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Ph ddefenseamri

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Ph ddefenseamri

  1. 1. Taoufik AMRI 1
  2. 2. Overview
  3. 3. Chapter IQuantum Description of Light Chapter II Quantum Protocols Chapter V Chapter VI Experimental Detector of Chapter III Illustration The Wigner’s « Schrödinger’s Cat » States Quantum States and Propositions Friend Of Light Chapter IV Quantum Properties of Measurements Chapter VII Application to Interlude Quantum Metrology 3
  4. 4. Introduction
  5. 5. The Quantum WorldThe “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 ! 5
  6. 6. The Quantum World AND ? “alive” “dead”The cat is actually a detector of the atom’s state • Result “dead” : the atom is disintegrated • Result “alive” : the atom is excited Entanglement 6
  7. 7. The Quantum World AND !? OR “alive” “dead”Quantum Decoherence : Interaction with the environmentleads to a transition into a more classical behavior, inagreement with the common intuition ! 7
  8. 8. The Quantum WorldMeasurement PostulateThe state of the measured system, just after a measurement, isthe state in which we measure the system.Before the measurement : the system can be in a superposition ofdifferent states. One can only make predictions about measurementresults.After the measurement : Update of the state provided by themeasurement … Measurement Problem ? 8
  9. 9. Quantum States of Light
  10. 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 10
  11. 11. Quantum States of LightDiscrete description of light : density matrix “Decoherence” Coherences Populations Properties required for calculating probabilities 11
  12. 12. Quantum States of LightContinuous description of light : Wigner Function Classical0 Vacuum Quantum Vacuum ≠ ( 0, 0 ) 0 12
  13. 13. Quantum States of LightWigner representation of a single-photon state 1 = Negativity is a signature of a strongly non-classical behavior ! 13
  14. 14. Quantum States of Light“Schrödinger’s Cat” States of Light (SCSL)Quantum superposition of two incompatible states of light + “AND” Wigner Interference structurerepresentation of is the signature of the SCSL non-classicality 14
  15. 15. Quantum States and Propositions
  16. 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). 16
  17. 17. Quantum States and PropositionsPhysical Properties and PropositionsA property about the system is a precise value for a given observable.Example : the light pulse contains exactly n photons n=3The proposition operator is ˆ Pn = n nFrom an exhaustive set of propositions ∑ ˆ ˆ Pn = 1 n 17
  18. 18. Quantum States and PropositionsGeneralized Observables and PropertiesA proposition can also be represented by a hermitian and positiveoperatorThe probability of checking such a property is given by Statement of Gleason-Bush’s Theorem 18
  19. 19. Quantum States and PropositionsReconstruction of a quantum state Quantum state Quantum state distributes the physical properties represented by hermitian and positive operators Statement of Gleason-Busch’s Theorem Exhaustive set of propositions 19
  20. 20. Quantum States and PropositionsPreparations and MeasurementsIn quantum physics, any protocol is based on state preparations,evolutions and measurements.We can measure the system with a given property, but we can alsoprepare the system with this same propertyTwo 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 20
  21. 21. Quantum States and Propositions Preparations Measurements Choice “m” Result “n” ? ? 21
  22. 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) 22
  23. 23. Quantum Properties of Measurements T. Amri et al., Phys. Rev. Lett. 106, 020502 (2011).
  24. 24. Properties of a measurementRetrodictive Approach answers to natural questions when we performa measurement : What kind of preparations could lead to such a result ? The properties of a measurement are those of its retrodicted state ! 24
  25. 25. Properties of a measurementNon-classicality of a measurementIt corresponds to the non-classicality of its retrodicted state Gaussian Entanglement λ →1Quantum state conditioned on Necessary condition ! an expected result “n” 25
  26. 26. Properties of a measurementProjectivity of a measurementIt can be evaluated by the purity of its retrodicted stateFor a projective measurementThe probability of detecting the retrodicted state Projective and Non-Ideal Measurement ! 26
  27. 27. Properties of a measurementFidelity of a measurementOverlap between the retrodicted state and a target stateMeaning in the retrodictive approachProposition operator For faithful measurements, the most probable preparation is the target state ! 27
  28. 28. Properties of a measurementDetectivity of a measurementProbability of detecting the target stateProbability of detecting the retrodicted state Probability of detecting a target state 28
  29. 29. Interlude
  30. 30. The Wigner’s Friend Amplification of Vital Signs Effects of an observation ? 30
  31. 31. Quantum properties of Human Eyes Wigner representation of the POVM element describing the perception of light Quantum state retrodicted from the light perception 31
  32. 32. Effects of an observationQuantum state of the cat (C), the light (D) and the atom (N)State conditioned on the light perception Quantum decoherence induced by the observation 32
  33. 33. Interests of a non-classical measurementLet us imagine a detector of “Schrödinger’s Cat” states of lightEffects of this measurement (projection postulate) “AND” Quantum coherences are preserved ! 33
  34. 34. Detector of “Schrödinger’s Cat” States of Light
  35. 35. Detector of “Schrödinger’s Cat” States of Light “We can measure the system with MainaIdea : property, but we can also given Predictive Version VS Retrodictive Version same prepare the system with this property !” 35
  36. 36. Detector of “Schrödinger’s Cat” States of Light Predictive Version : Conditional Preparation of SCS of light A. Ourjoumtsev et al., Nature 448 (2007) 36
  37. 37. Detector of “Schrödinger’s Cat” States of LightRetrodictive Version : Detector of “Schrödinger’s Cat” States Photon countingNon-classical Measurements Projective but Non-Ideal ! Squeezed Vacuum 37
  38. 38. Detector of “Schrödinger’s Cat” States of LightRetrodicted States and Quantum Properties : Idealized Case Projective but Non-Ideal ! 38
  39. 39. Detector of “Schrödinger’s Cat” States of LightRetrodicted States and Quantum Properties : Realistic Case 39 Non-classical Measurement
  40. 40. Applications in Quantum Metrology
  41. 41. Applications in Quantum MetrologyTypical 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 41
  42. 42. Applications in Quantum MetrologyEstimation of a phase-space displacement Predictive probability of detecting the target state 42
  43. 43. Applications in Quantum MetrologyGeneral scheme of the Predictive Estimation of a Parameter We must wait the results of measurements ! 43
  44. 44. Applications in Quantum MetrologyGeneral scheme of the Retrodictive Estimation of a Parameter 44
  45. 45. Applications in Quantum MetrologyFisher Information and Cramér-Rao Bound ε ε + δε Relative distance Fisher Information 45
  46. 46. Applications in Quantum MetrologyFisher Information and Cramér-Rao Bound Fisher Information is the variation rate of probabilities under a variation of the parameterAny estimation is limited by the Cramér-Rao bound Number of repetitions 46
  47. 47. Applications in Quantum MetrologyIllustration : Estimation of a phase-space displacement Optimal Minimum noise influence Fisher Information is optimal only when the measurement is projective and ideal 47
  48. 48. Applications in Quantum MetrologyPredictive and Retrodictive Estimations The Quantum Cramér-Rao Bound is reached … 48
  49. 49. Applications in Quantum Metrology Retrodictive Estimation of a Parameter Projective but Non-Ideal ! Predictive RetrodictiveThe result “n” is uncertain even The target state is the most though we prepare its target probable preparation leading to state the result “n” 49
  50. 50. Conclusions and PerspectivesQuantum Behavior of Measurement ApparatusSome quantum properties of a measurement are only revealed by itsretrodicted state.Foundations of Quantum Theory• The predictive and retrodictive approaches of quantum physics havethe same mathematical foundations.• The reconstruction of retrodicted states from experimental dataprovides a real status for the retrodictive approach and its quantumstates.Exploring the use of non-classical measurementsRetrodictive version of a protocol can be more relevant than itspredictive version. 50

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