Optimal protocols for quantum metrology with noisy measurements

2025-04-29 0 0 1.52MB 41 页 10玖币

侵权投诉

Sisi Zhou,1, 2, ∗Spyridon Michalakis,1, †and Tuvia Gefen1, ‡

1Institute for Quantum Information and Matter,

California Institute of Technology, Pasadena, CA 91125, USA

2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

(Dated: October 12, 2023)

Measurement noise is a major source of noise in quantum metrology. Here, we explore pre-

processing protocols that apply quantum controls to the quantum sensor state prior to the ﬁnal

noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the

estimation precision. We deﬁne the quantum preprocessing-optimized Fisher information, which de-

termines the ultimate precision limit for quantum sensors under measurement noise, and conduct a

thorough investigation into optimal preprocessing protocols. First, we formulate the preprocessing

optimization problem as a biconvex optimization using the error observable formalism, based on

which we prove that unitary controls are optimal for pure states and derive analytical solutions of

the optimal controls in several practically relevant cases. Then we prove that for classically mixed

states (whose eigenvalues encode the unknown parameter) under commuting-operator measure-

ments, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases.

Finally, we demonstrate that in multi-probe systems where noisy measurements act independently

on each probe, the noiseless precision limit can be asymptotically recovered using global controls for

a wide range of quantum states and measurements. Applications to noisy Ramsey interferometry

and thermometry are presented, as well as explicit circuit constructions of optimal controls.

I. INTRODUCTION

Quantum metrology is one of the pillars of quantum

science and technology [1–5]. This ﬁeld deals with fun-

damental precision limits of parameter estimation im-

posed by quantum physics. Notably, it seeks to use

non-classical eﬀects to enhance the estimation precision

of unknown parameters in quantum systems, which has

led to the development of improved sensing protocols

in various experimental platforms [6–11]. To character-

ize the metrological limit of quantum sensors, the quan-

tum Cram´er–Rao bound (QCRB) [12,13], which is sat-

urable for large number of experiments, is conventionally

used. It is deﬁned using the quantum Fisher information

(QFI) [14–16], which is one of the most useful and cele-

brated tools in quantum metrology, with a considerable

amount of research focused on developing better ways to

calculate and bound it [17–22].

Although the QCRB and the QFI apply extensively

in quantum sensing, they are deﬁned assuming that ar-

bitrary quantum measurements can be applied on quan-

tum states to extract information about the unknown

parameter. However, in actual experimental platforms,

such as nitrogen-vacancy centers [23–29], superconduct-

ing qubits [30], trapped ions [31,32], and more, mea-

surements are often noisy and time-expensive, render-

ing the sensitivity of practical quantum devices far from

the theoretical limits given by the QCRB. In particu-

lar, measurement noise remains a signiﬁcant source of

noise in quantum sensing experiments. Other sources of

∗sisi.zhou26@gmail.com

†spiros@caltech.edu

‡tgefen@caltech.edu

noise, such as system evolution and state preparation,

have been studied extensively, with methods developed

to mitigate their eﬀect [17,33–45].

To tackle the eﬀect of measurement noise on quantum

metrology, interaction-based readouts were proposed [46–

51] and demonstrated experimentally [52–54], where be-

spoke inter-particle interactions that enhance phase es-

timation precision in spin ensembles are applied before

the noisy measurement step and after the probing step.

The idea of employing unitary controls in a preprocessing

manner, i.e. after the unknown parameter has been im-

parted but prior to the ﬁnal measurement, was later for-

mulated as the imperfect (or noisy) QFI problem [50,55],

where the preprocessing is optimized over all unitary

operations. Classical post-processing methods, such as

measurement error mitigation [56–58], can then work in

complement to the quantum preprocessing method for

parameter estimation under noisy measurements.

Apart from a few speciﬁc cases, such as qubit sensors

with lossy photon detection [55], setting the metrological

limit under measurement noise by computing imperfect

QFI has been diﬃcult, limiting its practical application.

In this work, we propose a more general measurement

optimization scheme, where arbitrary quantum controls

(i.e., general quantum channels that can be implemented

utilizing unitary gates and ancillas) are applied before the

noisy measurement. The goal is to identify the FI opti-

mized over all quantum preprocessing channels for gen-

eral quantum states and measurements, that we call the

quantum preprocessing-optimized FI (QPFI) and quan-

tiﬁes the ultimate power of quantum sensors with mea-

surement noise, and to obtain the corresponding optimal

controls, that can be applied to achieve the optimal sen-

sitivity in practical experiments.

We systematically study the QPFI, along with the cor-

arXiv:2210.11393v3 [quant-ph] 11 Oct 2023

responding optimal preprocessing controls in this work.

In Sec. II, we ﬁrst deﬁne the QPFI and review related

concepts. We then introduce the concept of error ob-

servables in Sec. III, and use it to demonstrate that the

QPFI problem can be cast as a biconvex optimization

problem [59]. In turn, this allows us to ﬁnd analytical

conditions for optimality, and to identify optimal controls

saturating the QPFI in the setting of commuting mea-

surements applied to pure states (see Sec. IV). The case

of classically mixed states (i.e., states for which the un-

known parameter is encoded in the eigenvalues) is studied

in Sec. V. Besides analytical solutions, we also manage

to prove that unitary controls are optimal for pure states

under general measurements, and that coarse-graining

controls are optimal for classically mixed states under

commuting-operator measurements, with a counterexam-

ple illustrating the non-optimality of unitary controls.

For general mixed states, we further prove useful bounds

on the QPFI in Sec. VI. In terms of the asymptotic behav-

ior of identical local measurements acting on multi-probe

systems, in Sec. VII, we identify a suﬃcient condition for

the convergence of the QPFI to the QFI using an opti-

mal encoding protocol based on the Holevo–Schumacher–

Westmoreland (HSW) theorem [60,61]. We show that

the relevant condition is satisﬁed by a generic class of

quantum states, including low-rank states, permutation-

invariant states, and Gibbs states (with an unknown tem-

perature), while previously only the pure state case was

proven [55].

Our results provide a theoretically-accessible preci-

sion bound for quantum metrology under noisy measure-

ments, along with a roadmap towards preprocessing op-

timization in sensing experiments.

II. DEFINITIONS

Given a quantum state ρθas a function of an un-

known parameter θ, the procedure to estimate θgoes

as follows (see Fig. 1a): (1) Perform a quantum mea-

surement {Mi}on ρθ, which gives a measurement out-

come iwith probability pi,θ = Tr(ρθMi); (2) Infer the

value of θusing an estimator ˆ

θ, which is a function of

the measurement outcome i; (3) Repeat the above two

steps multiple times and use the average of ˆ

θover many

trials as the ﬁnal estimate of θ. Here, the quantum mea-

surement {Mi}is mathematically formulated as a posi-

tive operator-valued measure (POVM) [62] that satisﬁes

Mi≥0 and PiMi= (we use A≥0 to indicate an op-

erator Athat is positive semideﬁnite). We also assume in

this work that ρθand Milie in ﬁnite-dimensional Hilbert

spaces, with measurement outcomes contained in a ﬁnite

set.

In estimation theory, the Cram´er–Rao bound

(CRB) [63–65] provides a lower bound on the estima-

tion error for any locally unbiased estimator ˆ

θat a local

State

𝜌!

𝑖 ↦ #

𝜃(𝑖)𝑖

Measurement {𝑀!}

𝑝",! =Tr 𝓔(𝜌!)𝑀"

Estimator

Preprocessing

Channel: 𝜌!↦ 𝓔(𝜌!)

State

𝜌!

Preprocessing

Unitary: 𝜌!↦ 𝑼𝜌!𝑼$

𝑖 ↦ #

𝜃(𝑖)𝑖

Measurement {𝑀!}

𝑝",! =Tr 𝑼𝜌!𝑼$𝑀"

Estimator

State

𝜌!

(a)

(b)

(c)

𝑖 ↦ #

𝜃(𝑖)

𝑖

Measurement {𝑀!}

𝑝",! =Tr 𝜌!𝑀"

Estimator

FIG. 1. (a) Standard parameter estimation procedure of

a quantum state ρθusing a quantum measurement {Mi}.

The estimation of θis through an unbiased estimator ˆ

θas

a function of measurement outcomes i. The CRB states

∆ˆ

θ≥1/pNexprF(ρθ,{Mi}). (b) Preprocessing protocols

where the measurement device is ﬁxed, and the quantum con-

trol acting before the measurement is optimized over all quan-

tum channels. The CRB states ∆ˆ

θ≥1/pNexprFP(ρθ,{Mi}).

ﬁxed, and the quantum control acting before the measure-

ment is optimized over all unitary channels. The CRB states

∆ˆ

θ≥1/pNexprFU(ρθ,{Mi}). Diﬀerent types of FIs dis-

cussed in this work satisfy F(ρθ,{Mi})≤FU(ρθ,{Mi})≤

FP(ρθ,{Mi})≤J(ρθ) (and each inequality can be strict).

point θ0where ρθis diﬀerentiable, satisfying

E[ˆ

θ|θ0] = θ0,and ∂

∂θ E[ˆ

θ|θ]θ=θ0

= 1,(1)

where we use E[·|θ] to denote the conditional expectation

over the probability distribution {pi,θ }. The above condi-

tion indicates that locally unbiased estimators ˆ

θprovide

an unbiased estimation of θat the point θ0, which is also

precise up to ﬁrst order in its neighborhood. Note that

in the following we will implicitly use E[·] to represent

E[·|θ] and consider locally unbiased estimators at a local

point θ. The CRB states that the estimation error ∆ˆ

(i.e., the standard deviation of the estimator ˆ

θ) has the

following lower bound:

∆ˆ

θ:= (E[(ˆ

θ−θ)2])1

2≥1

pNexprF(ρθ,{Mi}),(2)

where Nexpr is the number of experiments performed,

and F(ρθ,{Mi}) is the FI of the probability distribution

{pi,θ = Tr(ρθMi)}[63–65], deﬁned by

F(ρθ,{Mi}) := X

i:Tr(ρθMi)̸=0

(Tr(∂θρθMi))2

Tr(ρθMi).(3)

The CRB is often saturable asymptotically (i.e., when

Nexpr → ∞) using the maximum likelihood estima-

tor [63–65] and therefore the FI, which is inversely pro-

portional to the variance of the estimator, serves as a

good measure of the degree of sensitivity of {pi,θ}with

respect to θ. One caveat is the CRB only applies to lo-

cally unbiased estimators and can be violated by biased

estimators. Additionally, there exist singular cases where

maximum likelihood estimators are no longer necessarily

asymptotically unbiased, e.g., when the support of {pi,θ}

varies in the neighborhood of θ, and the CRB may not

apply to them [66]. However, for self-consistency, this

paper will focus only on optimizing the FI, regardless of

the limitations of the CRB.

The QFI of ρθis the FI maximized over all possible

quantum measurements on ρθ(see Appx. A for further

details) and we will refer to the optimal measurements

as QFI-attainable measurements. Formally, the QFI is

deﬁned by [12–14]

J(ρθ) = max

{Mi}F(ρθ,{Mi}),(4)

giving rise to the QCRB

∆ˆ

θ≥1

pNexprJ(ρθ),(5)

which characterizes the ultimate lower bound on the es-

timation error. Going forward, we will also overload the

notation and write

J({pi,θ}) := X

i:pi,θ ̸=0

(∂θpi,θ)2

pi,θ

,(6)

to denote the FI of a classical probability distribution

{pi,θ}, satisfying pi,θ ≥0 and Pipi,θ = 1. Note that,

from now on, we will implicitly assume that the summa-

tion is taken over terms with non-zero denominators.

In practice, the optimal measurements achieving the

QFI are not always implementable, restricting the range

of applications of the QCRB. For example, the projec-

tive measurement onto the basis of the symmetric loga-

rithmic operators, which is usually a correlated measure-

ment among multiple probes, is known to be optimal [14],

while quantum measurements in experiments are usually

noisy and not exactly projective. Here, we consider a

metrological protocol in which arbitrary quantum con-

trols can be implemented, after the unknown parameter

θhas been imparted to the quantum sensor state ρθand

before a ﬁxed quantum measurement is performed (see

Fig. 1b). We call this additional step “preprocessing”,

“pre-measurement-processing” in full. Note that the idea

of implementing preprocessing quantum controls to im-

prove sensitivity goes beyond the FI formalism and ap-

plies to other ﬁgures of merit of quantum sensors [67].

This model eﬀectively describes quantum experiments

where the measurement error is dominant, while the gate

implementation error and the state preparation error is

relatively small, a noise model that arises naturally in

modern quantum devices such as nitrogen-vacancy cen-

ters [23–26] and superconducting qubits [30].

To quantify the sensitivity of estimating θon ρθwith

the measurement {Mi}ﬁxed, we deﬁne the FI optimized

over all preprocessing quantum channels, or the quan-

tum preprocessing-optimized Fisher information (QPFI),

to be

FP(ρθ,{Mi}) = sup

F(E(ρθ),{Mi}),(7)

where Eis an arbitrary quantum channel (or a CPTP

map [68]). See Appx. B for mathematical properties of

the QPFI. In particular, when the quantum measurement

is ﬁxed, the CRB induced by the QPFI, i.e.,

∆ˆ

θ≥1

pNexprFP(ρθ,{Mi}),(8)

provides a practical and tighter Cram´er–Rao-type bound,

compared to the QCRB, for parameter estimation un-

der noisy measurements. We assume in the following

discussions that all measurements are non-trivial (i.e.,

∃Mi̸∝ , for all {Mi}) and ∂θρθ̸= 0 so that the QPFI

is always positive.

Unless stated otherwise, we will denote the systems

that ρθand {Mi}act on by HSand HS′, respectively,

and we will refer to HSas the input system and HS′

as the output system. We do not assume HS∼

=HS′

here. This broader context is of particular interest when

the quantum state ρθcannot be directly measured (e.g.,

readout of superconducting qubits via a resonator [30]

and readout of nuclear spins via an electron spin in a

nitrogen-vacancy center [69–72]); or when the quantum

state is restricted to a subsystem of the entire system

while quantum measurement can be performed globally.

Note that for generic noisy measurements, the supre-

mum in Eq. (7) is usually attainable, i.e., there exists

an optimal Esuch that F(E(ρθ),{Mi}) is maximized

(see Appx. C). However, there exist singular cases where

F(E(ρθ),{Mi}) has no maximum, due to the singularity

of the FI at the point Tr(E(ρθ)Mi) = 0 (see Sec. IV C for

an example). In such cases, there still exist near-optimal

quantum controls that attain supEF(E(ρθ),{Mi})−ηfor

any small η > 0. In fact, we prove in Appx. C that:

Theorem 1. Let M(ϵ)

i= (1 −ϵ)Mi+ϵTr(Mi)d, where

d= dim(HS′)and 0<ϵ<1. Then

FP(ρθ,{Mi}) = lim

ϵ→0+FP(ρθ,{M(ϵ)

i}),(9)

and the QPFI FP(ρθ,{M(ϵ)

i})is attainable for any ϵ∈

(0,1].

In the following, we will focus mostly on the case where

the QPFI is attainable. We will discuss the behavior of

the QPFI, exploring numerical optimization algorithms

and analytical solutions to the optimal controls for cer-

tain practically relevant quantum states and measure-

ments.

We will also examine the FI optimized over all uni-

tary preprocessing channels, which we call the quan-

tum unitary-preprocessing-optimized Fisher information

(QUPFI) [50,55]

FU(ρθ,{Mi}) = sup

F(UρθU†,{Mi}),(10)

where Uis an arbitrary unitary gate. (Note that our

QUPFI is the same as the imperfect QFI in [55].) Unlike

the QPFI, we assume HS′∼

=HS(and do not distinguish

between S′and S) when we talk about the QUPFI, so

that it is well deﬁned. We note here that Theorem 1

holds for the QUPFI, as well.

The optimal preprocessing controls that attain the

QPFI and the QUPFI usually depend on θ, whose value

should be roughly known before the experiment. Other-

wise, one might use the two-step method by ﬁrst using

pNexpr states to obtain a rough estimate ˜

θ≈θ, and

then performing the optimal controls based on ˜

θon the

remaining Nexpr −pNexpr states [73–75]. The two-step

procedure introduces a negligible amount of error asymp-

totically.

Before we proceed, we prove a relation between the

QPFI and the QUPFI that will be useful later.

Proposition 2. Let HSand HS′be the input and out-

put systems of E. Suppose HA1and HA2are ancil-

lary systems such that HA1⊗ HS∼

=HA2⊗ HS′. If

dim(HA1)≥dim(HS′)2(or equivalently, dim(HA2)≥

dim(HS) dim(HS′)), then

FP(ρθ)S,{(Mi)S′}=

FU(ρθ)S⊗ |0A1⟩⟨0A1|,{(Mi)S′⊗A2},(11)

where we use subscripts to denote the systems the opera-

tors are acting on.

Proof. Any quantum channel E(·) = PrE

i=1 Ki(·)K†

ifrom

HSto HS′can be implemented by acting unitarily on HS

and an ancillary system HA1, and then tracing over an

auxiliary system HA2, if dim(HA2)≥rE(Stinespring’s

dilation [68]). For any quantum channel with the in-

put system HSand the output system HS′, there al-

ways exists a Kraus representation E(·) = PrE

i=1 Ki(·)K†

such that rE≤dim(HS′) dim(HS) [68]. Therefore, if

dim(HA2)≥dim(HS′) dim(HS), the unitary extension

should exist.

Let HS⊗ HA1∼

=HS′⊗ HA2be the enlarged,

isomorphic input and output Hilbert spaces, respec-

tively. If dim(HA1)≥dim(HS′)2, then dim(HA2) =

dim(HA1) dim(HS)

dim(HS′)≥dim(HS′) dim(HS). Thus, there is

a unitary UEmapping HS⊗HA1to HS′⊗HA2such that

E(σ) = TrA2(UE(σ⊗ |0⟩⟨0|)U†

E).(12)

From Eq. (3), it follows that:

F(E(ρθ),{Mi}) = F(TrA2(UE(ρθ⊗ |0⟩⟨0|)U†

E),{Mi})

=F(UE(ρθ⊗ |0⟩⟨0|)U†

E,{Mi⊗ }),

where we omit the subscripts for simplicity. Note that the

Stinespring’s dilation technique is also useful in relating

the QFI of a mixed state to the QFI of its puriﬁcation in

an extended Hilbert space [17,18]. Taking the supremum

over Ein the above equality, we have

FP(ρθ,{Mi})≤FU(ρθ⊗ |0⟩⟨0|,{Mi⊗ }).(13)

On the other hand, for any Ufrom HS⊗ HA1to HS′⊗

HA2, TrA2(U((·)⊗|0⟩⟨0|)U†) is a quantum channel from

HSto HS′, proving the other direction of Eq. (11).

III. ERROR OBSERVABLE FORMULATION

In this section, we will formalize the optimization of

FI over quantum preprocessing controls as a biconvex

optimization problem using the concepts of error observ-

ables. Using this new formulation, the preprocessing op-

timization problem becomes numerically tractable with

standard algorithms for biconvex optimization [59]; and

also analytically tractable for practically relevant quan-

tum states (see Sec. IV).

Here, we consider the preprocessing optimization prob-

lem in Eq. (7). On the surface, it may appear from

the deﬁnition of FI (Eq. (3)) that the target function

F(E(ρ),{Mi}) is mathematically formidable. To simplify

the target function, we introduce the error observable X

and the squared error observable X2, deﬁned by

X=X

xiMi,and X2=X

iMi,(14)

where xiis interpreted as the diﬀerence between the esti-

mator value ˆ

θ(i) and the true value θ, i.e., xi=ˆ

θ(i)−θ.

We assume there are rmeasurement outcomes and use x

to denote the vector (x1, . . . , xr). The local unbiasedness

conditions (Eq. (1)) for a single-shot measurement then

become

Tr(ρθX)=0,and Tr(∂θρθX)=1.(15)

It can be veriﬁed mathematically (which is essentially a

proof of the CRB) that the minimum of the variance of

the estimator under the local unbiasedness conditions is

the inverse of the FI; that is,

F(ρθ,{Mi})−1= min

xTr(ρθX2),s.t. Eq. (15). (16)

The problem above is a convex optimization over vari-

ables x, which can be solved using, e.g., the method of

Lagrange multipliers [76]. The optimal solution to xis

xi=

Tr(∂θρθMi)

Tr(ρθMi)

Pj:Tr(ρθMj)̸=0

(Tr(∂θρθMj))2

Tr(ρθMj)

,(17)

when Tr(ρθMi)̸= 0, and xi= 0 when Tr(ρθMi) = 0.

Note that the error observable formulation was previ-

ously used to derive the QCRB [77], where the QFI sat-

isﬁes

J(ρθ)−1= min

XTr(ρθX2),s.t. Eq. (15),(18)

and Xan arbitrary Hermitian matrix subject to the con-

straints in Eq. (15). This formulation has several useful

applications [78–80]. In particular, an algorithm was pro-

posed in [55] based on Eq. (18), to optimize the QFI of

quantum channels.

Combining Eq. (16) and Eq. (7), we have that

FP(ρθ,{Mi})−1= inf

(x,E)Tr(E(ρθ)X2),(19)

s.t. Tr(E(ρθ)X) = 0,

Tr(E(∂θρθ)X)=1.

Let HSand HS′be the input and output systems of

Eand let {|k⟩S}dim(HS)

k=1 and {|j⟩S′}dim(HS′)

j=1 be two sets

of orthonormal basis of HSand HS′, respectively. In

the rest of this section, we use matrix representations of

operators in the above bases. It is convenient to rep-

resent a CPTP map Eusing a linear operator acting on

HS′⊗HS. Let E(·) = PiKi(·)K†

ibe the Kraus represen-

tation of E. Then, the linear operator Ω = Pi|Ki⟩⟩⟨⟨Ki|

is usually called the Choi matrix of E[68], where |⋆⟩⟩ :=

Pjk(⋆)jk |j⟩S′|k⟩Sand (⋆)jk =⟨j|S′(⋆)|k⟩S. Ω cor-

responds to a CPTP map if and only if Ω ≥0 and

TrS′(Ω) = S.Eacting on any density operator σcan be

expressed using Ω through E(σ) = TrS(( ⊗σT)Ω) (we

use (·)Tto denote matrix transpose). Using the Choi

matrix representation in Eq. (19), we have:

Theorem 3. The optimal value of the following biconvex

optimization problem gives the inverse of the QPFI.

FP(ρθ,{Mi})−1= inf

(x,Ω) Tr((X2⊗ρT

θ)Ω),(20)

s.t. Ω≥0,TrS′(Ω) = S,

Tr((X⊗ρT

θ)Ω) = 0,

Tr((X⊗∂θρT

θ)Ω) = 1.

Eq. (20) is a biconvex optimization problem of vari-

ables xand Ω. Fixing Ω, Eq. (20) is a quadratic program

with respect to x, and ﬁxing x,Eq. (20) is a semideﬁnite

program with respect to Ω; each of which is eﬃciently

solvable when the system dimensions are moderate and

the domain of variables is compact.

Note that the domain of xis unbounded in Eq. (20). In

practice, one may impose a bounded domain on xso that

the minimum of Eq. (20) always exists. For cases where

the QPFI is attainable, the optimal value of the bounded

version will be equal to the one of Eq. (20) when the size

of the bounded domain is suﬃciently large. For singular

cases where the QPFI is not attainable, the optimal value

of the bounded version will approach the one of Eq. (20)

with an arbitrarily small error as the size of the domain

increases. We describe an algorithm called the global

optimization algorithm [81]inAppx. D that can solve

the bounded version of Eq. (20).

Finally, we note that Theorem 3 does not directly gen-

eralize to the case of QUPFI because the Choi matrices

of unitary operators do not form a convex set. On the

other hand, besides the set of quantum channels, our

approach is also useful in optimizing the FI over other

sets of quantum controls when the constraints on their

Choi matrices can be represented using semideﬁnite con-

straints, e.g., the set of quantum channels that act only

on a subsystem of the entire system.

IV. PURE STATES

In this section, we consider the special case where

ρθ=ψθ=|ψθ⟩⟨ψθ|is pure, which is most common

in sensing experiments. We ﬁrst consider the optimiza-

tion of the FI over the error vector xand the unitary

control U, and obtain two necessary conditions for the

optimality of (x, U). We use these conditions to prove

equality between the QPFI and the QUPFI for pure

states, showing that unitary controls are optimal for such

states (when HS∼

=HS′). We also obtain an analytical

expression of the QPFI for binary measurements (i.e.,

measurements with only two outcomes), and a semi-

analytical expression and analytical bounds for general

commuting-operator measurements (i.e., measurements

{Mi}that satisfy [Mi, Mj] = 0 for all i, j). In particular,

we prove that the optimal control is given by rotating

the pure state and its derivative into a two-dimensional

subspace spanned by two of the common eigenstates of

the commuting-operator measurements.

A. Necessary conditions for optimal controls

Proposition 2 shows that the optimization for the

QPFI can be reduced to an optimization for the QUPFI

using the ancillary system. Thus, here we ﬁrst focus on

the following optimization problem over the unitary con-

trol

FU(ρθ,{Mi})−1= inf

(x,U)Tr(U ρθU†X2),(21)

s.t. Tr(UρθU†X)=0,(22)

Tr(U∂θρθU†X)=1.(23)

We obtain necessary conditions for the optimality of

(x, U) that will be useful later.

Lemma 4. If (x, U)is an optimal point for Eq. (21), it

must satisfy

Tr(U∂θρθU†X)

Tr(UρθU†X2)[X2, UρθU†] = 2[X, U∂θρθU†].(24)

In particular, suppose ρθ=|ψθ⟩⟨ψθ|is pure. Let

|ψ⊥

θ⟩:= 1

√n(1 − |ψθ⟩⟨ψθ|)|∂θψθ⟩,(25)

|ϕ⟩:= U|ψθ⟩,|ϕ⊥⟩:= U|ψ⊥

θ⟩,(26)

where the normalization factor n=⟨∂θψθ|(1 −

|ψθ⟩⟨ψθ|)|∂θψθ⟩. Then Eq. (24) is equivalent to the fol-

lowing two conditions:

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

OptimalprotocolsforquantummetrologywithnoisymeasurementsSisiZhou,1,2,∗SpyridonMichalakis,1,†andTuviaGefen1,‡1InstituteforQuantumInformationandMatter,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA2PerimeterInstituteforTheoreticalPhysics,Waterloo,OntarioN2L2Y5,Canada(Dated:October12,2023)Measure...

展开>> 收起<<

Optimal protocols for quantum metrology with noisy measurements.pdf

共41页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Optimal protocols for quantum metrology with noisy measurements

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: