Journal ArticleDOI

# 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.
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
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)
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)
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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

Journal ArticleDOI
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.

39 citations

Journal ArticleDOI
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.

28 citations

Journal ArticleDOI
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.

27 citations

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.

13 citations

##### References
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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

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

[...]

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