Incompatibility in Quantum Parameter Estimation
IIncompatibility in Quantum Parameter Estimation
Federico Belliardo ∗ and Vittorio Giovannetti † NEST, Scuola Normale Superiore, I-56126 Pisa, Italy NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
A hierarchy of incompatibility measures, each corresponding to a set of allowed measurements, isintroduced. These quantify the genuine quantum incompatibility in the estimation task of multipleparameters, and have a geometric character which is backed by a clear operational interpretation.The analysis is then applied to some simple systems of one, two, and three qubits in order to trackthe effect of a local depolarizing noise on the incompatibility of the estimation task. A semidefiniteprogram is described and used to numerically compute the figure of merit when the analyticaltools are not sufficient, among these we include an upper bound computable from the symmetriclogarithmic derivatives only. Finally we generalize the results obtained for one and two qubits tothe case of a general unitary encoding on a finite dimensional probe.
I. INTRODUCTION
Quantum metrology [1–5] is a special branch of quan-tum information theory that focuses on the possibil-ity of using quantum effects for improving the accuracyof conventional estimation procedures. Thanks to thehuge variety of potential applications (which among otherinclude the probing of delicate biological systems [6],optical interferometry [7, 8], gravitational wave detec-tion [9, 10], magnetometry [11–15] and atomic clocks [16–18]), this research field is likely to play a fundamentalrole in the looming quantum technology revolution. Asevident from the seminal works of Holevo [19] and Hel-strom [20], this research field can be thought as a quan-tum counterpart of Experimental Design [21, 22]. Specif-ically the main goal of quantum metrology is to efficientlyplan different types of experiments by minimizing the in-vested effort to overcome noisy fluctuations that originateby fabrication errors, external fields, microscopic degreesof freedom that are only statistically taken into account,and intrinsic limitations related to the formal structureof the quantum theory itself (e.g. the Heisenberg un-certainty principle). In recent years many significant re-sults have accumulated in the domain of multi-parameterquantum metrology [23, 24], i.e. processes where an agenttries to recover two or more attributes of a physical sys-tem (modeled by real numbers) via properly chosen mea-surements. The first studies that lit up the experimentalinterest in this subject have been done on the joint esti-mation of phase and phase diffusion [25–29], on quantumimaging [30–37], and on magnetometry [38, 39]. Whatmakes the problem intriguing is that in a purely quantumsetting, due to constraints ultimately related to the in-compatibility of non-commuting observables [40], it couldbe that an efficient experiment for the determination ofone specific parameter leads to poor results in the preci-sion of the others (while this may also be true in classicalmechanics, since here the phenomenon is related to the ∗ [email protected] † [email protected] technological limits of the experimenter, there is no rea-son to believe it to be fundamental). Aim of the presentwork is to quantify the genuine quantum incompatibilityassociated with the estimation task of multiple param-eters. For this purpose, building up from the quantumCram´er-Rao (QCR) bound [3, 41], we introduce a hierar-chy of incompatibility measures each corresponding to aset of allowed measurements. The analysis is then appliedto some simple systems of one, two, and three qubits inorder to track the effect of a local depolarizing noise onthe incompatibility of the estimation task. A semidefi-nite program is described and used to numerically com-pute the figure of merit when the analytical tools are notsufficient. Finally we notice that the strategies that allowus to codify information without incompatibility in thetwo-qubits examples can be generalize to the case of ageneral unitary encoding on a finite dimensional probe.Before proceeding with the presentation, we add here aterminology clarification: with “quantum parameter” es-timation we denote the task of extracting a parameterencoded on a certain given fixed state of a quantum sys-tem, while if we use “quantum metrology” it means thatwe have the possibility of choosing the probe that willundergo the encoding process. In this perspective theproblem of parameter estimation is hence a sub-problemof quantum metrology. In this paper we will take theprobe to be fixed and therefore we will be dealing withparameter estimation.An outlook of the manuscript follows. In Sec. II we in-troduce the setting of quantum metrology, and isolate theform of incompatibility that we will characterize later on.In Sec. III a first figure of merit based on estimation pro-cedures which are locally unbiased, is defined and its geo-metrical meaning is presented. The exploration of its for-mal properties points toward a more workable expression,given in Sec. III B. In Sec. IV an alternative definition,based on the Locally Asymptotic Covariance condition ofRef. [42], is presented which admits the analytical upperbound of Sec. IV A and is numerically computable withthe semidefinite program presented in App. E. In Sec. Vwe study a different incompatibility figure of merit, de-fined for an estimation task in which only locally unbi- a r X i v : . [ qu a n t - ph ] F e b ased separable measurements on the probes are allowed(no entanglement). The fact that multiple incompatibil-ity measures could been defined is not incidental, indeedin Sec. VI we argument for the existence of a hierarchyof incompatibility figures of merit, each defined accord-ing to the capabilities of the experimenter. Sec. VII isdedicated to some examples with systems of one, two,and three qubits, possibly subject to local depolarizationnoise. Finally in Sec. VIII we analyze the scenario of ageneral unitary encoding on a finite dimensional quantumsystem. We compute the effect of the depolarizing noiseon the incompatibility, according to the upper bound ofEq. (43), and observe how a maximally entangled probe(with one or two uses of the encoding channel) alwaysgives full compatibility, for the figure of merit of Sec. IV.Furthermore we see how the anti-parallel spin strategy ofSec. VII B can be generalized to finite dimensional sys-tems. Conclusions and remarks are given in Sec. IX. II. MULTIPARAMETER QUANTUMESTIMATIONA. Setting and definitions
A prototypical example of multi-parameter quantummetrology is provided by magnetometry [11–15, 43]where a spin particle is used as a probe for evaluat-ing the three components of a magnetic field B :=( B x , B y , B z ). In the most basic scenario the evolu-tion of the particle is given by the unitary transforma-tion U B := exp [ i ( B x S x + B y S y + B z S z ) t ] where S i for i = x, y, z are the components of the spin. By measur-ing the evolved state of the probe we can hence try inferthe values of B x , B y , B z , following the post-processingof the measurements output. What makes this proce-dure truly quantum in nature is that, fixing the numberof experimental repetitions, due to the non-commutingnature of the generators S i , any attempt to improvethe estimation accuracy of one of the cartesian compo-nents of B will have a negative impact on the accura-cies of the other two [44]. An exact formalization ofthis problem can be obtained by considering a more gen-eral model where one is asked to determine d parameters θ := ( θ , θ , . . . , θ d ) ∈ Θ (an open subset of R d ) thathave been encoded into the input state ρ of a probingquantum system via a mapping of the form ρ → ρ θ := E θ ( ρ ) , (1)where now E θ is a completely positive, trace-preserving(CPT) transformation [45] which parametrically dependsupon θ and which, at variance with the simplified sce-nario detailed at the beginning of the section, might in-clude a noise disturbing the process. Given N copies of ρ θ we can now try to recover the needed information byperforming on them some (possibly joint) positive opera-tor valued measure (POVM) M N := { E ( N ) ˆ θ } ˆ θ whose ele-ments are labelled by a classical outcome variable ˆ θ that, without loss of generality [46], can be assumed to belongto the same set Θ of θ . Accordingly M N can hence bethought as a process which, starting from ρ ⊗ N θ , inducesa measure on Θ, defined by the conditional probabilitydistribution P M N ( ˆ θ | θ ) := Tr [ E ( N ) ˆ θ ρ ⊗ N θ ] , (2)with the stochastic outcome ˆ θ := (ˆ θ , ˆ θ , . . . , ˆ θ d ) ∈ Θplaying the role of the estimator of θ . The two most im-portant properties of the estimator ˆ θ are the bias vector b ( N ) ( θ ) := ( b ( θ ) , b ( θ ) , · · · b d ( θ )), of components b i ( θ ) := E [ˆ θ i ] − θ i , (3)and the mean square error (MSE) d × d matrix Σ ( N ) ( θ ),of elements Σ ( N ) ij ( θ ) := E [( ˆ θ i − θ i )(ˆ θ j − θ j )] , (4)with E representing the statistical average computed withthe probability measure in Eq. (2). Ideally we would liketo deal with estimators that are unbiased, meaning that b ( N ) ( θ ) = 0 for all θ ∈ Θ, but this may not alwaysbe possible. Accordingly in what follows we shall focuson sensing, i.e. we shall measure small variations of theparameters θ around a known value and assume that weare allowed to employ locally unbiased POVMs at leastfor such special point – see Appendix A for the formaldefinition of such property. For these measurements thequantum Cram´er-Rao (QCR) bound [3, 41] gives a limiton the precision of the sensing task, formulated as a lowerbound on the associated MSE matrix, i.e.Σ ( N ) ( θ ) ≥ F − ( θ ) N . (5)In this expression F ( θ ) is the so called quantum Fisherinformation (QFI) matrix which no longer depends uponthe selected POVM M N and whose elements can be com-puted as F ij ( θ ) := 12 Tr (cid:104) ρ θ ( L i ( θ ) L j ( θ ) + L j ( θ ) L i ( θ )) (cid:105) , (6)with L i ( θ ) the symmetric logarithmic derivative(SLD) [41] associated to the i th component of the pa-rameter vector θ , i.e. the operator (possibly dependentupon θ ) fulfilling the identity ∂ρ θ ∂θ i = 12 (cid:16) ρ θ L i ( θ ) + L i ( θ ) ρ θ (cid:17) . (7)For a pure state ρ θ = | ψ θ (cid:105)(cid:104) ψ θ | the above equation admitsas solution L i ( θ ) = 2 ∂ρ θ ∂θ i , (8)while in general a solution is [41] L i ( θ ) = 2 (cid:90) + ∞ e − sρ θ ∂ρ θ ∂θ i e − sρ θ ds . (9)Throughout the paper we will assume the QFI to be lim-ited (i.e. (cid:107) F ( θ ) (cid:107) < ∞ ) and non-singular (i.e. F ( θ ) > d we can allow in our study is upperbounded by D − D being the dimension of theHilbert space associated with the probing system (indeedvalues of d greater than such limit will necessarily force alinear dependence upon the SLD operators L i ( θ ) leadingto a singular QFI matrix). B. Achievability of the multi-parameter QCRbound
In general the multiparameter QCR bound of Eq. (5)cannot be saturated, meaning that there is no locallyunbiased POVM M N with a Σ ( N ) ( θ ) matrix equal to F − ( θ ) /N or, equivalently, which is capable of saturatingthe inequalityTr [ G · Σ ( N ) ( θ )] ≥ N Tr [ G · F − ( θ )] := C S ( G, θ ) N , (10)for all choices of a positive weight matrix G ≥
0. This isthe form of metrological incompatibility that will be ex-tensively studied in this paper. In order to better appre-ciate the meaning of this, suppose that we are interestedin the estimation of an analytic function f ∈ C ω (Θ) ofthe unknown parameters vector θ . The function f willbe evaluated on the estimator ˆ θ extracted from the ob-servations. By expanding to first order the expectationvalue of f ( ˆ θ ) − f ( θ ) we get the expression for the error ε := E [( f ( ˆ θ ) − f ( θ )) ] (11) (cid:39) (cid:88) i,j E [ ∂ i f ( θ )(ˆ θ i − θ i ) ∂ j f ( θ )(ˆ θ j − θ j )]= (cid:88) i,j ∂ i f ( θ ) ∂ j f ( θ ) E [(ˆ θ i − θ i )(ˆ θ j − θ j )] , (12)which can be equivalently written as: ε = (cid:104) ∂f ( θ ) | Σ ( N ) ( θ ) | ∂f ( θ ) (cid:105) = Tr [ G ( θ ) · Σ ( N ) ( θ )] , (13)where we introduced the rank-1 weight matrix G ij ( θ ) = ∂ i f ( θ ) ∂ j f ( θ ) = | ∂f ( θ ) (cid:105)(cid:104) ∂f ( θ ) | , with | ∂f ( θ ) (cid:105) ∈ R .Written in this form we can now use Eq. (10) to casta bound on the accuracy of the estimation of f ( θ ). Asa matter of fact a rank-1 G can always be thought asthe weight matrix of some function f ( θ ). We will seethat according to our definitions a rank-1 G manifestsno incompatibility, indeed the error along a certain di-rection | ∂f ( θ ) (cid:105) in the parameter space can saturate theultimate QFI (this can be understood e.g. from the up-per bound (39) discussed in Sec. IV A below, which, for G rank-1, collapses on C S ( G, θ )). On the contrary thegap manifests itself when the weight matrix G is at leastrank-2. This situations arises as we try to estimate atthe same time multiple functions of the parameters θ ,named f ( θ ), f ( θ ) , . . . , f K ( θ ), which could also just be the components θ , θ , . . . , θ d of the vector θ . To eachof the functions we associate a weight g i ≥
0, then thetotal error is the weighted sum of the errors for the esti-mation of each f i ( θ ), i.e. ε := K (cid:88) i =1 g i Tr [ | ∂f i ( θ ) (cid:105)(cid:104) ∂f i ( θ ) | Σ ( N ) ] = Tr [ G ( θ ) · Σ ( N ) ] , (14)with G ( θ ) := (cid:80) Ki =1 g i | ∂f i ( θ ) (cid:105)(cid:104) ∂f i ( θ ) | ≥ III. INCOMPATIBILITY FOR LOCALLYUNBIASED MEASUREMENTS
In this section we introduce a first figure of merit togauge the incompatibility of multi-parameter estimationprocedures, which is based on the assumption that theagent is allowed to perform on the probes arbitrary lo-cally unbiased POVM.
A. Definition
Given the encoding (1) and a generic weight matrix G ,from Eq. (10) it follows that a bona-fide evaluation of theprecision attainable with a locally unbiased POVM M N can be obtained by considering the ratio r N ( G, M N , θ ) := N Tr (cid:2) G · Σ ( N ) ( θ ) (cid:3) Tr [ G · F − ( θ )] ≥ , (15)where Σ ( N ) ( θ ) is the MSE matrix (4) associated withM N . As indicated by the notation the quantity (15) ex-hibits an explicitly functional dependence upon G andM N which we remove by considering the term r N ( θ ) := inf M N ∈M (LU) N sup G ≥ r N ( G, M N , θ ) , (16)where now M (LU) N indicates the set of locally unbiasedPOVM on N copies of the probes. For any given elementsM N of M (LU) N the sup G ≥ selects the weight matrix thathas the reachable precision Tr (cid:2) G · Σ ( N ) ( θ ) (cid:3) as far awayfrom the information content Tr (cid:2) G · F − ( θ ) (cid:3) /N as pos-sible. Then we minimize on M N ∈ M (LU) N to compute thebest worst case scenario, as in a typical min-max defini-tion [47]. The figure of merit r N ( θ ) quantifies the com-petition between optimal measurements for different pa-rameters, and has a clear operational meaning. Becauseof the QCR bound in Eq. (15) we have r N ( θ ) ≥ N probes are fully compatible only when r N ( θ ) = 1. Thisis true if and only if ∃ M N ∈ M (LU) N (possibly dependentupon θ ) for which in Eq. (15) equality holds ∀ G . On thecontrary r N ( θ ) > ∀ M N ∈ M (LU) N ∃ G ≥ r ( θ ) := lim inf N →∞ r N ( θ ) , (17)which always exists and from r N ( θ ) inherits the property r ( θ ) ≥
1. In particular in this case we have r ( θ ) = 1 ifonly if there exists a sequence of M N ∈ M (LU) N , which,for all G ≥
0, allows us to saturate the inequality (15)asymptotically in N . We now briefly show that r N ( θ )in Eq. (16) is invariant under reparametrization. Thistranslates to r ( θ ), which is therefore a well defined prop-erty of the statistical manifold (see Sec. III D). Considera reparametrization θ = θ ( η ) having an invertible Ja-cobian J ij := ∂θ i ( η ) ∂η j . The MSE matrix for the param-eters θ , defined in Eq. (4), can be written Σ ( N ) ( θ ) = J Σ ( N ) ( η ) J t , where Σ ( N ) ij ( η ) := E [( ˆ η i − η i )(ˆ η j − η j )] is theMSE matrix for the parameters η . Similarly we write theinverse of the QFI matrix as F − ( θ ) = JF − ( η ) J t , with F ( η ) computed from the symmetric logarithmic deriva-tives L i ( η ), which differ from the definition in Eq. (7)in the derivatives, which are taken with respect to η i .Its now easy to show that r N ( θ ) = r N ( η ). The ac-tion of the Jacobian matrix on the MSE matrix andon the QFI can be moved on G , that becomes J t GJ both at numerator and at denominator of the ratio inEq. (15), while multiplying respectively Σ ( N ) ( η ) and F − ( η ). Then we observe that the set of positive matri-ces is invariant under congruence for an invertible matrix,i.e. J t { G ≥ } J = { G ≥ } , and therefore we get r N ( θ ) = inf M N ∈M (LU) N sup G ≥ N Tr (cid:2) G · Σ ( N ) ( η ) (cid:3) Tr [ G · F − ( η )] := r N ( η ) . (18)It worth stressing that, by construction the quantity r ( θ )only depends upon the input probe state ρ , the encod-ing E θ , and the specific point of interest θ . Accordinglywith respect to previous results presented in the liter-ature [24], it is an intrinsic property of the statisticalmanifold defined by the trajectories (1). B. Formal developments
An alternative way to compute r ( θ ) can be obtained byeffectively inverting the role of the various optimizationsthat enter in Eq. (17). Since the derivation of this fact israther technical, we decided to report it in Appendix Bpresenting here just the final result of such analysis. Wefirst give some supporting definitions, i.e. r N ( G, θ ) := inf M N ∈M (LU) N r N ( G, M N , θ ) , (19)and r ( < ) n ( G, θ ) := min N ≤ n r N ( G, θ ) . (20) With this in mind we can then express r ( θ ) as r ( θ ) = sup G ≥ lim n →∞ r ( < ) n ( G, θ ) , (21)or equivalently as r ( θ ) = sup G ≥ C ( G, θ ) C S ( G, θ ) , (22)with C S ( G, θ ) defined in Eq. (10), and with the numer-ator given by the quantity C ( G, θ ) := lim n →∞ min N ≤ n inf M N ∈M (LU) N N Tr [ G · Σ ( N ) ( θ )] . (23) C. Bounds and semi-classical limit
While determining the exact value of r ( θ ) could bedemanding, it is relatively easy to show that it is alwayssmaller than or equal to d , i.e. r ( θ ) ≤ d . From Eqs. (15)–(17) it follows that such inequality can be proven by ex-hibiting a sequence of resource numbers { N n } n ∈ N forwhich lim n →∞ N n = ∞ , each with an associated locallyunbiased POVM (cid:101) M N n ∈ M (LU) N n having as MSE matrix (cid:101) Σ ( N n ) ( θ ) = d F − ( θ ) N n . (24)Indeed if such measurements exist we have r N n ( G, (cid:101) M N n , θ ) = d ∀ G ≥ , (25)and hence r N n ( θ ) ≤ r N n ( G, (cid:101) M N n , θ ) = d ∀ n ∈ N . (26)This whole construction implies that ∀ N ∃ N ≥ N suchthat r N ( θ ) ≤ d . This fact, according to the definition ofEq. (17), leads to r ( θ ) ≤ d . In order to show that suchsequence { N n } n ∈ N and POVMs M (LU) N n exist first of all,without loss of generality, we reparametrize the modelsuch that the QFI matrix F ( θ ) is diagonal, i.e. F ij ( θ ) = F ii ( θ ) δ ij . (27)The idea is hence to recover the value of θ by estimatingits components θ , θ , · · · , θ d one at a time with a set ofindependent measurements. For this purpose, taking N n to be a multiple of d (i.e. choosing N n = nd ), we organizethe total number of copies of ρ ( θ ) into d distinct groupsof n elements each: we then use the i th group to estimatethe value θ i through the outcomes of the projective mea-surement associated with the self-adjoint operator [47] (cid:98) O i ( θ ) := θ i + F − ii ( θ ) L i ( θ ) , (28)where L i ( θ ) is the corresponding i th SLD derivative en-tering Eq. (7), calculated via the formula (9). As shownin Appendix A the resulting estimation scheme is lo-cally unbiased so we can use it as a proper candidate for (cid:101) M N n . Notice next that since the various components of θ are determined via measurements performed on inde-pendent groups of copies of ρ θ the corresponding estima-tors are statistically uncorrelated leading to a diagonalMSE, i.e. (cid:101) Σ ( N n ) ij ( θ ) = (cid:101) Σ ( N n ) ii ( θ ) δ ij . The diagonal entriescan instead be computed by observing that they coincidewith the MSE associated with individual projective mea-surements performed on their corresponding block of n probes, i.e. (cid:101) Σ ( N n ) ii ( θ ) = 1 n Tr [ ρ θ ( (cid:98) O i ( θ ) − θ i ) ]= 1 n F − ii ( θ ) Tr [ ρ θ L i ( θ )] = 1 n F − ii ( θ ) , (29)which due to Eq. (27) and the fact that n = N n /d finallyleads to (24) and hence to (26). We conclude the sectionby observing that in the above example the incompati-bility signaled by the value r N n ( G, (cid:101) M N n , θ ) = d appearsbecause the measurements for each individual parameter θ i in general do not commute. Suppose that we are deal-ing with a classical system, i.e ρ θ is diagonal ∀ θ [24]. Inthis scenario the SLDs generators can be all chosen tobe diagonal an therefore commuting. This means thatthe measurements (cid:98) O i ( θ ) can be jointly performed on asingle probe ρ θ ( N = 1), and therefore we get a singleprobe POVM with MSE matrix Σ (1) ( θ ) = F − ( θ ) to berepeated on all the resources, leading to r ( θ ) = 1. D. Geometric interpretation
The parameters θ ∈ Θ can be interpreted as coordi-nates defining via Eq. (1) a submanifold of the spaceof states S ( H ), called the statistical manifold. TheQFI matrix, being a positive semidefinite matrix can bethought as a Riemannian metric on this manifold. Thismetric is generally non trivial as it explicitly dependson the coordinates θ and may have intrinsic curvature.The QFI is said to be a distinguishability metric [48, 49]:given two very near states ρ θ and ρ θ + d θ , their infinitesi-mal distance in the QFI metric is ds := 14 F ij ( θ ) dθ i dθ j = 2 (cid:16) − (cid:112) F ( ρ θ , ρ θ + d θ ) (cid:17) , (30)which is negatively correlated with the fidelity F ( ρ θ , ρ θ + d θ ) between ρ θ and ρ θ + d θ , defined as F ( ρ, σ ) := (cid:2) Tr (cid:0)(cid:112) √ ρσ √ ρ (cid:1)(cid:3) [50]. In order to gain in-formation about θ it is thus better to choose the probestate ρ such that in the statistical manifold the codi-fied state ρ θ is highly distinguishable from its neighbors ρ θ + d θ , and has therefore the highest statistical distancefrom them as possible. This picture clarifies why the in-verse of the distinguishability metric, i.e. F − ( θ ), givesthe precision to which a single point θ can be identifiedin Θ, given the quantum state ρ θ . For ρ ⊗ N θ the relevantmetric is F − ( θ ) /N . When a measurement is performedand an estimator ˆ θ is chosen there is a new Riemannian FIG. 1. Representation of a 2D statistical manifold with itstangent space at a point and two directions | v (cid:105) and | v (cid:105) onthis plane. metric insisting on the statistical manifold: the positivesemidefinite Σ ( N ) ( θ ) matrix. The key question is if onecan find a POVM M N ∈ M (LU) N with a MSE metric thatfully adapts to the underling quantum metric F − ( θ ) /N of the manifold, i.e. if the inequality (5) can be saturated(at a certain point θ ). In general this is not possible. Letus introduce a representation of G as a sum of projectors | v i (cid:105)(cid:104) v i | , each weighted with g i ≥
0, where | v i (cid:105) are direc-tions on the statistical manifold, i.e. G := (cid:80) i g i | v i (cid:105)(cid:104) v i | ,then1 N Tr [ G · F − ( θ )] = 1 N (cid:88) i g i (cid:104) v i | F − ( θ ) | v i (cid:105) . (31)According to the above expression, the information con-tent is a weighted combination of the distinguishabilityof the manifold in different directions, see Fig. 1 for a2D representation. This has to be compared with theexperimental weighted distinguishability, i.e.Tr [ G · Σ ( N ) ( θ )] = (cid:88) i g i (cid:104) v i | Σ ( N ) ( θ ) | v i (cid:105) , (32)given by a particular measurement. The whole pointof the non commutative nature of the manifold is theimpossibility to saturate the distinguishability in morethan one direction at the same time. By takingsup G ≥ N Tr [ G · Σ ( N ) ( θ ) ] Tr[ G · F − ( θ )] = sup g i ≥ , | v i (cid:105) N (cid:80) i g i (cid:104) v i | Σ ( N ) ( θ ) | v i (cid:105) (cid:80) i g i (cid:104) v i | F − ( θ ) | v i (cid:105) , (33)we measure the worst case fitting of the Σ ( N ) ( θ ) matrixon the metric F − ( θ ) /N at a point θ , spanning all pos-sible sets of directions and weights. Then we minimizeon the classical metric (and hence on the POVM) to findthe most adapt one. By taking the asymptotic limit ofinfinitely many probes (through the lim inf) we concludethe breakdown of Eq. (17) to its geometric meaning. Wesum up everything and say that r ( θ ) measures, in theasymptotic scenario, the failure of finding a metric onthe statistical manifold, stemming from a measurements,which fully adapts to the underlying quantum metric (inall directions) at a specific point θ . IV. INCOMPATIBILITY UNDER LOCALLYASYMPTOTIC COVARIANCE
Different lower bounds on the quantity C ( G, θ ) ofEq. (23) are known in the literature, of which thetighter one is the Holevo-Cram´er-Rao bound functional C H ( G, θ ) [51, 52]. Exploiting this facts we can useEq. (22) to deduce the following inequality r ( θ ) ≥ sup G ≥ C H ( G, θ ) C S ( G, θ ) . (34)The Holevo bound has been proved to be achievable [53],but this requires leaving the domain of locally unbiasedmeasurements, on which the minimization in Eq. (22) istaken. This means the result of [53] doesn’t allow us toconclude that the inequality (34) saturates. The bestapproach to overcome this issue is to introduce a newfigure of merit taking the infimum on a set of POVMfor which we know that an asymptotic precision boundholds, which is achievable within the class of chosen mea-surements. To this purpose we will consider the se-quences of POVM (M k ) k ∈ N ∈ M (LAC) C that satisfy localasymptotic covariance (LAC) at the point θ ∈ Θ, as de-fined in [42]. The k th measurement M k of a sequencein M (LAC) C acts on k probes and has Σ ( k ) ( θ ) as associ-ated MSE matrix. Because of the definition of LAC,(M k ) k ∈ N ∈ M (LAC) C admits a limiting MSE matrix, i.e.lim k →∞ k Σ ( k ) ( θ ) := Σ( θ ) (we also ask (cid:107) Σ( θ ) (cid:107) < ∞ ).The new figure of merit is hence defined as r ( θ ) := inf M (LAC) C sup G ≥ Tr [ G · Σ( θ )]Tr [ G · F − ( θ )] . (35)In [42] the authors have proven the validity of the Holevobound for the LAC measurements and its achievabilityin the same class (for a full rank ρ θ with non degeneratespectrum), that isinf M (LAC) C Tr [ G · Σ( θ )] = C H ( G, θ ) . (36)It is easy to prove that the set M (LAC) C is con-vex [54]. Furthermore given Σ ( θ ) and Σ ( θ ) the lim-iting MSE matrices of two sequences (cid:0) M k (cid:1) k ∈ N and (cid:0) M k (cid:1) k ∈ N , the asymptotic MSE of the convex combi-nation (cid:0) α M k + (1 − α ) M k (cid:1) k ∈ N is Σ( θ ) = α Σ ( θ ) +(1 − α ) Σ ( θ ). Consequently, just like in Appendix B,the Minimax theorem of Kneser [55, 56] can be appliedto swap the order of inf over M (LAC) C and sup G ≥ . It isunderstood that the argument of Eq. (35) can be castinto the same form of Eq. (B7), from which the linearityin (M k ) k ∈ N and G , and the continuity in G easily follow.This swap, together with Eq. (36), gives r ( θ ) = sup G ≥ inf M (LAC) C Tr [ G · Σ( θ )]Tr [ G · F − ( θ )] = sup G ≥ C H ( G, θ ) C S ( G, θ ) , (37) which on side provides an operational interpretation tothe right-hand-side term of Eq. (34), and on the otherside explicitly shows that (35) is a proper lower bound for r ( θ ) – see however Appendix F for an alternative proof ofthis fact. In [57] the upper bound C H ( G, θ ) ≤ C S ( G, θ )is given, which implies r ( θ ) ≤
2. Because of this it makessense to introduce I ( θ ) := r ( θ ) − , (38)as a proper quantifier of the LAC incompatibility: byconstruction it belongs to the interval [0 ,
1] with I ( θ ) = 0indicating full compatibility, while I ( θ ) = 1 maximalincompatibility. A. Upper bound on r ( θ ) In this section we propose an upper bound on r ( θ )that relies only on the computation of the symmetriclogarithmic derivatives of Eq. (7). It is essentially basedon C Z ( G, θ ) [58], a well know upper bound on C H ( G, θ ),which reads C H ( G, θ ) ≤ C Z ( G, θ ) (39):= Tr (cid:2) GF − ( θ ) (cid:3) + Tr Abs (cid:2) GF − ( θ ) A ( θ ) F − ( θ ) (cid:3) , where A ( θ ) contains the expectation values of the com-mutators of the SLDs: A ij ( θ ) := 12 i Tr [ ρ θ [ L i ( θ ) , L j ( θ )]] . (40)In writing Eq. (39) we used Tr Abs [ G · R ] :=Tr |√ GR √ G | , with | X | := √ XX † . Combining Eq. (39)and Eq. (37) we get r ( θ ) ≤ G ≥ Tr Abs (cid:2) G · F − ( θ ) A ( θ ) F − ( θ ) (cid:3) Tr [ G · F − ( θ )] := r (cid:63) ( θ ) . (41)The above inequality shows that a sufficient conditionto have compatibility is A ( θ ) = 0. In Appendix C wecompute explicitly sup G ≥ in Eq. (41) and obtain r (cid:63) ( θ ) = 1 + (cid:107) F − ( θ ) A ( θ ) F − ( θ ) (cid:107) , (42)which translates to an upper bound on I ( θ ), i.e. I ( θ ) ≤ I (cid:63) ( θ ) := r (cid:63) ( θ ) − (cid:107) F − ( θ ) A ( θ ) F − ( θ ) (cid:107) . (43)This strengthen the interpretation of A ( θ ) as a measureof incompatibility [44]. Notice also that the upper bound I (cid:63) ( θ ) is the quantity R ( θ ) defined in [24]. V. INCOMPATIBILITY FOR SEPARABLEMEASUREMENTS
We now go back to the first definition of a figure ofmerit presented in Eq. (17), but consider the minimiza-tion in Eq. (16) to be performed only on the locally un-biased separable measurements subset M (LU-S) N of M (LU) N which operate locally on ρ ⊗ N θ . This brings to the defini-tions r sN ( θ ) := inf M N ∈M (LU-S) N sup G ≥ r N ( G, M N , θ ) , (44)and r s ( θ ) := lim inf N →∞ r sN ( θ ) . (45)Now we apply the result of [59], which gives us a lowerbound on the precision of the estimation with N probeswhen we use a measurement M N ∈ M (LU-S) N . The boundreads N Tr [ G · Σ ( N ) ( θ )] ≥ (cid:18) Tr (cid:113) F − ( θ ) GF − ( θ ) (cid:19) D − , (46)where Σ ( N ) ( θ ) is the MSE matrix of M N and D is thesize of the Hilbert space of the single probe ρ θ . Thistranslates to a lower bound on r N ( G, M N , θ ) ∀ N , i.e. r N ( G, M N , θ ) ≥ (cid:18) Tr (cid:113) F − ( θ ) GF − ( θ ) (cid:19) ( D −
1) Tr[ F − ( θ ) GF − ( θ )] , (47)which propagates to the definition of r s ( θ ) r s ( θ ) ≥ sup G ≥ (cid:18) Tr (cid:113) F − ( θ ) GF − ( θ ) (cid:19) ( D −
1) Tr[ F − ( θ ) GF − ( θ )] = dD − , (48)where in the last passage follows from the AM-QM in-equality and its saturation. Observe that the inequal-ity (48) bares no reference to the details of the encodingprocess (1) and that it is non trivial only if the number d of parameters we have to estimate is larger than or equalto D − M (LU-S) N , because only the localunbiasedness is required in their proof. Therefore we canwrite r s ( θ ) = sup G ≥ C s ( G, θ ) C S ( G, θ ) , (49)where now C s ( G, θ ) := lim n →∞ min N ≤ n inf M N ∈M (LU-S) N N Tr [ G · Σ ( N ) ( θ )] . (50)At least for the case of a qubit probe ( D = 2) the aboveexpression allows us to exactly compute r s ( θ ). Indeed asshown in Refs. [51, 59] for this model one has C s ( G, θ ) = (cid:18) Tr (cid:113) F − ( θ ) GF − ( θ ) (cid:19) leading to the identity D = 2 = ⇒ r s ( θ ) = d , (51) which shows that the case of a single, multi-parameter es-timation always exhibit incompatibility for separable lo-cally unbiased measurements (remember that our analy-sis is explicitly restricted to the cases where d ≤ D − VI. HIERARCHY OF INCOMPATIBILITYMEASURES
It is self-evident that whether a certain estimation pro-cess is compatible or not depends on the set of measure-ments M N that we are allowed to perform. Consider ahierarchy of POVM sets M (1) N ⊆ M (2) N ⊆ · · · ⊆ M ( k ) N ∀ N , (52)we define the figure of merit r ( i ) N ( θ ) as in Eq. (16), buttaking M ( i ) N as the domain of the infimum. By construc-tion the r ( i ) N ( θ ) satisfy the following hierarchy of inequal-ities r (1) N ( θ ) ≥ r (2) N ( θ ) ≥ · · · ≥ r ( k ) N ( θ ) ∀ N , (53)which carries over to r (1) ( θ ) ≥ r (2) ( θ ) ≥ · · · ≥ r ( k ) ( θ ) , (54)when taking the proper N → ∞ limits (17). Notice thatif the allowed measurements are not locally unbiasedness,then the lower bound in Eq. (15) is not valid, and with itwe lose also the property r ( i ) ( θ ) ≥
1. Nevertheless evenin this situation the ratios entering in the definitions of r ( i ) N ( θ ) and r ( θ ) maintain their operational and geomet-rical interpretation. Two out of the three defined figuresof merit defined in the previous sections (i.e. r ( θ ) and r s ( θ )) are associated to set of measurements that ful-fill the condition (52) (i.e. M (LU-S) N ⊆ M (LU) N ): thereforewe can immediately conclude that r ( θ ) ≤ r s ( θ ). Thefigure of merit r ( θ ) defined in Eq. (35) refers instead di-rectly to measurement sequences and cannot be triviallycompared to r ( θ ) and r s ( θ ) using the above arguments.From Eqs. (34) and (37) we know however that r ( θ ) is al-ways smaller than or equal to r ( θ ) (see also Appendix F),leading us to the hierarchy r ( θ ) ≤ r ( θ ) ≤ r s ( θ ) . (55) VII. INCOMPATIBILITY OF A NOISYESTIMATION TASK
In this section, by using the previously defined figuresof merit r ( θ ) in Eq. (37) and r s ( θ ) in Eq. (45), we studythe incompatibility of the estimation process in a fewsimple cases concerning the sensing of two phases θ and θ encoded by the unitary transformation U θ := exp [ i ( θ σ y + θ σ z )] , (56)acting on individual qubits. The probes will be states ofone, two and three qubits that are highly capable of cod-ifying information. When we detect incompatibility weadd a depolarizing noise acting locally on each individualencoded qubit, which is given by the mapΛ λ ( ρ ) := λρ + (1 − λ ) , (57)with λ ∈ [ − / ,
1] being a characteristic parameter of themodel [60]. The transformation Λ λ induces a shrinking ofthe qubit Bloch vector by a factor given by the modulus | λ | which can be used to gauge the intensity of the noise.In particular for λ = 1 the map (57) corresponds to thenoiseless evolution, and for λ = 0 to the complete depo-larization process, while negative values of λ indicate thepresence of an inversion of the Bloch sphere with respectto the origin [61]. We are interested in investigating ifthe noise can force the system to a more classical behav-ior and therefore ensure compatibility in the estimationscenario, as it does for measurements [40]. Notice thatthe chosen noise is covariant and therefore in all our ex-amples it could be applied before or after the encodingwithout changing the final output ρ θ . A. Incompatibility for a one-qubit probe
First of all we analyze the case of a single qubit probe.After the encoding by U θ in Eq. (56), the probe under-goes the action of the noise map Λ λ , so that its final state ρ θ is described by the mapping (1) with E θ given by E θ ( · · · ) := Λ λ ( U θ · · · U † θ ) = U θ Λ λ ( · · · ) U † θ . (58)The purity of the encoded state ρ θ = E θ ( ρ ) is in-dependent on θ , this makes the statistical model D-invariant [58, 62]. Furthermore, at least for λ < ρ θ is guaranteed to be full rank and non degeneratein spectrum (unless we take as the input densityfor the probe). As explained in [42] these conditionsallow us to conclude that for the model in Eq. (58)the Holevo-Cram´er-Rao bound C H ( G, θ ) coincides with C Z ( G, θ ) defined in Eq. (39), therefore the inequality(41) is saturated and I ( θ ) = I (cid:63) ( θ ), this means thatthe incompatibility can be computed from the symmet-ric logarithmic derivatives only. We consider an arbi-trary qubit probe state ρ := ( + r · σ ). Its Blochvector is r := ( r x , r y , r z ), with Tr( ρ ) = (1 + (cid:107) r (cid:107) ).After the encoding, the Bloch vector of E θ ( ρ ) is r θ := λ ( r x ( θ ) , r y ( θ ) , r z ( θ )). We can perform an implicitly de-fined change of variables ( θ , θ ) → ( α, β ), that brings usto r ( α,β ) = λ (cid:112) ρ − α cos β, cos α sin β, sin α ).For this model we compute the matrices F ( α, β ) and the A ( α, β ), which are F ( α, β ) = (cid:18) λ λ cos α (cid:19) , (59) A ( α, β ) = (cid:18) − λ cos αλ cos α (cid:19) . (60) that substituted in Eq. (43) give [63] I ( θ , λ ) = I ( α, β, λ ) = (cid:112) ρ − | λ | ∀ θ . (61)Equation (61) reveals that the noise level intensity con-trols directly the compatibility. Indeed for fixed inputthe value of I ( θ , λ ) reaches its maximum in the noiselessscenario ( λ →
1) providing full incompatibility
I → ρ θ to the completely mixed state ( λ →
0) the codified in-formation is dissipated and the compatibility increases,indeed
I →
0. We finally remind the reader that, as an-ticipated at the end of Sec. V, for a single qubit we get r s ( θ ) = d = 2. Again this result is valid ∀ θ and for everyinput probe ρ . B. Incompatibility for a two-qubits probe
In this section we analyze the compatibility of threedifferent two qubit encoding scenarios in the absence ofnoise effects and for some special instances of the inputstates. Consider first the ancilla-aided model, in whichonly one of the two qubits is subject to the unitary en-coding (56). This means that the total evolution of thetwo qubits is ⊗ U θ . As input state for the probes wetake a Bell state, which is known in the literature to beoptimal for the estimation of SU (2) operations [64], forexample | ψ (cid:105) := | (cid:105) + | (cid:105)√ , (62)where | (cid:105) and | (cid:105) are the eigenvectors of σ z . From a directcomputation we see that for this state A ( θ ) = 0 ∀ θ ,which from Eq. (43) gives [65] I ( θ ) = 0 , ∀ θ . (63)leading to no incompatibility in terms of LAC estimationprocedures. Interestingly enough exactly the same resultcan be obtained also when operating on the maximallyentangled state Eq. (62) with the encoding U θ ⊗ U θ . In-deed by explicit computation we get again A ( θ ) = 0 ∀ θ that leads once more to (63). We will see in Sec. VIII Bthat these effects are just a special instance of a more gen-eral trend since a maximally entangled state of a finitedimensional probe always gives full compatibility, bothfor one and two uses of the encoding unitary channel.We now give a last example, which we here name“anti-parallel spin strategy” for future reference. Takethe input state going through the encoding U θ ⊗ U θ tobe | + (cid:98) n (cid:105) ⊗ |− (cid:98) n (cid:105) , where | + (cid:98) n (cid:105) and |− (cid:98) n (cid:105) have oppositeBloch vectors + (cid:98) n and − (cid:98) n . This state has the sameQFI of the state of two parallel spins | + (cid:98) n (cid:105) ⊗ | + (cid:98) n (cid:105) ,but has A ( θ ) = 0 ∀ θ (in contrast to | + (cid:98) n (cid:105) ⊗ | + (cid:98) n (cid:105) ),which means that it is fully compatible and a superiorprobe for the sensing task. This result can be obtainedfrom direct computation or thanks to the observationof Sec. VIII C, where we generalize this ideas to finitedimensional probes. This phenomenon was already ob-served by Gisin and Popescu in [66]. C. Three entangled qubits
Consider now the scenario in which we have at disposalmultiple copies of three entangled qubits and we codifythem through U θ ⊗ U θ ⊗ U θ , with U θ given in Eq. (56).In this example again we won’t be able to compute r ( θ )for every probe state, therefore we will concentrate on | ψ (cid:105) := | ψ z (cid:105) + | ψ y (cid:105)√ , (64)with | ψ z (cid:105) := 1 √ | (cid:105) + | (cid:105) ) , (65) | ψ y (cid:105) := 1 √ (cid:0) | φ + φ + φ + (cid:105) + | φ − φ − φ − (cid:105) (cid:1) , (66)where | φ + (cid:105) and | φ − (cid:105) are the eigenvectors of σ y corre-sponding to the positive and negative eigenvalue respec-tively. In [38] it is proved that the analogous state forthe estimation of three phases with N entangled qubitsreaches Heisenberg scaling in the QFI in all the threeparameters. At difference with the previous examples,here we are able to compute the figure of merit for theprobe | ψ (cid:105) only at the point θ = 0 through numericalevaluations via the semidefinite program reported in Ap-pendix E, these indicate a non-null I ( θ = 0). We adda local depolarization noise Λ λ on each qubit and com-pute I ( θ = 0 , λ ) and its upper bound I (cid:63) ( θ = 0 , λ ) asfunctions of λ to see if the noise increases compatibil-ity [65], the results are reported in Fig. 2. I ( θ = 0 , λ )and I (cid:63) ( θ = 0 , λ ) have been computed for 100 values of λ uniformly distributed in ( − / , I ( θ = 0 , λ ) and I (cid:63) ( θ = 0 , λ ) both display a non-monotonic behavior with respect to | λ | . This behavior ofthe incompatibility has already been observed in [67].The figure of merit I ( θ , λ ) appears to be not correlatedwith the information quantities F − ( θ , λ ) and F − ( θ , λ )or with the purity of the encoded state, as these measuresare all monotonic in the noise λ . Also because of this wethink of I ( θ ) as a genuine non trivial new property ofthe estimation process.Notice that for λ = 0, the state isunable to codify information ( F ( θ ) = 0). VIII. USING ENTANGLEMENT TO ENFORCELAC COMPATIBILITY FOR GENERALFINITE-DIMENSIONAL PROBES
In this section we generalize the results of Sec. VII Aand Sec. VII B to a generic unitary encoding of d param- FIG. 2. The orange dashed curve is the upper bound I (cid:63) ( θ =0), defined in Eq. (43), computed for the encoded three qubitsstate in Eq. (64) as a function of the local noise intensity λ .The blue solid curve is the figure of merit I ( θ = 0) of Eq. (38)referred to the same scenario and computed numerically asexplained in Appendix E. The curves are symmetric around λ = 0. The empty point in λ = 0 indicates that at this pointthe information quantities are not defined. eters on a D -dimensional probe in S ( H ), i.e. U θ := exp i d (cid:88) j =1 θ j H j , (67)where H j are null-trace hermitian operators acting on H . For the estimation around θ = 0, these operators arethe infinitesimal generators of the encoding. However fora generic point θ (cid:54) = 0 this is not necessarily true. Asexplained in Appendix D, for a given probe state, thesensing procedure around a point θ (cid:54) = 0 can however bedescribed in terms of an effective set of new generators H eff j ( θ ). Accordingly, since the results of the preset sec-tion are valid for estimations around θ = 0 for all possiblechoices of H j , we can conclude that they hold true also ∀ θ in the encoding (67). Finally as for the noise modelwe replace Eq. (57) withΛ λ ( ρ ) := λρ + (1 − λ ) D , (68)which for λ ∈ [ − D − ,
1] is a proper generalization of thedepolarization channel for a D -dimensional system [60,61]. A. Incompatibility rescaling under the noise andits asymmetric behavior around λ = 0 As in Sec. VII A let us start considering a single-probescenario where the state of the system is described by thedensity matrix ρ θ = Λ λ ( U θ | ψ (cid:105)(cid:104) ψ | U † θ ) = U θ Λ λ ( | ψ (cid:105)(cid:104) ψ | ) U † θ , = λ | ψ θ (cid:105)(cid:104) ψ θ | + (1 − λ ) D , (69)with | ψ (cid:105) being the pure input state of the system, andwith | ψ θ (cid:105) := U θ | ψ (cid:105) . If we now call L i ( θ ) the symmetric0logarithmic derivative associated to the parameter θ i inthe absence of noise, i.e. the SLD of | ψ θ (cid:105) , given in Eq. (8),then it can be seen that for λ (cid:54) = 1 L i ( θ , λ ) = λD λ ( D − L i ( θ ) , (70)is the SLD in the noisy scenario. We obtain this ex-pression by substituting ρ θ of Eq. (69) in Eq. (9). Fromthis result the QFI matrix F ( θ , λ ) and the commuta-tor matrix A ( θ , λ ) are both found to be proportionalto their noiseless counterparts F ( θ ) and A ( θ ) computedfrom L i ( θ ), i.e. F ( θ , λ ) = λ D λ ( D − F ( θ ) , (71) A ( θ , λ ) = λ D [2 + λ ( D − A ( θ ) . (72)Replaced into Eq. (43) the above expression finally leadto I (cid:63) ( θ , λ ) = | λ | D λ ( D − I (cid:63) ( θ ) , (73)with I (cid:63) ( θ ) being the noiseless incompatibility figure ofmerit. Notice that this expression is not symmetricaround λ = 0, i.e. I (cid:63) ( θ , λ ) (cid:54) = I (cid:63) ( θ , − λ ) for λ ≥
0. Wedefine the asymmetry factor as κ (cid:63) ( λ ) := |I (cid:63) ( θ , λ ) − I (cid:63) ( θ , − λ ) |I (cid:63) ( θ , λ ) = 2 | λ | ( D − − λ ( D − . (74)The presence of an asymmetry in the properties of the D -dimensional depolarizing channel around λ = 0 wasalready pointed out in the context of communicationin [61]. We show through a numerical example that thisasymmetry exists not only for the upper bound I (cid:63) ( θ , λ )but also for the actual figure of merit I ( θ , λ ). Considerthe encoding of two phases ( d = 2) on a qutrit ( D = 3)via the unitary in Eq. (67). The generators are chosen tobe H = − i i , H = − . (75)The probe state is | ψ (cid:105) = √ (cid:0) − (cid:1) t , and the sensingis performed around the point θ = 0. As in Sec. VII C thefigure of merit has been computed with the semidefiniteprogram presented in Appendix E, for 500 equally spacedvalues of λ in the allowed region. In Fig. 3 the plot of I ( θ = 0 , λ ) is reported for λ ∈ ( − / , λ ∈ ( − / , / λ < λ >
0. Ithas been plotted in order to highlight the presence of theasymmetry. For this model, at the point θ = 0, the upperbound I (cid:63) ( θ = 0 , λ ) and the figure of merit I ( θ = 0 , λ )coincide. FIG. 3. This curve is the incompatibility figure of merit ofEq. (38), for the asymptotic covariant measurements, numer-ically computed for the qutrit example of Sec. (VIII A) for λ ∈ ( − / , λ = 0 indicates thatat this point the information quantities are not defined. Thedashed curve in the small insert is the mirrored figure of meritfor λ > B. Compatibility of the maximally entangled states
In this subsection we will show that the results ofSec. VII B are only a particular case of this more gen-eral observations, by proving that the use of an ancilla,maximally entangled with the probe, can completely re-move the incompatibility as measured by the LAC mea-sure leading to the identity I ( θ ) = 0 , ∀ θ , (76)(the analysis however does not necessarily holds for otherfigures of merit in the hierarchy which may still exhibitnon trivial dependence upon θ ). For the sake of sim-plicity we shall focus here only on the noiseless scenario,i.e. neglect the presence of the depolarization channel.Consider hence as input the following pure state [65] | ψ (cid:105) := 1 √ D d (cid:88) i =1 | i (cid:105) ⊗ | i (cid:105) ∈ H ⊗ H , (77)on which the evolution ⊗ U θ acts to produce the outputstate | ψ θ (cid:105) := ⊗ U θ | ψ (cid:105) , (78)(i.e. the Choi–Jamio(cid:32)lkowski state of the channel U θ [68,69]). Following the analysis of Appendix D the associatedsymmetric logarithmic derivatives, given by Eq. (8) with ρ θ := | ψ θ (cid:105)(cid:104) ψ θ | , can be expressed as L k ( θ ) = 2 iD (cid:88) ij | i (cid:105)(cid:104) j | ⊗ (cid:0) H eff k ( θ ) | i (cid:105)(cid:104) j | − | i (cid:105)(cid:104) j | H eff k ( θ ) (cid:1) , (79)which lead to the following expressions for the QFI and A ( θ ) matrices: F lm ( θ ) = 2 Tr (cid:0) { H eff l ( θ ) , H eff m ( θ ) } (cid:1) /D ,A lm ( θ ) = − i Tr (cid:0)(cid:2) H eff l ( θ ) , H eff m ( θ ) (cid:3)(cid:1) /D = 0 , (80)1where in the last identity we used the fact that (cid:2) H eff l ( θ ) , H eff m ( θ ) (cid:3) is a traceless operator. Accordinglythe upper bound (43) imposes (76) hence the thesis: theaddition of a sufficiently large ancilla indeed permits toremove entirely the quantum incompatibility for the LACmeasurements.A similar result holds true also when we let evolve themaximally entangled state of Eq. (77) through U θ ⊗ U θ .Indeed in this case Eq. (78) gets replaced by | ψ θ (cid:105) := U θ ⊗ U θ | ψ (cid:105) , (81)which leads to F lm ( θ ) = 8 Tr (cid:0) { H eff l ( θ ) , H eff m ( θ ) } (cid:1) /D , (82) A lm ( θ ) = − i Tr (cid:0)(cid:2) H eff l ( θ ) , H eff m ( θ ) (cid:3)(cid:1) /D = 0 , (83)so we have again full compatibility (76). C. Anti-parallel spin strategy and generalizations
Now we generalize the “anti-parallel spin” strategy ofsection Sec. VII B. Suppose that we only have to param-eter to estimate ( d = 2) and we take for the probes theseparable input state | ψ (cid:105) ⊗ | ψ (cid:105) that we evolve throughthe mapping induced by U θ ⊗ U θ (again we do not in-clude the noise effects). Observe next that a sufficientcondition for attaining full LAC compatibility at θ (i.e. I ( θ ) = 0) is to force the upper bound (43) to nullifye.g. by imposing A ( θ ) = 0. By explicit computation weget the latter constraint can be ensured by forcing thecondition (cid:104) ψ | [ H eff1 ( θ ) , H eff2 ( θ )] | ψ (cid:105) + (cid:104) ψ | [ H eff1 ( θ ) , H eff2 ( θ )] | ψ (cid:105) = 0 . (84)The operator [ H eff1 ( θ ) , H eff2 ( θ )] is skew-hermitian andtherefore is diagonalizable and has purely imaginaryeigenvalues ± ia j , where a j >
0, for j = 1 , , . . . , (cid:98) D (cid:99) ,each associated to an eigenvector |± a j (cid:105) . If the dimension D is odd, then we have an extra unique zero eigenvalue.Let’s denote with V the unitary operator that performssuch diagonalization, i.e. V † (cid:2) H eff1 ( θ ) , H eff2 ( θ ) (cid:3) V =diag ( ± ia , ± ia , · · · ). Let’s define S ⊆ { , , . . . , (cid:98) D (cid:99)} ,then a couple of states that guarantees compatibility is | ψ (cid:105) := 1 |S| (cid:88) j ∈S e iϕ j V | i ( − s j a j (cid:105) , (85) | ψ (cid:105) := 1 |S| (cid:88) j ∈S e iϕ j V |− i ( − s j a j (cid:105) , (86)where ϕ j and ϕ j are arbitrary phases, s j ∈ { , } , and |S| is the cardinality of S . Notice that we are also free toadd in the definition of whichever | ψ (cid:105) or | ψ (cid:105) the state V | (cid:105) , with | (cid:105) being the eigenvector with null eigenvalue(in case there is one). With the above choice of probesthe compatibility condition of Eq. (84) is realized. Noticealso that the QFI matrix of | ψ (cid:105) ⊗ | ψ (cid:105) is the sum of the QFIs of | ψ (cid:105) and | ψ (cid:105) . This two states taken individu-ally manifest incompatibility, but jointly they retain theirQuantum Fisher Information while gaining full compat-ibility. In this sense this construction is the analogue ofthe “anti-parallel spin strategy” of Sec. (VII B). We alsoobserve that a state | ψ (cid:105) , being an equal superposition of | ψ (cid:105) and | ψ (cid:105) contains both the eigenvectors associatedto the positive and to the negative eigenvalues, and it ishence fully compatible. The condition of Eq. (84) justifiesalso the compatibility of | + (cid:98) n (cid:105) ⊗ |− (cid:98) n (cid:105) ∀ θ in Sec. VII B.These two states are an orthogonal basis for the qubitHilbert space, therefore (cid:104) + (cid:98) n | [ H eff1 ( θ ) , H eff2 ( θ )] | + (cid:98) n (cid:105) + (cid:104)− (cid:98) n | [ H eff1 ( θ ) , H eff2 ( θ )] | − (cid:98) n (cid:105) = Tr (cid:0) [ H eff1 ( θ ) , H eff2 ( θ )] (cid:1) = 0 , hence condition (84) is satisfied and (76) holds. Inciden-tally we observe also that for a d -parameter estimation,the state | ψ (cid:105) ⊗ | ψ (cid:105) ⊗ · · · ⊗ | ψ D (cid:105) its compatible when {| ψ i (cid:105)} Di =1 is base for the Hilbert space H , because thenwe would have A ij ( θ ) ∝ Tr (cid:0)(cid:2) H eff i ( θ ) , H eff j ( θ ) (cid:3)(cid:1) = 0 . (87) IX. CONCLUSIONS
One of the defining properties of quantum mechanicsis the intrinsic incompatibility between the possible ex-periments that could be carried out on a quantum sys-tem. This causes the appearance of information limitson the precision to which different characteristics of acertain quantum system can be known. Formally, thiscomes always from the non-commutativity of quantumoperations. The main role of this paper is to give a the-oretical foundation to the measure of incompatibility inthe quantum estimation task. For this purpose we definein Sec. III a figure of merit having a well defined opera-tional and geometrical meaning. We gave three differentversions of our figure of merit, in Eq. (17), Eq. (45), andEq. (35). They differ from one another in the set of quan-tum measurements that are allowed. For r ( θ ) we considerlocally unbiased measurements around the point of inter-est, and for r s ( θ ) we also require them to be separable.The last figure of merit, r ( θ ), is based on the sequences ofmeasurements that satisfy LAC [42]. In general multiplenested measurement sets give rise to a hierarchy of figuresof merit, as explained in Sec. VI. It makes sense that thedefinition of incompatibility depends on the operationsthat we are able to perform, which is our level of controlover the system. The figure of merit r ( θ ) in Eq. (35) isbuilt to be easily liked to the asymptotic results of es-timation theory [42]. This allows us to easily computeit, as explained in Appendix E, and to give the upperbound in Sec. IV A. In Sec. VII the estimation is stud-ied in the scenario where single qubit depolarizing noiseacts, this is a form of disturbance which is often used tomodel the decoherence dynamics in metrology [70, 71].The incompatibility turns out to be non monotonic in2the noise strength, despite the noise being structurallysimple. As a further development it would be interestingto determine which state gives the maximum and min-imum incompatibility for a certain encoding at a fixedpoint θ . This optimization is hard because the figure ofmerit is a complicated non linear function of the state.In a sense the probe that maximizes incompatibility isthe one which captures at most the quantum propertiesof the encoding process. Another question that couldbe worth tackling is determining (if it exists) the pointin the hierarchy (54) from which the incompatibility iseliminated ( r ( i ) ( θ ) = 1), in other words which class ofmeasurements is sufficient to achieve compatibility. It isnot clear whether this information will be significant ornot, as it will depend on the estimation point and on theprobe state, not only on the structure on the encodingchannel. Finally we would like to look for a link betweenthe incompatibility and the Heisenberg scaling. In thiscontext the only relevant measures are the one that as-sume no constraints in the separability properties of theinput, because a single giant entangled probe would beused. ACKNOWLEDGMENTS
We thank Yuxiang Yang for discussions. We ac-knowledges support by MIUR via PRIN 2017 (Progettodi Ricerca di Interesse Nazionale): project QUSHIP(2017SRNBRK).
Appendix A: Locally unbiased estimation procedures
Formally a POVM M N operating on ρ ⊗ N θ is said tobe locally unbiased at the point θ ∈ Θ if its associatedbias vector field b ( N ) ( θ ) : Ω → R d defined as in Eq. (3),fulfills the conditions b ( N ) ( θ ) (cid:12)(cid:12)(cid:12) θ = θ = 0 , ∂∂θ j b ( N ) ( θ ) (cid:12)(cid:12)(cid:12) θ = θ = 0 , ∀ j = 1 , · · · , d . (A1)An example of a POVM which belongs to such set isthe measurement (cid:102) M N introduced in Sec. III C. To seethis we compute the bias vector b ( N ) ( θ ) of the projectivemeasurement associated with the observable (cid:98) O i ( θ (cid:48) ) ofEq. (28), with θ (cid:48) ∈ Θ fixed, and we get b i ( θ ) = Tr (cid:104) ρ θ (cid:98) O i ( θ (cid:48) ) (cid:105) − θ i = θ (cid:48) i − θ i + F − ii Tr (cid:2) ρ θ L i ( θ (cid:48) ) (cid:3) , (A2)which, thanks to the identityTr [ ρ θ L i ( θ )] = Tr (cid:20) ∂ρ θ ∂θ i (cid:21) = 0 , (A3)and the diagonal form of the QFI matrix (27) clearlyfulfills Eq. (A1) for θ = θ (cid:48) . Appendix B: Proof of the alternative definition
We will arrive at Eq. (21) through a series of lemmas.
Lemma 1 . The function r N ( G, M N , θ ) is continuousin G ∈ G at fixed M N ∈ M (LU) N .The denominator of r N ( G, M N , θ ) can be bounden asTr [ G · F ( θ ) − ] ≥ λ min (cid:0) F ( θ ) − (cid:1) = λ max ( F ( θ )) − = (cid:107) F ( θ ) (cid:107) , this means r N ( G, M N , θ ) ≤ N (cid:107) F ( θ ) (cid:107) Tr [ G · Σ ( N ) ( θ )] , (B1)it follows | r N ( G , M N , θ ) − r N ( G , M N , θ ) |≤ N (cid:107) F ( θ ) (cid:107)(cid:107) Σ ( N ) ( θ ) (cid:107)(cid:107) G − G (cid:107) . (B2)Its important to assume that the set M N ∈ M (LU) N con-tains only measurements with bounded MSE matrices,i.e. (cid:107) Σ ( N ) ( θ ) (cid:107) ≤ C . Therefore we have | r N ( G , M N , θ ) − r N ( G , M N , θ ) |≤ N C (cid:107) F ( θ ) (cid:107)(cid:107) G − G (cid:107) , (B3)which means the function r N ( G, M N , θ ) is Lip-schitz continuous with constant N C (cid:107) F ( θ ) (cid:107) andtherefore continuous. It will be useful in the lat-ter to notice that r N ( F ( θ ) GF ( θ ) , M N , θ ) = N Tr [ G · F ( θ ) Σ ( N ) ( θ ) F ( θ ) ] is also continuous in G ∈ G , because is the composition of the continuousfunctions r N ( G, M N , θ ) and G → F ( θ ) GF ( θ ) .In this paper only the upper semicontinuity of r N ( G, M N , θ ) is actually used. Lemma 2 . The figure of merit r N ( θ ) defined in Eq. (16)can be equivalently expressed as r N ( θ ) = sup G ≥ inf M N ∈M (LU) N r N ( G, M N , θ ) . (B4)This Lemma is based on the application of the Mini-max theorem of Kneser [55, 56]. First of all we need tocast r N ( θ ) in a suitable form. We start from the observa-tion that the set of positive weight matrices G is invariantunder congruence for the positive matrix F ( θ ) , i.e. { G ≥ } = F ( θ ) { G ≥ } F ( θ ) = { F ( θ ) GF ( θ ) ≥ } . (B5)This means that the sup can be taken on the matrices3 F ( θ ) GF ( θ ) ≥ r N ( θ ), so we have r N ( θ ) = (B6)= inf M N ∈M (LU) N sup F ( θ ) GF ( θ ) ≥ N Tr [ F ( θ ) GF ( θ ) · Σ ( N ) ( θ )]Tr [ F ( θ ) GF ( θ ) · F ( θ ) − ]= inf M N ∈M (LU) N sup G ≥ N Tr [ G · F ( θ ) Σ ( N ) ( θ ) F ( θ ) ]Tr [ G ]= inf M N ∈M (LU) N sup G ∈G N Tr [ G · F ( θ ) Σ ( N ) ( θ ) F ( θ ) ] . (B7)In the second line, in the domain of the supremum, wehave again used that every F ( θ ) GF ( θ ) ≥ G ≥
0. In the last equation the sup isrestricted to the set G = { G ≥ , Tr G = 1 } , which iscompact and convex. The set of locally unbiased mea-surements M N ∈ M (LU) N can be thought as a (non-empty) subset of a dual vector space [72, 73], whichis a convex set because the locally unbiased measure-ments are stable under convex combination. The func-tion N Tr [ G · F ( θ ) Σ ( N ) ( θ ) F ( θ ) ] is linear in both itsarguments. The linearity in G is self evident, so we onlyshow linearity in the measurement. Suppose that we aregiven two POVMs denoted by M and M , characterizedrespectively by the effects E ˆ θ and E ˆ θ associated to theoutcome ˆ θ . We have dropped for simplicity the subscript N in M and M and we will also drop the superscript( N ) in the MSE matrix Σ( θ ). Consider the POVM be-ing the linear combination M := λ M + (1 − λ ) M . Bydefinition its effects areE ˆ θ := λ E ˆ θ + (1 − λ ) E ˆ θ . (B8)The Σ( θ ) matrix associated to M is computed as expec-tation value on the probability distribution p ( ˆ θ ) := Tr( ρ θ E ˆ θ )= α Tr( ρ θ E ˆ θ ) + (1 − α ) Tr( ρ θ E ˆ θ )= αp ( ˆ θ ) + (1 − α ) p ( ˆ θ ) . (B9)The linearity of p ( ˆ θ ) translates to the linearity of Σ( θ ),i.e. Σ ij ( θ ) = α Σ ij ( θ ) + (1 − α )Σ ij ( θ ) . (B10)This means that the whole argument of the inf sup inEq. (B7) is linear in the POVM, and it is additionallyupper semicontinuous in G at fixed M N (In Lemma 1 weproved continuity, which implies upper semicontinuity).We have now proved all the required hypothesis for theMinimax theorem of Kneser [55, 56], which allows us towrite r N ( θ ) = sup G ≥ inf M N ∈M (LU) N N Tr (cid:2) G · Σ ( N ) ( θ ) (cid:3) Tr [ G · F ( θ ) − ] . (B11)It is worth stressing that without such argument thequantity r ( ↓ ) N ( θ ) := sup G ≥ inf M N ∈M (LU) N N Tr [ G · Σ ( N ) ( θ ) ] Tr[ G · F ( θ ) − ] is by construction always smaller than or equal to r ( ↑ ) N ( θ ) := inf M N ∈M (LU) N sup G ≥ N Tr [ G · Σ ( N ) ( θ ) ] Tr[ G · F ( θ ) − ] , i.e. r ( ↓ ) N ( θ ) ≤ r ( ↑ ) N ( θ ). Lemma 3 . r N ( θ ) ≥ r N N ( θ ) . (B12)Let N = N N with N , N integers. Given N copiesof the probe we can organize them into N distinct sub-groups, each of them containing N probes. We nowperform exactly the same measurement M N on eachgroup and use the N outcomes to estimate θ by tak-ing their arithmetic mean. Calling this measurementM (cid:63)N it follows that its MSE matrix Σ (cid:63)N ( θ ) correspondsto Σ N ( θ ) /N , being Σ N ( θ ) the MSE matrix of M N .This holds true because the estimators are unbiased at θ . Therefore we have r N ( G, M (cid:63)N , θ ) = N Tr [ G · Σ (cid:63)N ( θ )]Tr [ G · F ( θ ) − ] (B13)= N Tr [ G · Σ N ( θ )] N Tr [ G · F ( θ ) − ] (B14)= N Tr [ G · Σ N ( θ )]Tr [ G · F ( θ ) − ] (B15)= r N ( G, M N , θ ) . (B16)We now need to introduce a new quantity: r N ( G, θ ) := inf M N ∈M (LU) N r N ( G, M N , θ ) , (B17)where the supremum on G ≥ εN such that r N ( G, M εN , θ ) ≤ inf M N ∈M (LU) N r N ( G, M , θ ) + ε , with ε >
0. Then wehave r N ( G, θ ) ≤ r N ( G, M (cid:63)N , θ ) = r N ( G, M εN , θ ) ≤ inf M N ∈M (LU) N r N ( G, M N , θ ) + ε = r N ( G, θ ) + ε . where r N ( G, θ ) has been defined in Eq. (B17). Becauseof the arbitrariness of ε we have r N ( G, θ ) ≤ r N ( G, θ ).Taking sup G ≥ on both the members of this inequalitygives finally the thesis. Lemma 4. r ( θ ) = inf N ≥ r N ( θ ) = lim n →∞ min N ≤ n r N ( θ ) . (B18)Recall the definition of r ( θ ) in Eq. (17), it can be ex-pressed as r ( θ ) = lim n →∞ r n ( θ ), where r n ( θ ) := inf N ≥ n r N ( θ ) , (B19)which is by construction non-decreasing in n , i.e. r m ( θ ) ≥ r n ( θ ) ∀ m ≥ n . (B20)4Our goal is to show that due to Lemma 3 , the inequalityin the above expression is always saturated, or equiva-lently that r m ( θ ) = r ( θ ) = inf N ≥ r N ( θ ) ∀ m ≥ , (B21)which will lead automatically to r ( θ ) = r ( θ ) =inf N ≥ r N ( θ ). We can prove Eq. (B21) by contradiction:assume that there exists m such that r m ( θ ) > r ( θ ).This implies that there must exist k < m integer suchthat r ( θ ) = r k ( θ ) < r m ( θ ) . (B22)This however can’t be true because thanks to Lemma 3 we must have r k ( θ ) ≥ r mk ( θ ) ≥ inf N ≥ m r N ( θ ) = r m ( θ ) . (B23)For the second equality in Eq. (B18) it is easy to noticethat the sequence defined by r ( < ) n ( θ ) := min N ≤ n r N ( θ ) , (B24)is explicitly decreasing, i.e. r ( < ) n +1 ( θ ) ≤ r ( < ) n ( θ ) , (B25)therefore its limit exists and it is easy to show it coincideswith inf N ≥ r N ( θ ) as we see in the following. Take ε > ∃ N ε such that r N ε ( θ ) ≤ inf N ≥ r N ( θ ) + ε , (B26)therefore ∀ ε , ∃ n ε := N ε such that r ( < ) n ε ( θ ) ≤ r N ε ( θ ) ≤ inf N ≥ r N ( θ ) + ε , (B27)furthermore r ( < ) n ( θ ) ≥ inf N ≥ r N ( θ ) , (B28)because in r ( < ) n ( θ ) the domain of minimization is smaller.Together Eq. (B27) and Eq. (B28) meanlim n →∞ r ( < ) n ( θ ) = inf N ≥ r N ( θ ) = r ( θ ) . (B29)Before proceeding further let us make yet another obser-vation: the above construction can be applied even in theabsence of the optimization over G . Specifically, for all G ≥ N ∈ N , we define the the quantities r n ( G, θ ) := inf N ≥ n r N ( G, θ ) , (B30) r ( < ) n ( G, θ ) := min N ≤ n r N ( G, θ ) , (B31)with r N ( G, θ ) defined in Eq. (B17). By construction, forall given G , r n ( G, θ ) is explicitly non-decreasing, while r ( < ) n ( G, θ ) is explicitly non-increasing, i.e. r n +1 ( G, θ ) ≥ r n ( G, θ ) , (B32) r ( < ) n +1 ( G, θ ) ≤ r ( < ) n ( G, θ ) . (B33) Moreover following the same arguments presented in Lemma 4 it turns out that r n ( G, θ ) is indeed constant,i.e. r n ( G, θ ) = r ( G, θ ) ∀ n . (B34)and r ( G, θ ) = inf N ≥ r N ( G, θ ) (B35)= lim n →∞ min N ≤ n r N ( G, θ ) (B36)= lim n →∞ r ( < ) n ( G, θ ) . (B37) Lemma 5 . The function r N ( G, θ ) is upper semicon-tinuous in G ∈ G ∀ N .The function r N ( G, M N , θ ), defined in Eq. (15),is continuous in G ∈ G for fixed M N because of Lemma 1 , and in particular upper semicontinuous.The function r N ( G, θ ) defined in Eq. (B17) is uppersemicontinuous in G ∈ G because the infimum ofa family of upper semicontinuous functions (here la-beled by the measurement M N ) is upper semicontinuous.Consider next the supremum over G of r ( G, θ ), thiscan be evaluated assup G ≥ r ( G, θ ) = sup G ≥ inf N ≥ r N ( G, θ ) (B38)= sup G ≥ inf N ≥ inf M N ∈M (LU) N r N ( G, M N , θ )= inf N ≥ inf M N ∈M (LU) N sup G ≥ r N ( G, M N , θ )= inf N ≥ r N ( θ ) = r ( θ ) = r ( θ ) . (B39)Going from Eq. (B38) to Eq. (B39) requires the applica-tion of two different versions of the Minimax theorem.First of all we need to commute inf N ≥ and sup G ≥ (sup G ). This can be accomplished with the Ky Fan Min-imax theorem [56, 74]. In order to use this result it mustbe proved that r N ( G, θ ) = inf M N ∈M (LU) N r N ( G, M N , θ ) isKy Fan concave-convex on G × N . This condition isequivalent to having Ky Fan concavity in G for everyfixed N and Ky Fan convexity in N for every fixed G .Let us fix an arbitrary N ∈ N . Given the combination G α := αG + (1 − α ) G with α ∈ [0 , ∀ ε ∃ M εN suchthat r N ( G α , θ ) = inf M N ∈M (LU) N r N ( G α , M N , θ ) (B40) ≥ r N ( G α , M εN , θ ) − ε . (B41)By expanding G α in r N ( G α , M εN , θ ) we have r N ( G α , M εN , θ ) = α r N ( G , M εN , θ )+ (1 − α ) r N ( G , M εN , θ ) , (B42)5which thanks to the definition of inf M N ∈M (LU) N becomes r N ( G α , M εN , θ ) ≥ α inf M N ∈M (LU) N r N ( G , M N , θ )+ (1 − α ) inf M N ∈M (LU) N r N ( G , M N , θ ) , (B43)finally, substituting r N ( G, θ ), we get r N ( G α , M εN , θ ) ≥ αr N ( G , θ ) + (1 − α ) r N ( G , θ ) . (B44)Putting together Eq. (B41) and Eq. (B44) gives r N ( G α , θ ) ≥ αr N ( G , θ ) + (1 − α ) r N ( G , θ ) − ε . (B45)which for (cid:15) → r N ( G, θ ). Let’s now prove the Ky Fan convexity in N .Consider N , N ∈ N and an arbitrary G ∈ G , we have r N N ( G, θ ) ≤ α r N ( G, θ ) + (1 − α ) r N ( G, θ ) ∀ α ∈ [0 , . (B46)This is true because thanks to Lemma 3 we have r N N ( G, θ ) ≤ r N ( G, θ ) and r N N ( G, θ ) ≤ r N ( G, θ ). Lemma 5 proves that r N ( G, θ ) is upper semicontinuousin G for every fixed N , this concludes the hypothesischeck for the application of the Ky Fan Minimax theo-rem, according to whichsup G inf N ≥ r N ( G, θ ) = inf N ≥ sup G r N ( G, θ ) . (B47)In order to get Eq. (B39) from Eq. (B38) we still needsup G inf M N ∈M (LU) N r N ( G, M N , θ )= inf M N ∈M (LU) N sup G r N ( G, M N , θ ) . (B48)This is the content of Lemma 2 . By putting togetherEq. (B37) and Eq. (B39) we get the expression r ( θ ) = sup G ≥ lim n →∞ r ( < ) n ( G, θ ) , (B49)which concludes our proof. Appendix C: Explicit computation of r (cid:63) In this section we prove that sup G ≥ in the definition of r (cid:63) ( θ ) in Eq. (41) can be computed exactly and we obtainthe explicit expression for r (cid:63) ( θ ) in Eq. (42). First of allwe define A (cid:48) ( θ ) := F ( θ ) − A ( θ ) F ( θ ) − . This means wecan write Eq. (41) as r (cid:63) ( θ ) − G ∈G Tr Abs [ G · A (cid:48) ( θ )]= sup G ∈G Tr |√ GA (cid:48) ( θ ) √ G | = sup G ∈G Tr (cid:20)(cid:113) √ G ( − A (cid:48) ( θ ) GA (cid:48) ( θ )) √ G (cid:21) , (C1) with the sup taken on G . Because A ( θ ) † = − A ( θ ) itholds that − A ( θ ) (cid:48) GA ( θ ) (cid:48) ≥
0. The trace in Eq. (C1)can be associated to the definition of the (squared) fi-delity between the states identified by the matrices G and − A ( θ ) (cid:48) GA ( θ ) (cid:48) . Notice that this last state must benormalized. Therefore we write r (cid:63) − G ∈G √F ( G, − A ( θ ) (cid:48) GA ( θ ) (cid:48) ) (C2)= (cid:112) Tr [ − A ( θ ) (cid:48) GA ( θ ) (cid:48) ] · (C3) · sup G ∈G √F (cid:18) G, − A ( θ ) (cid:48) GA ( θ ) (cid:48) Tr [ − A ( θ ) (cid:48) GA ( θ ) (cid:48) ] (cid:19) . We will prove that there is a choice of G that gives boththe maximum of Tr [ − A ( θ ) (cid:48) GA ( θ ) (cid:48) ] and of the squaredfidelity. Let’s write A ( θ ) (cid:48) in the form A ( θ ) (cid:48) = QM Q T where M is a block diagonal matrix having 2 × M i := (cid:18) λ i − λ i (cid:19) , with 0 ≤ λ i +1 ≤ λ i ∈ R . If A (cid:48) ( θ ) is ofodd size the matrix M has the last row and column fullof 0. We have Tr [ − A ( θ ) (cid:48) GA ( θ ) (cid:48) ] = Tr [ − M (cid:101) GM ], with (cid:101) G = Q T GQ , which explicitly readsTr [ − M (cid:101) GM ] = λ (cid:16) (cid:101) G + (cid:101) G (cid:17) + λ (cid:16) (cid:101) G + (cid:101) G (cid:17) + λ (cid:16) (cid:101) G + (cid:101) G (cid:17) + · · · . (C4)The maximum of the above expression is λ , realized for a (cid:101) G having (cid:101) G + (cid:101) G = 1 and all the other matrix elementsnull. Notice that (cid:107) F − A ( θ ) F − (cid:107) = (cid:107) A ( θ ) (cid:48) (cid:107) = λ . Forthe square fidelity to reach its maximum ( √F = 1) itmust be G = − A ( θ ) (cid:48) GA ( θ ) (cid:48) Tr [ − A ( θ ) (cid:48) GA ( θ ) (cid:48) ] , (C5)this is realized for (cid:101) G = (cid:101) G = . Therefore we havebuild implicitly a matrix G that saturates the sup andgives Eq. (42). Appendix D: Effective generators for θ (cid:54) = 0 Consider the single qubit encoding given in Eq. (56), inorder to compute the relevant metrological quantities, forexample the QFI and the Holevo-Cram´er-Rao bound, itis necessary to take the derivatives of this evolution, eval-uated at θ , i.e. ∂U θ ∂ θ (cid:12)(cid:12)(cid:12) θ . If we are sensing small deviationsof the phases around θ = 0, then these expressions arefairly easily computable, they are indeed ∂U θ ∂θ (cid:12)(cid:12)(cid:12) θ = iσ y ,and ∂U θ ∂θ (cid:12)(cid:12)(cid:12) θ = iσ z . But if the base point of the sensingprocess is θ (cid:54) = 0, then these derivatives became cumber-some, and can hinder the derivation of simple analyticalresults. To overcome this issue we show in this sectionthat the metrological properties of the estimation at apoint θ (cid:54) = 0 are equivalent to that of a sensing process6around zero, where the encoding has the effective gener-ators H eff1 ( θ ) and H eff2 ( θ ), which are null-trace hermitianoperators depending on the non null point θ and are ingeneral not simply σ y and σ z . We write explicitly thesmall variations δ θ from the base point θ in the encod-ing (56), i.e. U θ + δ θ := e i [( θ + δθ ) σ y +( θ + δθ ) σ z ] := e iH + iδH . (D1)The variables δ θ are now the unknown parameters, while θ is known and fixed. The Hamiltonians H := θ σ y + θ σ z and δH := δθ σ y + δθ σ z have been defined. We expandthe expression for U θ + δ θ in terms of H and δH withthe Baker-Campbell-Hausdorff formula, and keep onlythe first order terms in the infinitesimal variation δ θ ,obtaining U θ + δ θ (cid:39) U θ e iδH − [ iH,iδH ]+ [ iH, [ iH,iδH ]]+ ... . (D2)Now, the idea is to perform the rotation U − θ on the probeafter the encoding with U θ + δ θ , in such way we compen-sate for the know component of the rotation U θ + δ θ andleave only a term depending on the new unknown vari-ables δ θ . U − θ U θ + δ θ (cid:39) e iδH − [ iH,iδH ]+ [ iH, [ iH,iδH ]]+ ... (D3)= e i [ δθ H eff1 ( θ )+ δθ H eff2 ( θ ) ] . (D4)In the last expression we have collected the terms multi-plied by θ and θ respectively, which have been named H eff1 ( θ ) and H eff2 ( θ ). Notice that the commutator of twoskew-hermitian operators like iH and iδH is again skew-hermitian, this applies to all the elements of the exponen-tiated sum in Eq. (D3), and means that the right handside of Eq. (D3) is a unitary operator even if we have ne-glected higher order terms in δ θ . The exponentiated sumis either equal to iδθ H eff1 ( θ ) or to iδθ H eff2 ( θ ) when weset either δθ = 0 or δθ = 0, therefore the effective gen-erators are also hermitian operators. Consider a probe ρ codified by U θ + δ θ , i.e. ρ θ + δ θ (cid:39) U θ + δ θ ρU † θ + δ θ . All the in-formational quantities remain the same if a know unitaryis applied to the state, indeed its effects can be always ab-sorbed at the measurement stage (if the selected measure-ments set allows to do so). By choosing U − θ to be thisunitary we get U − θ U θ + δ θ ρU † θ + δ θ U †− θ = U δ θ ρU † δ θ , with U δ θ := U − θ U θ + δ θ . We observe that the traces of H eff1 ( θ )and H eff1 ( θ ) can be neglected without consequences, in-deed they contribute only to a global phase. Also if thegate U θ is used multiple times on an entangled state, sothat the encoding is U θ ⊗ U θ ⊗ · · · ⊗ U θ , the traces of thegenerators only give an irrelevant global phase. We nowfurther manipulate the encoding and look for a parame-terization in which the generators are orthogonal. Twoqubits operators H and H are said to be orthogonal if { H , H } = 0. As null-trace hermitian operators on aqubit H eff1 ( θ ) and H eff2 ( θ ) can be written H eff1 ( θ ) = α ( θ ) σ x + β ( θ ) σ y + γ ( θ ) σ z , (D5) H eff2 ( θ ) = α ( θ ) σ x + β ( θ ) σ y + γ ( θ ) σ z , (D6) with α ( θ ) := ( α ( θ ) , α ( θ ) , α ( θ )) ∈ R and β ( θ ) :=( β ( θ ) , β ( θ ) , β ( θ )) ∈ R . The orthogonality conditionis then { H eff1 ( θ ) , H eff2 ( θ ) } = 2 α ( θ ) · β ( θ ) . We can de-compose H eff2 ( θ ) in a term proportional to H eff1 ( θ ) andone orthogonal as following H eff2 ( θ ) := α ( θ ) · β ( θ ) (cid:107) α ( θ ) (cid:107) H eff1 ( θ ) + H ⊥ eff2 ( θ ) , (D7)this is the definition of H ⊥ eff2 ( θ ), which satisfies { H eff1 ( θ ) , H ⊥ eff2 ( θ ) } = 0. We define x ( θ ) := α ( θ ) · β ( θ ) (cid:107) α ( θ ) (cid:107) for ease of notation and substitute Eq. (D7) in Eq. (D4),thus getting U δ θ = e i [( δθ + x ( θ ) δθ ) H ⊥ eff1 ( θ )+ δθ H ⊥ eff2 ( θ )] , (D8)where H eff1 ( θ ) has been renamed H ⊥ eff1 ( θ ). The fi-nal step is to normalize the generators, thus defining (cid:101) H ⊥ eff1 ( θ ) := H ⊥ eff1 ( θ )Tr [ H ⊥ eff1 ( θ ) ] , and (cid:101) H ⊥ eff1 ( θ ) := H ⊥ eff2 ( θ )Tr [ H ⊥ eff2 ( θ ) ] .Going from H eff i ( θ ) to (cid:101) H ⊥ eff i ( θ ) corresponds to the fol-lowing reparametrization (cid:40) δθ (cid:48) = Tr (cid:2) H ⊥ eff2 ( θ ) (cid:3) ( δθ + x ( θ ) δθ ) ,δθ (cid:48) = Tr (cid:2) H ⊥ eff2 ( θ ) (cid:3) δθ . (D9)A rotation of the reference system can align (cid:101) H ⊥ eff i ( θ )with σ y and σ z , remember thought that the probe statemust also be transformed. Let us introduce the unitary V θ such that V θ (cid:101) H ⊥ eff1 ( θ ) V † θ = σ y and V θ (cid:101) H ⊥ eff2 ( θ ) V † θ = σ z , then V θ U δ θ V † θ = e i ( δθ (cid:48) σ y + δθ (cid:48) σ z ), while the probestate becomes V θ ρV † θ . Appendix E: Formulation of the semidefiniteprogram
We start from Eq. (37) and write r = sup G ≥ C H ( G, θ )Tr [ G · F ( θ ) − ]= sup G ∈G C H (cid:16) F ( θ ) GF ( θ ) , θ (cid:17) . (E1)The semidefinite program for C H ( G, θ ) reported in [67]is C H ( G, θ ) = minimize V ∈ S n ,X ∈ R ˜ d × n Tr [ G · V ]subject to (cid:18) V X T R † θ R θ X ˜ r (cid:19) ≥ ,X T ∂ s θ ∂ θ = n . (E2)See the work [67] for the definitions of all the objects ap-pearing in this program, it is not necessary to understand7them in order to follow our derivation. Eq. (E1) becomes r ( θ ) = sup G ∈G minimize V ∈ S n ,X ∈ R ˜ d × n Tr [ F ( θ ) GF ( θ ) · V ]subject to (cid:18) V X T R † θ R θ X ˜ r (cid:19) ≥ ,X T ∂ s θ ∂ θ = n . (E3)The objective function Tr [ F ( θ ) GF ( θ ) · V ] is linearand continuous in both G and V . The domain of the supand the min are both convex, with G being compact. Wecan therefore apply again the Mimimax theorem of [55]as done in Appendix B. Having sup G ∈G as the innermostoperation we can solve it and writesup G ∈G Tr [ F ( θ ) GF ( θ ) · V ] = (cid:107) F ( θ ) V F ( θ ) (cid:107) . (E4)Now the objective of the minimization is the spectralnorm of F ( θ ) V F ( θ ). We can introduce a dummy vari-able t and write the program as r ( θ ) = minimize V ∈ S n ,X ∈ R ˜ d × n t subject to (cid:107) F ( θ ) V F ( θ ) (cid:107) ≤ t, (cid:18) V X T R † θ R θ X ˜ r (cid:19) ≥ ,X T ∂ s θ ∂ θ = n . (E5)The condition on (cid:107) F ( θ ) V F ( θ ) (cid:107) can be written as λ max ( F ( θ ) V F ( θ ) V F ( θ ) ) ≤ t = ⇒ F ( θ ) V F ( θ ) V F ( θ ) ≤ t = ⇒ t − F ( θ ) V F ( θ ) ( t ) − F ( θ ) V F ( θ ) ≥ . Because of the Schur complement condition for the posi-tive semidefinite matrices [75] the optimization becomes r ( θ ) = minimize V ∈ S n ,X ∈ R ˜ d × n t subject to (cid:18) t F ( θ ) V F ( θ ) F ( θ ) V F ( θ ) t (cid:19) ≥ , (cid:18) V X T R † θ R θ X ˜ r (cid:19) ≥ ,X T ∂ s θ ∂ θ = n , (E6)From which we compute I ( θ ) according to Eq. (38). Thissemidefinite program is solved by means of the modelingsystem CVX developed on Matlab [76]. Appendix F: Relation between r ( θ ) and r ( θ ) In this section we present an alternative proof that r ( θ ) ≤ r ( θ ) without asking any condition on ρ θ , i.e.without using (36). Consider the expression for r ( θ )given in Eq. (B18) ( Lemma 4 , Appendix B), then ∀ n ∃ N (cid:63) such that r N (cid:63) ( θ ) = min N ≤ n r N ( θ ), with N (cid:63) ( n ) ≤ n dependent on n . We define the n th element ofthe sequence { ˜ r n ( θ ) } n ∈ N as ˜ r n ( θ ) := r N (cid:63) ( θ ). Eq. (B18)tells us that lim n →∞ ˜ r n ( θ ) = r ( θ ) . (F1)Let us fix ε >
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