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### Quantum Properties of Measurements

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

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 !

“dead”

The Quantum WorldAND ?

- The cat is actually a detector of the atom’s state
- Result “dead” : the atom is disintegrated
- Result “alive” : the atom is excited

Entanglement

“dead”

The Quantum WorldAND ?

OR !

Quantum Decoherence : Interaction with the environment leads to a transition into a more classical behavior, in agreement with the common intuition!

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

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 Light

Discrete description of light : density matrix

Coherences

Populations

Properties required for calculating probabilities

Quantum States of Light

Wigner representation of a single-photon state

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

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

- 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 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 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 state distributes the physical properties represented by hermitian and positive operators

Statement of Gleason-Busch’s Theorem

Quantum States and PropositionsReconstruction of a quantum state

Quantum state

Exhaustive set of propositions

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 Propositions

POVM Elements describing any measurement apparatus

Quantum state corresponding to the proposition checked by the measurement

Born’s Rule (1926)

- 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 !

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 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 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 measurement

Detectivity of a measurement

Probability of detecting the target state

Probability of detecting the retrodicted state

Probability of detecting a target state

Wigner representation of the POVM element describing the perception of light

Quantum state retrodicted from the light perception

Quantum properties of Human EyesEffects 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

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 “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 “Schrödinger’s Cat” States of Light

Predictive Version : Conditional Preparation of SCS of light

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

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 “Schrödinger’s Cat” States of Light

Retrodicted States and Quantum Properties : Idealized Case

Projective but Non-Ideal !

Detector of “Schrödinger’s Cat” States of Light

Retrodicted States and Quantum Properties : Realistic Case

Non-classical Measurement

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 Metrology

Estimation of a phase-space displacement

Predictive probability of detecting the target state

Applications in Quantum Metrology

General scheme of the Predictive Estimation of a Parameter

We must wait the results of measurements !

Applications in Quantum Metrology

General scheme of the Retrodictive Estimation of a Parameter

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 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 Metrology

Predictive and Retrodictive Estimations

The Quantum Cramér-Rao Bound is reached …

Retrodictive

Applications in Quantum MetrologyRetrodictive 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

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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