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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« Schrödinger’s Cat » States
The Wigner’s Friend
Quantum Properties of Measurements
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 !
Light behaves like a wave or/and a packet
Wigner representation of a single-photon state
Negativity is a signature of a strongly non-classical behavior !
“Schrödinger’s Cat” States of Light (SCSL)
Quantum superposition of two incompatible states of light
Wigner representation of the SCSL
Interference structure is the signature of non-classicality
P. Busch, Phys. Rev. Lett. 91, 120403 (2003).
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
POVM Elements describing any measurement apparatus
Quantum state corresponding to the proposition checked by the measurement
Born’s Rule (1926)
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 !
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 !
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 !
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 !
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 perceptionQuantum properties of Human Eyes
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
Let us imagine a detector of “Schrödinger’s Cat” states of light
Effects of this measurement (projection postulate)
Quantum coherences are preserved !
“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
Predictive Version : Conditional Preparation of SCS of light
Retrodictive Version : Detector of “Schrödinger’s Cat” States
Projective but Non-Ideal !
Retrodicted States and Quantum Properties : Idealized Case
Projective but Non-Ideal !
Retrodicted States and Quantum Properties : Realistic Case
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
Estimation of a phase-space displacement
Predictive probability of detecting the target state
General scheme of the Predictive Estimation of a Parameter
We must wait the results of measurements !
General scheme of the Retrodictive Estimation of a Parameter
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
Illustration : Estimation of a phase-space displacement
Minimum noise influence
Fisher Information is optimal only when the measurement is projective and ideal
Predictive and Retrodictive Estimations
The Quantum Cramér-Rao Bound is reached …
Quantum Behavior of Measurement Apparatus
Some quantum properties of a measurement are only revealed by its retrodicted state.
Exploring the use of non-classical measurements
Retrodictive version of a protocol can be more relevant than its predictive version.