QUANTUM METROLOGY IN REALISTIC SCENARIOS
Janek Kolodynski
Faculty of Physics, University of Warsaw, Poland
PART I – QUANTUM METROLOGY WITH UNCORRELATED NOISE
•
Rafal Demkowicz-Dobrzanski, JK, Madalin Guta –
”The elusive Heisenberg limit in quantum metrology”, Nat. Commun. 3, 1063 (2012).
•
JK, Rafal Demkowicz-Dobrzanski –
”Efficient tools for quantum metrology with uncorrelated noise”, New J. Phys. 15, 073043 (2013).
PART II – BEATING THE SHOT NOISE LIMIT DESPITE THE UNCORRELATED NOISE
•
Rafael Chaves, Jonatan Bohr Brask, Marcin Markiewicz, JK, Antonio Acin –
”Noisy metrology beyond the Standard Quantum Limit”, Phys. Rev. Lett. 111, 120401 (2013).
(CLASSICAL) QUANTUM METROLOGY
ATOMIC SPECTROSCOPY: “PHASE” ESTIMATION
N two-level atoms (qubits) in a separable state
unitary rotation
output state
(separable)
uncorrelated measurement – POVM:
independent processes
unbiased
estimator
with
shot noise
Classical Fisher Information
Quantum Fisher Information
(measurement independent)
always saturable
in the limit N → ∞
QUANTUM FISHER INFORMATION
• QFI of a pure state
:
• QFI of a mixed state
:
– Symmetric Logarithmic Derivative
Evaluation requires eigen-decomposition of the density matrix, which size grows exponentially, e.g. d=2N for N qubits.
• Geometric interpretation – QFI is a local quantity
Two PDFs
Two q. states
:
:
The necessity of the asymptotic limit of repetitions N → ∞ is a consequence of locality.
• Purification-based definition of the QFI
[Escher et al, Nat. Phys. 7(5), 406 (2011)] –
[Fujiwara & Imai, J. Phys. A 41(25), 255304 (2008)]
–
(IDEAL) QUANTUM METROLOGY
ATOMIC SPECTROSCOPY: “PHASE” ESTIMATION
GHZ state
unitary rotation
output state
measurement on all probes:
estimator:
repeating the procedure k times
• Atoms behave as a “single object” with N times greater phase
change generated (same for the N00N state and photons).
• N → ∞ is not enough to achieve the ultimate precision.
“Real” resources are kN and in theory we require k → ∞.
Heisenberg limit
• A source of uncorrelated decoherence acting independently on each atom will “decorrelate” the atoms, so that
we may attain the ultimate precision in the N → ∞ limit with k = 1, but at the price of scaling …
(REALISTIC) QUANTUM METROLOGY
ATOMIC SPECTROSCOPY: “PHASE” ESTIMATION
with dephasing
noise added:
optimal pure state
In practise need to optimize for
particular model and N
distorted
unitary rotation
mixed
output state
measurement on all probes
complexity of computation
grows exponentially with N
estimator
OBSERVATIONS
• Infinitesimal uncorrelated disturbance forces asymptotic
(classical) shot noise scaling.
• The bound then “makes sense” for a single shot (k = 1).
• Does this behaviour occur for decoherence of a generic type ?
constant factor improvement
over shot noise
achievable with k = 1 and spinsqueezed states
The properties of the single use of a channel –
– dictate the
asymptotic ultimate scaling of precision.
EFFICIENT TOOLS FOR DETERMINING
LOWER-BOUNDING
D
In order of their power and range of applicability:
o Classical Simulation (CS) method
o Stems from the possibility to locally simulate quantum channels via classical probabilistic mixtures:
o Optimal simulation corresponds to a simple, intuitive, geometric representation.
o Proves that almost all (including full rank) channels asymptotically scale classically.
o Allows to straightforwardly derive bounds (e.g. dephasing channel considered).
o Quantum Simulation (QS) method
o Generalizes the concept of local classical simulation, so that the parameter-dependent state does not
need to be diagonal:
o Proves asymptotic shot noise also for a wider class of channels (e.g. optical interferometer with loss).
o Channel Extension (CE) method
o Applies to even wider class of channels, and provides the tightest lower bounds on
(e.g. amplitude damping channel)
.
o Efficiently calculable numerically by means of Semi-Definite Programming even for finite N !!!.
CLASSICAL/QUANTUM SIMULATION OF A CHANNEL
as a Markov chain:
CLASSICAL/QUANTUM SIMULATION OF A CHANNEL
as a Markov chain:
CLASSICAL/QUANTUM SIMULATION OF A CHANNEL
as a Markov chain:
CLASSICAL/QUANTUM SIMULATION OF A CHANNEL
as a Markov chain:
shot noise scaling !!!
But how to verify if this construction is possible and what
is the optimal (“worse”) classical/quantum simulation
giving the tightest lower bound on the ultimate precision?
THE ”WORST” CLASSICAL SIMULATION
The set of quantum channels (CPTP maps) is convex
Locality:
Quantum Fisher Information at a
given :
depends only on:
We want to construct the ”local classical simulation” of the form:
The ”worst” local classical simulation:
Does not work for
-extremal channels, e.g unitaries
.
GALLERY OF DECOHERENCE MODELS
CONSEQUENCES ON REALISTIC SCENARIOS
“PHASE ESTIMATION”
IN ATOMIC SPECTROSCOPY WITH DEPHASING (η = 0.9)
GHZ strategy
S-S strategy
finite-N bound
asymptotic bound
worse than
classical region
better than
Heisenberg Limit
region
input correlations dominated region
uncorrelated decoherence dominated region
worth investing in GHZ states
e.g. N=3
[D. Leibfried et al, Science, 304 (2004)]
worth investing in S-S states
e.g. N=105 !!!!!!!
[R. J. Sewell et al, Phys. Rev. Lett. 109, 253605 (2012)]
CONSEQUENCES ON REALISTIC SCENARIOS
PERFORMANCE OF GEO600 GRAVITIONAL-WAVE INTERFEROMETER
/ns
coherent beam
squeezed vacuum
[Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)]
FREQUENCY ESTIMATION IN RAMSEY SPECTROSCOPY
Estimation of
t - extra free parameter
Parallel dephasing:
two Kraus operators – non-full rank channel – SQL-bounding Methods apply for any t
“Ellipsoid” dephasing:
four Kraus operators – full rank channel – SQL-bounding Methods apply for any t
four Kraus operators – full rank channel – SQL-bounding Methods apply, but…
as t → 0 up to O(t2) – two Kraus operators SQL-bounding Methods fail !!!
Transversal dephasing:
RAMSEY SPECTROSCOPY - RESOURCES: Total time of the experiment
, number of particles involved
:
PRECISION:
⁞
⁞
⁞
⁞ ⁞
⁞
⁞
⁞
⁞
⁞
optimize over t
⁞
RAMSEY SPECTROSCOPY WITH TRANSVERSAL DEPHASING
Beyond the shot noise !!!
(a) Saturability with the GHZ states
(dotted) – parallel dephasing
(dashed) – transversal dephasing without t-optimisation
(solid) – transversal dephasing with t-optimisation
(b) Impact of parallel component
[Chaves et al, Phys. Rev. Lett. 111, 120401 (2013)]
CONCLUSIONS
•
Classically, for separable input states, the ultimate precision is bound to shot noise
scaling 1/√N, which can be attained in a single experimental shot (k=1).
•
For lossless unitary evolution highly entangled input states (GHZ, N00N) allow for
ultimate precision that follows the Heisenberg scaling 1/N, but attaining this limit may
in principle require infinite repetitions of the experiment (k→∞).
•
The consequences of the dehorence acting independently on each particle:
•
The Heisenberg scaling is lost and only a constant factor quantum enhancement
over classical estimation strategies is allowed.
•
(?) The optimal input states in the N → ∞ limit achieve the ultimate precision in
a single shot (k=1) and are of a simpler form:
spin-squeezed atomic - [Ulam-Orgikh, Kitagawa, Phys. Rev. A 64, 052106 (2001)]
squeezed light states in GEO600 - [Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)].
•
However, finding the optimal form of input states is still an issue.
Classical scaling suggests local correlations:
Gaussian states – [Monras & Illuminati, Phys. Rev. A 81, 062326 (2010)]
MPS states – [Jarzyna et al, Phys. Rev. Lett. 110, 240405 (2013)].
CONCLUSIONS
•
We have formulated three methods:
Classical Simulation, Quantum Simulation and Channel Extension;
that may efficiently lower-bound the constant factor of the quantum asymptotic
enhancement for a generic channel by properties of its single use form
(Kraus operators).
•
The geometrical CS method proves the cQ/√N for all full-rank channels and more
(e.g. dephasing).
•
The CE method may also be applied numerically for finite N as a semi-definite
program.
•
After allowing the form of channel to depend on N, what is achieved by the
(exprimentally-motivated) single experimental-shot time period (t) optimisation,
we establish a channel that, despite being full-rank for any finite t, achieves the
ultimate super-classical 1/N5/6 asymptotic – the transversal dephasing.
Application to atomic magnetometry: [Wasilewski et al, Phys. Rev. Lett. 104, 133601 (2010)].
THANK YOU FOR YOUR ATTENTION