detecting “schrödinger’s cat” states of light : insights from the retrodictive approach
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Detecting “Schrödinger’s Cat” States of Light : Insights from the Retrodictive Approach. Taoufik AMRI and Claude FABRE Quantum Optics Group, Laboratoire Kastler Brossel, France. INTERNATIONAL CONFERENCE ON QUANTUM INFORMATION OTTAWA, JUNE 2011. Introduction. Preparations. Measurements. - PowerPoint PPT PresentationTRANSCRIPT
Detecting “Schrödinger’s Cat” States of Light : Insights from the Retrodictive Approach
Taoufik AMRI and Claude FABRE
Quantum Optics Group,Laboratoire Kastler Brossel, France
INTERNATIONAL CONFERENCE ON QUANTUM INFORMATION
OTTAWA, JUNE 2011
Introduction
Result “n”
?
Preparations Measurements
Choice “m”
?
Born’s Rule (1926)
Predictive and Retrodictive Approaches
Quantum state corresponding to the property checked by the measurement
POVM Elements describing any measurement apparatus
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 !
Properties of a measurement
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 !
1
Gaussian Entanglement
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 !
Properties of a measurement
Preparation operator
Detector of “Schrödinger’s Cat” States of Light
Detector of “Schrödinger’s Cat” States of Light
Scheme of the detector
Non-classical Measurements
Projective but Non-Ideal !
Photon counting
Squeezed Vacuum
Detector of “Schrödinger’s Cat” States of Light
Retrodicted States and Quantum Properties : Idealized Case
Projective but Non-Ideal !
Applications in Quantum Metrology
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
Relative distance
Fisher Information
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 retrodictive probabilities under a variation of the parameter
Number of repetitions
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 !
Applications in Quantum Metrology
Predictive and Retrodictive Estimations of a phase-space displacement
The Quantum Cramér-Rao Bound is reached …
Conclusions and Perspectives
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.
T. Amri et al., Phys. Rev. Lett. 106, 020502 (2011).
T. Amri et al., in preparation (2011).
Acknowledgements
Many thanks to Stephen M. Barnett and Luiz Davidovichfor fruitful discussions !
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 !”
Detector of “Schrödinger’s Cat” States of Light
Predictive Version : Conditional Preparation of SCS of light
A. Ourjoumtsev et al., Nature 448 (2007)
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
Retrodictive Estimation of a Parameter
Maximally mixed !
Von Neumann
Entropy
Concavity
No Pain, No Gain !