Uncertainty and Trade-offs in Quantum Multiparameter Estimation

Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In Quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cram\'er--Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state independent uncertainty relation between the parameters of a qubit system.

where σ A = (A − A ) 2 is the standard deviation of an observable A in a given state ψ ( · ≡ ψ| · |ψ ). For canonically conjugate observables such as position and momentum the right hand side of Eq. (1) equals /2. This relation implies that it is impossible to prepare a particle in a state with arbitrarily sharp statistics for both position and momentum. Note that such uncertainty relations do not tell anything about statistics of joint measurements of both observables. Rather, the standard deviations on the left hand side of Eq. (1) correspond to measurements of A and B on two independent ensembles of identical copies of the state |ψ . The preparation uncertainty relation between position and momentum is tight, as equality is achieved for specific states [5]. The relation Eq. (1) hence quantifies an attainable trade off between the sharpness of the position and momentum measurement statistics. Subsequent works formulated preparation uncertainty relations which involve other measures for the spread of a distribution [6][7][8][9].
The development of quantum measurement theory [10][11][12] allowed to formulate accuracy-disturbance uncertainty relations which quantify the disturbance caused by a positive operator valued measure (POVM) measurement to the statistics of a subsequent measurement of another POVM [13][14][15][16][17][18][19]. Joint measurement uncertainty relations have been discussed by many authors [20][21][22][23][24] and most recently in Ref. [25]. They describe the deviation of the statistics in a joint approximate measurement of two quantities from their statistics when measured separately. Many more authors have considered these two notions of uncertainty, for a more complete list see references in Refs. [18,19]. There is still debate between the proponents of the most recent approaches regarding which of them most properly captures Heisenberg's qualitative considerations [26][27][28][29][30].
Quantum parameter estimation theory provides yet another way to quantify quantum uncertainty. In this framework one considers a family of quantum states parametrized by real numbers, and the task is to estimate the parameters corresponding to a given state by performing measurements on identical copies of the state. In the one parameter case, the quantum Fisher information (QFI) Cramér-Rao bound provides a lower bound on the asymptotic scaling of the variance of an unbiased estimator [31,32]. The bound is achievable in the asymptotic limit of many copies of the state with a separable measurement [33,34]. Of particular importance is the case when the parameter to be estimated is elapsed time t for a state |ψ(t) = exp(−itH)|ψ(0) evolved with a given Hamiltonian; in that case, the quantum Cramér-Rao bound is proportional to the expectation value of the Hamiltonian [35], and hence yields the well known time-energy uncertainty relation [36]. Results of this type can be seen as hybrid preparation-measurement uncertainty Finally, we compute the Holevo Cramér-Rao bound for a three level system model and describe the structure of its trade-off surface which is generic to any d-dimensional quantum system. The paper is organized as follows. In Section I we briefly review the required background and set up our notation; in Section II we show how to obtain trade-off curves from Cramér-Rao type bounds for two parameters; in Section III we present our result for the qubit model which include a state independent trade-off surface; Section IV describes the structure of the trade-off surface of the Holevo Cramér-Rao bound for a qutrit; we conclude with a discussion in Section V.

I. PRELIMINARIES
We start by reviewing estimation theory. In classical estimation theory [40] we are given a family of probability distributions with probability density p(θ) parametrized by a vector of parameters θ = (θ 1 , . . . , θ K ). The task is to estimate the unknown values θ 0 by sampling from p(θ 0 ). In order to do so we shall pick an estimator, a function that produces an estimated valueθ(x 1 , x 2 , . . . , x N ) given the N samples drawn {x i }. The estimation statistics are then described by the random variableθ(X 1 , X 2 , . . . , X N ), where the random variables X i are distributed according to p(x i |θ 0 ) := p(X = x i |θ 0 ). An estimator is called locally unbiased if Eθ = θ 0 , where E is the expectation value is with respect to p(θ 0 ).

A. The Cramér-Rao Bound
Let θ be a single parameter. The Cramér-Rao bound is a lower bound on the variance of the estimator Var(θ) = E(θ − θ 0 ) 2 . When the estimator is unbiased the bound is given by the inverse of the Fisher information f(θ 0 ) := with N the number of samples.
In the multiparameter case we define the covariance matrix of the estimators (we shall suppress the θ 0 dependence in the notation)

and the Fisher information matrix
The Cramér-Rao bound then takes the form of an inequality between positive semidefinite matrices This bound is achievable asymptotically by the maximum likelihood method. More precisely, it is shown that there is a locally unbiased estimator for which the rescaled covariance matrix N V ≈ F −1 in the limit of large N [40]. To compensate for the overall 1/N improvement in precision due to the use of many copies of the source p(θ), we pick the rescaled covariance matrix N V as the figure of merit for the precision of the estimator in the asymptotic regime. We keep the N explicit in the notation as a reminder.

B. Quantum Parameter Estimation
In quantum parameter estimation, instead of a probability distribution we are given a quantum state ρ(θ) (satisfying ρ ≥ 0, Trρ = 1) which depends on θ. For a given measurement M with POVM elements {M i } (satisfying M i ≥ 0, M i = I) we obtain a probability distribution for the outcomes p M (i|θ) = TrM i ρ(θ) which depends on θ through the state ρ(θ). Classical estimation theory can now be applied to the estimation of θ from p M . The problem of quantum parameter estimation is hence equivalent to the one of finding the measurement which maximizes this classical Fisher information. The Fisher information associated with the measurement M is where dρ dθ is evaluated at θ 0 . The symmetric logarithmic derivative quantum Fisher information (SLD-QFI) is defined as where L(θ 0 ) is the symmetric logarithmic derivative (SLD) defined implicitly by When ρ is of full rank, solutions to Eq. (6) are unique as the only matrix that anti-commutes with ρ is the zero matrix. We will always assume that this is the case. For treatment of the case of degenerate states see Refs. [33,52,53]. The SLD-QFI bound [33] states that for any measurement M (we shall suppress the θ 0 dependence from now on). The proof is obtained by the use of the Cauchy-Schwarz inequality: where we used the definition of the SLD in the second equality and M i = I in going to the last line. Braunstein and Caves [33] proved that equality in Eq. (7) is attained when M is a projective measurement in the basis which diagonalizes L, hence identifying the optimal measurement strategy.
One can also define the right logarithmic derivative (RLD) and corresponding to it is the right logarithmic derivative quantum Fisher information (RLD-QFI) bound. This bound will be discussed later.
In the case of multiple parameters, the Fisher information matrix of the measurement M is defined according to Eq. (3) as The quantum Fisher information matrix is defined as where L i is the symmetric logarithmic derivative with respect to θ i . The multiparameter SLD-QFI bound is an inequality in the sense of semidefinite matrices: This bound is a consequence of the one parameter bound Eq. (7). To see this, let v be a vector in the space of parameters R K 1 , let θ v := i v i θ i . From linearity of the definition of the SLD Eq. (6), it follows that the corresponding symmetric logarithmic derivative is 1 A note on notation: to reduce confusion between state vectors in Hilbert space and vectors in parameter space we will stick to Dirac notation ψ|O|φ for the former and vector notation v M v for the latter.
We then have and h v denote the one parameter Fisher information of the measurement M and the quantum Fisher information for the estimation of θ v respectively.
In other words, in the multi parameter setting, the SLD-QFI bound Eq. (8) can be stated as the following: for any linear combination of the parameters θ v = v i θ i , a one parameter SLD-QFI bound f v ≤ h v applies. In addition, for any v the bound is attainable with a projective measurement in the basis diagonalizing L v 2 . Further notice that because of their quadratic forms, the covariance matrix V , the Fisher information matrix F, and the quantum Fisher information matrix H all transform in the same way under linear coordinate transformations. When θ i →θ i := j R ij θ j , all three matrices transform as (·) → R(·)R . This implies that matrix inequalities between them are invariant under rotations of coordinates 3 .

II. TRADE-OFF
If two linear combinations of the parameters {θ i } defined by the vectors u and v result in commuting SLDs [L u , L v ] = 0, then optimal estimation of the two parameters θ u and θ v can be achieved simultaneously by performing a measurement in the basis which diagonalizes both of them.
However, [L u , L v ] = 0 will typically not be satisfied. In general, we expect there to be a trade-off between the achievable precision in the two parameters in the following sense. Let M (λ), λ ∈ [0, 1] be a family of measurements with POVM elements {M i (λ)} such that M (0) is the optimal measurement for θ v and M (1) is the optimal measurement for θ u . For intermediate values of λ the precisions of the estimators for θ x (which we quantify by Var(θ x )) will take values larger than optimal.
Trade-off curves are commonly used in detection theory. In particular receiver operating characteristic curves (ROC curves) are a convenient way to represent how the probability for false positive detection increases as one increases the sensitivity [54]. In the context of uncertainty relations, a similar representation was used in [55] for preparation uncertainties of angular momentum components. As we will now show, trade-off curves (or surfaces) are a convenient representation of the data which is typically encoded in uncertainty relations.
The known bounds on precision in parameter estimation are most often stated as lower bounds on the expected cost, resulting from a given positive definite K × K cost matrix G [32,46,51] 4 . In general these are bounds of the form where V is the covariance matrix of the estimatorθ and f is a real scalar function on semidefinite matrices. This family of inequalities defines a region in R K of allowed values for the vector of variances (Var(θ 1 ), Var(θ 2 ), . . . , Var(θ K )). The boundary of this region is the trade-off surface. We now show how this is obtained by considering specific examples.
A. Classical Trade-off Curves: the Quantum Fisher Information Cramér-Rao Bound By classical we refer to the situation when the optimal precision values for the different parameters are independent of each other. This is automatically the case in classical parameter estimation where the maximum likelihood method asymptotically achieves the optimal values for all the variances Var(θ i ) simultaneously [40].
Let us begin by plotting the trade-off curve resulting from the SLD-QFI bound Eq. (8). As discussed above, this bound can be interpreted as the assertion that for every direction in parameter space, the single parameter bound applies. Therefore we do not expect to be able to extract nontrivial trade-off relations from it.
The matrix inequalities Eq. (4) and Eq. (8) imply Consider the case of two parameters and let G = t 1−t for t ∈ (0, 1). This form of cost matrix corresponds to a fixed total cost of 1 which is divided between θ 1 and θ 2 with proportion t/(1 − t). Let H −1 = u1 b b u2 . Equation (10) becomes where V i is the variance of θ i . This implies that for every value of t ∈ [0, 1] the points in the (N V 1 , N V 2 ) plane which are not excluded by Eq. (10) lie above the line N V 2 = u 2 + (u 1 − N V 1 ) t 1−t . All of these lines pass through the point (u 1 , u 2 ) and as t varies between 0 and 1 the slope of the line varies between 0 and −∞. The allowed region (not excluded by any value of t) is In particular, the bound Eq. (10) does not exclude the point (N V 1 = u 1 , N V 2 = u 2 ), which corresponds to optimal precision for both θ 1 and θ 2 simultaneously. This classical-or trivial-trade-off bound is plotted in Fig. 1 as the blue dotted curve.

B. Non-trivial Trade-off Curves: the Gill-Massar Bound
To demonstrate nontrivial trade-off we shall introduce the bound proved by Gill and Massar in Ref. [51]. They showed that for separable measurements on N identical copies of finite, d-dimensional quantum systems the following holds: This bound implies [51] that for any G ≥ 0 We will refer to Eq. (12) as the Gill-Massar (GM) bound. The non-linear dependence of the right hand side of Eq. (12) on G results in a non-trivial trade-off curve. Let G and H be parametrized as before. Using the following expression for the fidelity of 2 × 2 matrices [56] which appears in the right hand side of Eq. (12) we obtain the following family of lines in the (N V 1 , N V 2 ) plane: To obtain a formula for the trade-off curve fix V 1 and maximize V 2 with respect to t. This results in the following parametrization of the curve in terms of t ∈ (0, 1): Figure 1 shows the trade-off curves obtained for fixed values of u 1 , u 2 and for d = 2, 3, . . . , 6. In addition the trivial trade-off curve resulting from the SLD-QFI bound is shown. The figure clearly shows that for d > 2 the GM bound is unattainable as it allows a higher precision for each of the parameters than that allowed by the SLD-QFI bound.
Furthermore, Fig. 1 shows that for d > 2 the GM bound does not exclude any region above the trivial trade-off curve. This is in agreement whit Ref. [51] where it was concluded that when the number of parameters K satisfies K ≤ d − 1, the SLD-QFI bound is stronger then the GM bound.  14), several of which are plotted (green) for the d = 2 case. For d = 2 the GM trade-off curve is asymptotic to the SLD-QFI curve, whereas for d > 2 the GM curves are below the SLD-QFI curve.

C. The Right Logarithmic Derivative Quantum Fisher Information Bound and the Holevo Cramér-Rao Bound
We shall now introduce the right logarithmic derivative quantum Fisher information (RLD-QFI) bound. This bound exhibits nontrivial trade-off, with the 'strength' of the trade-off between the variances of θ i and θ j depending directly on the expectation value of the commutator of the corresponding SLDs Trρ The right logarithmic derivative (RLD) is defined implicitly by The RLD-QFI matrix is then defined by Just as the SLD-QFI matrix, the RLD-QFI matrix bounds the covariance matrix of any locally unbiased estimator [46]: This bound implies, as before, a lower bound on the expected cost associated with any positive cost matrix G > 0, which, due to the fact that R is a Hermitian matrix (whereas H is real and symmetric) takes the form [32, Lemma 6.6.1] where | · | is the absolute value function defined for Hermitian matrices via their spectral decomposition; and Re and Im refer to the real and imaginary parts of a matrix taken entry-wise. The imaginary part results in a non-trivial trade-off curve. To see this, consider the case of two parameters. Because R −1 is Hermitian, its imaginary part is anti-symmetric. Let , and G = t 1−t . Equation (15) becomes The right hand side has the same functional dependence on t as in Eq. (14) with d = 2. From this we conclude that this bound results in a non-trivial trade-off curve which is asymptotic to the lines N V 1 = r 1 and N V 2 = r 2 .
In certain cases, it is possible to express the RLD-QFI matrix in terms of the SLDs. In the case of what is called a D-invariant model 5 [32,46] the following holds: where D is a matrix whose entries are proportional to the expectation values of the commutators of the SLDs: As H and D are real, the imaginary part of R −1 is H −1 DH −1 /2, which together with Eq. (15) implies Comparing to Eq. (16) we see that in this case Trρ [L i , L j ] determines how much area the trade-off curve excludes above the trivial curve resulting from the SLD-QFI bound Eq. (10) (which has only the TrGH −1 term). We mention the Holevo Cramér-Rao bound, which is in general stronger than both the SLD-QFI and the RLD-QFI bounds [46]. In the D invariant case the Holevo bound coincides with the RLD-QFI bound [32,46]. As we will be dealing only with such cases, we shall not present the Holevo Cramér-Rao bound here and only mention results we will need for our discussion 6 . The Holevo Cramér-Rao bound was shown to be equal to the SLD-QFI bound iff the expectation values of the commutators between all SLDs vanish [41]. In Gaussian state shift models where one estimates the displacement parameters, it has been shown that the Holevo Cramér-Rao bound is attainable [32]. The theory of local asymptotic normality maps any quantum estimation problem involving many copies of the same state to a Gaussian shift model [47]. This implies asymptotic attainability of the Holevo Cramér-Rao bound with collective measurements [41,48].

III. THE QUBIT MODEL
Let us next move to the estimation of the most general density matrix of a qubit, which is parametrized by three parameters. This problem is also known as quantum state tomography [60]. In order to observe trade-off relations between more than two parameters, it is enough to consider a qubit system. In the qubit case, the GM bound is attainable with a measurement performed on single copies of the state [51,61].
In this section we compare the GM bound and the Holevo Cramér-Rao bound (which in this case is equal to the RLD bound) through the resulting trade-off surfaces. We also investigate the set of optimal measurements which saturate the inequalities. We characterize this set in two cases: when the parametrization is aligned with ρ 0 (when the z axis is pointing in the direction of the Bloch vector of ρ 0 ); and when it is not aligned. Finally we use the trade-off surfaces computed for different parametrizations to obtain a state independent trade-off surface, and derive state independent uncertainty relations.
We work in the Bloch sphere parametrization, using Pauli matrices as a basis, and with ρ 0 = [I + z 0 σ z ]/2, the full parametrization is ρ(θ) = ρ 0 + θ i σ i . Note that the initial state can always be brought to this form by rotating the Bloch sphere and working in the appropriate basis. We will call this coordinate system the adjusted one, and later-in Section III C-we shall return to describe things in a general coordinate system. We will identify θ 1 ≡ x, θ 2 ≡ y and θ 3 ≡ z. When the state is full rank (z 0 < 1) the solution to the equation defining the SLDs is unique and given by The resulting SLD-QFI is diagonal and takes the form

A. Comparison Between Gill-Massar and Holevo Cramér-Rao Bounds
Let us take a cost matrix parametrized as The GM bound is given by Eq. (12): The Holevo Cramér-Rao bound is equal to the RLD bound because the model is D-invariant (this is verified by a direct computation). Computing the matrix According to Eq. (19) the RLD-QFI bound is then given by From this expression one can already guess that the RLD bound exhibits nontrivial trade-off only between the x and y parameters as r appears only in the term coming from TrGH −1 on the right hand side. This is a generic feature of the RLD-QFI bound for finite dimensional quantum systems. We will show that this is the case in a 3-level system in Section IV. Using Eqs. (23) and (25) we find for each bound the smallest allowed value of N V z for a grid of values of N V x , N V y (for fixed N V x , N V y we can find N V z by requiring equality in Eqs. (23) and (25) and maximizing over a grid of values for s and t). The results are plotted in figure 2. For states with z 0 < 1, the Holevo Cramér-Rao (=RLD) bound is strictly weaker than the GM bound. Recall that the GM bound is attainable with single copy measurements whereas the Holevo Cramér-Rao bound with collective measurements. This conforms with our expectation that collective entangled measurements should provide an advantage over separable ones. As the state ρ 0 tends towards a pure state, the GM bound tends towards the Holevo Cramér-Rao bound, as can be seen from Eqs. (23) and (25)

B. Measurements Attaining the Gill-Massar Bound
The bound Eq. (11) is achievable for qubits. Gill and Massar show that for a qubit system (d=2), every matrix F that satisfies Eq. (11) is obtainable as the Fisher information matrix of a measurement M F . M F is a probabilistic mixture of three projective measurements along the directions which diagonalize F (seen as Bloch vectors). By probabilistic mixture we mean combining measurements in the following way: let M (1) and M (2) be measurements with POVM elements {M j } J j=1 . We say that M is a probabilistic mixture of M (1) and M (2) if M has I + J POVM elements M k = λM In the rest of this section, we will require more detailed notation. We denote the Fisher information matrix corresponding to ρ n = (I + n · σ)/2, the estimation of parameters θ, and to a projective measurement M = P v along a Bloch vector v as F(ρ n , θ, P v ). The following calculation shows that this matrix equals In Ref. [51] it is shown, as part of the proof of the bound Eq. (11), that the optimal rescaled covariance matrix for a given cost matrix G is given by Plugging in d = 2, H from Eq. (21) and a cost matrix parametrized as in Eq. (22) we obtain Comparing this with the Fisher information of a probabilistic mixture with proportions (α, β, Those are given bȳ This gives a simple characterization of the optimal measurements, i.e. the measurements for which the obtained variances lie on the trade-off surface. They are probabilistic mixtures of projective measurements in the x, y and z directions, with different proportions optimizing for different cost matrices. These projective measurements happen to be the optimal ones in the one-parameter estimation scenario as the SLDs are diagonal in the x,y and z bases respectively (see Eq. (20)). Note, however, that we have thus far been working in the adjusted coordinate system, where the z axis is aligned with ρ 0 . In the next paragraph we analyze the case of a general coordinate system.

C. General Coordinates
So far we have considered a general state ρ 0 but in order to simplify the analysis, we adjusted our coordinate system such that the z axis was aligned with the Bloch vector of ρ 0 . In the last paragraph we saw that in this adjusted coordinate system the optimal trade-off is attained by probabilistic mixtures of the Pauli operators (rotated to the adjusted basis). We are not always free to choose the coordinate system we work in, and it is likely that we would like to optimize our measurement for a cost matrix which is diagonal in a different coordinate system than the one adjusted to ρ 0 . We now look at the trade-off surface in a coordinate system which is not aligned with the state, and investigate the measurements which achieve the trade-off surface. This will turn out to be useful for deriving our state independent result in Section III D.
Changing the coordinate system rotates the covariance matrix, the SLDs, and the quantum Fisher information matrix as described in Section I B. In Ref. [51] it is described how to find the measurement which achieves a desired Fisher information matrix satisfying Eq. (11) (this is achieved by mixing the projective measurements corresponding to the Bloch vectors which constitute the eigenbasis of the desired Fisher information matrix). We used their method to compute the optimal measurements by finding the measurements which result in the inverse of Eq. (27) for different diagonal costs G. The result is that the optimal measurements no longer belong to an easily characterizable family. As G is varied, the three Bloch vectors describing the projectors of which the measurement is composed travel around the Bloch sphere. The SLD measurements, which when working in the adjusted coordinates could be mixed in different proportions to get variances anywhere on the trade-off surface, no longer play a role. Below we demonstrate that they are far from optimal even in the case of a pure cost matrix (one which assigns all the cost to one parameter), and that in fact, the optimal for such a cost matrix is to measure the corresponding Pauli operator.
In the following we fix an arbitrary coordinate system and test the performance of two families of measurementsone consisting of probabilistic mixtures of the SLD measurements, and the other of the Pauli operators in the chosen coordinate system-and see how they fare compared to the measurements attaining the GM bound. Let ρ 0 = (I + z 0 σ z )/2 as before and let θ be the adjusted coordinate system as before. Any orthogonal coordinate system is related to the adjusted one by a rotation. Letθ i = j R ij θ j be the coordinates in which we would like to work, where R ∈ O(3) is a rotation matrix. The state ρ in these coordinates reads We denote the Pauli matrices in the chosen coordinates byσ i := (Rσ) i = j R ij σ j . As explained in Section I B, the quantum Fisher information matrix now takes the form where H diag is the QFI matrix in the adjusted coordinates given in Eq. (21). According to Eq. (9), the SLDs corresponding to this coordinate system Lθ i are given by linear combinations of the SLDs in the adjusted coordinates: Each Lθ i is diagonal in a basis consisting of two pure states corresponding to two antipodal points on the Bloch sphere. For a general rotation R, the three bases diagonalizing Lθ i , i = 1, 2, 3 no longer correspond to three mutually orthogonal lines through the center of the Bloch sphere (because of the non zero I component in L z , see Eq. (20)).
We will now show that the Pauli measurements in the chosen coordinates achieve the optimal expected cost for a pure cost matrix, i.e. G i := e i e i (where e 1 = (1, 0, 0) etc.). The optimal cost according to Eq. (27) is given by where the last equality is due to orthogonality of R. The Fisher information for a Pauli measurement is given by where we used Eq. (26) for the calculation of the Fisher information of a projective measurement in the adjusted coordinates (θ), and the transformation rule for F under change of coordinates. Taking probabilistic mixtures of the three Pauli measurements and inverting the resulting Fisher information matrix we obtain the following family of covariance matrices: whereM (α, β) is a probabilistic mixture with proportions (α, β, 1 − α − β) of the measurements in the Pauli bases corresponding to our chosen coordinates. We see this achieves the optimal cost for pure cost matrices Eq. (29) for i = 1, 2, 3 in the limits α → 1, β → 1 and α, β → 0 respectively.
For randomly sampled diagonal cost matrices as in Eq. (22) we computed the optimal covariance matrix using Eqs. (21), (27) and (28). In addition we computed the covariance matrices corresponding to random probabilistic mixtures of the Pauli measurements, and to random probabilistic mixtures of SLD measurements. Figure 3 shows the resulting trade-off surfaces between the variances of the three parametersθ i . It is clearly seen that the Pauli measurements lie above the optimal surface, and that the SLD measurements preform significantly worse than the other two. This is due to correlations between the parameters in the SLD measurements. In further numerical calculations we performed it was observed that the separation between the Pauli measurements and the optimal measurements is noticeable for states closer to the sphere of pure states (z 0 > 0.5), and that it vanished when one of the coordinate axes came close to alignment with ρ 0 . It is clearly seen that the SLD measurements perform much worse than the rest, and that the variances of the rotated Pauli measurements lie above, but close to, the trade-off surface. The rotated Pauli measurements approach the trade-off surface far away from the origin as shown in the main text.

D. State Independent Trade-Off Surface
So far we have always considered state-dependent bounds. Indeed, all the bounds we used in order to plot our trade-off surfaces involved explicit dependence on the state ρ 0 (recall that the quantum Fisher information matrix H always depends on ρ 0 ). A state independent trade-off surface can be obtained as the boundary of the union over all states ρ 0 of the attainable regions-the regions laying above the trade-off surface (equivalently, as the boundary of the intersection of the unattainable regions). To obtain a graphical representation of this state independent trade-off surface, we would need to plot the trade-off surfaces corresponding to different state ρ 0 all on the same plot, and see what region remains uncovered.
We fix our standard coordinate system to be in terms of the usual Pauli operators ρ(θ) = ρ 0 + θ i σ i , and for every state ρ 0 in the Bloch sphere we use Eqs. (21), (27) and (28) to sample points from the trade-off surface corresponding to the GM bound with that state. More precisely, we compute the quantum Fisher information matrix H(z 0 ) in the coordinates aligned with ρ 0 (z 0 is the length of the Bloch vector of ρ 0 ) and then rotate it back to the standard coordinates with the appropriate rotation R ∈ O(3). We plug the result into Eq. (27) and plot the diagonal entries of V opt (G) for randomly sampled diagonal cost matrices G. We do this for a grid of values of z 0 ∈ [0, 1] and of the angles parameterizing the rotation (R = R x (α)R y (β)R z (γ), where R x (α) is a rotation around the x axis by an angle α). This procedure is equivalent to running over a grid of states ρ 0 .
The result is shown in Fig. 4. The figure shows that a non-trivial state independent trade-off relation holds between the three parameters of a qubit state. This result relies on the GM bound Eq. (12) and therefore applies whenever the parameters are estimated from the outcomes of separable measurement strategies. The shape of the trade-off surface in Fig. 4(a) has features similar to the boundary of the preparation uncertainty regions found in [55,Figures 6,7]. Its projection to the x, z plane shown in Fig. 4(b) suggests that the following uncertainty relation holds for the rescaled variances 7 : This bound coincides with the preparation uncertainty relation ∆(σ x /2) 2 + ∆(σ z /2) 2 ≥ 1/4 proven in [55,62]. We now prove Eq. (30). It is enough to prove the case i = 1, j = 2, this will become clear from Eq. (31) below, where we have the freedom to rotate H −1 . We therefore prove V 1 + V 2 = TrV P 2 ≥ 1/4, where P 2 is the following matrix: Denote the optimal covariance matrix for a state ρ 0 with Bloch vector of length z 0 and the cost matrix P 2 in a coordinate frame rotated by a rotation R ∈ O(3) with respect to ρ 0 as V opt (P 2 , R, z 0 ). As explained above, minimizing the expected cost TrV P 2 over all states ρ 0 is equivalent to minimizing TrV opt (P 2 , R, z 0 )P 2 over all choices of coordinate systems (specified by R ∈ O(3)) and all z 0 ∈ [0, 1]. According to Eq. (27) and Eq. (28) we have TrN V opt (P 2 , R, z 0 )P 2 = Tr 7 Recall that in our parametrization θ i is the deviation of σ i /2 from its true value, if we were to parametrize the state as ρ(θ) = ρ 0 +θ·σ/2 the lower bounds in Eqs. (30), (34) and (35) would be 4 times bigger.
We now proceed to minimize Eq. (31) over R ∈ O(3) and z 0 ∈ [0, 1]. First notice that the minimum is always obtained for pure states (z 0 = 1) because: where we used the operator monotonicity of the square root function (A ≥ B ≥ 0 ⇒ √ A ≥ √ B) going to the second line. We would now like to perform the minimization over R ∈ O(3). A convenient parametrization of R for this purpose is given by R( u, φ) where u is a unit vector and φ is the angle of rotation. Using the Rodrigues' rotation formula [63], R( u, φ) is given explicitly by where we used the shorthand c := cos φ and s := sin φ and where * stands in place of entries we will not use. Plugging this into Eq. (31) and setting z 0 = 1 we obtain which we performed numerically to obtain the value 1 4 . The two-parameter relations Eq. (30) fully characterize the attainable region for two parameters, as seen from Fig. 4(b). As a partial characterization of the shape of the region attainable for all three parameters in Fig. 4(a) we prove the following 8 : i.e. that the plane N V (θ x ) + N V (θ y ) + N V (θ z ) = 1 is a supporting plane of the attainable region, as can be seen in Fig. 4(a). As before, the minimum of TrV opt (I, R, z 0 )I is obtained when z 0 = 1. There is no need to minimize over R as we can use the cyclicity of the trace to eliminate R with R . We obtain We conclude this section by mentioning that the same reasoning can be applied to the Holevo Cramér-Rao bound. Starting from Eq. (19) with a rotated QFI matrix and a rotated D matrix (Eq. (18)): and setting G = I we obtain where we used Eq. (24). This is minimized when z 0 = 0 and we obtain the bound 9 which is a state independent bound implied by the Holevo Cramér-Rao bound and therefore holds for collective measurements. It is saturated in the case of a maximally mixed state. In this case the commutation condition Trρ [L i , L j ] = 0 is satisfied, which means that the Holevo bound coincides with the SLD-QFI bound [41] and is attainable due to local asymptotic normality [47,48]. This also shows that the state independent trade-off surface for estimation using collective measurements is different from the one for separable measurements shown in Fig. 4, as with collective measurements Eq. (34) can be violated. It would be interesting to compute the state independent trade-off surface implied by the Holevo Cramér-Rao bound. We leave this for future works.

IV. THE HOLEVO CRAMÉR-RAO BOUND IN THE QUTRIT MODEL
In this section, we compute the Holevo Cramér-Rao bound for a qutrit or three level system. We use a model for which the Holevo Cramér-Rao bound is equal to the RLD bound and can, therefore, be computed by Eq. (19). As in the qubit case, the obtained bound exhibits both trivial and non trivial trade-offs between various parameters.
We compute the SLDs for a parametrization of a state of a 3-level system in terms of the Gell-Mann matrices.
The diagonal entries k i are related to θ 0 3 , θ 0 8 by The derivatives of ρ, evaluated at θ = 0 are In Ref. [53] the SLDs for a three level system were computed in a more general setting. For simplicity, assume the state is full rank, and using the structure constants of su(3) (given in Ref. [53]), simply verify that the SLDs are given by When all the k i are different, the model is D-invariant (see Footnote 5; this also follows from the su(3) structure constants). We compute the expectation values of the commutators of the SLDs Tr[L i , L j ]ρ 0 to obtain the matrix elements of D (Eq. (18)). The only non-zero elements are The quantum Fisher information matrix has only two non-zero entries off from the diagonal (H 83 = H 38 = 0). Combining these observations we can use Eq. (19) to understand the trade-offs which the Holevo Cramér-Rao bound exhibit in this model. We can treat the matrices appearing in Eq. (19) as block diagonal. In H −1 the only block which contains off diagonal terms corresponds to the parameters θ 3 , θ 8 . Since in this block, D is zero, we do not need to consider its contribution. In the other blocks (corresponding to (θ 1 , θ 2 ), (θ 4 , θ 5 ) and (θ 6 , θ 7 )), the result of taking the absolute value of the restriction of D to this block conjugated with a diagonal positive matrix (the restriction of √ GH −1 to the same block), results in a functional dependence of the right hand side of Eq. (19) which is a sum of three terms similar to Eq. (16). More precisely, for G ij = δ ij g i with g i ≥ 0 and g i = 1 TrGV (θ) ≥TrGH −1 + 1 2 Tr where a, b and c are the values of H −1 in the corresponding blocks (which we do not compute explicitly as we just want to demonstrate the qualitative behavior). The functional dependence of the above on {g i } implies that non-trivial trade-off appears only between pairs of parameters corresponding to the x and y Pauli matrices within each of the 3 su(2) sub-algebras of su(3), and within each sub-algebra the trade-off is as the RLD bound in Fig. 2 (trivial trade-off with the diagonal element).

V. DISCUSSION
This paper illustrated the fact that the unsaturability of the Quantum Fisher information Cramér-Rao bound for multiparameter estimation gives rise to a rich variety of quantum uncertainty relations in the form of trade-off curves. Those trade-off curves relate to each other the prefactors c ij of the covariances V ij = c ij /N of the optimal estimators for the unknown parameters {θ i } in the limit when a large number of copies N of the state are available. This can be seen as a parameter estimation analogue of the quantum Chernoff and Hoeffding bounds [64] in quantum hypothesis testing, where trade-off curves are obtained for the error exponents α i for the error of the first versus the second kind -scaling as exp (−α i N ).
Trade-off curves bring into direct view the property which distinguishes quantum multiparameter estimation from its classical statistics counterpart-the unattainability of simultaneous optimal precision. This property is often discussed in the literature, however, we have never seen such trade-off curves plotted for the known tight bounds. Ref . [45] provides a comparison between the Holevo Cramér-Rao bound and the Gill-Massar bound by comparing the bounds they put on the expected cost for a single (although state-dependent) cost matrix. In another work, Ref. [65] the authors present the difference between the regions of variances excluded by bounds on their arithmetic, geometric and harmonic means. What distinguishes our approach from the above works is that to obtain the trade-off curve we use the bound on the expected cost for a family of different costs all at once. This is best illustrated in Fig. 1 which shows how the trade-off curve is obtained as the point-wise maximum over a family of lines. We can also apply this in the reverse to obtain tight bounds on the expected cost given a convex region of attainable variances, as the latter is determined by its supporting hyperplanes.
Trade-offs in quantum parameter estimation belong to the joint-measurement type of uncertainty relations. They show that when we wish to estimate certain parameters by performing a measurement on a quantum state, increased precision in one parameter will typically come at the cost of increased uncertainty in other parameters.
Investigation of the trade-off surfaces implied by the Gill-Massar bound led us to our main result-a state independent uncertainty relation between the three parameters of a qubit system. We provided numerical evidence for this trade-off relation ( Fig. 4(a)) and proved an additive bound Eq. (34) which forms part of the trade-off surface. In addition, we proved two-parameter additive uncertainty relations Eq. (30) which coincide with the uncertainty relation for state preparation proven in Refs. [55,62]. We showed that the Holevo Cramér-Rao bound also implies an additive uncertainty relation with a smaller lower bound than in Eq. (34). Our method for deriving state independent trade-off surfaces from state-dependent bounds could be applied to the Holevo Cramér-Rao bound for a qubit to obtain a trade-off surface for estimation with collective measurements.
The attainability of the symmetric logarithmic derivative quantum Fisher information (SLD-QFI) bound, which exhibits classical (or trivial) trade-off, with collective measurements has recently been shown to be equivalent to the commutation condition Tr[L i , L j ]ρ 0 = 0, the vanishing of the expectation values of the commutators between all SLDs [41]. The degree to which this fails to be the case has been suggested in Ref. [58] as a measure of incompatibility between parameters. In Section II C we demonstrated this by relating the algebraic form of the Holevo Cramér-Rao bound to the strength of the resulting trade-off curve. Equations (16) and (19) show how as the expectation value Tr[L i , L j ]ρ 0 approaches zero, the corresponding trade-off curve becomes closer and closer to the trivial one. We have also provided two examples of systems-the qubit (Section III) and qutrit (Section IV) models-where the commutation condition is satisfied only between some pairs of parameters, and demonstrated how this reflects in their trade-off surfaces.
The attainable bounds we dealt with in this paper pertain to two different measurement scenarios. The Gill-Massar bound Eq. (12), is attainable for qubit ensembles (d = 2) with separable measurements, whereas the Holevo Cramér-Rao bound Eq. (17) is attainable for finite dimensional systems with collective entangled measurements. From the algebraic form of the Gill-Massar bound Eq. (12) and the Holevo Cramér-Rao bound Eq. (17) the trade-off structure is not immediately visible. In Fig. 2 we plotted the trade-off surfaces for each of the bounds for the qubit case to show the qualitative difference between the two. The Holevo Cramér-Rao bound allows for higher precision and exhibits non-trivial trade-off only between the x and y parameters, whereas the GM bound-between all three parameters.
The attainability of the Holevo Cramér-Rao bound for finite dimensional systems relies on the theory of quantum local asymptotic normality. As described in [47], in the asymptotic limit the statistical model of a finite dimensional quantum system splits into a product of a classical Gaussian shift model corresponding to the diagonal elements of the density matrix, and independent harmonic oscillator models for the off-diagonal elements. The trade-off surfaces of the Holevo Cramér-Rao bound, which we described for the qubit (Fig. 2) and qutrit (Section IV) systems, are exactly what one would expect to find in the corresponding asymptotic models. In both cases the parameters corresponding to the diagonal components behave like classical systems, i.e. they have trivial trade-off with any other parameter. In the three level system we observe the splitting of the off-diagonal parameters into independent pairs that have nontrivial trade-off within the pair, and trivial trade-off with elements of other pairs. The trade-off structure described in Section IV is therefore generic to finite dimensional systems when collective measurements can be implemented.
Finally, we studied the optimal single copy measurements in the qubit model. We showed that the strategy of measuring different SLD operators on parts of an ensemble of identical states, which is optimal for the case of a coordinate system aligned with the state ρ 0 , is far from optimal in the case of general coordinates. We further demonstrated that measuring the Pauli operators (rotated to the coordinate frame) achieves the optimal cost when all the cost is assigned to one parameter. Our numerical calculations Fig. 3 further showed that the rotated Pauli measurements are not very far from the optimal for general cost matrices.