detecting “schrödinger’s cat” states of light : insights from the retrodictive approach

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 !