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1 Taoufik AMRI

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Overview

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

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Introduction

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5 The Quantum World The Schrödingers 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 !

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6 The Quantum World The cat is actually a detector of the atoms state Result dead : the atom is disintegrated Result alive : the atom is excited dead alive AND ? Entanglement

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

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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 ?

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Quantum States of Light

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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

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

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

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

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14 Quantum States of Light Schrödingers 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

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Quantum States and Propositions

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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 Gleasons 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).

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17 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 Quantum States and Propositions n=3

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18 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 Quantum States and Propositions Statement of Gleason-Bushs Theorem

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

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20 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 Quantum States and Propositions

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21 Quantum States and Propositions Result n ? PreparationsMeasurements Choice m ?

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

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

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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 !

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

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26 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

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27 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 ! Properties of a measurement Proposition operator

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

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Interlude

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30 The Wigners Friend Effects of an observation ? Amplification of Vital Signs

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

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

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33 Let us imagine a detector of Schrödingers Cat states of light Effects of this measurement (projection postulate) Interests of a non-classical measurement Quantum coherences are preserved ! AND

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Detector of Schrödingers Cat States of Light

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

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36 Detector of Schrödingers Cat States of Light Predictive Version : Conditional Preparation of SCS of light A. Ourjoumtsev et al., Nature 448 (2007)

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37 Detector of Schrödingers Cat States of Light Retrodictive Version : Detector of Schrödingers Cat States Non-classical Measurements Projective but Non-Ideal ! Photon counting Squeezed Vacuum

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38 Detector of Schrödingers Cat States of Light Retrodicted States and Quantum Properties : Idealized Case Projective but Non-Ideal !

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39 Detector of Schrödingers Cat States of Light Retrodicted States and Quantum Properties : Realistic Case Non-classical Measurement

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Applications in Quantum Metrology

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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

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

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

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

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

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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

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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

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

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

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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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