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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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Taoufik amri

Taoufik AMRI


Overview

Overview


Taoufik amri

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


Introduction

Introduction


The quantum world

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 !


The quantum world1

“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


The quantum world2

“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!


The quantum world3

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 ?


Quantum states of light

Quantum States of Light


Quantum states of light1

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


Quantum states of light2

“Decoherence”

Quantum States of Light

Discrete description of light : density matrix

Coherences

Populations

Properties required for calculating probabilities


Quantum states of light3

Classical Vacuum

Quantum Vacuum

Quantum States of Light

Continuous description of light : Wigner Function


Quantum states of light4

Quantum States of Light

Wigner representation of a single-photon state

Negativity is a signature of a strongly non-classical behavior !


Quantum states of light5

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


Quantum states and propositions

Quantum States and Propositions


Quantum states and propositions1

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).


    Quantum states and propositions2

    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


    Quantum states and propositions3

    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


    Quantum states and propositions4

    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


    Quantum states and propositions5

    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


  • Quantum states and propositions6

    Quantum States and Propositions

    Preparations

    Measurements

    Result “n”

    ?

    Choice “m”

    ?


    Quantum states and propositions7

    Quantum States and Propositions

    POVM Elements describing any measurement apparatus

    Quantum state corresponding to the proposition checked by the measurement

    Born’s Rule (1926)


    Quantum properties of measurements

    Quantum Properties of Measurements

    • T. Amri et al., Phys. Rev. Lett. 106, 020502 (2011).


    Properties of a measurement

    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 measurement1

    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 measurement2

    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 measurement3

    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


    Properties of a measurement4

    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


    Interlude

    Interlude


    The wigner s friend

    Amplification of Vital Signs

    The Wigner’s Friend

    Effects of an observation ?


    Quantum properties of human eyes

    Wigner representation of the POVM element describing the perception of light

    Quantum state retrodicted from the light perception

    Quantum properties of Human Eyes


    Effects of an observation

    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


    Interests of a non classical measurement

    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 !


    Detector of schr dinger s cat states of light

    Detector of “Schrödinger’s Cat” States of Light


    Detector of schr dinger s cat states of light1

    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 schr dinger s cat states of light2

    Detector of “Schrödinger’s Cat” States of Light

    Predictive Version : Conditional Preparation of SCS of light

    • A. Ourjoumtsev et al., Nature 448 (2007)


    Detector of schr dinger s cat states of light3

    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


    Detector of schr dinger s cat states of light4

    Detector of “Schrödinger’s Cat” States of Light

    Retrodicted States and Quantum Properties : Idealized Case

    Projective but Non-Ideal !


    Detector of schr dinger s cat states of light5

    Detector of “Schrödinger’s Cat” States of Light

    Retrodicted States and Quantum Properties : Realistic Case

    Non-classical Measurement


    Applications in quantum metrology

    Applications in Quantum Metrology


    Applications in quantum metrology1

    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


    Applications in quantum metrology2

    Applications in Quantum Metrology

    Estimation of a phase-space displacement

    Predictive probability of detecting the target state


    Applications in quantum metrology3

    Applications in Quantum Metrology

    General scheme of the Predictive Estimation of a Parameter

    We must wait the results of measurements !


    Applications in quantum metrology4

    Applications in Quantum Metrology

    General scheme of the Retrodictive Estimation of a Parameter


    Applications in quantum metrology5

    Relative distance

    Applications in Quantum Metrology

    Fisher Information and Cramér-Rao Bound

    Fisher Information


    Applications in quantum metrology6

    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


    Applications in quantum metrology7

    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 metrology8

    Applications in Quantum Metrology

    Predictive and Retrodictive Estimations

    The Quantum Cramér-Rao Bound is reached …


    Applications in quantum metrology9

    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”


    Conclusions and perspectives

    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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