Quantum Feedback Control of Entanglement

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Quantum Feedback Control of Entanglement
University of Camerino, Italy

• Stefano Mancini

in collaboration with H. M. Wiseman, Griffith
University, Brisbane AU
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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.

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Outline

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

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An intuitive approach for Gaussian States
Unconditional generation of single mode squeezing
p
q
What is about two-mode squeezing control, i.e.
entanglement control?
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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
  rc that on average reproduces r.
• The master eq is unravelled into stochastic
  quantum trajectories, with different measurements
  leading to different unravellings.

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Continuous Markovian unravelling
In the limit of infinitesimal jumps occuring
infinitely frequently a diffusive unravelling
results, i.e. an evolution for rc that is
continuous and Markovian
Infinitesimal complex Wiener increments
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Y is a symmetric complex matrix constrained by
Unravelling matrix
Measurement Current (measurement
results upon which the evolution of rc is
conditioned)
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Linear systems (with N degrees of freedom)
Feedback Hamiltonian as far as u is related to J
Gaussian states
Moments equations
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Conditional Linear Dynamics
Additional noisy term ? dw
Additional positive term
Denote as WU a stabilizing solution Vcss
The set of WUs (from detectable unravellings)
is determined by
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Feedback control
In fb control, u(t) depends on the history of the
measurement record y(s) for sltt. The typical aim
over some interval time is to minimize the
expected value of a cost function
(Bayesian feedback)
LQG
Direct feedback (Markovian feedback) entails
making the time dependent H linear in the
instataneous output y(t)
LQG
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Optimal feedback control
In case of time-independent cost functions, we
wish to minimize mEh in the steady state
with no control cost!
Bayesian feedback, allows to set , hence
SemiDefinite Program Minimize m with constraints
on WU
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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)
A proper choice of BF, allows to set at
ss, hence
SemiDefinite Program Minimize m with constraints
on WU
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The case of NDPO
(amplitude damping)
The systems dynamics leads to the steady state
covariance matrix
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Degree of entanglement
is the lowest symplectic eigenvalue of the PT
Gaussian state V
L
?
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Optimal feedback control
We wish to make q1 q2 and p1 -p2 like in EPR
state
Applying the semidefinite program
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L
?
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What kind of measurements?
Given WU it is possible to find the (actually
optimal) unravelling U, hence C
Measurement of q1-q2 and p1p2 ( non-local! )
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Local feedback action
System 1
System 2
Env 1
Env 2
Local Operations (Measurements Drivings)
Classical Communication
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Local feedback action single quadrature
measurements
Due to the symmetry of NDPOs steady state, there
are no preferred quadratures to be locally
measured provided their angles sum up to p
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
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As a consequence of feedback action
Maximize the Log Neg over feedback parameters ?,
?- with the stability constraint A(?, ?-)lt0
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Distingushing different cases
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Max of Log Neg over ?
Max of Log Neg over ?
A)
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Local feedback action joint quadratures
measurements
Choose the feedback action BF so that
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Max of Log Neg over ?
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Coherence recovery ?
?? are the symplectic eigenvalues of the Gaussian
steady state
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From the results about Log Neg, and purity we may
conclude that
is the optimal feedback LOCC for NDPO
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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
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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)!

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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!

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Some References

• 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).