Download Quantum States and Propositions

Taoufik AMRI
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Overview
Chapter I
Quantum Description
of Light
Chapter II
Quantum Protocols
Chapter III
Quantum States
and Propositions
Chapter V
Chapter VI
Experimental
Illustration
Detector of
The Wigner’s
Friend
« Schrödinger’s Cat » States
Of Light
Chapter IV
Quantum Properties of
Measurements
Interlude
Chapter VII
Application to
Quantum Metrology
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Introduction
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 !
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The Quantum World
AND ?
“dead”
“alive”
The cat is actually a detector of the atom’s state
• Result “dead” : the atom is disintegrated
• Result “alive” : the atom is excited
Entanglement
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The Quantum World
AND
OR !?
“dead”
“alive”
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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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
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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Quantum States of Light
Discrete description of light : density matrix
“Decoherence”
Coherences
Populations
Properties required for calculating probabilities
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Quantum States of Light
Continuous description of light : Wigner Function
Classical0 Vacuum
Quantum Vacuum
  0, 0 
0
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Quantum States of Light
Wigner representation of a single-photon state
1 
Negativity is a signature of a strongly non-classical behavior !
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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
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Quantum States and Propositions
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).
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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
Pˆn  n n
n=3
From an exhaustive set of propositions
ˆ
ˆ
P

1
 n
n
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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
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Quantum States and Propositions
Reconstruction 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
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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
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Quantum States and Propositions
Preparations
Measurements
Choice “m”
Result “n”
?
?
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Quantum States and Propositions
POVM Elements describing
any measurement apparatus
Quantum state corresponding
to the proposition checked by
the measurement
Born’s Rule (1926)
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Quantum Properties of Measurements
 T. Amri et al., Phys. Rev. Lett. 106, 020502 (2011).
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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Properties of a measurement
Non-classicality of a measurement
It corresponds to the non-classicality of its retrodicted state
Gaussian Entanglement
 1
Quantum state conditioned on
an expected result “n”
Necessary condition !
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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 !
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Properties of a measurement
Fidelity of a measurement
Overlap between the retrodicted state and a target state
Meaning in the retrodictive approach
Proposition operator
For faithful measurements, the most probable preparation
is the target state !
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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
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Interlude
The Wigner’s Friend
Amplification of Vital Signs
Effects of an observation
?
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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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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
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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 !
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Detector of “Schrödinger’s Cat” States of Light
Detector of “Schrödinger’s Cat” States of Light
“We can measure the system with
MainaIdea
:
given property, but we can also
prepare
the systemVersion
with this same
Predictive Version VS
Retrodictive
property !”
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Detector of “Schrödinger’s Cat” States of Light
Predictive Version : Conditional Preparation of SCS of light
 A. Ourjoumtsev et al., Nature 448 (2007)
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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
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Detector of “Schrödinger’s Cat” States of Light
Retrodicted States and Quantum Properties : Idealized Case
Projective but Non-Ideal !
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Detector of “Schrödinger’s Cat” States of Light
Retrodicted States and Quantum Properties : Realistic Case
Non-classical Measurement
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Applications in Quantum Metrology
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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Applications in Quantum Metrology
Estimation of a phase-space displacement
Predictive probability
of detecting the
target state
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Applications in Quantum Metrology
General scheme of the Predictive Estimation of a Parameter
We must wait the results of measurements !
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Applications in Quantum Metrology
General scheme of the Retrodictive Estimation of a Parameter
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Applications in Quantum Metrology
Fisher Information and Cramér-Rao Bound

  
Relative distance
Fisher Information
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Applications in Quantum Metrology
Fisher Information and Cramér-Rao Bound
Fisher Information is the variation rate of probabilities
under a variation of the parameter
Any estimation is limited by the Cramér-Rao bound
Number of repetitions
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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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Applications in Quantum Metrology
Predictive and Retrodictive Estimations
The Quantum Cramér-Rao Bound is reached …
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Applications in Quantum Metrology
Retrodictive Estimation of a Parameter
Projective but Non-Ideal !
Predictive
The result “n” is uncertain even
though we prepare its target
state
Retrodictive
The target state is the most
probable preparation leading to
the result “n”
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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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