter III Quantum States and Propositions The Wigner’s Friend Chapter IV Quantum Properties of Measurements Chapter VII Application to Quantum Metrology Interlude

  4. Introduction

  5. The Quantum World The “Schrödinger’s Cat” Experiment (1935) The cat is isolated from the environment The state of the cat is entangled to the one of a typical quantum system : an atom !

  6. “alive” “dead” The Quantum World AND ? • The cat is actually a detector of the atom’s state • Result “dead” : the atom is disintegrated • Result “alive” : the atom is excited Entanglement

  7. “alive” “dead” The Quantum World AND ? OR ! Quantum Decoherence : Interaction with the environment leads to a transition into a more classical behavior, in agreement with the common intuition!

  8. The Quantum World • Measurement Postulate • The state of the measured system, just after a measurement, is the state in which we measure the system. • Before the measurement : the system can be in a superposition of different states. One can only make predictions about measurement results. • After the measurement : Update of the state provided by the measurement … • Measurement Problem ?

  9. Quantum States of Light

  10. Quantum States of Light Light behaves like a wave or/and a packet “wave-particle duality” • Two ways for describing the quantum state of light : • Discrete description : density matrix • Continuous description : quasi-probability distribution

  11. “Decoherence” Quantum States of Light Discrete description of light : density matrix Coherences Populations Properties required for calculating probabilities

  12. Classical Vacuum Quantum Vacuum Quantum States of Light Continuous description of light : Wigner Function

  13. Quantum States of Light Wigner representation of a single-photon state Negativity is a signature of a strongly non-classical behavior !

  14. Quantum States of Light “Schrödinger’s Cat” States of Light (SCSL) Quantum superposition of two incompatible states of light + “AND” Wigner representation of the SCSL Interference structure is the signature of non-classicality

  15. Quantum States and Propositions

  16. Quantum States and Propositions • Back to the mathematical foundations of quantum theory • The expression of probabilities on the Hilbert space is given by the recent generalization of Gleason’s theorem (2003) based on • General requirements about probabilities • Mathematical structure of the Hilbert space • Statement : Any system is described by a density operator allowing predictions about any property of the system. P. Busch, Phys. Rev. Lett. 91, 120403 (2003).

  17. n=3 Quantum States and Propositions Physical Properties and Propositions A property about the system is a precise value for a given observable. Example : the light pulse contains exactly n photons The proposition operator is From an exhaustive set of propositions

  18. Quantum States and Propositions Generalized Observables and Properties A proposition can also be represented by a hermitian and positive operator The probability of checking such a property is given by Statement of Gleason-Bush’s Theorem

  19. Quantum state distributes the physical properties represented by hermitian and positive operators Statement of Gleason-Busch’s Theorem Quantum States and Propositions Reconstruction of a quantum state Quantum state Exhaustive set of propositions

  20. Quantum States and Propositions • Preparations and Measurements • In quantum physics, any protocol is based on state preparations, evolutions and measurements. • We can measure the system with a given property, but we can also prepare the system with this same property • Two approaches in this fundamental game : • Predictive about measurement results • Retrodictive about state preparations • Each approach needs a quantum state and an exhaustive set of propositions about this state

  21. Quantum States and Propositions Preparations Measurements Result “n” ? Choice “m” ?

  22. Quantum States and Propositions POVM Elements describing any measurement apparatus Quantum state corresponding to the proposition checked by the measurement Born’s Rule (1926)

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

  24. 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 !

  25. 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 !

  26. 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 !

  27. 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 ! Proposition operator

  28. Properties of a measurement Detectivity of a measurement Probability of detecting the target state Probability of detecting the retrodicted state Probability of detecting a target state

  29. Interlude

  30. Amplification of Vital Signs The Wigner’s Friend Effects of an observation ?

  31. Wigner representation of the POVM element describing the perception of light Quantum state retrodicted from the light perception Quantum properties of Human Eyes

  32. Effects of an observation Quantum state of the cat (C), the light (D) and the atom (N) State conditioned on the light perception Quantum decoherence induced by the observation

  33. Interests of a non-classical measurement Let us imagine a detector of “Schrödinger’s Cat” states of light Effects of this measurement (projection postulate) “AND” Quantum coherences are preserved !

  34. Detector of “Schrödinger’s Cat” States of Light

  35. 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

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

  37. Detector of “Schrödinger’s Cat” States of Light Retrodictive Version : Detector of “Schrödinger’s Cat” States Photon counting Non-classical Measurements Projective but Non-Ideal ! Squeezed Vacuum

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

  39. Detector of “Schrödinger’s Cat” States of Light Retrodicted States and Quantum Properties : Realistic Case Non-classical Measurement

  40. Applications in Quantum Metrology

  41. Applications in Quantum Metrology Typical Situation of Quantum Metrology Sensitivity is limited by the phase-space structure of quantum states Estimation of a parameter (displacement, phase shift, …) with the best sensitivity

  42. Applications in Quantum Metrology Estimation of a phase-space displacement Predictive probability of detecting the target state

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

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

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

  46. 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 probabilities under a variation of the parameter Number of repetitions

  47. 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

  48. Applications in Quantum Metrology Predictive and Retrodictive Estimations The Quantum Cramér-Rao Bound is reached …

  49. Predictive Retrodictive Applications in Quantum Metrology Retrodictive Estimation of a Parameter Projective but Non-Ideal ! 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”

  50. Conclusions and Perspectives Quantum Behavior of Measurement Apparatus Some quantum properties of a measurement are only revealed by its retrodicted state. • Foundations of Quantum Theory • The predictive and retrodictive approaches of quantum physics have the same mathematical foundations. • The reconstruction of retrodicted states from experimental data provides a real status for the retrodictive approach and its quantum states. Exploring the use of non-classical measurements Retrodictive version of a protocol can be more relevant than its predictive version.

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