1. INTERNATIONAL CONFERENCE ON QUANTUM INFORMATION
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

2. Introduction
Preparations
Measurements
Choice “m” ?
Result “n” ?

3. Predictive and Retrodictive Approaches
POVM Elements describing any measurement apparatus
Quantum state corresponding to the property checked by the measurement
Born’s Rule (1926)

4. Quantum Properties of Measurements
T. Amri et al., Phys. Rev. Lett. 106, (2011).

5. 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 !

6. 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 !

7. 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 !

8. 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 !
Preparation operator

9. Detector of “Schrödinger’s Cat” States of Light

10. Detector of “Schrödinger’s Cat” States of Light
Scheme of the detector
Photon counting
Non-classical Measurements
Projective but Non-Ideal !
Squeezed Vacuum

11. Detector of “Schrödinger’s Cat” States of Light
Retrodicted States and Quantum Properties : Idealized Case
Projective but Non-Ideal !

12. Applications in Quantum Metrology

13. Applications in Quantum Metrology
General scheme of the Predictive Estimation of a Parameter
We must wait the results of measurements !

14. Applications in Quantum Metrology
General scheme of the Retrodictive Estimation of a Parameter

15. Applications in Quantum Metrology
Fisher Information and Cramér-Rao Bound
Relative distance
Fisher Information

16. 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

17. Applications in Quantum Metrology
Retrodictive Estimation of a Parameter
Projective but Non-Ideal !
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”

18. Applications in Quantum Metrology
Predictive and Retrodictive Estimations of a phase-space displacement
The Quantum Cramér-Rao Bound is reached …

19. Conclusions and Perspectives
Quantum Behavior of Measurement Apparatus
Some quantum properties of a measurement are only revealed by its retrodicted state.
T. Amri et al., Phys. Rev. Lett. 106, (2011).
Exploring the use of non-classical measurements
Retrodictive version of a protocol can be more relevant than its predictive version.
T. Amri et al., in preparation (2011).

20. Acknowledgements Many thanks to Stephen M. Barnett and Luiz Davidovich
for fruitful discussions !

22. 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

23. Detector of “Schrödinger’s Cat” States of Light
Predictive Version : Conditional Preparation of SCS of light
A. Ourjoumtsev et al., Nature 448 (2007)

24. 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

25. Applications in Quantum Metrology
Retrodictive Estimation of a Parameter
No Pain, No Gain !
Maximally mixed !
Von Neumann
Entropy
Concavity