Fermat's Library | Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment annotated/explained version.

Restoring Heisenberg scaling in noisy quantum metrology by

monitoring the environment

Francesco Albarelli

1,2

, Matteo A. C. Rossi

1,3

, Dario Tamascelli

1

, and Marco G. Genoni

1

Quantum Technology Lab, Dipartimento di Fisica “Aldo Pontremoli”, Universit

`

a degli Studi di Milano, IT-20133, Milan, Italy

2

Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

3

QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun

Yliopisto, Finland

November 30, 2018

We study quantum frequency estimation for N

qubits subject to independent Markovian noise

via strategies based on time-continuous moni-

toring of the environment. Both physical intu-

ition and the extended convexity of the quan-

tum Fisher information (QFI) suggest that these

strategies are more eﬀective than the standard

ones based on the measurement of the uncondi-

tional state after the noisy evolution. Here we

focus on initial GHZ states subject to parallel

or transverse noise. For parallel, i.e., dephas-

ing noise, we show that perfectly eﬃcient time-

continuous photodetection allows us to recover

the unitary (noiseless) QFI, and hence obtain

Heisenberg scaling for every value of the moni-

toring time. For ﬁnite detection eﬃciency, one

falls back to noisy standard quantum limit scal-

ing, but with a constant enhancement due to

an eﬀective reduced dephasing. In the trans-

verse noise case, Heisenberg scaling is recov-

ered for perfectly eﬃcient detectors, and we ﬁnd

that both homodyne and photodetection-based

strategies are optimal. For ﬁnite detector eﬃ-

ciency, our numerical simulations show that, as

expected, an enhancement can be observed, but

we cannot give any conclusive statement regard-

ing 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.

1 Introduction

Quantum metrology is one of the most promising ﬁelds

within the realm of quantum technologies, with appli-

Marco G. Genoni: marco.genoni@ﬁsica.unimi.it

cations ranging from spectroscopy and magnetometry

to interferometry and gravitational waves detection [1–

4]. While N uncorrelated (classical) probe states lead

to an estimation precision scaling as 1/

√

N, typically

referred to as standard quantum limit (SQL), quantum

probes made of N entangled particles allow to design

schemes with precision scaling as 1/N , reaching the so-

called Heisenberg limit (HL) [5].

The above result relies on the assumption of noise-

less unitary dynamics, but the unavoidable interaction

with the surrounding environment can have dramatic

consequences on the performances of these protocols.

When the dynamics is described by a dephasing dy-

namical semigroup, the estimation precision is bounded

to follow a SQL scaling, and thus only a constant en-

hancement can be obtained by exploiting quantum re-

sources [6–9]. Several attempts to circumvent these no-

go theorems and obtain a superclassical scaling in the

presence of noise have been pursued, exploiting time-

inhomogeneous dynamics [10–14], noise with a partic-

ular geometry [15, 16], dynamical decoupling [17], and

quantum error-correction protocols (or, more generally,

the possibility to implement control Hamiltonians) [18–

26].

Our goal is to attack the problem of noisy quan-

tum metrology by exploiting time-continuous monitor-

ing [27, 28]. Time-continuous measurements have been

often studied as a tool for quantum parameter/state

estimation [29–36], with a particular focus on classical

time-dependent signals [37–39]. More recently, methods

to calculate classical and quantum Cram´er-Rao bounds

for parameter estimation via time-continuous monitor-

ing have been proposed [40–44], and put into action [45–

48].

In this paper, we apply such techniques to the prob-

lem of frequency estimation for an ensemble of two-level

atoms (qubits), each subjected to independent Marko-

vian noise. In particular, we consider estimation strate-

gies based on time-continuous (weak) measurements of

Accepted in Quantum 2018-11-29, click title to verify 1

arXiv:1803.05891v3 [quant-ph] 29 Nov 2018

each qubit via the monitoring of the corresponding envi-

ronment, and a ﬁnal (strong) measurement on the con-

ditional state of the N qubits. Few similar attempts in

this sense have already been discussed in the literature,

but they all present substantial diﬀerences compared to

our approach. For instance, in [48–50] the same problem

of frequency estimation (magnetometry) is addressed,

but in the presence of collective transverse noise. There,

a single environment collectively interacts with all the

qubits and the corresponding noise is not detrimental

for the Heisenberg scaling; time-continuous monitoring

is however extremely promising as, thanks to the cor-

responding back-action on the system, a Heisenberg-

limited precision can be achieved even by starting the

dynamics with an uncorrelated spin-coherent state. An

estimation problem more similar to ours is presented

in [51], where independent environments interact with

each probe, but the analysis is restricted to a single (dis-

crete) step of the dynamics and to interaction Hamilto-

nians commuting with the generator of the phase rota-

tion (i.e., in the presence of pure dephasing only).

Here we consider a proper continuous evolution, de-

scribed by a Markovian master equation in the Lind-

blad form, leading to a dynamics satisfying the semi-

group property. We focus in particular on independent

noise, either parallel or transverse to the generator of

the phase rotation to be estimated. In the former case,

which physically corresponds to pure dephasing, the

unconditional dynamics leads to a standard quantum

limited precision, even for an inﬁnitesimal amount of

noise [8]. In the latter, it was shown that, by optimiz-

ing over the evolution time, it is possible to restore a

super-classical scaling between SQL and HL [15, 16].

Our goal is to study whether in both cases time-

continuous monitoring will allow for the restoration of

the HL, and to analyze in detail the eﬀect of the mon-

itoring eﬃciency on the performance of the estimation

schemes. First, we will derive the ultimate limit on

estimation precision, optimizing over the most general

measurements on the joint degrees of freedom of sys-

tem and environment. We will then restrict ourselves to

the strategies brieﬂy described above, focusing on time-

continuous photodetection and homodyne-detection.

To achieve those aims, we develop a stable and com-

pact numerical algorithm that allows us to calculate

the eﬀective quantum Fisher information characteriz-

ing this kind of measurement strategy, and that will ﬁnd

application in quantum metrology problems beyond the

ones considered in this paper.

The manuscript is structured as follows: In Sec. 2

we introduce the basic concepts of quantum estimation

theory, with a particular focus on measurement strate-

gies based on time-continuous measurements. In Sec. 3

we ﬁrst introduce the problem of noisy quantum fre-

quency estimation and then we present our original re-

sults for parallel and transverse Markovian noise. The

following sections are devoted to describing the meth-

ods exploited to obtain such results. In Sec. 4 we show

how to evaluate analytically the ultimate limits posed

by quantum mechanics for strategies optimizing over

all the the possible global measurements on system and

environment. In Sec. 5 we present in detail the numer-

ical algorithm we have developed for the calculation of

the eﬀective quantum Fisher information, pertaining to

strategies based on time-continuous monitoring of the

environment. Sec. 6 concludes the paper with some ﬁnal

remarks and discussion of possible future directions.

2 Quantum estimation via time-

continuous measurements

A classical estimation problem is typically described in

terms of a conditional probability p(x|ω) of obtaining

the measurement outcome x, given the value of the pa-

rameter ω that one wants to estimate. The classical

Cram´er-Rao bound states that the precision of any un-

biased estimator is lower bounded as

δω ≥

1

p

MF[p(x|ω)]

, (1)

where M is the number of measurements performed,

F[p(x|ω)] =

p

[(∂

ω

ln p(x|ω))

2

] is the classical Fisher

information, and

p

[·] denotes the average over the

probability distribution p(x|ω).

In the quantum setting, the probability is obtained

via the Born rule, p(x|ω) = Tr[%

ω

Π

x

], where %

ω

is a

family of quantum states parametrized by ω, and Π

x

is an element of the positive-operator valued measure

(POVM) describing the measuring process. It is then

possible to optimize over all the possible POVMs, ob-

taining the quantum Cram´er-Rao inequality [52]

δω ≥

1

p

MF[p(x|ω)]

≥

1

p

MQ[%

ω

]

, (2)

where

Q[%

ω

] = lim

→0

8 (1 − F [%

ω

, %

ω+

])



2

(3)

is the quantum Fisher information (QFI), expressed

in terms of the ﬁdelity between quantum states

F [%

1

, %

2

] = Tr



p

√

%

1

%

2

√

%

1



. The QFI Q[%

ω

] clearly

depends only on the quantum statistical model, namely

the ω-parametrized family of quantum states %

ω

, and

one can always ﬁnd the optimal POVM saturating the

bound., i.e., such that F[p(x|ω)] = Q[%

ω

].

In several quantum parameter estimation problems,

including the case of standard frequency estimation, %

ω

Accepted in Quantum 2018-11-29, click title to verify 2

corresponds to the unconditional state evolved accord-

ing to a Markovian master equation of the form

d%

dt

= L

ω

% = −i[

ˆ

H

ω

, %] +

X

j

D[ˆc

j

]% , (4)

where the parameter is encoded in the Hamiltonian

ˆ

H

ω

, the operators ˆc

j

denote diﬀerent independent

noisy channels, and we have deﬁned the superoperator

D[

ˆ

A]% =

ˆ

A%

ˆ

A

†

− (

ˆ

A

†

ˆ

A% + %

ˆ

A

†

ˆ

A)/2.

The master equation above can be obtained assum-

ing that the system is interacting with a sequence

of input operators ˆa

(j)

in

(t), one for each noise opera-

tor ˆc

j

, that satisfy the bosonic commutation relation

[ˆa

(j)

in

(t), ˆa

(k)†

in

(t

0

)] = δ

jk

δ(t − t

0

). Some information on

the parameter ω is then contained in the corresponding

output operators ˆa

(j)

out

(t), i.e., the environmental modes,

just after the interaction with the system. One can

imagine to perform a joint measurement on all output

modes and the quantum system itself. The ultimate

limit on the estimation of ω that takes into account all

these strategies based on measurements of system and

output, is quantiﬁed by the QFI introduced in [42]. It

depends only on the initial state of the system and on

the superoperator L

ω

describing the evolution. In gen-

eral, it can be evaluated as

Q

L

ω

= 4∂

ω

1

∂

ω

2

log |Tr[¯%]|



ω

1

=ω

2

=ω

, (5)

where, in the case we are considering (i.e., a Hamilto-

nian parameter), the operator % is the one solving the

generalized master equation

d¯%

dt

= L

ω

1

,ω

2

¯%

= −i



ˆ

H

ω

1

¯% − ¯%

ˆ

H

ω

2



+

X

j

D[ˆc

j

]¯% . (6)

This QFI corresponds to an optimization of the classi-

cal FI over all the possible measurement strategies de-

scribed above, including the possibility of performing

non-separable (entangled) measurements over the sys-

tem and all the output modes at diﬀerent times.

However, one can restrict to more feasible strategies,

where the outputs are sequentially measured continu-

ously in time, and a ﬁnal strong measurement is per-

formed on the conditional state of the system. Depend-

ing on the type of measurement performed on the out-

puts, one obtains diﬀerent stochastic master equations

for the conditional state of the system. In the following

we will focus on the two paradigmatic cases of photode-

tection (PD) and homodyne detection (HD).

In the ﬁrst case, assuming time-continuous PD on

each output with eﬃciency η

j

, the evolution is described

by the stochastic master equation [27]

d%

(c)

= −i[

ˆ

H

ω

, %

(c)

] dt +

X

j

(1 − η

j

)D[ˆc

j

]%

(c)

dt

−

η

j

2

(ˆc

†

j

ˆc

j

%

(c)

+ %

(c)

ˆc

†

j

ˆc

j

) dt + η

j

Tr[%

(c)

ˆc

†

j

ˆc

j

]%

(c)

dt

+

ˆc

j

%

(c)

ˆc

†

j

Tr[%

(c)

ˆc

†

j

ˆc

j

]

− %

(c)

!

dN

j

, (7)

where dN

j

denote Poisson increments taking value 0

(no-click event) or 1 (detector click event), and having

average value [dN

j

] = η

j

Tr[ˆc

†

j

ˆc

j

%

(c)

] dt.

Likewise, for time-continuous HD on each output, the

stochastic master equation reads [27, 53]

d%

(c)

= −i[

ˆ

H

ω

, %

(c)

] dt +

X

j

D[ˆc

j

]%

(c)

dt

+

X

j

√

η

j

H[ˆc

j

]%

(c)

dw

j

, (8)

where H[ˆc]% = ˆc% + %ˆc

†

− Tr[%(ˆc + ˆc

†

)]%, and dw

j

=

dy

j

−

√

η

j

Tr[%

(c)

(ˆc

j

+ ˆc

†

j

)] represents a standard Wiener

increment (s.t. dw

j

dw

k

= δ

jk

dt), operationally cor-

responding to the diﬀerence between the measurement

result dy

j

and the rescaled average value of the operator

(ˆc

j

+ ˆc

†

j

).

In general, diﬀerent measurement strategies mathe-

matically correspond to diﬀerent possible unravellings

of the Markovian master equation: any quantum state,

solution of Eq. (4) at a certain time t, can be written

as %

unc

=

P

traj

p

traj

%

(c)

, where p

traj

is the probability

of a certain trajectory leading to the conditional state

%

(c)

, and where the sum is replaced by an integral in

the case of time-continuous measurements with a con-

tinuous spectrum (e.g. homodyne).

One can then deﬁne an eﬀective QFI, which poses the

ultimate bounds for these kinds of estimation strate-

gies [39, 48, 51], and that depends both on the speciﬁc

unravelling and on the monitoring eﬃciencies η

j

:

e

Q

unr,η

j

= F[p

traj

] +

X

traj

p

traj

Q[%

(c)

] . (9)

As it is apparent from the formula,

e

Q

unr,η

is equal to

the sum of the classical Fisher information of the trajec-

tories probability distribution F[p

traj

] =

P

traj

(∂

ω

p

traj

)

2

p

traj

,

plus the average of the QFIs of the corresponding con-

ditional states Q[%

(c)

]. The ﬁrst term quantiﬁes the

amount of information obtained from the measurement

of the output modes after the interaction with the sys-

tem, while the second term quantiﬁes the amount of in-

formation obtained from the ﬁnal strong measurement

on the conditional states of the system itself. We point

Accepted in Quantum 2018-11-29, click title to verify 3

out that an analogous ﬁgure of merit has been recog-

nized as the appropriate one for metrology with post-

selection [54–56].

A method for the calculation of F[p

traj

] has been

ﬁrstly proposed in [41] for generic quantum states, and

then in [44] for continuous-variable Gaussian states. In

Sec. 5, we describe in detail a more compact and sta-

ble numerical algorithm for the calculation of the terms

present in Eq. (9), for both homodyne and photodetec-

tion measurements.

It is reasonable to expect that, thanks to the informa-

tion obtained from the time-continuous monitoring, and

thanks to the, typically beneﬁcial, eﬀects of quantum

measurements on the conditional states, one will obtain

more information on the parameter via these strate-

gies, than by solely measuring the unconditional state

solution of the Markovian master equation (4). This

intuition is conﬁrmed by the extended convexity prop-

erty of the QFI, recently proved in [39, 57]. In fact, the

following chain of inequalities holds:

Q[%

unc

] ≤

e

Q

unr,η

j

≤ Q

L

ω

. (10)

We also conjecture that, for a ﬁxed unravelling, and

for ﬁxed eﬃciency on all the output channels (i.e., for

η

j

= η ∀j), the eﬀective QFI is monotonic with respect

to the eﬃciency parameter, that is:

e

Q

unr,η

≤

e

Q

unr,η

0

⇐⇒ η ≤ η

0

(conjecture) .

(11)

3 Quantum frequency estimation in the

presence of noise

We consider a system of N qubits, described by a Hamil-

tonian

ˆ

H

ω

= (ω/2)

P

N

j=1

σ

(j)

z

, where ω is the unknown

frequency to be estimated, and σ

(j)

z

is the Pauli-z oper-

ator acting on the j-th qubit. The system is interacting

also with a Markovian environment, so that the evolu-

tion is described by the master equation

d%

dt

= L

ω

% = −i[

ˆ

H

ω

, %] +

κ

2

N

X

j=1

D[σ

(j)

α

]% , (12)

where α ∈ {x, z}, i.e., we only consider noise parallel

or transverse to the Hamiltonian. The master equa-

tion (12) can be easily mapped to Eq. (4) by consider-

ing the noise operators ˆc

j

=

p

κ/2σ

(j)

α

. In what follows

we will study all the quantities described in the pre-

vious section in order to assess the estimation of the

parameter ω.

In quantum frequency estimation strategies, one con-

siders the number of qubits N and the total time of

the experiment T as the resources of the protocol. The

quantum Cram´er-Rao bound (2) is then more eﬃciently

rewritten as

δω

√

T ≥

1

p

Q/t

≥

1

p

max

t

[Q/t]

(13)

where t = T/M corresponds to the duration of each

round, over which one can perform a further optimiza-

tion, and where Q corresponds here to the proper QFI

charaterizing the particular estimation strategy consid-

ered.

In the rest of the manuscript we are restricting

to initial entangled GHZ states |ψ

GHZ

i = (|0i

⊗N

+

|1i

⊗N

)/

√

2. It is well known that in the noiseless

case, i.e., for κ = 0, the corresponding QFI is Heisen-

berg limited, Q

HL

= N

2

t

2

. This leads to a quadratic

enhancement w.r.t. the “standard quantum limited”

QFI, Q

SQL

= Nt

2

(obtained in the case of a factorized

coherent-spin initial state |ψ

coh

i = [(|0i + |1i)/

√

2]

⊗N

).

In what follows, we will consider the noisy case (κ >

0); the generic QFI Q in Eq. (13) will correspond to

either: i) the QFI of the unconditional state Q[%

unc

]

corresponding to the master equation (12); ii) the ulti-

mate QFI Q

L

ω

obtained optimizing over all the possible

measurements on system and environmental outputs;

iii) the eﬀective QFI

e

Q

unr,η

corresponding to a spe-

ciﬁc time-continuous (sequential) measurement of the

output modes and a ﬁnal strong measurement on the

conditional state of the system. In particular we are

will focus on time-continuous photodetection and ho-

modyne detection, with measurement eﬃciency η and

we are labelling the respective eﬀective QFIs as

e

Q

pd,η

and

e

Q

hom,η

. It is important to remark that, given the

master equation (12), one is assuming that N diﬀerent

(homodyne or photo-) detectors are monitoring the en-

vironment of each qubit. We will however ﬁnd instances

where this assumptions may be relaxed.

In the next subsections we address separately the

two diﬀerent cases of parallel and transverse noise. For

each case we start by reviewing known results for the

QFI of the unconditional states; then we will present

our original results, regarding the ultimate QFI Q

L

ω

and eﬀective QFIs

e

Q

pd,η

and

e

Q

hom,η

(corresponding to

the schemes pictorially represented in Fig. 1), without

dwelling into the details of their calculation, which will

be left to Secs. 4 and 5.

3.1 Parallel noise

Parallel noise, corresponding to the master equa-

tion (12) with σ

(j)

α

= σ

(j)

z

, is typically considered the

most detrimental noise for frequency estimation since

the evolution induces dephasing in the eigenbasis of

the Hamiltonian

ˆ

H

ω

. For an initial GHZ state, the

Accepted in Quantum 2018-11-29, click title to verify 4

measurement

…

PD:

HD:

%

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Figure 1: Schematic representation of the metrological ap-

proach we consider in this paper. A N-qubit (possibly entan-

gled) input state |ψ

0

i interacts with N independent environ-

ments which are monitored by N detectors, either by photode-

tection (PD) or homodyne detection (HD). In the former case

each output is a binary valued detection record N

j

(t), in the

latter a real valued photo-current y

j

(t). The map E

(c)

ω,j

(t) repre-

sents the corresponding conditional dynamics, i.e., the solution

of the stochastic master equation for the single j-th qubit only,

either for PD (7) or HD (8), and thus depending on the corre-

sponding measurement output N

j

(t) or y

j

(t). The N-qubit

quantum state %

(c)

(t), conditioned on all the measurement

outcomes acquired during the evolution, is further collectively

measured at the end of the conditional dynamics.

QFI of the unconditional state can be evaluated ana-

lytically [6, 58], obtaining Q[%

unc

] = N

2

t

2

e

−2κNt

.

By optimizing over the single-shot duration t, one ob-

tains that the optimal QFI is standard quantum limited,

max

t



Q[%

unc

]

t



=

N

2eκ

. (14)

This result is equivalent to the one obtainable via a fac-

torized initial state. Only a constant enhancement can

in fact be gained, by optimizing over the initial entan-

gled state [6]; by doing this one saturates the ultimate

noisy bound derived in [7, 8], which dictates that, as

soon as some parallel noise is present in the dynamics,

the ultimate precision is standard quantum limited.

On the other hand, the ultimate QFI Q

L

ω

can be

easily evaluated (see Sec. 4 for details), and it turns out

to be equal to the noiseless QFI, i.e.

Q

k

L

ω

= N

2

t

2

. (15)

This shows that, by measuring also the output modes, it

is in principle possible to recover not only a Heisenberg

scaling for the error δω, but also the whole information

on the parameter.

For both time-continuous homodyne and photodetec-

tion, the evolution of an initial GHZ state, under par-

allel noise and conditioned on the measurement results,

is restricted to a two-dimensional Hilbert space. As

a consequence, the results obtained for N = 1 qubit

can be readily used to infer the results for generic N

qubits, by simply rescaling the evolution time t → Nt

(see Sec. 5 for details). We can thus obtain numerically

exact values of the eﬀective QFIs for any value of N ; in

particular we have been able to numerically verify that

the strategy based on time-continuous photodetection

with perfect eﬃciency η = 1 is indeed optimal, i.e.

e

Q

k

pd,η=1

= Q

k

L

ω

= N

2

t

2

, (16)

showing that the noiseless Heisenberg-limited result can

be recovered, without the need to perform complicated

(non-local in time) measurement strategies on system

and environment.

The physical explanation of this result can be eas-

ily understood by studying the action of the two Kraus

operators describing the conditional dynamics for ini-

tial GHZ states (see Sec. 5 for more details on the

Kraus operators). The no-jump evolution is ruled by

the inﬁnitesimal Kraus operator M

0

= − i

ˆ

H

ω

dt +

(κN/4) dt; as a consequence it easy to check that the

GHZ state evolves as in the unitary case, apart from an

irrelevant normalization factor that can be ignored.

On the other hand, when a jump occurs due to a pho-

ton detected in the output corresponding to one of the

N qubits, the only eﬀect of the corresponding Kraus

operator, M

(j)

1

= σ

(j)

z

p

(κ/2)dt, is to induce a relative

minus sign between the |0i

⊗N

and the |1i

⊗N

compo-

nents of the quantum conditional state, independently

on the index j of the corresponding qubit. In general,

after a time t and m detected photons from all the the

output channels, the conditional state reads:

|ψ

m|GHZ

i =

1

√

2



|0i

⊗N

+ e

i(Nωt+mπ)

|1i

⊗N



. (17)

It is important to remark that, thanks to the symme-

try of the GHZ state, and given that jumps may occur

only one by one [27], it is not necessary to know ex-

actly from which qubit channel the photon has been

detected; as a consequence one may obtain the same

result by using a single perfect photo-detector monitor-

ing jointly all the N output modes. Once the number

of detected photons m is known, and the correspond-

ing “GHZ-equivalent” conditional state is prepared, one

can estimate the frequency ω at the Heisenberg limit.

Accepted in Quantum 2018-11-29, click title to verify 5

As soon as the photodetection monitoring is not

perfectly eﬃcient (we always consider equal eﬃciency

for all the qubits, η

j

= η < 1, ∀j), the Heisen-

berg scaling is immediately lost. Our numerical re-

sults clearly show that the eﬀective QFI is equal to

e

Q

k

pd,η

= N

2

t

2

e

−2κ(1−η)Nt

, and the optimized QFI reads

max

t

"

e

Q

k

pd,η

t

#

=

N

2eκ(1 − η)

. (18)

Time-continuous ineﬃcient photodetection thus leads

to a constant enhancement, compared to the uncondi-

tional/classical case (14), which physically corresponds

to have a reduced eﬀective dephasing parameter κ

eﬀ

=

κ(1 −η). We remark that also in this case it is not nec-

essary to monitor every output channel separately, and

that a single-photo detector can be employed. More-

over, in this picture, the overall eﬃciency parameter

η corresponds to the product between the factual eﬃ-

ciency of the detectors and the fraction of qubits that

are eﬀectively monitored.

We also notice that, for every value of ω, all the infor-

mation is contained in the conditional quantum states:

the classical Fisher information F[p

traj

] is in fact iden-

tically equal to zero. This follows from the unitarity of

the Pauli matrices, leading to ˆc

†

j

ˆc

j

= κ

2

/2, and thus

to a parameter-independent Poisson increment with av-

erage value [dN

j

] = (ηκ/2) dt in Eq. (7). Neverthe-

less, we remark that the output from the photodetection

measurement is in fact essential to know the correspond-

ing conditional state, and thus to extract the whole in-

formation on ω via the ﬁnal strong measurement.

These results can be readily extended to the scenario

of non-linear quantum metrology, where k-body Hamil-

tonian and p-body dissipators are considered [59]. If

we focus on the case where k and p are odd numbers, by

preparing the initial state in a GHZ state, and by moni-

toring via photodetection each dissipation channel, one

can show by a simple rescaling of the variables ω and κ

that for unit monitoring eﬃciency, the ultimate (noise-

less) scaling ∼ N

k

can be restored; on the other hand, as

soon as the eﬃciency is smaller than one, one goes back

to the noisy scaling ∼ N

k−p/2

, with only a constant

enhancement, due to the ﬁnite monitoring eﬃciency.

The results for the case of continuous homodyne mon-

itoring are not reported here since the corresponding

eﬀective QFI, at ﬁxed eﬃciency η, is always lower than

the one obtained for continuous photodetection, and

Heisenberg scaling is not recovered even in the case of

perfect monitoring.

3.2 Transverse noise

Thanks to the symmetry of the GHZ state, the QFI

for the unconditional dynamics with arbitrary collapse

operators can be obtained without the need to diago-

nalize the full density matrix [15]. The corresponding

optimized QFI can be then numerically obtained and

the scaling is found to be intermediate between SQL

and Heisenberg: max

t



Q



%

⊥

unc



/t



≈ N

5/3

.

On the other hand, the ultimate QFI can be com-

puted analytically (see Sec. 4 for details on the calcula-

tion), yielding

Q

⊥

L

ω

=

N

2

(1 − e

−κt

)

2

+ N

h

2κt + 1 − (2 − e

−κt

)

2

i

κ

2

.

(19)

Two main observations are in order here: on the one

hand we observe that Q

⊥

L

ω

depends on the noise pa-

rameter κ and is always smaller than the noiseless QFI

Q

HL

= N

2

t

2

. This shows how, unlike in the parallel

noise case, for transverse (non-commuting) noise, part

of the information leaking into the environment is ir-

retrievably lost, and cannot be recovered even if one

has at disposal all the environmental degrees of free-

dom. On the other hand, this expression does explic-

itly show Heisenberg scaling, and can be further op-

timized over the evolution time t. In contrast to the

unconditional case, the optimal time t

opt

(N) does not

go to zero for N → ∞, instead it tends to a constant:

lim

N→∞

t

opt

(N) = c/κ, where c ≈ 1.26 . The very same

results can be obtained by computing the unconditional

QFI in the limit ω → 0 [60], which is already known to

give rise to Heisenberg scaling (see Appendix D of [16]),

i.e.

Q

⊥

L

ω

= lim

ω→0

Q



%

⊥

unc



. (20)

Notice that it is necessary to take the limit, instead

of using directly ω = 0, because for this value of the

parameter the density matrix changes its rank and this

gives rise to a discontinuity in the QFI [61, 62]. The ef-

fective QFI for photodetection and homodyne detection

is obtained numerically with the methods presented in

Sec. 5. From our results we observe that for unit eﬃ-

ciency η = 1 the eﬀective QFI saturates the ultimate

bound in both cases

e

Q

⊥

hd,η=1

=

e

Q

⊥

pd,η=1

= Q

⊥

L

ω

. (21)

This has been checked up to N = 14, but we conjec-

ture that this equality holds in general. The two terms

that contribute to

e

Q

⊥

hd,η=1

in Eq. (9) always sum up to

Q

⊥

L

ω

but with ω-dependent behaviors. This is shown

for a particular set of parameters in Fig. 2. On the

other hand, as explained before for the parallel case,

e

Q

⊥

pd,η=1

is only equal to the average QFI of the con-

ditional states, since the classical FI F[p

traj

] is always

identical to zero. As expected, for η < 1, we observe

Accepted in Quantum 2018-11-29, click title to verify 6

0 1 2 3 4 5

t

0

10

20

30

˜

Q

⊥

hd|η=1

/t

˜

Q

⊥

hd|η=1

F[p

traj

] Q[%

(c)

]

Figure 2: Contributions of the classical FI F[p

traj

] and the QFI

of the ﬁnal strong measurement Q[%

(c)

] to the eﬀective QFI

˜

Q

hd|η=1

for homodyne with N = 7 and ω = 1, in the transverse

noise case.

that the eﬀective QFI lies between the unconditional

QFI and the ultimate QFI, conﬁrming the chained in-

equalities (10) for both detection strategies and also

the conjecture regarding the monotonicity with the ef-

ﬁciency η.

As an example, in Fig. 3 we show the eﬀective QFI

over time for photodetection and homodyne detection

at diﬀerent eﬃciencies, for three diﬀerent values of N .

We can see that at lower eﬃciencies the curves tend

to the unconditional QFI Q[%

⊥

unc

], while for perfect eﬃ-

ciency they coincide with Q

L

ω

. Notice that in general

one cannot deﬁne a hierarchy between the two strate-

gies, and that in particular, in the case of homodyne

detection, the curves become constant at large t, due to

the non-vanishing contribution of the classical Fisher

information F[p

traj

] that is linear in t.

Since the numerical method for non-unit eﬃciency

requires using the full Hilbert space, the complexity of

the algorithm is exponential in N and we have been able

to obtain results only up to N = 7. Consequently, we

cannot explicitly witness a diﬀerent scaling from the un-

conditional case. As a matter of fact the diﬀerence be-

tween max

t

h

Q

⊥

L

ω

/t

i

and max

t



Q



%

⊥

unc



/t



is not very

signiﬁcant for N ≤ 7. This is shown in Fig. 4 for pho-

todetection, in the case ω = κ. As we can see, the

two quantities have a similar scaling in this range of N ,

with the eﬀective QFI lying between them with optimal

values that are monotonous with η. The optimal mea-

surement time decreases with N, and it increases with

increasing η.

4 Evaluation of the ultimate QFI

In this section we show how to derive the analytical

formulas for the ultimate QFI Q

L

ω

for both parallel

and transverse noise.

4.1 Parallel noise

We start by showing that, for parallel noise, the ulti-

mate QFI is equal to the QFI of the noiseless case: the

proof is however more general and is valid whenever

the collapse operators ˆc

j

commute with the free Hamil-

tonian

ˆ

H

ω

.

The master equation (4) is obtained by con-

sidering the following interaction Hamiltonian be-

tween the system and the input modes:

ˆ

H

int

(t) =

P

N

j=1



ˆc

j

ˆa

(j)†

in

(t) + ˆc

†

j

ˆa

(j)

in

(t)



. We remark that t is

merely a parameter that labels which mode interacts

with the system at time t and for each t we have a dif-

ferent operator acting on a diﬀerent Hilbert space. The

total time-dependent Hamiltonian for the system and

environment is thus

ˆ

H

SE

(t) =

ˆ

H

ω

+

ˆ

H

int

(t) and each

input mode interacts with the main system for an in-

ﬁnitesimal time dt. However, for the sake of clarity,

we will consider a discretization with a ﬁnite interac-

tion time δt, so that the evolution over a total time

T = Mδt involves a ﬁnite number M of input modes.

We also assume that the state of the input modes is the

vacuum |0i and that the initial state of the system |ψ

0

i

is pure.

Under these assumptions the joint state of system and

environment evolves as

|ψ

SE

(ω)i =

ˆ

U

t

M

. . .

ˆ

U

t

2

ˆ

U

t

1



|ψ

0

i ⊗ |0i

⊗M



, (22)

where t

j

= j ·δt and U

t

j

= exp

h

−iδt



ˆ

H

ω

+

ˆ

H

int

(t

j

)

i

.

This joint state is connected to the operator ¯% ap-

pearing in Eq. (6), since its trace represents the ﬁdelity

for two diﬀerent values of the parameter [42, 43], i.e.

hψ

SE

(ω

1

)|ψ

SE

(ω

2

)i = Tr [¯%] . (23)

When all the collapse operators ˆc

j

commute with the

free Hamiltonian

ˆ

H

ω

,

ˆ

H

int

commutes as well and we

have

ˆ

U

t

i

= exp

h

−iδt

ˆ

H

int

(t

i

)

i

· exp

h

−iδt

ˆ

H

ω

i

. (24)

Therefore, in the computation of the overlap (23) the

terms due to the interaction cancel out and we have

hψ

SE

(ω

1

)|ψ

SE

(ω

2

)i = hψ

0

|exp [−iT (H

ω

2

− H

ω

1

)] |ψ

0

i,

so that Eq. (5) gives the QFI of the unitary case. In

particular, this is true for parallel noise, i.e., for ˆc

j

=

Accepted in Quantum 2018-11-29, click title to verify 7

0 1 2 3 4 5

0.0

0.5

1.0

1.5

˜

Q

⊥

pd|η

/t

N = 1

0 1 2 3 4 5

0

2

4

6

N = 3

0 1 2 3 4 5

0

10

20

N = 7

0 1 2 3 4 5

t

0.0

0.5

1.0

1.5

˜

Q

⊥

hd|η

/t

0 1 2 3 4 5

t

0

2

4

6

0 1 2 3 4 5

t

0

10

20

η

0.25

0.50

0.75

0.90

0.99

1.00

Q

L

ω

Q[%

unc

]

η

0.25

0.50

0.75

0.90

0.99

1.00

Q

L

ω

Q[%

unc

]

η

0.25

0.50

0.75

0.90

0.99

1.00

Q

L

ω

Q[%

unc

]

Figure 3: The ﬁgure shows the eﬀective QFI for photodetection

e

Q

⊥

pd,η

/t (top row) and for homodyne

e

Q

⊥

ph,η

/t (bottom row) with

transverse noise over time, for diﬀerent eﬃciencies η and for diﬀerent values of N . The eﬀective QFI is compared with the ultimate

bound Q

⊥

L

ω

(black dashed line), and the QFI for the unconditional evolution Q[%

⊥

unc

] (black dot-dashed line). Here we take ω = κ.

The colored curves are obtained numerically as explained in Sec. 5, by simulating a large number > 10 k trajectories.

1 2 3 4 5 6 7

N

0

5

10

15

20

max

t

Q/t

1 2 3 4 5 6 7

N

0

1

2

3

4

t

opt

η

0.25

0.50

0.75

0.90

0.99

Q

L

ω

Q[%

unc

]

Figure 4: The maximum over time of the quantity Q/t (left panel) and the corresponding optimal time (right panel) for ω = κ,

where Q =

e

Q

⊥

pd|η

for various values of η for the colored dots, Q = Q

⊥

L

ω

for the black circles, and Q = Q[%

unc

] for the black

stars. The gray shade shows the region individuated by the bounds in Eq. (10). Notice that Q

⊥

L

ω

/t is monotonically increasing for

N ≤ 3, and it reaches its maximum value for t → ∞, and that the optimal measurement time N ≤ 2 for η = 0.99 is out of scale.

Accepted in Quantum 2018-11-29, click title to verify 8

p

k/2σ

(j)

z

, and, for any pure initial state of the system

|ψ

0

i, we have:

Q

k

L

ω

= Q

h

e

−i

ˆ

H

ω

t

|ψ

0

ihψ

0

|e

i

ˆ

H

ω

t

i

(25)

= 4



hψ

0

|



∂

ω

ˆ

H

ω



2

|ψ

0

i − hψ

0

|



∂

ω

ˆ

H

ω



|ψ

0

i

2



.

4.2 Transverse noise

When the collapse operators do not commute with the

generator, as in the case of frequency estimation with

transverse noise, we can simplify the computation by

using the assumption of an identical and independent

noise acting on each qubit. Since Eq. (6) is linear in ¯%

and the coeﬃcients are time-independent, we can still

write the solution as a linear map, formally

˜

E

ω

1

,ω

2

(t) =

exp



t

˜

L

ω

1

,ω

2



. This map is only guaranteed to be linear

and in general it is not even positive.

Since the map acts independently on every qubit,

we can still write the global action on the N -qubit

state as the tensor product

˜

E

N

ω

1

,ω

2

(t) =

˜

E

ω

2

,ω

2

(t)

⊗N

,

where

˜

E

ω

2

,ω

2

(t) is the single-qubit solution. The ulti-

mate bound is thus obtained as

Q

L

ω

[|ψ

0

i] = 4 ∂

ω

1

∂

ω

2

log Tr[

˜

E

N

ω

1

,ω

2

(t) %

0

]



ω

1

=ω

2

=ω

,

(26)

where %

0

is the initial pure state. Given our choice

%

0

= |ψ

GHZ

ihψ

GHZ

|, the computation can be greatly sim-

pliﬁed. We ﬁnd that

˜

E

N

ω

1

,ω

2

(t) |ψ

GHZ

ihψ

GHZ

| = (27)

=

1

2

"



˜

E

ω

1

,ω

2

(t) |0ih0|



⊗N

+



˜

E

ω

1

,ω

2

(t) |1ih1|



⊗N

+



˜

E

ω

1

,ω

2

(t) |0ih1|



⊗N

+



˜

E

ω

1

,ω

2

(t) |1ih0|



⊗N

#

.

For transverse noise and for a single qubit, the equa-

tion to solve to compute Q is the following

d¯%

dt

=

˜

L

⊥

ω

1

,ω

2

[¯%] =

= −

i

2

(ω

1

σ

z

¯% −ω

2

˜%σ

z

) +

κ

2

(σ

x

¯%σ

x

− ¯%) .

(28)

The solution is obtained in the same way as for canon-

ical master equations: by choosing a basis of opera-

tors and using a matrix representation of superopera-

tors [63] (see also the Supplementary Material of [42]).

By making use of the normalized Pauli operators ˜σ

i

=

σ

i

/

√

2 (where σ

0

= ), so that Tr [˜σ

i

˜σ

j

] = δ

ij

, we

can ﬁnd the matrix associated to the single qubit map

˜

E

ω

1

,ω

2

(t), obtained by matrix exponentiation. We can

write the generalized density operator ¯% in Bloch form

as ¯% =

1

√

2



a

0

˜σ

0

+ ~a ·

~

˜σ



such that its trace is simply

Tr [¯%] = a

0

. In this notation, the initial states |0ih0| and

|1ih1| correspond to the vectors e

00

=

1

√

2

(1, 0, 0, 1)

T

and

e

11

=

1

√

2

(1, 0, 0, −1)

T

, while the oﬀ-diagonal elements

are e

01/10

=

1

√

2

(0, 1, ±i, 0)

T

.

Since we are only interested in the coeﬃcient a

0

, we

only need the ﬁrst row of the matrix representation of

˜

E

⊥

ω

1

,ω

2

(t), which turns out to be



˜

E

⊥

ω

2

,ω

2

(t)



0,∗

= e

−

κt

2



cosh



t

2

q

κ

2

− (ω

1

− ω

2

)

2



+

κ sinh



t

2

√

κ

2

−(ω

1

−ω

2

)

2



2

√

κ

2

−(ω

1

−ω

2

)

2

, 0, 0,

i(ω

2

−ω

1

) sinh



t

2

√

κ

2

−(ω

1

−ω

2

)

2



√

κ

2

−(ω

1

−ω

2

)

2



. (29)

We can see that the oﬀ-diagonal terms are kept oﬀ-

diagonal by the map, therefore they do not contribute

to the trace and we can further simplify the calculation

as

Tr



˜

E

⊥N

ω

1

,ω

2

(t)|ψ

GHZ

ihψ

GHZ

|



= (30)

=

1

2



Tr



˜

E

⊥

ω

1

,ω

2

(t)|0ih0|



N

+ Tr



˜

E

⊥

ω

1

,ω

2

(t)|1ih1|



N



=

1

√

2

(



[

˜

E

⊥

ω

1

,ω

2

(t)]

0,∗

· e

00



N

+



[

˜

E

⊥

ω

1

,ω

2

(t)]

0,∗

· e

11



N

)

.

We can thus plug this result into (26) and ﬁnally obtain

Eq. (19).

5 Numerical algorithm for the calcu-

lation of the eﬀective QFI for time-

continuous strategies

We can now describe the numerical algorithm we have

implemented to calculate the eﬀective QFI

e

Q

unr,η

. The

numerical results presented in this manuscript are ob-

tained with the code available online at [64], written in

the Julia language [65].

We will focus on the case of homodyne detection and

then we will discuss the small changes needed for pho-

todetection. We ﬁrst review the existing method to cal-

culate one of the two key quantities in Eq. (9), namely

Accepted in Quantum 2018-11-29, click title to verify 9

the classical Fisher information F[p

traj

]. In [41] it is

shown that it can be calculated as

F[p

traj

] =



Tr[τ]

2



, (31)

where we have deﬁned the operator

τ =

∂

ω

˜%

(c)

Tr[˜%

(c)

]

, (32)

in terms of the unormalized conditional state ˜%

(c)

evolv-

ing according to the SME

d˜%

(c)

= −i[

ˆ

H

ω

, ˜%

(c)

] dt +

X

j

D[ˆc

j

]˜%

(c)

dt

+

X

j

√

η

j



ˆc

j

˜%

(c)

+ ˜%

(c)

ˆc

†

j



dy

j

, (33)

and such that %

(c)

= ˜%

(c)

/Tr[˜%

(c)

].

One can then numerically integrate simultaneously

the two SMEs, the one for the conditional state %

(c)

in

Eq. (8), and the one above for τ, and then evaluate the

corresponding FI, by averaging over a certain number

of trajectories. However, it is computationally very ex-

pensive to obtain a guaranteed Hermitian %

(c)

by using

standard methods, such as Euler-Maruyama or Euler-

Milstein, and still these methods do not guarantee the

positivity of corresponding density operator.

An alternative method to integrate the SME, which

is able to circumvent these problems, has been intro-

duced in [53, 66], by exploiting the Kraus operators

corresponding to the (weak) measurement performed at

each instant of time. After an inﬁnitesimal time dt, the

evolved state corresponding to (8) can in fact be written

as

%

(c)

t+dt

=

M

dy

%

(c)

t

M

†

dy

+

P

j

(1 − η

j

)ˆc

j

%

(c)

t

ˆc

†

j

dt

Tr[M

dy

%

(c)

t

M

†

dy

+

P

j

(1 − η

j

)ˆc

j

%

(c)

t

ˆc

†

j

dt]

,

(34)

where we have explicitly put the time dependence of the

density operators and where we have deﬁned the Kraus

operator

M

dy

= − i

ˆ

H

ω

dt −

1

2

X

j

ˆc

†

j

ˆc

j

dt +

X

j

√

ηˆc

j

dy

j

,

(35)

with dy = {dy

j

} being a vector of measurement results,

corresponding to each output channel,

dy

j

=

√

η

j

Tr[%

(c)

t

(ˆc

j

+ ˆc

†

j

)] dt + dw

j

. (36)

Equation (34) can be used for numerical purposes,

where the inﬁnitesimal time dt is replaced by a ﬁnite

time step ∆t, the Wiener increments dw

j

are replaced

by Gaussian random variables ∆w

j

centered in zero and

with variance equal to ∆t, and where one can also imple-

ment the second order Euler-Milstein corrections. The

(numerical) Kraus operators in this case read

M

∆y

= − i

ˆ

H

ω

∆t −

1

2

X

j

ˆc

†

j

ˆc

j

∆t +

X

j

√

η

j

ˆc

j

∆y

j

+

X

j,k

η

2

ˆc

j

ˆc

k

(∆y

j

∆y

k

− δ

j,k

∆t) , (37)

with

∆y

j

=

√

η

j

Tr[%

(c)

t

(ˆc

†

j

+ ˆc

j

)] ∆t + ∆w

j

, (38)

denoting the (ﬁnite) increments of the measurement

records.

In the following we will show how to extend this

method to obtain an eﬃcient and numerically stable

calculation of both F[p

traj

] and Q[%

c

]. We will prove

everything in terms of the “inﬁnitesimal” Kraus opera-

tors, which will have to be replaced by Eq. (37) when

implementing the numerical algorithm. We will start by

showing the results for the most general case of SME

and ineﬃcient detection; we will then describe the more

eﬃcient algorithm that can be implemented in the case

of perfect detection (η

j

= 1), i.e., when the dynamics

can be described by a stochastic Schr¨odinger equation.

5.1 Non-unit eﬃciency detection (stochastic

master equation)

We start by observing that the evolution of the unnor-

malized conditional state and of its derivative can be

written in terms of the Kraus operators (in what fol-

lows we will omit the superscript (c) used to denote

conditional states):

˜%

t+dt

= M

dy

˜%

t

M

†

dy

+

X

j

(1 − η

j

)ˆc

j

˜%

t

ˆc

†

j

dt ,

∂

ω

˜%

t+dt

= M

dy

(∂

ω

˜%

t

)M

†

dy

+ (∂

ω

M

dy

)˜%

t

M

†

dy

+

+ M

dy

˜%

t

(∂

ω

M

†

dy

)+

+

X

j

(1 − η

j

)ˆc

j

(∂

ω

˜%

t

)ˆc

†

j

dt , (39)

where ∂

ω

M

dy

= −i(∂

ω

ˆ

H

ω

) dt. The trace of the unnor-

malized state reads

Tr[˜%

t+dt

] = Tr[M

dy

˜%

t

M

†

dy

+

X

j

(1 − η

j

)ˆc

j

˜%

t

ˆc

†

j

dt]

= Tr[˜%

t

]Tr[M

dy

%

t

M

†

dy

+

X

j

(1 − η

j

)ˆc

j

%

t

ˆc

†

j

dt] ,

(40)

Accepted in Quantum 2018-11-29, click title to verify 10

where we have used the relation %

t

= ˜%

t

/Tr[˜%

t

]. We can

now use these formulas to obtain the evolution for the

operator τ

t+dt

just in terms of the operators %

t

and τ

t

at the previous time step:

τ

t+dt

=

1

Tr[˜%

t+dt

]





(∂

ω

M

dy

)˜%

t

M

†

dy

+ M

dy

(∂

ω

˜%

t

)M

†

dy

+ M

dy

˜%

t

(∂

ω

M

†

dy

) +

X

j

(1 − η

j

)ˆc

j

(∂

ω

˜%

t

)ˆc

†

j

dt





=

(∂

ω

M

dy

)%

t

M

†

dy

+ M

dy

τ

t

M

†

dy

+ M

dy

%

t

(∂

ω

M

†

dy

) +

P

j

(1 − η

j

)ˆc

j

τ

t

ˆc

†

j

dt

Tr

h

M

dy

%

t

M

†

dy

+

P

j

(1 − η

j

)ˆc

j

%

t

ˆc

†

j

dt

i

. (41)

One can thus evaluate the trace of this operator

at each time t, and evaluate accordingly the classical

Fisher information F[p

traj

] as in Eq. (31).

Notice that the evolution for the derivative operator

∂

ω

%

t

can be now written in terms of the renormalized

operators %

t

and τ

t

:

∂

ω

%

t

=

∂

ω

˜%

t

Tr[˜%

t

]

−

Tr[∂

ω

˜%

t

]

Tr[˜%

t

]

2

˜%

t

= τ

t

− Tr[τ

t

]%

t

. (42)

The QFI Q[%

t

] can then be evaluated at each time t

by using the formula [67]:

Q[%

t

] = 2

X

λ

s

+λ

t

6=0

|hψ

s

|∂

ω

%

t

|ψ

t

i|

2

λ

s

+ λ

t

, (43)

upon writing %

t

in its eigenbasis, i.e., %

t

=

P

s

λ

s

|ψ

s

ihψ

s

|.

We would like to underline the key features of our

algorithm. The relevant ﬁgures of merit could naively

be derived from the evolution of the unnormalized con-

ditional state ˜%

t

, described in Eq. (39). However Tr[˜%

t

]

becomes very small during the evolution, leading to nu-

merical instabilities in the evaluation of both F[p

traj

]

and Q[%

t

]. Thanks to Eqs. (41) and (42), we are able

to express the above quantities only in terms of the nu-

merically stable operators %

t

and τ

t

. Besides, the for-

mulation in terms of Kraus operators, following [53, 66],

ensures that the density operator remains positive, as

opposed to standard numerical integration of the SME.

5.2 Unit eﬃciency detection (stochastic

Schr

¨

odinger equation)

The above calculations are greatly simpliﬁed when the

dynamics starts with a pure initial state and the eﬃ-

ciency parameters are equal to one, η

j

= 1. In fact

the quantum conditional state |ψ

t

i remains pure during

the whole evolution and the dynamics is described by a

stochastic Schr¨odinger equation. We can thus work with

state vectors, instead of density matrices, with a conse-

quent reduction of complexity of the numerical simula-

tion, which allows us to reach higher values of N with

a given amount of memory.

In terms of Kraus operators, the unnormalized and

normalized conditional states are obtained respectively

as:

|

e

ψ

t+dt

i = M

dy

|

e

ψ

t

i, (44)

|ψ

t+dt

i =

M

dy

|ψ

t

i

q

hψ

t

|M

†

dy

M

dy

|ψ

t

i

=

M

dy

|

e

ψ

t

i

q

h

e

ψ

t

|M

†

dy

M

dy

|

e

ψ

t

i

.

The operator τ

t

in this case can be written as

τ

t

=

∂

ω

(|

e

ψ

t

ih

e

ψ

t

|)

h

e

ψ

t

|

e

ψ

t

i

=

|∂

ω

e

ψ

t

ih

e

ψ

t

| + |

e

ψ

t

ih∂

ω

e

ψ

t

|

h

e

ψ

t

|

e

ψ

t

i

, (45)

and its trace is equal to

Tr[τ

t

] =

h

e

ψ

t

|∂

ω

e

ψ

t

i + h.c.

h

e

ψ

t

|

e

ψ

t

i

= hψ

t

|φ

t

i + hφ

t

|ψ

t

i. (46)

In the last equation we have introduced the vector

|φ

t

i =

|∂

ω

e

ψ

t

i

q

h

e

ψ

t

|

e

ψ

t

i

. (47)

At time t + dt, the vector |φ

t+dt

i can be obtained as

|φ

t+dt

i =

(∂

ω

M

dy

)|

e

ψ

t

i + M

dy

|∂

ω

e

ψ

t

i

q

h

e

ψ

t

|M

†

dy

M

dy

|

e

ψ

t

i

=

(∂

ω

M

dy

)|ψ

t

i + M

dy

|φ

t

i

q

hψ

t

|M

†

dy

M

dy

|ψ

t

i

, (48)

where we have exploited the identity

h

e

ψ

t

|M

†

dy

M

dy

|

e

ψ

t

i = h

e

ψ

t

|

e

ψ

t

ihψ

t

|M

†

dy

M

dy

|ψ

t

i. (49)

Accepted in Quantum 2018-11-29, click title to verify 11

We notice that as in Eq. (41), the evolution equation

for the vector |φ

t+dt

i depends only on the vectors |ψ

t

i

and |φ

t

i and not on the unnormalized state |

e

ψ

t

i, and

one can readily evaluate the classical Fisher information

in terms of these two vectors via Eqs. (46) and (31).

Regarding the QFI of the conditional state, we ﬁrst

observe that

|∂

ω

ψ

t

i =

|∂

ω

e

ψ

t

i

q

h

e

ψ

t

|

e

ψ

t

i

−

h

e

ψ

t

|∂

ω

e

ψ

t

i + h∂

ω

e

ψ

t

|

e

ψ

t

i|

e

ψ

t

i

2h

e

ψ

t

|

e

ψ

t

i

3/2

= |φ

t

i −

hψ

t

|φ

t

i + hφ

t

|ψ

t

i

2

|ψ

t

i. (50)

As we are dealing with pure states, the QFI of the con-

ditional state can be simply evaluated as [67]

Q[|ψ

t

i] = 4



h∂

ω

ψ

t

|∂

ω

ψ

t

i + (h∂

ω

ψ

t

|ψ

t

i)

2



. (51)

The algorithms above have been derived for the case

of time-continuous homodyne detection. Nonetheless

one can easily extend them to time-continuous photode-

tection as follows.

The Kraus operators M

dy

have to be replaced at each

time step with one of the following Kraus operators cor-

responding to “no detector click” and “detector click”

(in one of the output channels) events, respectively

M

0

= − i

ˆ

H

ω

dt −

1

2

X

j

ˆc

†

j

ˆc

j

dt , (52)

M

(j)

1

=

p

η

j

dt ˆc

j

. (53)

At each time step, each Kraus operator M

(j)

1

has to

be applied with probabilities p

(j)

1

= η

j

Tr[%

t

ˆc

†

j

ˆc

j

] dt and,

correspondingly, the Kraus operator M

0

has to be ap-

plied with probability p

0

= 1 −

P

j

p

(j)

1

[27].

We ﬁnally remark that in the case of frequency es-

timation with parallel noise and initial GHZ state, the

numerical calculations are incredibly simpliﬁed thanks

to the symmetry of the dynamics. In fact the whole

(both conditional and unconditional) evolution is equiv-

alently described in two two-dimensional Hilbert space

spanned by the vectors |

¯

0i = |0i

⊗N

and |

¯

1i = |1i

⊗N

.

As regards continuous photodetection, the Kraus oper-

ators (52) and (53) are replaced by the eﬀective Kraus

operators

M

0

=

2

− iNωσ

z

dt − (Nκ/4)

2

dt , (54)

M

1

=

p

(ηκ/2) dt N σ

z

. (55)

where

2

and σ

z

are respectively the identity operator

and the Pauli-z matrix in the Hilbert space spanned

by |

¯

0i and |

¯

1i, and where the operators M

1

and M

0

have to be applied respectively with probabilities ¯p

1

=

N(ηκ/2) dt (i.e., the total probability of observing a

photon in one of the N output channels) and ¯p

0

= 1−¯p

1

.

More practically the results for generic N qubits can be

readily obtained by exploiting the results obtained for

one qubit, and rescaling the time as t → Nt. In this

case it is also easy to show that the eﬃciency param-

eter η corresponds to the product between the factual

eﬃciency of the photo-detectors (that we assume to be

equal) and the fraction of qubits that are eﬀectively

monitored.

6 Conclusions and remarks

We have discussed for the ﬁrst time the usefulness

of time-continuous monitoring in the context of noisy

quantum metrology. We have proven that the desired

Heisenberg scaling can in fact be restored by exploiting

these schemes and we have obtained several conceptu-

ally and practically relevant results.

We have shown the fundamental diﬀerence between par-

allel and transverse noise with regards to the ultimate

limit achievable when the environmental degrees of free-

dom can be measured. In the ﬁrst case, i.e., when the

Hamiltonian and the noise generator commute, having

access to every degree of freedom of the environment

allows us to restore the unitary (noiseless) Heisenberg

limit. In the latter case, i.e., in the presence of trans-

verse noise, the ultimate QFI still presents a Heisenberg

(quadratic) scaling in the number of qubits N; however,

it is always lower than the unitary QFI, thus showing

that some information is irremediably lost due to the

interaction with the environment. These ﬁndings com-

plement the results obtained for frequency estimation

with open quantum systems [68], where the geometry

of the noise is indeed crucial in determining the diﬀerent

achievable scalings.

The second non-trivial and conceptually relevant re-

sult is that estimation strategies based on (sequential)

time-continuous monitoring and ﬁnal strong measure-

ments on the system are optimal, i.e., the correspond-

ing eﬀective QFIs

e

Q

unr,η

j

are equal to the ultimate QFIs

Q

L

ω

. This is true for both parallel and transverse

noise, showing that no complicated estimation strate-

gies based on “entangled measurements” on the envi-

ronment and system are needed to achieve the ultimate

limit on the precision. This result, which was not sug-

gested by any mathematical or physical intuition, is par-

ticularly important from a practical point of view, as it

greatly relaxes the assumptions and demands that are

requested to achieve the ultimate limits.

We have also discussed in detail the case of ﬁnite ef-

ﬁciency monitoring. In the presence of parallel noise

we have shown that the estimation precision is sub-

ject to the standard quantum limit, with a behaviour

Accepted in Quantum 2018-11-29, click title to verify 12

ruled by a reduced eﬀective dephasing: one can still

obtain a constant enhancement, which can be made ar-

bitrarily high by increasing the monitoring eﬃciency. In

the presence of transverse noise, we have observed the

expected enhancement with respect to the optimized

(super-classical) unconditional results [15]. However,

due to the computational complexity of the algorithm

needed to obtain the eﬀective QFI, we could not infer

any conclusion regarding the scaling with the number

of probes N.

It is important to remark that the use of continu-

ous measurements and feedback techniques has long

been recognized as a useful tool for ﬁghting decoherence

and to prepare non-classical quantum states [27, 69–75].

Our metrological scheme follows this line of thought,

but with the great advantage that it is based on con-

tinuous monitoring only, without error correction steps

(or feedback), diﬀering it from other recent approaches

in noisy quantum metrology [21, 22].

Moreover, while most of the literature on parame-

ter estimation with continuous measurements focuses

on the information gained from the continuous signal,

the crucial part that makes our protocol able to recover

Heisenberg scaling is the ﬁnal strong measurement on

the conditional state. A great deal of eﬀort has been

also devoted to studying the asymptotic properties of

estimation via repeated/continuous measurements [76–

78]. In this approach one is usually interested in per-

forming a single run of the experiment and thus observes

the system for a long time. Our point of view is radi-

cally diﬀerent and rooted in previous works on quantum

frequency estimation. As a matter of fact, we are in-

terested in the initial part of the dynamics, long before

reaching the steady state. For this reason the asymp-

totic regime of the statistical model is reached by run-

ning the experiment many times, instead of observing

the system for a long time.

In order to obtain these results we have developed

a stable algorithm for the evaluation of the eﬀective

quantum Fisher information that quantiﬁes the perfor-

mance of the proposed strategies. The ﬂexibility of this

algorithm and the originality of our protocols pave the

way for a series of future investigations that will exploit

techniques and ideas that so far have only been applied

to standard noisy metrology schemes. For example, one

can investigate the performances of these protocols with

optimized [79] and noisy quantum probes [14], or by

considering entangled ancillary systems [80–82]. Fur-

thermore, one can also study in detail the role played

by quantum coherence [83] and the possibility of con-

sidering unravellings of non-Markovian master equa-

tions [84].

Acknowledgments

The authors acknowledge useful discussions with A.

Del Campo, M. Paris, P. Rouchon, A. Smirne and T.

Tufarelli. MACR was supported by the Horizon 2020

EU collaborative project QuProCS (Grant Agreement

No. 641277) and by the Academy of Finland Centre of

Excellence program (project 312058). MGG acknowl-

edges support from Marie Skodowska-Curie Action

H2020-MSCA-IF-2015 (project ConAQuMe, grant no.

701154) and from a Rita Levi-Montalcini fellowship of

MIUR.

FA and MACR have equally contributed to this work.

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