N - Faculty of Physics University of Warsaw

Quantum metrology in realistic scenarios
Jan Kolodynski, Rafal Demkowicz-Dobrzanski
Institute of Theoretical Physics, University of Warsaw, Poland
Sketch of the proof of the CS/QS methods
Introduction
Quantum metrology offers an enhanced performance in experiments such as gravitational-wavesdetection, as well as atomic clocks, magnetometry and spectroscopy. Classically, as dictated by the
“central limit theorem” of statistics, precision in these experiments is limited by the 1/N½ Shot Noise
factor with N being the number of particles (photons, atoms) employed. In principle, quantum features
such as entanglement or squeezing, yielding “stronger-than-classical” inter-particle correlations, allow to
overcome this limit and attain the so-called Heisenberg Scaling of precision – 1/N. We show that when
even infinitesimal uncorrelated noise is taken into account, the maximal possible quantum
enhancement in the asymptotic limit of N → ∞ amounts to a constant factor rather than quadratic
improvement [2]. We provide efficient tools for deriving bounds on precision in the regimes of finite and
infinite N, which are determined only by the form of quantum channel representing the single-particle
evolution. In this poster, we present these tools applied to phase estimation models with decoherence
of metrologically relevant types: depolarization, dephasing, particle losses and spontaneous emission.
Their application to other models such as decoherence-strength and frequency estimation may be found
in [1], where in the latter case they have been also recently shown to certify the super-classical 1/N^(5/6)
precision scaling to be the ultimate one allowed by the perpendicular-dephasing noise [5].
General quantum estimation scheme .
output state
input state
⁞
⁞
⁞
estimator
SHOT NOISE SCALING!
But how to find for a given channel the necessary
which lead to the tightest bounds i.e. the smallest
METROLOGICAL SCENARIO [3]:
quantum
measurement
quantum C-R bound
• Consider (entangled) pure input states of
N atoms/photons:
.
• Assume the evolution of each particle to be
independent  N independent channels.
• Design a strategy of estimating as close
as possible to , which gives on average
the minimal error:
.
• Seek for the optimal: input state,
measurement scheme and the estimator.
“The” ultimate theoretical bound on precision
(QS) or, forcing it to be diagonal,
(CS)
and
respectively [1,2]?
The optimal local Classical Simulation (CS)
• The -parameterised family of channels
forms a curve in the convex set of all Completely
Positive Trace Preserving maps,
, of common input and output spaces.
• The QFI of the output state
at given
depends only on single channel’s
and
.
Hence, the QFI calculated for channel
coincides with the QFIs of all channels
that are locally
equivalent, i.e. such that for
:
• Using this fact and the convexity of space, one can prove that the optimal simulation for noisy unitary
rotation corresponds to a two-point
comprising of channels
lying at the two outermost points
along the tangent at
[1,2]:
Then,
with
The ultimate upper (lower) bound on precision (error) is given by the quantum Cramer-Rao bound [4]:
where FQ is the
IMPLYING ASYMPTOTIC SN SCALING :
Quantum Fisher Information (QFI)
Is this bound theoretically saturable (hence, possibly achievable in an experiment)? YES, but… :
• An optimal POVM exists, but may be very hard to find. (and realize in an experiment…)
• Only when the estimation is local, i.e. we estimate the deviations of from a known value
, so
that the prior distribution of is fully localized (
). (real ”priors” can only do worse…)
• However, the locality condition may always be met if we allow to make infinitely many repetitions of
the experiment, k → ∞. (then an example of an efficient estimator is the max. likelihood one…)
N.B. For the bound to be practical Fcl < ∞,
is
i.e. all extremal channels, in particular unitary ones
FURTHERMORE, QFI IS VERY HARD TO MAXIMIZE OVER THE INPUT FOR A GENERAL MIXED OUTPUT.
The impact of uncorrelated noise
e.g.
PHASE ESTIMATION IN ATOMIC SPECTROSCOPY WITH QUBITS AND UNCORRELATED DEPHASING [1,2]:
must be regular in the tangential “direction of
, are
“ at
, that is
-non-extremal
-extremal and the CS method does not apply.
The Channel Extension (CE) method
By allowing the channel to act in a trivial
way on an extended input space, one can
only improve the precision of estimation.
This leads to an upper bound on QFI that goes around the input state optimization and is defined via
the minimization over channel Kraus representations
in
:
In the
asymptotic
limit of
N→∞
Required
Ultimate
number
Input
precision
dictated locally of reps,
by the QFI
k
GHZ strategy
SS strategy
finite-N bound
asymptotic bound
noiseless (η=1)
classically
quantum
1
HS
worse than
classical region
∞
1
quantum
1
SN
Importantly, any channel that admits a Kraus representation for
which the second term vanishes asymptotically scales like SN !!
In the asymptotic limit of N → ∞, the optimal CE bound then reads
where the Hermitian matrices h are any generators of unitary Kraus operators rotations u,
that satisfy the necessary condition:
• A numerical minimization over h may be efficiently performed by recasting the problem into a
Semi-Definite Programming optimization task.
• The optimal local QS corresponds then to a further constraint
.
• Relaxing the
condition we get an even tighter bound that varies smoothly with N, so that it
may be also efficiently applied in the finite N regime. By rewriting it similarly to the asymptotic case
in an SDP form, we obtain the more effective finite-N CE method.
noisy (η<1)
classically
denotes the operator norm and
Achievable precision vs. bounds for η=0.9 dephasing
prohibited better
than Heisenberg
Scaling region
, GHZ – Greenberger-Horne-Zeilinger & SS – Spin-Squeezed states
Examples and asymptotic results
Efficient tools to upper-bound the precision
Depolarization
Dephasing
Losses (e.g. interferometry)
Spontaneous emission
inside the set of quantum channels
full rank → -non-extremal
on the boundary, non-extremal,
-non-extremal
on the boundary, non-extremal,
but -extremal
on the boundary,
extremal
o Classical Simulation (CS) method [1]
• Stems from the possibility to simulate locally quantum channels via classical probabilistic mixtures:
• Optimal simulation corresponds to a simple, intuitive, geometric interpretation.
• Allows to simply derive the asymptotic bound via the Choi-Jamiolkowski representation of
• Proves that almost all (including full rank) channels asymptotically scale classically.
.
o Quantum Simulation (QS) method [1]
• Generalization of local CS by allowing the
parameter-dependent state to be non-diagonal.
• Proves asymptotic shot noise for a wider class of channels (e.g. phase-estimation with loss). Optimal
local QS may be efficiently found numerically by means of Semi-Definite Programming.
N/A
o Channel Extension (CE) method [1,2]
• Stems from the idea that by extending each channel
with an auxiliary ancilla one can only improve the precision.
• Applies to even wider class of quantum channels (e.g. phase-estimation with noise due to spontaneous
emission) and provides the tightest asymptotic bounds. Numerically, it corresponds to the same
Semi-Definite Program as in QS, but with some of the constraints relaxed, which may be improved
and applied also to establish the bound for finite N – finite-N CE method.
• The finite-N CE method may be applied to channels which form varies with N, so that
, what
may be utilized to ”beat” the Shot Noise despite the decoherence, e.g. [5].
N/A
References
1.
2.
3.
4.
5.
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V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
S. L. Braunstein, C. M. Caves, Phys. Rev. Lett. 72, 3439-3443 (1994).
R. Chaves, J. B. Brask, M. Markiewicz, J. Kolodynski, A. Acin, Phys. Rev. Lett. 111, 120401 (2013).