1. Quantum Feedback Control of Entanglement in collaboration with H. M. Wiseman, Griffith University, Brisbane AU University of Camerino, Italy Stefano Mancini

2. The problem of system control Systems do not always behave the way we would! Add a “controller” to the system to improve its behavior! “Open loop” controllers act on the system without obtaining information, e.g. “bang bang”; “Closed loop” controllers act on the system after obtaining information, e.g. feedback; Quantum feedback arises when the environment of an open quantum system is engineered so that the information lost into that environment comes back to affect the system; The feedback loop consists of a quantum system, a classical detector and a classical actuator.

3. Outline An intuitive approach for Gaussian states An intuitive approach for Gaussian states General theory of feedback control in linear quantum systems General theory of feedback control in linear quantum systems Optimal quantum control Optimal quantum control Application to NDPO Application to NDPO Local vs non-local feedback control actions Local vs non-local feedback control actions Coherence recovery (i.e. purity of controlled state) Coherence recovery (i.e. purity of controlled state) Related issues Related issues Conclusions Conclusions

4. An intuitive approach for Gaussian States Unconditional generation of single mode squeezing What is about two-mode squeezing control, i.e. entanglement control? p q

5. General theory Master eq derived under the sys-env weak coupling assumption. Then, it is possible to measure the environment continually. Monitoring the bath yields information about the system, producing a stochastic conditioned state  c that on average reproduces  The master eq is unravelled into stochastic quantum trajectories, with different measurements leading to different unravellings.

6. Continuous Markovian unravelling In the limit of infinitesimal jumps occuring infinitely frequently a diffusive unravelling results, i.e. an evolution for  c that is continuous and Markovian Infinitesimal complex Wiener increments

7. Y is a symmetric complex matrix constrained by Measurement Current (measurement results upon which the evolution of  c is conditioned) Unravelling matrix

8. Gaussian states Feedback Hamiltonian as far as u is related to J Moments equations Linear systems (with N degrees of freedom)

9. Conditional Linear Dynamics Denote as W U a stabilizing solution V c ss Additional noisy term  dw Additional positive term The set of W U s (from detectable unravellings) is determined by

10. Feedback control In fb control, u(t) depends on the history of the measurement record y(s) for s<t. The typical aim over some interval time is to minimize the expected value of a cost function (Bayesian feedback) Direct feedback (Markovian feedback) entails making the time dependent H linear in the instataneous output y(t) LQG:

11. Optimal feedback control In case of time-independent cost functions, we wish to minimize m=E[h] in the steady state Bayesian feedback, allows to set, hence SemiDefinite Program: Minimize m with constraints on W U with no control cost!

12. A proper choice of BF, allows to set at ss, hence SemiDefinite Program: Minimize m with constraints on W U What is about direct feedback? y has unbounded variation, so doing direct fb is no less onerous than doing optimal Bayesian fb (with no control cost)

13. The case of NDPO (amplitude damping) The system’s dynamics leads to the steady state covariance matrix

14. Degree of entanglement is the lowest symplectic eigenvalue of the PT Gaussian state V L 

15. Optimal feedback control We wish to make q 1 = q 2 and p 1 = -p 2 like in EPR state Applying the semidefinite program

16. L 

17. What kind of measurements? Measurement of q 1 -q 2 and p 1 +p 2 ( non-local! ) Given W U it is possible to find the (actually optimal) unravelling U, hence C

18. Local feedback action Local Operations (Measurements & Drivings) + Classical Communication Env 1 System 1 Env 2 System 2

19. Local feedback action: single quadrature measurements Due to the symmetry of NDPO’s steady state, there are no preferred quadratures to be locally measured provided their angles sum up to  By measuring qs quadratures, the feedback action only makes sense on the conjugate quadratures. Then, we choose BF so to get the most general quadratic form of local feedback Hamiltonian like

20. As a consequence of feedback action Maximize the Log Neg over feedback parameters +, - with the stability constraint A’( +, - )<0

21. Distingushing different cases

22. Max of Log Neg over A) Max of Log Neg over

23. Local feedback action: joint quadratures measurements Choose the feedback action BF so that

24. Max of Log Neg over

25. Coherence recovery ?   are the symplectic eigenvalues of the Gaussian steady state

26. From the results about Log Neg, and purity we may conclude that is the optimal feedback LOCC for NDPO

27. A possible experiment The feedback action is extremly robust against non unit efficiency: An overall  =0.7 gives 250% enhancement of entanglement close to the threshold

28. Related issues The optimal local feedback action shows that Gaussian LOCC are able to distill entanglement if they continuously happen when the interaction is “on”; generalization of the results by Eisert, et al. PRL (2002); Fiurasek PRL (2002)? Quantum Feedback allows us to recover quantum information lost into environments; for what kind of channels is that possible (F. Buscemi et al. PRL 2005)? Extracting classical information from the environment and exploiting it as additional amount of side information may improve quantum communication performances (Gregoratti & Werner, 2003; Hayden & King 2004)!

29. Conclusions Entanglement can be controlled via quantum feedback; Semidefinite program for optimal control in LS; Optimal local feedback for NDPO; Find a general recipe for LS? What is about out of feedback loop fields? What is about collecting information from one environment to control information leakage into another? Quantum Feedback: an arena still to be explored!

30. S. M. & H. W., “Optimal control of entanglement via quantum feedback”, PRA 75, 012330 (2007). S. M., “Markovian feedback to control CV entanglement”, PRA 73, 010304(R) (2006). H. W. & A. Doherty, “Optimal unravellings for feedback control in linear quantum systems”, PRL 94, 070405 (2005). J. Wang & S. M., “Towards feedback control of entanglement”, EPJD 32, 257 (2005). H. W., S. M. & J. Wang, “Bayesian feedback vs Markovian feedback”, PRA 66, 013807 (2002). L. Thomsen, S. M. & H. W., “Spin squeezing via quantum feedback”, PRA 65, 061801(R) (2002). H. W. & G. Milburn, “Squeezing via feedback”, PRA 49, 1350 (1994). Some References