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Cramér-Rao bound for time-continuous measurements in linear Gaussian quantum systems

17 Jan 2017-Physical Review A (American Physical Society)-Vol. 95, Iss: 1, pp 012116

Abstract: We describe a compact and reliable method to calculate the Fisher information for the estimation of a dynamical parameter in a continuously measured linear Gaussian quantum system. Unlike previous methods in the literature, which involve the numerical integration of a stochastic master equation for the corresponding density operator in a Hilbert space of infinite dimension, the formulas here derived depend only on the evolution of first and second moments of the quantum states and thus can be easily evaluated without the need of any approximation. We also present some basic but physically meaningful examples where this result is exploited, calculating analytical and numerical bounds on the estimation of the squeezing parameter for a quantum parametric amplifier and of a constant force acting on a mechanical oscillator in a standard optomechanical scenario.
Topics: Quantum operation (63%), Quantum state (63%), Quantum process (62%), Quantum algorithm (62%), Master equation (59%)

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PHYSICAL REVIEW A 95,012116(2017)
Cram
´
er-Rao bound for time-continuous measurements in linear Gaussian quantum systems
Marco G. Genoni
Quantum Technology Lab, Dipartimento di Fisica, Universit
`
a degli Studi di Milano, 20133 Milano, Italy
(Received 7 September 2016; published 17 January 2017)
We describe a compact and reliable method to calculate the Fisher information for the estimation of a dynamical
parameter in a continuously measured linear Gaussian quantum system. Unlike previous methods in the literature,
which involve the numerical integration of a stochastic master equation for the corresponding density operator in
a Hilbert space of infinite dimension, the formulas here derived depend only on the evolution of first and second
moments of the quantum states and thus can be easily evaluated without the need of any approximation. We
also present some basic but physically meaningful examples where this result is exploited, calculating analytical
and numerical bounds on the estimation of the squeezing parameter for a quantum parametric amplifier and of a
constant force acting on a mechanical oscillator in a standard optomechanical scenario.
DOI: 10.1103/PhysRevA.95.012116
I. INTRODUCTION
Parameter estimation via quantum probes and quantum
measurements will lead to a new generation of detectors
characterized by sensitivities not achievable through only
classical means [1,2]. The promised quantum enhancement
is typically however lost as soon as some decoherence affects
the system [3,4]. On the other hand, the information leaking
into the environment can be in principle used for parameter
estimation as well, in particular via time-continuous monitor-
ing of the environment itself [5,6]. While several strategies
based on time-continuous measurements and feedback have
been proposed for quantum state engineering, in particular
with the main goal of generating steady-state squeezing and
entanglement [5,715]ortostudyandexploittrajectories
of superconducting qubits [16,17], less attention has been
devoted to parameter estimation. Notable exceptions are the
estimation of a magnetic field via a continuously monitored
atomic ensemble [18], the tracking of a varying phase
[1921], the estimation of Hamiltonian and environmental
parameters [2229], and optimal state estimation for a cavity
optomechanical system [30].
The ultimate precision achievable by quantum metrolog-
ical strategies is determined by the classical and quantum
Cram
`
er-Rao bounds [3133], which are expressed in terms
of, respectively, classical and quantum Fisher information
(FI). Recently, methods have been proposed to calculate
these quantities in the stationary regime for certain relevant
setups [22,23,28]orinthedynamicalregimeinthecase
of time-continuous homodyne and photon-counting mea-
surements [24,25]. In particular, in order to evaluate the
FI corresponding to a continuous homodyne detection, the
method presented in Ref. [24]reliesontheintegrationof
stochastic master equations for operators characterizing the
quantum state and the measurement performed. While this
can be straightforwardly accomplished in the case of finite-
dimensional quantum systems, such as two-level atoms and
superconducting qubits, the method becomes computationally
very expensive and less reliable in the case of large or
even infinite-dimensional systems, such as the electromagnetic
field, atomic ensembles, and mechanical oscillators. In fact, in
these cases, one has to truncate the corresponding Fock space,
posing a constraint on the maximum energy of the system.
The goal of this article is to provide an efficient and
reliable method to calculate the FI for parameter estimation via
time-continuous measurements in linear Gaussian quantum
systems. Gaussian systems represent a subclass of infinite-
dimensional bosonic systems, whose properties and dynamics
can be univocally described in terms of first and second mo-
ments only [3436]. In order to observe such a dynamics, one
has to consider Hamiltonians at most quadratic in the canonical
operators, a linear coupling with the environment, and time-
continuous monitoring via Gaussian measurements [3638].
This restriction allows one to greatly simplify the analysis
of infinite-dimensional quantum systems and at the same time
describe several state-of-the-art experimental setups in the area
of quantum optics, optomechanics, trapped ions, and atomic
ensembles. I n particular, we remind the reader how optimal
state estimation via time-continuous monitoring has been very
recently accomplished for a Gaussian cavity optomechanical
system [30], showing the timeliness and relevance of this
approach.
In detail, the manuscript is structured as follows. In Sec. II
we provide a basic introduction to Gaussian systems and their
diffusive and conditional dynamics, while in Sec. III we revise
the Cram
´
er-Rao bound, with a focus on a posteriori Gaussian
distributions. In Sec. IV we present the main result of the
manuscript, that is a method for the calculation of the FI for
Gaussian systems depending only on the evolution of first and
second moments and that does not need any approximation or
limit on the energy of the quantum states in exam. To show the
potential of our results, in Sec. V we provide two examples,
calculating numerical and analytical bounds on the estimation
precision for the squeezing parameter in a quantum parametric
amplifier and for a constant force acting on a mechanical
oscillator in a standard opto-mechanical setup.
II. DIFFUSIVE AND CONDITIONAL DYNAMICS
IN LINEAR GAUSSIAN SYSTEMS
We consider a set of n bosonic modes described by a vector
of quadrature operators
ˆ
r
T
= (
ˆ
x
1
,
ˆ
p
1
,...,
ˆ
x
n
,
ˆ
p
n
), satisfying
the canonical commutation relation [
ˆ
r,
ˆ
r
T
] = i! (with !
jk
=
δ
k,j+1
δ
k,j1
)[39]. We define a quantum state ϱ Gaussian,
if and only if can be written as a ground or thermal state of a
2469-9926/2017/95(1)/012116(8) 012116-1 ©2017 American Physical Society

MARCO G. GENONI PHYSICAL REVIEW A 95,012116(2017)
quadratic Hamiltonian, i.e.,
ϱ =
exp{β
ˆ
H
G
}
Z
,β R, (1)
where
ˆ
H
G
= (1/2)
ˆ
r
T
H
G
ˆ
r and H
G
! 0[34,36]. Gaussian
states can be univocally described by the vector of first
moments R and the covariance matrix σ [39]:
R = Tr[ϱ
ˆ
r], σ = Tr[ϱ{
ˆ
r R,(
ˆ
r R)
T
}]. (2)
We also recall that, in order to describe a proper Gaussian
quantum state, the covariance matrix has to satisfy the
physicality condition σ + i! ! 0[40].
We now consider a dynamics generated by a Hamiltonian
with linear and quadratic terms of the form
ˆ
H
s
=
1
2
ˆ
r
T
H
s
ˆ
r
ˆ
r
T
!u, (3)
where H
s
is a matrix of dimension 2n × 2n,whileu is a
2n-dimensional vector. We also assume that the system is
coupled to a large Markovian environment described by a
train of incoming modes
ˆ
r
b
(t), each of which interacts with
the system at a given time t.Thecorrelationscharacterizingthe
environment are specified through the white-noise condition
!"
ˆ
r
b
(t),
ˆ
r
T
b
(t
)
#$
= σ
b
δ(t t
), σ
b
+ i! ! 0. (4)
The interaction in time dt between the s ystem and the
environment is ruled by the Hamiltonian
ˆ
H
C
dt =
ˆ
r
T
C
ˆ
r
b
(t)dt =
ˆ
r
T
Cd
ˆ
r
b
(t) =
ˆ
r
T
C
ˆ
r
b
(t)dW.
Here we have introduced the so-called quantum Wiener
increment [41]
d
ˆ
r
b
(t) =
ˆ
r
b
(t)dt =
ˆ
r
b
(t)dW,
with dW being a real Wiener increment such that dW
2
=
dt,andwhere
ˆ
r
b
(t)isavectorof“proper”dimensionless
field operators (that can be associated with detector clicks
in the laboratory and formally with positive-operator valued
measure (POVM) operators in the Hilbert space) satisfying
the canonical commutation relation [
ˆ
r
b
(t),
ˆ
r
b
(t)
T
] = i! (we
refer to Ref. [36]andtoAppendixB for more details on these
definitions and on the derivation of the following formulas).
By tracing out the degrees of freedom of the environment,
the dynamics of the Gaussian state is then described by the
following equations.
dR
t
dt
= AR
t
+ u, (5)
dσ
t
dt
= Aσ
t
+ σ
t
A
T
+ D, (6)
where we have introduced the drift matrix A = !H
s
+
(!C!C
T
)/2 and the diffusion matrix D = !Cσ
b
C
T
!
T
.
One should notice that, as expected, the linear term in the
Hamiltonian (3)isresponsibleforonlyadisplacementofthe
first moment vector, while the evolution of the covariance
matrix is not affected.
We now assume that the environment is continuously
monitored at each time via a Gaussian measurement described
by a matrix σ
m
(such that σ
m
+ i! ! 0) and whose measure-
ment outcome corresponds to a vector x
m
.Inthisscenario,
the conditional state is still Gaussian, and the dynamics is
described by a stochastic equation for the first moment vector
and by a deterministic Riccati equation for the covariance
matrix [36](seealsoAppendixB):
dR
t
= AR
t
dt + udt +
%
σ
t
B + N
2
&
dw,
(7)
dσ
t
dt
= Aσ
t
+ σ
t
A
T
+ D (σ
t
B + N )(σ
t
B + N )
T
,
where dw is a vector of independent Wiener increments (such
that dw
j
dw
k
= δ
jk
dt)andwehaveintroducedthematrices
B = C!
T
(σ
b
+ σ
m
)
1/2
and N = !Cσ
b
(σ
b
+ σ
m
)
1/2
.
The dynamics we have just presented in terms of first and
second moments for Gaussian states can be equivalently de-
scribed by the following family of stochastic master equations
for the (infinite-dimensional) density operator:
dϱ =i[
ˆ
H,ϱ]dt +
L
'
j=1
D[
ˆ
c
j
]ϱdt + dz
%
ˆ
cϱ + ϱ%
ˆ
c
dz,
(8)
where
ˆ
c =
(
C
ˆ
r, D[
ˆ
o]ϱ =
ˆ
oϱ
ˆ
o
{
ˆ
o
ˆ
o,ϱ}/2, %
ˆ
o =
ˆ
o Tr
[ϱ
ˆ
o], and dz is a vector of complex Wiener increments [37,38].
For our purposes is important to recall that the outcomes x
m
of the measurement performed on the bath operators
ˆ
r
b
(t)are
distributed according to a Gaussian multivariate distribution
with mean value
¯
x
m
= !C
T
R
t
dW and covariance matrix
(σ
b
+ σ
m
)/2. Typically, the results of time-continuous mea-
surements are formulated as a real current with uncorrelated
noise [38], i.e.,
dy := (σ
b
+ σ
m
)
1/2
x
m
dW (9)
=B
T
R
t
dt +
dw
2
. (10)
III. FISHER INFORMATION FROM A MULTIVARIATE
GAUSSIAN DISTRIBUTION
Let us consider an a posteriori distribution p(x|θ), where
the vector x corresponds to the outcomes of the measurement
performed and θ is a parameter we want to estimate. In
particular we assume p(x|θ ) to be a multivariate Gaussian
distribution with mean value
¯
x
θ
and covariance matrix ',
where only the mean value depends on the parameter θ.The
ultimate limit on how accurate we can estimate θ is determined
by the Cram
`
er-Rao bound [42]
Var (
ˆ
θ) !
1
MF (θ)
, (11)
where Var(
ˆ
θ)isthevarianceofanunbiasedestimator
ˆ
θ, M is
the number of measurements, and
F (θ) = E
p
)
%
ln p(x|θ )
∂θ
&
2
*
(12)
is the FI corresponding to the distribution p(x|θ ). As it is clear
from the inequality (11), the FI F (θ )quantieshowwellwe
can infer the value of the parameter θ from the measurement
outcomes. By explicitly writing the square of the derivative
of the likelihood function l(x|θ ) = ln p(x|θ )andexploiting
the property E
p
[(x
¯
x
θ
)
j
(x
¯
x
θ
)
k
] = "
jk
, one can easily
012116-2

CRAM
´
ER-RAO BOUND FOR TIME-CONTINUOUS . . . PHYSICAL REVIEW A 95,012116(2017)
prove that the FI corresponding to a Gaussian a posteriori
distribution reads
F (θ) = (
θ
¯
x
θ
)
T
"
1
(
θ
¯
x
θ
). (13)
IV. FISHER INFORMATION FOR TIME-CONTINUOUS
MEASUREMENTS IN LINEAR GAUSSIAN
QUANTUM SYSTEMS
Let us assume that we want to estimate the value of a
parameter θ that characterizes the dynamic of a Gaussian linear
quantum system described by Eqs. (7). In particular we also
assume that only the system Hamiltonian
ˆ
H
s
depends on θ,
and thus only the drift matrix A
θ
and/or the vector u
θ
depends
on the parameter [43].
We stated above that the probability distribution p(x
m
|θ)
corresponding to the measurement performed at time t + dt
on the environment is a Gaussian distribution centered in
!C
T
R
t
dW and with covariance matrix (σ
b
+ σ
m
)/2. As only
the mean value depends on the parameter θ,viathefirst
moment vector R
t
,wecanexploitEq.(13)andwritethe
corresponding infinitesimal FI as
dF
(traj)
t
(θ) = 2(
θ
R
t
)
T
C!
T
(σ
b
+ σ
m
)
1
!C
T
(
θ
R
t
)dt.
(14)
Notice that this FI corresponds to a specific trajectory, as it is
calculated via the vector (
θ
R
t
)whoseevolutionisinprinciple
stochastic and described by the equations
d(
θ
R
t
) = (
θ
A)R
t
dt + A(
θ
R
t
)dt+(
θ
u)dt +
(
θ
σ
t
)B
2
dw,
d(
θ
σ
t
)
dt
= (
θ
A)σ
t
+ σ
t
(
θ
A)
T
+ A(
θ
σ
t
) + (
θ
σ
t
)A
T
(
θ
σ
t
)B(σ
t
B + N )
T
(σ
t
B + N )[(
θ
σ
t
)B]
T
.
(15)
As a consequence, the actual FI for the measurement per-
formed at time t + dt is evaluated by averaging over all the
possible trajectories, i.e.,
dF
t
(θ) = E
dw
+
dF
(traj)
t
(θ)
,
. (16)
Finally, if one consider the whole data set, i.e., the
continuous stream of measurement outcomes d ={x
m
}
t
t
=0
obtained up to time t,theFIcorrespondingtothewholea
posteriori distribution p(d|θ )canbecalculated,exploiting
its additive property [44], by integrating it (numerically or
analytically) as
F
t
(θ) =
-
t
t
=0
dF
t
(θ). (17)
V. E X A M P L E S
A. Estimation of the squeezing parameter for a quantum
parametric amplifier
Let us consider the Hamiltonian for a degenerate parametric
amplifier, which in the cavity-mode rotating frame reads
ˆ
H
s
=χ(
ˆ
x
ˆ
p +
ˆ
p
ˆ
x)/2, and assume that the corresponding
cavity mode is weakly interacting with a bath at zero
temperature, such that the interaction matrix corresponds to
FIG. 1. FI F
t
(χ)forcontinuoushomodyneandheterodyne
detection as a function of time and for χ =0.2κ (numerical
evaluation with 2000 trajectories). Green dotted line: Homodyne
detection of quadrature
ˆ
x.Reddashedline:Homodynedetection
of quadrature
ˆ
p.Bluesolidline:Heterodynedetection.
a beam splitter with C =
κ! and the correlation of the bath
is described by σ
b
= 1
2
.Thecorrespondingmasterequation
for the density operator reads ˙ϱ =i[
ˆ
H
s
,ϱ] + κD[
ˆ
a]ϱ,where
ˆ
a = (
ˆ
x + i
ˆ
p)/
2isthebosonicannihilationoperator.Inour
formalism the dynamics is described by the following drift
and diffusion matrices: A = diag(χ κ/2,χ κ/2) and
D = κ1
2
.
We now assume to perform a time-continuous measurement
on the cavity output, that is on the environmental modes after
the interaction with the cavity mode, via a Gaussian measure-
ment, in order to estimate the squeezing coupling constant χ .In
our analysis we focus on two types of measurements: a time-
continuous homodyne measurement of a quadrature
ˆ
r(φ) =
(cos φ
ˆ
x + sin φ
ˆ
p)/
2andatime-continuousheterodyne
detection, that is the projection on single-mode coherent states
(the details on the Gaussian description of these measurements
can be found in Appendix A). Under these assumptions,
Eq. (14)fortheinnitesimalFIsimpliestothefollowing
equations:
dF
t
= 2κ E
dw
[(
χ
ˆ
r
φ
t
)
2
]dt, homodyne quadrature
ˆ
r
φ
,
dF
t
= κ E
dw
[(
χ
ˆ
x
t
)
2
+ (
χ
ˆ
p
t
)
2
]dt, heterodyne.
In Fig. 1 we plot the FI F
t
(χ)asafunctionoftime.Wenotice
that the best performances, between the three measurements
here considered, are obtained via a time-continuous homodyne
measurement of the quadrature
ˆ
x,which,fortheparameters
we have chosen, is the quadrature being antisqueezed by the
Hamiltonian
ˆ
H
s
.Onecanalsoobservehow,atlongtimes,
all three curves present a linear behavior, indicating that the
infinitesimal FI takes the form dF
t
(χ) = Kdt.
B. Estimation of a constant force on a mechanical oscillator
We consider a standard cavity optomechanical setup where
amechanicaloscillatoroscillatingatfrequencyω
m
is coupled
to a cavity mode characterized by resonance frequency ω
c
and driven with a laser at frequency ω
l
= % + ω
c
.The
012116-3

MARCO G. GENONI PHYSICAL REVIEW A 95,012116(2017)
interaction Hamiltonian is linearized as
ˆ
H
int
= g
ˆ
x
m
ˆ
x
c
,and
as usual we consider the cavity decay rate κ, while the
mechanical oscillator is coupled to a phononic Markovian bath
characterized by n
th
thermal phonons, and the decoherence
rate γ [45]. A constant force is exerted on the mechanical
oscillator, described by the Hamiltonian
ˆ
H
λ
= λ
ˆ
x
m
,where
λ is the parameter we want to estimate. The details of
the master equation for the two-mode density operator and
of the corresponding Gaussian description can be found in
Appendix C.
In order to estimate the force parameter λ,theenvironment
of the cavity field, i.e., the cavity output field, is measured
continuously as in Ref. [30] (it is possible to include also the
continuous measurement on the oscillator environment, which
can be performed experimentally in certain optomechanical
setups, for example, by monitoring the light scattered from a
levitating nanosphere [14,46]).
Here we consider continuous homodyne detection with
finite efficiency η,whoseFIcanbeeasilyevaluatedviathe
formulas presented in the previous example. It is important
to notice how, in this case, only the vector u depends on the
parameter. As a consequence, since
λ
A = 0, also the matrix
λ
σ
t
= 0atanytime,andtheevolutionofthevector
λ
R
t
is
completely deterministic. It is not necessary then to average the
infinitesimal FI in Eq. (14), and, for some range of parameters,
it is possible to calculate its value analytically. For example, for
aresonantlaserdriving(% = 0) and for homodyne detection
of a quadrature
ˆ
r
c
(φ), one obtains
dF
t
(λ) = η ( g sin φ)
2
G(γ,κ,ω
m
,t)dt, (18)
where the function G(γ,κ,ω
m
,t)isreportedinAppendixC.
Remarkably, the FI grows quadratically with the coupling
constant g,and,asexpected,theoptimalperformancesare
obtained for the monitoring of the quadrature
ˆ
p
c
,which,
in turn, corresponds to the measurement of the mechanical
momentum
ˆ
p
m
.
The formula for the total FI F
t
(λ)istoocumbersometo
be reported here, and we plot its behavior as a function of the
cavity decay rate κ and total time t in Fig. 2.Oneobserves
how, for a small monitoring time, one has larger F
t
(λ)for
larger decay rate κ,asmoreinformationontheparameterλ is
leaking into the environment. On the other hand, by increasing
the monitoring time t,asymptoticbetterperformancesare
obtained for smaller values of κ:infactforlargerlosses,a
steady-state of the cavity field will be approached faster and
thus less information on the force acting on the mechanical
oscillator will be available in the cavity output.
This example clearly shows the potential of our method: in
fact, in order to evaluate the FI of this estimation problem
with the method described in Ref. [24], it would have
been necessary to integrate numerically a stochastic master
equation, over around thousands trajectories, for two-mode
operators and thus corresponding to approximated matrices
of dimension (d
m
d
c
) × (d
m
d
c
)(d
m
and d
c
being the truncated
dimensions of the Fock space for, respectively, the mechanical
oscillator and the cavity field).
FIG. 2. FI F
t
(λ) for continuous homodyne detection of the cavity
field quadrature
ˆ
p
c
as a function of time and for different values of the
cavity decay rate: κ = ω
m
/2, red solid line; and κ = ω
m
/10, green
dashed line; κ = ω
m
/20, blue dotted line (the other parameters are
chosen as g = ω
m
/2, γ = ω
m
/3, and η = 1). The inset shows the
behavior of the FI at small times.
VI. DISCUSSION AND OUTLOOK
We have presented a reliable method for the calculation
of the FI for dynamical parameter estimation in continuously
measured linear Gaussian quantum systems. As shown in the
two examples here described, our method greatly simplifies,
in terms of computation complexity, the calculation of bounds
on the estimation of parameters for such a rich and physically
relevant family of quantum systems, compared to both the
method presented in Ref. [24]andtheonesderivedfor
general classical Gaussian systems [4750]. It can also provide
analytical results, allowing one to investigate in more detail
the role played by the different physical parameters and
to compare easily the efficiency of different measurement
strategies. Furthermore, as Eqs. (7)areformallyequivalent
to the classical continuous-time Kalman filter, our method can
be generalized to the classical case (a more detailed discussion
can be found in Appendix D).
Our results will find applications in assessing the perfor-
mances of quantum sensors in several physical systems. It is
worth mentioning the estimation of a magnetic field, in cases
where an atomic spin ensemble can be approximated by a
bosonic field via the Holstein-Primakovv approximation [18]
and in several other quantum optomechanics setups and
estimation problems, in particular with the aim of testing
fundamental theories as corrections to Newtonian gravity [51]
or to the Schr
¨
odinger equation [15,52]. Moreover this approach
can be generalized to the estimation of stochastic parameters,
e.g., stochastic forces on mechanical oscillators, and of
parameters characterizing the interaction with the environment
and its temperature.
ACKNOWLEDGMENTS
The author thanks S. Conforti, M. Paris, and A. Serafini for
discussions and constant support and A. Mari for useful discus-
sions regarding the additive property of the Fisher information.
012116-4

CRAM
´
ER-RAO BOUND FOR TIME-CONTINUOUS . . . PHYSICAL REVIEW A 95,012116(2017)
The author acknowledges support from Marie Skłodowska-
Curie Action H2020-MSCA-IF-2015 under Grant No. 701154.
APPENDIX A: GAUSSIAN (GENERAL-DYNE)
MEASUREMENTS
We here briey present the parametrization of Gaussian
measurements that are discussed and used in this article. A
Gaussian measurement is univocally described by a matrix σ
m
,
such that σ
m
+ i! ! 0, and the corresponding measurement
outcomes are described by a vector x
m
.Westartbyfocusingon
single-mode projective Gaussian measurements, which thus
correspond in the Hilbert space to projection onto a single-
mode state |ψ
G
.Asthemostgeneralsingle-modeGaussian
state is a displaced squeezed vacuum state, the corresponding
general matrix σ
m
can be written as
σ
m
(s,φ) = R(φ)
%
s 0
01/s
&
R(φ)
T
, (A1)
with
R(φ) =
%
cos φ sin φ
sin φ cos φ
&
. (A2)
In the case of homodyne measurement of the quadrature
operator
ˆ
r
φ
= cos φ
ˆ
x + sin φ
ˆ
p,onehastoevalulatethelimit
σ
(hom)
m
= lim
s0
σ
m
(s,φ), while for heterodyne detection, i.e.,
projection onto coherent states, one has σ
(het)
m
= σ
m
(1,φ).
In order to take into account inefficient detection, the
measurement matrix σ
(η)
m
is calculated via the action of the dual
noisy map on the projective measurement covariance matrix
of Eq. (A1)as[36]
σ
(ineff)
m
= X
σ
m
X
T
+ Y
, (A3)
where
X
= 1
2
/
η,
Y
=
1 η
η
1
2
,
and η quantifies the detection efficiency.
Notice that if one wants to consider a two-mode (or
even multimode) local measurement, the overall measurement
matrix σ
m
is obtained by taking the direct sum of the single-
mode measurements, e.g., σ
m
= σ
m,(a)
σ
m,(b)
.Ifoneofthe
modes is not monitored, then one has to use the inefficient
measurement matrix σ
(ineff)
m
as in Eq. (A3)andtakethelimit
for the corresponding efficiency parameter η 0.
APPENDIX B: DIFFUSIVE AND CONDITIONAL
EVOLUTION UNDER TIME-CONTINUOUS
MEASUREMENTS
In this section we briefly describe the formalism and the
calculations developed in Ref. [36]thatleadtotheequations
describing the evolution of Gaussian states under time-
continuous general-dyne measurements on the environment.
To keep the presentation easier, we consider the case where
no Hamiltonian for the system is present, i.e., H
s
= 0, and
we focus on the interaction between the system and the
bath degrees of freedom. This interaction is described by the
Hamiltonian
ˆ
H
C
=
ˆ
r
T
C
ˆ
r
b
(t) =
1
2
ˆ
r
T
sb
H
C
ˆ
r
sb
=
1
2
ˆ
r
T
sb
%
0 C
C
T
0
&
ˆ
r
sb
,
(B1)
where
ˆ
r
b
(t)arethebathoperators,denedbythewhite-noise
condition
!"
ˆ
r
b
(t),
ˆ
r
T
b
(t
)
#$
= σ
b
δ(t t
), (B2)
and
ˆ
r
T
sb
= [
ˆ
r
T
,
ˆ
r
b
(t)
T
]. From Eq. (B2)onenoticesthatthe
operators
ˆ
r
b
(t)havethedimensionsofthesquarerootofa
frequency. One can then define the so-called quantum Wiener
increment d
ˆ
r
b
(t)as[5,41]
d
ˆ
r
b
(t) =
ˆ
r
b
(t) dt =
ˆ
r
b
(t) dW (B3)
and impose that
ˆ
r
b
(t)isavectorofdimensionlessfield
operators (that can be associated with detector clicks in the
laboratory and formally with POVM operators in the Hilbert
space), satisfying the canonical commutation relations
[
ˆ
r
b
(t),
ˆ
r
b
(t)] = i!. (B4)
By observing that
[d
ˆ
r
b
(t),d
ˆ
r
b
(t)] = [
ˆ
r
b
(t)dW,
ˆ
r
b
(t)dW ] = i!d W
2
, (B5)
= [
ˆ
r
b
(t)dt,
ˆ
r
b
(t)dt] = i!dt, (B6)
one obtains the relationship dW
2
= dt.Onecantheninterpret
dW as a stochastic Wiener increment, which is indeed
responsible for the diffusive behavior of the dynamics (in the
Heisenberg picture the system and bath operators show in fact
arandom-walk-likeevolution).Itisimportanttoremarkthat
the properties of dW are a consequence of the white-noise
condition describing the input operators
ˆ
r
b
(t)(othertypesof
correlations would lead to a different stochastic behavior).
We can now apply the Gaussian formalism to the vector
of (well-defined) canonical operators
ˆ
r
T
sb
= [
ˆ
r
T
,
ˆ
r
b
(t)
T
]. Under
the coupling described in Eq. (B1), the dynamics over
an interval dt is generated by the operator
ˆ
r
T
sb
H
C
ˆ
r
sb
dt =
ˆ
r
T
sb
H
C
ˆ
r
sb
dW .Byexpandingthecorrespondingsymplectic
transformation as
e
!H
C
dW
%
1 + !H
C
dW +
(!H
C
)
2
2
dt
&
, (B7)
one calculates the evolution of the system-bath covariance
matrix as
e
!H
C
dW
(σ σ
b
)e
(!H
C
)
T
dW
(σ σ
b
) + (Aσ + σ A
T
+ D)
×⊕
˜
σ
b
dt + σ
sb
dW, (B8)
where A = (!C!C
T
)/2andD = !Cσ
b
C
T
!
T
are typically
addressed as drift and diffusion matrices, while the other
matrices are
σ
sb
=
%
0 !Cσ
b
+ σ C!
T
σ
b
C
T
!
T
+ !C
T
σ 0
&
, (B9)
˜
σ
b
=
!C
T
!Cσ
b
+ σ
b
C
T
!C!
2
+ !
T
C
T
σ C!. (B10)
012116-5

Citations
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Journal ArticleDOI
03 Dec 2018
Abstract: We study quantum frequency estimation for $N$ qubits subjected to independent Markovian noise, via strategies based on time-continuous monitoring of the environment. Both physical intuition and an extended convexity property of the quantum Fisher information (QFI) suggest that these strategies are more effective than the standard ones based on the measurement of the unconditional state after the noisy evolution. Here we focus on initial GHZ states and on parallel or transverse noise. For parallel noise, i.e. dephasing, we show that perfectly efficient time-continuous photo-detection allows to recover the unitary (noiseless) QFI, and thus to obtain a Heisenberg scaling for every value of the monitoring time. For finite detection efficiency, one falls back to the noisy standard quantum limit scaling, but with a constant enhancement due to an effective reduced dephasing. Also in the transverse noise case we obtain that the Heisenberg scaling is recovered for perfectly efficient detectors, and we find that both homodyne and photo-detection based strategies are optimal. For finite detectors efficiency, our numerical simulations show that, as expected, an enhancement can be observed, but we cannot give any conclusive statement regarding the scaling. We finally describe in detail the stable and compact numerical algorithm that we have developed in order to evaluate the precision of such time-continuous estimation strategies, and that may find application in other quantum metrology schemes.

43 citations


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Abstract: We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjected to collective transverse noise. By preparing an initial spin coherent state, for any measurement performed after the evolution, the mean-square error of the estimate is known to scale as $1/J$, i.e. no quantum enhancement is obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclusively show that strategies based on time-continuous non-demolition measurements followed by a final strong measurement may achieve Heisenberg-limited scaling $1/{J^2}$ and also a monitoring-enhanced scaling in terms of the interrogation time. We also find that time-continuous schemes are robust against detection losses, as we prove that the quantum enhancement can be recovered also for finite measurement efficiency. Finally, we analytically prove the optimality of our strategy.

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Abstract: We demonstrate that a Kalman filter applied to estimate the position of an optically levitated nanoparticle, and operated in real-time within a field programmable gate array, is sufficient to perform closed-loop parametric feedback cooling of the center-of-mass motion to sub-Kelvin temperatures. The translational center-of-mass motion along the optical axis of the trapped nanoparticle has been cooled by 3 orders of magnitude, from a temperature of 300 K to a temperature of $162\ifmmode\pm\else\textpm\fi{}15$ mK.

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Abstract: We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjected to collective transverse noise. By preparing an initial spin coherent state, for any measurement performed after the evolution, the mean-square error of the estimate is known to scale as $1/J$, i.e. no quantum enhancement is obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclusively show that strategies based on time-continuous non-demolition measurements followed by a final strong measurement may achieve Heisenberg-limited scaling $1/{J^2}$ and also a monitoring-enhanced scaling in terms of the interrogation time. We also find that time-continuous schemes are robust against detection losses, as we prove that the quantum enhancement can be recovered also for finite measurement efficiency. Finally, we analytically prove the optimality of our strategy.

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Journal ArticleDOI
Yao Ma, Mi Pang, Li-Bo Chen1, Wen YangInstitutions (1)
Abstract: Quantum-enhanced parameter estimation has widespread applications in many fields. An important issue is to protect the estimation precision against the noise-induced decoherence. Here we develop a general theoretical framework for improving the precision for estimating an arbitrary parameter by monitoring the noise-induced quantum trajectories (MQT) and establish its connections to the purification-based approach to quantum parameter estimation. Monitoring the noise-induced quantum trajectory can be achieved in two ways: (i) Any quantum trajectories can be monitored by directly monitoring the environment, which is experimentally challenging for realistic noises, and (ii) certain quantum trajectories can also be monitored by frequently measuring the quantum probe alone via ancilla-assisted encoding and error detection. This establishes an interesting connection between MQT and the full quantum-error-correction protocol. Application of MQT to estimate the level splitting and decoherence rate of a spin 1/2 under typical decoherence channels demonstrate that it can avoid the long-time exponential loss of the estimation precision and, in special cases, recover the Heisenberg scaling.

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References
More filters

Book
Carl W. Helstrom1Institutions (1)
01 Jan 1969
Abstract: A review. Quantum detection theory is a reformulation, in quantum-mechanical terms, of statistical decision theory as applied to the detection of signals in random noise. Density operators take the place of the probability density functions of conventional statistics. The optimum procedure for choosing between two hypotheses, and an approximate procedure valid at small signal-to-noise ratios and called threshold detection, are presented. Quantum estimation theory seeks best estimators of parameters of a density operator. A quantum counterpart of the Cramer-Rao inequality of conventional statistics sets a lower bound to the mean-square errors of such estimates. Applications at present are primarily to the detection and estimation of signals of optical frequencies in the presence of thermal radiation.

3,781 citations


01 Jan 2005

192 citations


Additional excerpts

  • ...where ĤG = (1/2)r̂HG r̂ and HG ≥ 0 [34, 36]....

    [...]


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