Generalized-Mean Cramér-Rao Bounds for Multiparameter

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Generalized-mean Cram´er-RaoBounds for Multiparameter Quantum Metrology

Xiao-Ming Lu,1, 2, ∗ Zhihao Ma,3, † and Chengjie Zhang4, ‡ 1Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China 2Key Laboratory of Quantum Optics, Chinese Academy of Sciences, Shanghai 200800, China 3School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China 4School of Physical Science and Technology, Soochow University, Suzhou, 215006, China In multiparameter quantum metrology, the weighted-arithmetic-mean error of estimation is often used as a scalar cost function to be minimized during design optimization. However, other types of mean error can reveal different facets of permissible error combinations. By defining the weighted f-mean of estimation error and quantum Fisher information, we derive various quantum Cramér- Rao bounds on mean error in a very general form and also give their refined versions with complex quantum Fisher information matrices. We show that the geometric- and harmonic-mean quantum Cramér-Raobounds can help to reveal more forbidden region of estimation error for a complex signal in coherent light accompanied by thermal background than just using the ordinary arithmetic-mean version. Moreover, we show that the f-mean quantum Fisher information can be considered as information-theoretic quantities and is useful in quantifying asymmetry and coherence as quantum resources.

I. INTRODUCTION to geometric-mean error, has been widely adopted to for- mulate uncertainty relations for observables that are in- The random nature of quantum measurement imposes compatible in quantum mechanics [16, 20–26]. In fact, fundamental limits to estimation error of unknown pa- different types of mean error can manifest different facets rameters in quantum systems. To reveal these funda- of permissible error combinations for multiparameter es- mental limits, a variety of lower bounds on estimation timation. This motives us to generalize the arithmetic- error have been developed [1–8]. As the most popular er- mean estimation errors and study their fundamental lim- ror bound, the quantum Cramér-Raobound (QCRB) on its imposed by the random nature of quantum measure- unbiased estimator with any quantum measurement has ment. To do this, we first define the f-mean estimation been widely used in quantum metrology [9–13]. For mul- errors, which includes the ordinary arithmetic-mean er- tiparameter estimation, the QCRB is given as a matrix ror, geometric-mean error, and harmonic-mean error as inequality that restricts the possible error-covariance ma- special cases. We show that each f-mean estimation error trix of any unbiased estimation strategy, by the inverse of is bounded from below by an f-mean version of QCRB quantum Fisher information (QFI) matrix [1–3]. Unfor- with a corresponding f-mean QFI. We also give a re- tunately, this multiparameter QCRB cannot be in gen- fined f-mean QCRB, which is tighter than the f-mean eral saturated [1–3, 14, 15], meaning that there might not QCRB with the complex QFI matrix defined by Yuen exist an optimal measurement simultaneously minimizing and Lax [3]. Furthermore, we show that the f-mean the estimation errors of all parameter of interest. Due to QFIs have monotonicity under quantum operations and Heisenberg’s uncertainty principle [16], there would be a thus can be considered as information-theoretic quanti- trade-off between individual estimation errors, when the ties. We demonstrate that they are useful in quantum optimal measurements for different parameters are not resource theory, e.g., in quantifying asymmetry [27] and compatible in quantum mechanics. Since simultaneously coherence [28–34] as quantum resources. minimizing all estimation errors of individual parameters is generally infeasible, a scalar error is in practice de- manded as a cost function for optimization design. The II. QUANTUM CRAMER-RAO´ BOUND arXiv:1910.06035v3 [quant-ph] 5 Feb 2020 weighted-arithmetic-mean error is the most commonly used mean error in many previous applications of the Let us first introduce the QCRB and QFI, which play QCRB as well as other stronger bounds like the Holevo a pivot role in quantum parameter estimation [1,2]. The bound [1–3,5, 17–19]. task considered here is to estimate an unknown vector T Despite the usefulness of weighted-arithmetic-mean er- parameter θ := (θ1, θ2, . . . , θn) from observations on a ror, many other types of mean error exist and there are quantum system, where T denotes matrix transposition. no hard-and-fast rules for which mean should be used. The state of the quantum system is described by a density For example, the product of errors, which is equivalent operator %θ, which depends on the value of θ. A quantum measurement can be described by a positive-operator- P valued measure {My|My ≥ 0, y My = 11} with y denot- ing measurement outcomes and 11 the identity operator. ∗ [email protected]; http://xmlu.me Denote the estimator for the j-th unknown parameter θj † ˜ [email protected] by θj, which is a map from measurement outcomes y to ‡ [email protected] estimates for θj. The estimation error of multiple param- 2 eters can be characterized by the error-covariance matrix form of the classical weighted f-mean [38] of the eigen- E defined by its entries values xj of X as

Z −1 ˜ ˜ Mf,G(X) = f ( [f(xj)]), (3.2) Ejk := θj(y) − θj θk(y) − θk p(y|θ)dy, (2.1) E

where the expectation E is taken regarding xj with the where p(y|θ) := Tr My%θ with Tr being trace operation is probabilities pj = Tr GPj and Pj is the eigen-projection the conditional probability of obtaining a measurement of X corresponding to the eigenvalue xj. outcome y for a given true value of θ. For any unbiased The weighted f-mean error of estimation is given by estimator and any quantum measurement, the estimation Mf,G(E). For simplicity, we will use f-mean error to de- error obeys the following QCRB [1,3]: note both the weighted and unweighted versions. This f-mean error includes as special cases the arithmetic, ge- E ≥ F −1, (2.2) ometric and harmonic mean error, which we will discuss in detail later. For the case of single parameter estima- where F is the so-called QFI matrix [35]. Note that the tion (n = 1), Mf,G(E) is always reduced to the ordi- matrix inequality Eq. (2.2) means that E−F −1 is positive nary mean-square error, no matter what the function f semi-definite. is taken to be. There exist two versions of QFI matrix in the QCRB. We now derive the generalized QCRBs on the f-mean The first one is based on the symmetric logarithmic error. Assuming that the function f is either an opera- derivative (SLD) operator [36, 37] and the second one tor monotone or anti-monotone [39], we give the f-mean is based on the right logarithmic (RLD) operator [3]. QCRB as follows (see AppendixA for a detailed proof): The SLD-based QFI matrix FS is defined by [FS]jk := Re Tr LjLk%θ, where Re denotes the real part and Lj, 1 Mf,G(E) ≥ (3.3) the SLD operator for θj, is a Hermitian operator satis- Mf◦ζ,G(F ) fying ∂%θ/∂θj = (Lj%θ + %θLj)/2. The RLD-based QFI † where ζ : x 7→ 1/x is the reciprocal function. Note that matrix FR is defined by [FR] := Tr R %θRk, where Rj, jk j a real-valued continuous function f is called operator the RLD operator for θj, satisfies ∂%θ/∂θj = %θRj. The SLD-based QFI matrix is real symmetric while the RLD- monotone if f(A) ≥ f(B) always holds, and is called based QFI matrix is in general Hermitian. operator anti-monotone if f(A) ≤ f(B) always holds, whenever the two Hermitian operators A and B satisfy As the diagonal elements of E—estimation errors of in- A ≥ B ≥ 0. Furthermore, assuming that the weighted dividual parameters—might not be simultaneously min- f mean has homogeneity, i.e., M (tX) = t M (X) imized, one often use the weighted-mean error Tr GE as f,G f,G holds for any positive number t and any positive matrix the cost function to be minimized for optimizing quantum X, we can get a classical scaling E ≥ 1/ν M (F ) with estimation strategies, where the given weight matrix G f◦ζ,G respect to the number ν of repetition of the experiment, is real-symmetric and positive. It is easy to see that the due to the additivity of the QFI matrix. weighted-mean error is bounded as Tr GE ≥ Tr GF −1, according to the QCRB. To establish concrete f-mean QCRBs with the classical scaling 1/ν, we need to find the operator monotone or anti-monotone functions that result in homogeneous f means. The reader is directed to Hiai and Petz [39, III. GENERALIZED-MEAN QCRB Chapter 4] for discussions on operator monotone functions and to Hardy, Littlewood, and Pólya [38, Chapter To generalize the mean error of estimation, we first III] for discussions on the homogeneity of the f mean. define the weighted f-mean for a positive matrix X as In short, the functions x 7→ xs for s ∈ [−1, 1] \{0} and x 7→ ln(x) are either operator monotone or anti- −1 Mf,G(X) := f (Tr Gf(X)) , (3.1) monotone (see AppendixB) and give homogeneous f- means (see AppendixC); Therefore, they are qualified where f is a real-valued, continuous, and strictly mono- for the f-mean QCRBs. By a little abuse of notation, tonic function on the interval (0, +∞) and the weight ma- we adopt the convention of the generalized mean [38] trix G is real-symmetric and positive semi-definite. Note to denote by Ms,G the weighted generalized mean for s that whenever f is applied on a positive matrix X, it f : x 7→ x with s ∈ [−1, 1]\{0} and specifically set M0,G † means that f(X) = U diag{f(x1), f(x2), . . . , f(xn)} U , to the case of f : x 7→ ln(x) as lims→0 Ms,G = M0,G. † where U is a unitary matrix diagonalizing X as U XU = Also, we will use Ms for the corresponding unweighted f diag{x1, x2, . . . , xn}. Without loss of generality, we means. With this notation, the generalized QCRB reads henceforth set the weight matrix G to be normalized, i.e., −1 −1 Tr G = 1. The unweighted f-mean is given by substitut- Ms,G(E) ≥ ν M−s,G(F ) (3.4) ing G = In/n into Eq. (3.1) and will be simply denoted by Mf (X), where In is the n × n identity matrix. It is with s ∈ [−1, 1]. The generalized QCRBs in Eq. (3.3) worthy to mention that Mf,G(E) can be written in the and Eq. (3.4) are our first main result. 3

We use Holevo’s approach [2] to reﬁne the lower bound TABLE I. Three primary instances of the unweighted f-mean on the f-mean estimation error when the error-covariance errors Ms(E) and their reciprocal mean-QFIs M−s(F ). Here, matrix E is known to be bounded by a Hermitian matrix n is the number of parameters to be estimated. (i.e., the RLD-based QFI matrix). To do this, ﬁrst note s f(x) Ms(E) M−s(F ) that for a real-symmetric matrix A and a Hermitian ma- 1 x Tr E/n n/ Tr F −1 trix B satisfying A ≥ B, it holds that [2, Chapter 6] 0 ln x (det E)1/n (det F )1/n −1 −1 1/x n/ Tr E Tr F/n Tr A ≥ Tr Re B + kIm Bk1, (3.6) √ † where kOk1 := Tr O O is the Schatten 1-norm of an op- Arithmetic Mean Geometric Mean Harmonic Mean erator O. Suppose that f is an operator monotone func- 10 8.0 8.0 8.0 tion. Then, it follows from the ordinary QCRB Eq. (2.2)

6.0 6.0 6.0 and the non-negativity of the weight matrix G that 2

E 5 4.0 4.0 4.0 √ √ √ √ −1 2.0 2.0 2.0 Gf(E) G ≥ Gf(F ) G. (3.7) √ √ √ √ 0 Substituting A = Gf(E) G and B = Gf(F −1) G 0 5 10 0 5 10 0 5 10 into Eq. (3.6) and then applying f −1, we get 1 1 1 E E E Mf,G(E) ≥ Rf,G(F ) := √ √ FIG. 1. The (unweighted) arithmetic, geometric, and har- −1 −1 −1 f Tr G Re f F + k G Im f(F ) Gk1 . (3.8) monic means as functions of individual errors. Here, E1 and E2 are the eigenvalues of the error-covariance matrix E. It is easy to see that the above inequality still holds when f is an operator anti-monotone function. We here briefly discuss the f-mean estimation errors For the concrete f functions considered in this work, and their properties. Since the f-means have the classical i.e., f : x 7→ xs with s ∈ [−1, 1] \{0} and f : x 7→ ln x, representation as Eq. (3.2), they inherit the comparabil- the above-mentioned refined lower bound also has the ity [38, see Theorem 16], namely, classical scaling ν−1 with the number ν of repetitions of the experiments. To see this, note that when the ex- Mr,G ≤ Ms,G for − 1 ≤ r ≤ s ≤ 1. (3.5) periment was repeated ν times, the QFI matrix is given This comparability is a property of the f-means them- by νF due to the additivity of the QFI matrix. Substi- −1 −1 −s −s s selves and thus can be applied to both the f-mean esti- tuting f(ν F ) = ν F for the case of f(x) = x and ln(ν−1F −1) = (ln ν−1)I + ln(F −1) for the case of mation errors Ms,G(E) and the f-mean QFIs Ms,G(F ). Three primary instances of the generalized means are f(x) = ln x into Eq. (3.8), we obtain the weighted arithmetic, geometric, and harmonic means, −1 Mf,G(E) ≥ ν Rf,G(F ). (3.9) which are M1,G, M0,G, and M−1,G, respectively. We list in Tab.I the corresponding unweighted versions of The refined bound in Eqs. (3.8) and (3.9) with the f-mean estimation errors and the f-mean QFIs giving RLD-based QFI matrix is the second main result of this lower bounds on the estimation errors. The difference work. This bound is tighter than the f-mean QCRB between these three f-mean errors becomes obvious in Eq. (3.3) with the RLD-based QFI matrix, because the the regions where the eigenvalues of the error-covariance √ √ term k G Im f(F −1) Gk is nonnegative and will be re- matrix have a large fluctuation, e.g., one of the eigenval- 1 duced to Eq. (3.4) for Hermitian QFI matrices. ues is very small while the others are considerably large, as shown in Fig.1. The f-mean QCRBs in Eqs. (3.3) and (3.4) holds for IV. APPLICATION TO THE ESTIMATION OF both the SLD- and RLD-based QFI matrices. Never- A COHERENT SIGNAL theless, only the real part of f(F −1) is relevant to the f-mean QFI. To see this, note that f(F −1) is Her- mitian so that its imaginary part Im f(F −1) is anti- Now, let us consider the estimation of a complex coher- symmetric. Since the trace of the product of a symmetric ent signal µ accompanied by thermal background light. matrix and an anti-symmetric matrix must vanish, we get Following Helstrom [1], the parametric family of density Tr G Im f(F −1) = 0. As shown by Yuen and Lax [3], and operators is given by the GlauberSudarshan P represen- also by Holevo [2], via some elaborate mathematical ma- tation nipulations, the imaginary part of the RLD-based QFI 1 Z % = exp −|α − µ|2/η |αihα| d2α, (4.1) matrix in fact can be used to establish a tighter bound θ πη on the weighted-arithmetic-mean estimation error than the ordination QCRB. We shall generalize these result where η is the mean number of photons induced by the to the f-mean estimation error in what follows. background, |αi is a coherent state, and µ is a complex 4 number. Let us take the real and imaginary parts of µ η = 0 η = 1.5 as the parameters θ1 and θ2 to be estimated, i.e., 3 5 s = 1 s = 1 θ1 = Re µ and θ2 = Im µ. (4.2) s = 0 4 s = 0 2 s = 1 s = 1 The QFI matrices based on SLD and RLD have already − 3 − 2 been given in Ref. [1] and Ref. [3], respectively, that is, E 2 1 4 1 0 F = , (4.3) 1 S 2η + 1 0 1 1 2η + 1 −i 0 0 F = . (4.4) 0 1 2 3 0 1 2 3 4 5 R η(η + 1) i 2η + 1 1 1 E E We now calculate the f-mean versions of QCRB. The unweighted f-mean QFI only depends on the eigenvalues FIG. 2. Permissible combinations of the eigen-errors of es- of the QFI matrix, which are {4/(1 + 2η), 4/(1 + 2η)} for timating a complex coherent signal accompanied by thermal E E FS and {2/(1 + η), 2/η} for FR. Since the eigenvalues background light. Here 1 and 2 are the eigenvalues of the error-covariance matrix and η is the mean number of pho- of FS are the same, the unweighted f-mean SLD-based QFIs tons induced by the background. Only the regions above the curves are permissible by the corresponding unweighted f- 4 mean QCRBs with different values of s. The grey color de- M (F ) = ∀s ∈ [−1, 1]. (4.5) notes the forbidden region optimized over s. s S 1 + 2η

For the RLD-based QFI, the f-mean QFI is given by According to the refined bound Eq. (3.8) with G = I2/2, the unweighted f-mean error is then bounded as 1 2 s 1 2 s1/s Ms(FR) = + (4.6) 1/s 2 1 + η 2 η 1 s 1 s −1 −1 Ms(E) ≥ Tr (Re FR ) + (Im FR ) 2 2 1 p for s ∈ [−1, 1] \{0} and M0(FR) = 2/ η(η + 1). Due s s 1/s 1η s 1 η + 1 1 η s η + 1 to the comparability Eq. (3.5) of the generalized means, = + + − we have Ms(FR) ≥ M−1(FR). Moreover, it is easy to 2 2 2 2 2 2 2 ( show that M−1(FR) equals to Ms(FS), where the latter η , −1 ≤ s < 0 is in fact independent of s. Therefore, we get M (F ) ≥ = 2 (4.11) s R η+1 , 0 < s ≤ 1. Ms(FS) for any s ∈ [−1, 1], meaning that the f-mean 2 QCRB Eq. (3.4) with the SLD gives the tighter bound For the case of s = 0, we straightforwardly calculate the for this case than that with the RLD. refined bound as follows. Due to the eigenvalue decom- Next, we calculate the refined f-mean QCRB with position Eq. (4.7), it follows that RLD. Note that the inverse of the RLD-based QFI matrix has the following eigenvalue decomposition: η I + σ η + 1 I − σ ln f(F −1) = ln 2 2 + ln 2 2 R 2 2 2 2 −1 η I2 + σ2 η + 1 I2 − σ2 FR = + , (4.7) I η(η + 1) σ η 2 2 2 2 = 2 ln + 2 ln , (4.12) 2 4 2 η + 1 0 −i where σ = is the Pauli-y matrix. Since (I ± 2 i 0 2 from which we get M0(E) ≥ (η + 1)/2. In this example, we can see that the refined RLD-based σ2)/2 are projections, we get f-mean QCRB is tighter than the SLD-based one, when s s 0 ≤ s ≤ 1, and looser when −1 ≤ s < 0. In short, the s η I2 + σ2 η + 1 I2 − σ2 (F −1) = + . (4.8) f-mean error of estimating a complex coherent signal ac- R 2 2 2 2 companied by thermal background light is bounded from below by By noting that the matrices I2 and σ2 are purely real and imaginary, respectively, we get ( 2η+1 4 , −1 ≤ s < 0 s Ms(E) ≥ η+1 (4.13) s 1 η s η + 1 , 0 ≤ s ≤ 1. Re (F −1) = + I , (4.9) 2 R 2 2 2 2 We plot in Fig.2 the permissible combinations of s s −1 s 1 η η + 1 the eigen-errors—the eigenvalues of the error-covariance i Im (FR ) = − σ2. (4.10) 2 2 2 matrix—through the f-mean QCRBs derived in this 5 work. It can be seen from Fig.2 that the geometric-mean According to the definition in Eq. (5.1), this is equivalent and harmonic-mean QCRBs can reveal more forbidden to region than the ordinary arithmetic-mean QCRB, when F (Φ(% )) ≤ F (% ), (5.3) one of the eigen-errors is small and the other is consider- f,G θ f,G θ ably large. which is the monotonicity of the f-mean QFI. To reveal more forbidden regions of error combinations, The monotonicity under quantum operations is an we can optimize over the f functions (i.e., s ∈ [−1, 1] in important property of information-theoretic quantities; the above example) and the weight matrices G. In fact, This makes the f mean QFIs useful in quantum resource the optimization with respect to s can be substantially theory [31, 33, 40–43]. The essentials of quantum re- simplified when the lower bounds on f-mean estimation source theory are free states and free operations. A re- error are independent of s within an interval of s, due source measure is a function of quantum states that is to the comparability of the f-mean error as shown in monotonically non-increasing under free operations and Eq. (3.5). Let us take the bounds given in Eq. (4.13) as vanishes for all free states [40]. We show in the following an example. When −1 ≤ s < 0, Ms(E) have the com- that the f-mean QFIs can be used as resource measures mon lower bound (2η + 1)/4. From the comparability for asymmetry [27] and coherence [29]. Eq. (3.5), it can be seen that Ms(E) ≥ M−1(E) ≥ (2η + Asymmetry measure was proposed by Marvian and 1)/4, implying that all error-covariance matrices E per- Spekkens to quantify how much a symmetry of inter- missible by the harmonic-mean QCRB must be permit- est is broken for a given quantum state [27]. Follow- ted by the other f-mean QCRBs with s ∈ (−1, 0). Anal- ing Ref. [27], the symmetry is described by a group G, ogously, it can be seen that the geometric-mean QCRB the free states are taken to be the symmetric states that is tighter than other f-mean QCRBs with s ∈ (0, 1]. are invariant under the action of all group elements in Therefore, in Fig.2 the union of the forbidden regions by G, and the free operations are taken to be the symmet- † the harmonic-mean QCRB and that by the geometric- ric quantum operations Φ that satisfy Φ(U(g)ρU(g) ) = mean QCRB has already excluded the maximal regions U(g)Φ(ρ)U(g)† for all quantum states ρ and all g ∈ G, of eigen-error combination forbidden by Eq. (4.13). where U(g) is the unitary representation of g. Now we consider the symmetry described by a Lie group whose elements are parametrized by θ ∈ Θ ⊆ Rn. It can be † V. f-MEAN QFI AS shown that Ff,G(U(gθ)ρU(gθ) ) vanishes for symmetric INFORMATION-THEORETIC QUANTITY states and monotonically non-increasing under symmetric quantum operations due to the monotonicity of the Besides giving lower bounds on the f mean error for f-mean QFI. Thus, this quantity can be viewed as an quantum multiparameter estimation, the f-mean QFIs asymmetry measure. A potential application of this type can also be considered as information-theoretic quanti- is the scenario of estimating the angles of a collective ties and may have applications on quantum information SU(2) rotation on the spins of atom, e.g., see Ref. [44– field. For this purpose, let us treat the QFI matrix and 46]. the f-mean QFI as functions of the parametric density Besides, the f-mean QFIs can also be used to quantify quantum coherence [28, 29, 33, 34]. The quan- operators. We thereby denote by F (%θ) the QFI matrix of % and write tum resource theory of coherence is formulated for a θ fixed orthonormal basis {|ji}, which will be called the reference basis. The free states are incoherent states F (% ) := M (F (% )) (5.1) f,G θ f,G θ whose density matrices are diagonal with the reference basis. The definition of free operations, however, is not as the f-mean QFI of % . θ unique, leading to different frameworks of quantifying co- We here demonstrate that, like the ordinary QFI herence [33, 34]. In the seminal work by Baumgratz, matrix, the f-mean QFIs are also monotonically non- Cramer, and Plenio (BCP) [29], the free operations are increasing under quantum operation. Let Φ denote a given by the incoherent Kraus operators K , which sat- quantum operation, which is mathematically described l isfy the requirement that K ρK†/ Tr(K ρK†) are inco- by a completely positive and trace-preserving linear map l l l l herent whenever ρ is incoherent. A nonnegative func- on density operators. It is known that the QFI matrix it- tion C of ρ is said to be a coherence measure in the self has the monotonicity under quantum operations [39, BCP framework, if it vanishes only when ρ is incoher- Theorem 7.34], i.e., F (Φ(% )) ≤ F (% ). Now, suppose θ θ ent and possesses the strong monotonicity and convexity. that F (% ) and F (Φ(% )) are both non-degenerate. It θ θ The strong monotonicity means that C(ρ) ≥ P p C(ρ ) follows that F (% )−1 ≤ F (Φ(% ))−1, for the inverse func- l l l θ θ with p = Tr(K ρK†) and ρ = K ρK†/p holds for any tion is operator anti-monotone. Since f is an operator l l l l l l l set {Kl} of incoherent Kraus operators. The convexity monotone/anti-monotone function, it can be shown that P P means that l plC(ρl) ≥ C( l plρl) holds for any prob- −1 −1 −1 −1 ability distribution {pl} and density operators ρl. f Tr Gf(F (%θ) ) ≤ f Tr Gf(F (Φ(%θ)) ) . To account for superpositions among all reference basis (5.2) of a n-dimensional quantum system through the f-mean 6

QFI, we consider the following parametric density oper- According to the definition of arithmetic-mean QFI, ator we have F1(%θ) = (1/n) Tr F (%θ). For pure states, it can n n be shown that X X %θ = exp i θjZj ρ exp − i θjZj , (5.4) P P i j θj Zj −i j θj Zj j=1 j=1 F1 e |ψlihψl| e n where Zj’s are a set of n Hermitian operators that are all 4 X 2 2 = hψl|Z |ψli − hψl|Zj|ψli . (5.7) diagonal with the reference basis and satisfy Tr ZjZk = n j j=1 δjk. Such Zj’s are commuting with each other. In fact, at most n−1 independent parameters can be sensed into an Due to Lemma1, F1 is invariant under the transforma- n-dimensional quantum system by the commuting gen- tion Z → Z0 = P S Z , so we can always choose erators, for a global phase transformation does not af- j j k jk k Zj = |jihj|. Substituting Zj = |jihj| into Eq. (5.7), we fect the density operators. Consequently, the QFI ma- get the equality in Eq. (5.5). trix about n parameters must be degenerated. To make To show that C is a coherence measure in the BCP the f-mean QFI F usable in quantifying coherence, it b s framework, we resort to the work by Du et al.[48] and must require s ∈ (0, 1]. In this work, we focus on the Zhu et al.[49]: They proved that arithmetic-mean QFI F1. Intuitively, the superposition between the basis states X Cf (ρ) := min plf(µ(ψl)) (5.8) is a necessary resource for estimating the unknown pa- {p ,|ψ i} l l l rameters imprinted by Zj’s, for incoherent states are invariant under the sensing transformation exp(i P θ Z ). T j j j with µ(ψ) := (|h1|ψi|2, |h2|ψi|2,..., |hn|ψi|2) satisfies We show in what follows that the convex roof of the the strong monotonicity and convexity in the BCP frame- arithmetic-mean QFI, as defined in the following, is a work, as long as f is a real symmetric concave function. coherence measure in the BCP framework [29]: When we say f is symmetric, it means that f is invariant P P X i θj Zj −i θj Zj under any permutation of the elements of µ. It is easy to Cb(ρ) := min plF1 e j |ψlihψl| e j {pl,|ψli} see that C(ρ) is of the form Eq. (5.8) with l b n   4 X X 4 n = 1 − max pl |hj|ψli| , (5.5) 4 X 2 n {pl,|ψli} f(µ) = 1 − µj  , (5.9) l j=1 n j=1 where the minimization in the first line is taken over all ensemble decompositions {pl, |ψli} implementing ρ as which is a real-valued, symmetric, and concave function P ρ = l pl |ψlihψl| and |ji for j = 1, 2, . . . , n are the state Therefore, Cb possesses the strong monotonicity and con- vectors in the reference basis. Analogous to the nomen- vexity in the BCP framework according to Ref. [48, 49]. clature for the entanglement of formation [47], we can Moreover, f(µ) in Eq. (5.9) vanishes only when µ is a call Cb(ρ) the arithmetic-mean QFI of formation. sharp probability distribution such that one probability We first prove the equality in Eq. (5.5), for which the is unit and all others are zero. As a result, Cb vanishes following Lemma is needed. only when there exists an ensemble implementation of ρ Lemma 1. The unweighted f-mean QFI for n parame- such that all |ψli’s are in the reference basis, meaning that ρ is incoherent. We thus have proved that C(ρ) is a ters sensed by the commuting generators Zj’s is the same b as that sensed by another set of commuting generators coherence measure in the BCP framework. 0 Pn Although the convex roof involved in Cb(ρ) is difficult given by Zj = k=1 SjkZk, where S is an arbitrary n×n orthogonal matrix. to evaluate, we furthermore give an analytic result for 2-dimensional quantum systems (i.e., qubits). That is, Proof. It is known that the QFI matrix is transformed as F 7→ SFST under an orthogonal transformation 2 2 Cb(ρ) = (Tr σ1ρ) + (Tr σ2ρ) , (5.10) θ 7→ θ0 = Sθ of unknown parameters. Since the unweighted f-mean QFIs depend only on the eigenvalues of where σ1 and σ2 are the Pauli-x and -y matrices, respec- the QFI matrix, which does not change under orthogonal tively. In order to prove Eq. (5.10), we use Lemma1 to T √ √ transformations, we have Mf (SFS ) = Mf (F ). On the choose Z1 = σ3/ 2 and Z2 = 11/ 2, where σ3 is the other hand, the orthogonal transformation performed on Pauli-z matrix. We then get the unknown vector parameter can be moved to the set X 2 2 of generators, as Cb(ρ) = min pl hψl|σ3|ψli − hψl|σ3|ψli {p ,|ψ i} n n n l l l X X X θ0 Z = S θ Z = θ Z0 (5.6) 1 k k kj j k j j = F eiθσ3 ρe−iθσ3 , (5.11) k=1 j,k=1 j=1 4 0 Pn with Zj = k=1 SkjZk. We thus have proved the above where the last equality is due to the equivalence be- Lemma. tween the QFI and the convex roof of variance [50, 51]. 7

With the spectrum decomposition of the density opera- QCRB, the f-mean error of unbiased quantum estima- P tors ρθ = α λa |αihα|, it is known that the QFI can be tion is bounded from below by the inverse of a corre- given by [12] sponding reciprocal-f-mean QFI. We have also reﬁned the f-mean QCRB for complex QFI matrices given by 2(λ − λ )2 the RLD approach. Our f-mean versions of QCRB can iθσ3 −iθσ3 X α β 2 F (e ρe ) = | hα|σ3|βi | . be used in practical application for optimization prob- λα + λβ α,β|λa+λb6=0 lems in multiparameter quantum metrology. By applying (5.12) our f-mean QCRBs on the scenario of estimating a complex coherent signal accompanied by background thermal Write the density matrix in the Bloch representation P3 light, we have demonstrated that the f-mean QCRBs ρ = (11 + i=1 riσi)/2, where ri := Tr σiρ is the ith can reveal more forbidden regions of error combinations component of the Bloch vector r. The eigenvalues and than the ordinary one. We hope the method developed in eigen-projections of the density matrix are λ± = (1±r)/2 this work can help us better understand the fundamental and quantum limit of multiparameter estimation. 3 ! Moreover, we have showed that the emerged f- 1 X riσi |±ih±| = 11 ± , (5.13) mean QFIs themselves can be considered as a class of 2 |r| i=1 information-theoretic quantities. Like the ordinary QFI, the f-mean QFIs are monotonically non-increasing un- p 2 2 2 der quantum operations, which is an important property respectively, where |r| := (r1 + r2 + r3) is the length of the Bloch vector r. Substituting these eigenvalue and in quantum information theory. We have demonstrated eigenstates into Eq. (5.12), we get that the f-mean QFIs as well as its convex roof are useful for quantifying asymmetry and coherence in quan- iθσ3 −iθσ3 2 2 F (e ρe ) = 4r1 + 4r2, (5.14) tum resource theory. Considering the role of the f-mean QFI in quantum multiparameter estimation, the resource from which Eq. (5.10) immediately follows. measured in such a manner can be interpreted as being Moreover, we show that for any n-dimensional quan- valuable for the metrological purpose. tum system, Cb(ρ) is bounded as 4(n − 1) ACKNOWLEDGMENTS 0 ≤ Cb(ρ) ≤ , (5.15) n2 This work is supported by the National Natural and the upper bound is attained if ρ = |ψihψ| with √ Pn Science Foundation of China (Grant Nos. 61871162, |ψi = (1/ n) j=1 |ji up to arbitrary relative phases 11805048, 11935012 11571313, and 11734015), and the between |ji’s. The lower bound is obvious due to the Natural Science Foundation of Zhejiang Province, China non-negativity of the arithmetic-mean QFI. The up- (Grant No. LY18A050003). per bound can be obtain from Eq. (5.5) by noting that Pn 2 j=1 µj ≥ 1/n for any probability distribution {µj} with 2 µj = | hj|ψli | . Appendix A: Proof of the generalized QCRB It is worthy to mention that Yu proposed in Ref. [52] a coherence measure that is analogous to Cb given in We here prove the f-mean version of QCRB Eq. (3.3). Eq. (5.5) but using the Wigner-Yanase skew information The real-valued function f used in this work is supposed instead of the QFI and its convex roof. Yu also showed to be continuous, strictly monotonic, and either operator that reciprocal of the coherence measure thereof gives a monotone or anti-monotone. If f is operator monotone, lower bound on the harmonic-mean estimation error [52]. it follows from the ordinary QCRB that Since the Wigner-Yanase skew information is not larger E ≥ F −1 =⇒ f(E) ≥ f(F −1) than QFI [53], the reciprocal of Cb(ρ) in this work will give a tighter lower bound on the harmonic-mean estimation =⇒ Tr Gf(E) ≥ Tr Gf(F −1), (A1) error than that given by the Wigner-Yanase skew infor- where the weight matrix G is real-symmetric and positive mation. Besides, the convex roof of arithmetic-mean QFI semi-deﬁnite. Since f is continuous and strictly mono- has also been used by Kwon et al. in Ref. [54] to quantify tonic, its inverse function f −1 exists and must be mono- the non-classicality as a resource for quantum metrology. tonically increasing. Therefore,

−1 Mf,G(E) = f (Tr Gf(E)) VI. CONCLUSION 1 ≥ f −1 Tr Gf(F −1) = , (A2) M (F ) Summarizing, we have generalized the QCRB by in- f◦ζ,G troducing the concepts of f-mean estimation error and where ζ : x 7→ 1/x is the reciprocal function. On the f-mean QFI. We show that, analogous to the ordinary other hand, if f is operator anti-monotone, then f and 8 f −1 are both monotonically decreasing. Therefore, Besides, another important operator monotone function is the logarithm, for which the reader is directed to E ≥ F −1 =⇒ f(E) ≤ f(F −1) Ref. [39, Chapter 4]. =⇒ Tr Gf(E) ≤ Tr Gf(F −1) −1 −1 −1 =⇒ f (Tr Gf(E)) ≥ f Tr Gf(F ) . (A3) Appendix C: Homogeneity of the generalized means

We then still get the f-mean QCRB Eq. (3.3). Remind that an f-mean is said to be homogeneous if

−1 X −1 X f pjf(tλj) = tf pjf(λj) (C1) Appendix B: Operator monotone function j j

+ We here give some concrete instances of operator holds for any t ∈ R , λj ≥ 0, and any probability distri- s monotone or anti-monotone functions. Remind that a bution {pj}. When f(x) = x , it can be shown that function f :(a, b) → R is called operator monotone if 1/s A ≥ B always implies f(A) ≥ f(B), where A and B −1 X X s f pjf(tλj) = pj(tλj) are self-adjoint operators whose eigenvalues belongs to j j (a, b). Similarly, f is called anti-monotone if A ≥ B al- 1/s X s ways implies f(A) ≤ f(B). The LöwnerHeinzinequality = t pjλj . (C2) s s states that A ≥ B ≥ 0 implies A ≥ B for all s ∈ (0, 1]. j s Therefore, f : x 7→ x with s ∈ [0, 1] is an operator When f(x) = ln x, we have monotone on positive semi-definite matrices. The func- X X tion f : x 7→ 1/x on (0, ∞) is operator anti-monotone, f −1 p f(tλ ) = exp p ln(tλ ) as j j j j j j −1/2 −1/2 X X A ≥ B > 0 =⇒ B AB ≥ 11 = exp ln t pj + pj ln λj j j =⇒ B1/2A−1B1/2 ≤ 11 =⇒ A−1 ≤ B−1, (B1) X = t exp pj ln λj , (C3) where the second “ =⇒ ” can be seen by simulta- j neously diagonalizing B1/2A−1B1/2 and 11. Combin- P ing the Löwer-Heinze inequality with the operator anti- where we have used j pj = 1 in the last equality. There- monotonicity of f : x 7→ 1/x, we can see that f : x 7→ xs fore, we have shown that the f means for f : x 7→ xs and for s ∈ [−1, 0) are operator anti-monotones on (0, ∞). f : x 7→ ln x are homogeneous.

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