Optimal time for sensing in open quantum systems Zain H. Saleem Mathematics and Computer Science Division Argonne National Laboratory 9700 S Cass Ave Lemont IL 60439

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Optimal time for sensing in open quantum systems

Zain H. Saleem∗

Mathematics and Computer Science Division, Argonne National Laboratory, 9700 S Cass Ave, Lemont IL 60439

Anil Shaji†

School of Physics, IISER Thiruvananthapuram, Kerala, India 695551

Stephen K. Gray‡

Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA

(Dated: June 12, 2023)

We study the time-dependent quantum Fisher information (QFI) in an open quantum system sat-

isfying the Gorini-Kossakowski-Sudarshan-Lindblad master equation. We also study the dynamics

of the system from an eﬀective non-Hermitian dynamics standpoint and use it to understand the

scaling of the QFI when multiple probes are used. A focus of our work is how the QFI is maximized

at certain times suggesting that the best precision in parameter estimation can be achieved by

focusing on these times. The propagation of errors analysis allows us to conﬁrm and better under-

stand this idea. We also propose a parameter estimation procedure involving relatively low resource

consuming measurements followed by higher resource consuming measurements and demonstrate it

in simulation.

I. INTRODUCTION

Quantum sensing and metrology [1–8] involve the ex-

ploration of subtle quantum eﬀects to increase the preci-

sion of parameter estimation. Quantum sensing has be-

come one of the most promising applications of quantum

technologies, involving single- or multi-parameter estima-

tion. In this work we will use quantum Fisher informa-

tion (QFI) as a tool to study quantum sensing for open

quantum systems [9–14]. The QFI [1] quantiﬁes the the-

oretical bound on the achievable precision in estimating

a parameter using a quantum state as a probe and can

be regarded as a performance measure of a quantum sys-

tem as a quantum sensor. The open quantum systems

we will study in this work are dynamic, i.e., evolve with

time, and therefore it makes more sense to study the time

dependence of the QFI.

We consider as quantum probes one or more two-level

systems or qubits and employ two diﬀerent approaches

to study their environmental interactions or open sys-

tem dynamics [1]. The ﬁrst is based on the Gorini-

Kosskowski-Sudarshan-Lindblad (GKSL) master equa-

tion [15–18] where we assume a Markovian interaction

of the probe with its environment and integrate out the

degrees of freedom of the environment to derive a dy-

namical equation for the probe. The second approach

is based on a non-Hermitian extension of quantum me-

chanics [19] and allows us to investigate sensors with a

large number of probes. Here the Hamiltonian describing

the evolution of the probe is assumed to acquire an anti-

Hermitian part which can be associated with dissipative

eﬀects. One of our results is to show how the two-level

∗zsaleem@anl.gov

†shaji@iisertvm.ac.in

‡gray@anl.gov

non-Hermitian systems can be extended to display the

GKSL dynamics.

In the non-Hermitian approach one also can encounter

exceptional points that mark the transition of the Hamil-

tonian from a PT-symmetric form to one that is not PT-

symmetric [20]. Possible quantum advantages in sens-

ing and metrology facilitated by such exceptional points

have been of interest recently [21]. Framing the metrol-

ogy scheme using the non-Hermitian as well as GKSL

master equation based approaches also allow us to ad-

dress the question of metrological advantage around the

exceptional points. We ﬁnd no such advantage at the ex-

ceptional point, consistent with several previous studies

of related systems [21,22].

In the absence of dissipation, the QFI for the sys-

tem we consider increases monotonically with time. This

means that the achievable precision in the estimate of the

parameter of interest will improve with increased dura-

tion of the measurement. However, when dissipation is

present, this is not the case. We ﬁnd that for the param-

eter estimation problem we are considering, there is an

optimal time at which QFI is largest and consequently

one can expect to get best possible measurement preci-

sion at this time. With dissipation, it is important to also

verify whether the bound on the measurement precision

given by the QFI is achievable in practice.

In section II we will introduce classical and quantum

Fisher information. We investigate the time dependent

QFI for the open quantum systems via the GKSL formal-

ism in section III and discuss the extension to N-probes

case via the non-Hermitian approach. In section IV we

compare our result obtained via the time dependent QFI

with the propagation of error in the variance of the pa-

rameter and show that they both match to a high accu-

racy. We suggest an experimental procedure for param-

eter estimation making use of the optimum time concept

in section V. Finally in section VI we give conclusions

arXiv:2210.10926v2 [quant-ph] 9 Jun 2023

and future directions.

II. QUANTUM FISHER INFORMATION AND

PARAMETER ESTIMATION

Let us consider the problem of simultaneously estimat-

ing nparameters xi={x1, x2,···xn}in a quantum ex-

periment and denote the respective estimators of these

parameters by ˆ

xi={ˆx1,ˆx2,··· ˆxn}. The uncertainty in

the the estimator ˆ

xiis quantiﬁed by the covariance ma-

trix Cov(ˆ

xi,ˆ

xj) and is upper bounded by the quantum

Cram´er-Rao Bound [23],

Cov(ˆ

xi,ˆ

xj)≥1

MFi,j

(1)

where Mstands for the total number of experiments

and Fi,j is the quantum Fisher information matrix which

quantiﬁes the responsiveness of the quantum state of the

probe to changes in the measured parameters xi. The

coeﬃcients of the QFI matrix for a given initial state of

the probe given by the density matrix ρare given by the

formulae,

Fi,j = TrLi

∂ρ

∂xj,(2)

where Liis the symmetric logarithmic derivative [24] for

the parameter xiand is deﬁned implicitly by,

∂ρ

∂xi

2(ρLi+Liρ).(3)

Upon writing the density matrix in its eigenbasis ρ=

Paλa|λa⟩⟨λa|and substituting in the above equation we

get,

Li=X

{ab|λa+λb̸=0}

λa+λb⟨λa|∂iρ|λb⟩|λa⟩⟨λb|,(4)

This formula for the SLD when substituted in (2) gives

us the QFI for a mixed state [25],

Fij =X

{ab|λa+λb̸=0}

Fij (a, b) (5)

where,

Fij (a, b) = 2Re⟨λa|∂iρ|λb⟩⟨λb|∂jρ|λa⟩

λa+λb

,(6)

In the above formula one needs to ensure that the sum-

mation is performed only over those eigenvalues for which

λa+λb̸= 0. With a little bit of work one can also

show that if we have pure parameterized quantum state

|ψ⟩:= |ψ(⃗x)⟩then the formula for the QFI matrix be-

comes,

Fij = 4Re⟨∂iψ|∂jψ⟩−⟨∂iψ|ψ⟩⟨ψ|∂jψ⟩.(7)

The Fisher information for a particular parameter xiis

deﬁned as F≡Fi,i. Since we will be exclusively consid-

ering single parameter estimation we use Ffor QFI and

F(a, b) for the individual elements in the mixed state QFI

formula, Eq. (6).

III. TIME DEPENDENT QUANTUM FISHER

INFORMATION

A. Single Probe Case

Let us consider a three-level system with a Hamiltonian

in the interaction picture or rotating frame given by,

H=ℏg(|e⟩⟨f|+|f⟩⟨e|) + ℏ∆(|f⟩⟨f|−|e⟩⟨e|) (8)

where |e⟩and |f⟩are the ﬁrst two excited states of the

system, gis the coupling between these excited states

and ∆ is the detuning parameter. The third state which

is the ‘sink’ |s⟩enters the dynamics through the GKSL

master equation,

˙ρ=1

iℏ[H, ρ] + LeρL†

e−1

2{L†

eLe, ρ}.(9)

The jump operator describing the dissipation is

Le=√γe|s⟩⟨e|

and it is responsible for the loss of energy from state |e⟩

to the sink, |s⟩. We note that Eqs. (8) and (9) and non-

Hermitian variations have previously been considered by

Murch and co-workers [11,26] as models of a three-level

superconducting transmon qubit. Of course, this simple

model could also describe many other situations, for ex-

ample, the single excitation manifold of a two-level qubit

coupled to a bosonic cavity in a Jaynes-Cummings model.

It is instructive to connect this model to the phys-

ical scenario of estimating the interaction between an

atom and a cavity mentioned earlier. Imagine a stream

of atoms passing through the cavity one by one. Assume

that initially the atom is in a state |f⟩and that the cav-

ity induces coherent transitions between |f⟩and another

atomic level |e⟩. In the lossless case, the probability of

the atom exchanging a photon with the cavity and com-

ing to the state |e⟩during its transit through the cavity

is proportional to the coupling gas well as the detuning

∆. We assume that one of these two is the parameter to

be estimated and for simplifying the following discussion

we assume that this parameter is g. Once the transit

time of the atoms through the cavity is ﬁxed, counting

the number of atoms that emerge from the cavity in the

states |e⟩and |f⟩respectively will yield an estimate of

the value of g. This estimate is, in practice, obtained by

performing a straightforward one parameter ﬁt of the ob-

served statistics to the probability of de-excitation of the

atom obtained from the atom-cavity interaction model.

Let us assume that atomic transitions out of the state

|e⟩to levels other than |f⟩due to various reasons is the

FIG. 1. The Quantum Fisher Information Fcorresponding to estimation of the coupling gare plotted as functions of time

for diﬀerent values of g. The initial state of the quantum probe is |f⟩and the collapse operator used is Le. The values of g

are shown in the respective ﬁgures while the other parameters used are: ℏγe= 0.150 eV and ℏ∆ = 0.0 eV. Also shown in the

ﬁgures are the density elements ρee,ρﬀand ρss of the quantum probe.

main noise in the system. In order to estimate gin such

a scenario with losses, a model that takes into account

the relevant noise processes is required. The dissipation

operator Lein introduced to account for this noise. The

action of the jump operator, Le, takes the atom-cavity

system from the state |e⟩to a state |s⟩which serves as a

placeholder for all other atomic states except |e⟩and |f⟩.

To understand the dynamics described by Eqs. (8) and

(9), we start by identifying the three relevant basis states

in the Hilbert space of the atom-cavity system as,

|s⟩=





|f⟩=





|e⟩=





(10)

We assume that the probe starts in the atom excited

state |f⟩, i.e., ρﬀ(0) = 1 to get the following analytical

result for the evolution of the elements of the density

matrix of the quantum probe:

ρﬀ(t) = e−γet/2

α28g2+ (8g2−γ2

e) cos αt

2

+γeαsin αt

2

ρee(t) = e−γet/2

α216g2sin2αt

4

ρss(t)=1−e−γet/2

α216g2−γ2

ecos αt

2

+γeαsin αt

2

ρfe(t) = ρ∗

ef = 2ig e−γet/2

α2γe−γecos αt

2

+αsin αt

2 (11)

where α=p16g2−γ2

eand all other density matrix com-

ponents are zero. The expressions for the density matrix

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OptimaltimeforsensinginopenquantumsystemsZainH.Saleem∗MathematicsandComputerScienceDivision,ArgonneNationalLaboratory,9700SCassAve,LemontIL60439AnilShaji†SchoolofPhysics,IISERThiruvananthapuram,Kerala,India695551StephenK.Gray‡CenterforNanoscaleMaterials,ArgonneNationalLaboratory,Lemont,Illinois60439...

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Optimal time for sensing in open quantum systems Zain H. Saleem Mathematics and Computer Science Division Argonne National Laboratory 9700 S Cass Ave Lemont IL 60439

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