PHYSICAL REVIEW A 95,012116(2017)
Cram
´
erRao bound for timecontinuous 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 timecontinuous monitor
ing of the environment itself [5,6]. While several strategies
based on timecontinuous measurements and feedback have
been proposed for quantum state engineering, in particular
with the main goal of generating steadystate squeezing and
entanglement [5,7–15]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
[19–21], the estimation of Hamiltonian and environmental
parameters [22–29], 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
`
erRao bounds [31–33], 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 timecontinuous homodyne and photoncounting 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 twolevel atoms and
superconducting qubits, the method becomes computationally
very expensive and less reliable in the case of large or
even inﬁnitedimensional 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
timecontinuous 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 [34–36]. 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 [36–38].
This restriction allows one to greatly simplify the analysis
of inﬁnitedimensional quantum systems and at the same time
describe several stateoftheart 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 timecontinuous 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
´
erRao 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 optomechanical 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,j−1
)[39]. We deﬁne a quantum state ϱ Gaussian,
if and only if can be written as a ground or thermal state of a
24699926/2017/95(1)/012116(8) 0121161 ©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
2ndimensional 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 whitenoise 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 socalled 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 positiveoperator 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 (inﬁnitedimensional) 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 timecontinuous 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(xθ), 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(xθ ) 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
`
erRao 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(xθ )
∂θ
&
2
*
(12)
is the FI corresponding to the distribution p(xθ ). As it is clear
from the inequality (11), the FI F (θ )quantiﬁeshowwellwe
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(xθ ) = ln p(xθ )andexploiting
the property E
p
[(x −
¯
x
θ
)
j
(x −
¯
x
θ
)
k
] = "
jk
, one can easily
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CRAM
´
ERRAO BOUND FOR TIMECONTINUOUS . . . 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 TIMECONTINUOUS
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(dθ )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 cavitymode 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 timecontinuous 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)/
√
2andatimecontinuousheterodyne
detection, that is the projection on singlemode coherent states
(the details on the Gaussian description of these measurements
can be found in Appendix A). Under these assumptions,
Eq. (14)fortheinﬁnitesimalFIsimpliﬁestothefollowing
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 timecontinuous 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
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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 twomode 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
steadystate 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 twomode
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 [47–50]. 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 continuoustime 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 HolsteinPrimakovv 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.
0121164
CRAM
´
ERRAO BOUND FOR TIMECONTINUOUS . . . PHYSICAL REVIEW A 95,012116(2017)
The author acknowledges support from Marie Skłodowska
Curie Action H2020MSCAIF2015 under Grant No. 701154.
APPENDIX A: GAUSSIAN (GENERALDYNE)
MEASUREMENTS
We here brieﬂy 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
singlemode projective Gaussian measurements, which thus
correspond in the Hilbert space to projection onto a single
mode state ψ
G
⟩.AsthemostgeneralsinglemodeGaussian
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
s→0
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 twomode (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 TIMECONTINUOUS
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 generaldyne 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,deﬁnedbythewhitenoise
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 socalled 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
arandomwalklikeevolution).Itisimportanttoremarkthat
the properties of dW are a consequence of the whitenoise
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 (welldeﬁned) 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 systembath 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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