Taoufik AMRI
50
1 Taoufik AMRI
description
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 - PowerPoint PPT Presentation
Transcript of Taoufik AMRI
1
Taoufik AMRI
Overview
3
Chapter II
Quantum Protocols
Chapter III
Quantum States
and Propositions
Chapter VI
Detector of
« Schrödinger’s Cat » States
Of Light
Chapter IV
Quantum Properties of Measurements Chapter VII
Application to
Quantum MetrologyInterlude
Chapter V
Experimental Illustration
Chapter I
Quantum Description
of Light
The Wigner’s Friend
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
The Quantum World
The cat is actually a detector of the atom’s state
• Result “dead” : the atom is disintegrated
• Result “alive” : the atom is excited
“dead”“alive”
AND ?
Entanglement
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 !
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 ?
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
Quantum States of Light
Discrete description of light : density matrix
PopulationsCoherences
“Decoherence”
Properties required for calculating probabilities
12
Quantum States of Light
0
Continuous description of light : Wigner Function
Classical Vacuum Quantum Vacuum
0,0
0
13
Quantum States of LightWigner representation of a single-photon state
1
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
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
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
n̂P n nn=3
ˆˆ 1nn
P
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-Bush’s Theorem
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-Busch’s Theorem
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
21
Quantum States and Propositions
Result “n”
?
Preparations Measurements
Choice “m”
?
22
Born’s Rule (1926)
Quantum States and Propositions
Quantum state corresponding to the proposition checked by the measurement
POVM Elements describing any measurement apparatus
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 measurementNon-classicality of a measurement
It corresponds to the non-classicality of its retrodicted state
Quantum state conditioned on an expected result “n” Necessary condition !
1
Gaussian Entanglement
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
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
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
Interlude
30
The Wigner’s Friend
Effects of an
observation ?
Amplification of Vital Signs
31
Quantum properties of Human Eyes
Wigner representation of the POVM element describing the perception of light
Quantum state retrodicted from the light perception
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
33
Let us imagine a detector of “Schrödinger’s Cat” states of light
Effects of this measurement (projection postulate)
Interests of a non-classical measurement
Quantum coherences are preserved !
“AND”
Detector of “Schrödinger’s Cat” States of Light
35
Detector of “Schrödinger’s 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 !”
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
Non-classical Measurements
Projective but Non-Ideal !
Photon counting
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
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
Applications in Quantum Metrology
Fisher Information and Cramér-Rao Bound
Relative distance
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
Applications in Quantum Metrology
Retrodictive Estimation of a Parameter
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”
Projective but Non-Ideal !
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.
