Achieving quantum precision limit in adaptive qubit state tomography

The precision limit in quantum state tomography is of great interest not only to practical applications but also to foundational studies. However, little is known about this subject in the multiparameter setting even theoretically due to the subtle information tradeoff among incompatible observables. In the case of a qubit, the theoretic precision limit was determined by Hayashi as well as Gill and Massar, but attaining the precision limit in experiments has remained a challenging task. Here we report the first experiment which achieves this precision limit in adaptive quantum state tomography on optical polarization qubits. The two-step adaptive strategy employed in our experiment is very easy to implement in practice. Yet it is surprisingly powerful in optimizing most figures of merit of practical interest. Our study may have significant implications for multiparameter quantum estimation problems, such as quantum metrology. Meanwhile, it may promote our understanding about the complementarity principle and uncertainty relations from the information theoretic perspective.

The precision limit in quantum state tomography is of great interest not only to practical applications but also to foundational studies. However, little is known about this subject in the multiparameter setting even theoretically due to the subtle information tradeoff among incompatible observables. In the case of a qubit, the theoretic precision limit was determined by Hayashi as well as Gill and Massar, but attaining the precision limit in experiments has remained a challenging task. Here we report the first experimental verification of this precision limit in adaptive quantum state tomography on optical polarization qubits. The two-step adaptive strategy employed in our experiment is very easy to implement in practice. Yet it is surprisingly powerful in optimizing most figures of merit of practical interest. Our study may have significant implications for multiparameter quantum estimation problems, such as quantum metrology. Meanwhile, it may promote our understanding about the complementarity principle and uncertainty relations from the information theoretic perspective. Quantum state tomography is a procedure for inferring the state of a quantum system from quantum measurements and data processing [1][2][3]. It is a primitive of various quantum information processing tasks, such as quantum computation, communication, cryptography, and metrology [4][5][6][7][8]. In sharp contrast with the classical world, any measurement on a generic quantum system necessarily induces a disturbance, limiting further attempts to extract information from the system. Therefore, many identically prepared systems are usually required for reliable state determination. Conversely, the precision limit in quantum state tomography offers a perfect window for understanding the distinction between quantum physics and classical physics [9][10][11][12].
Recently, great efforts have been directed to improving the tomographic efficiency given limited quantum resources [13,14]. For example, adaptive measurements have been realized in experiments, which may improve the scaling of the infidelity in certain scenarios [14,15]. However, most studies have been tailored to deal with specific figures of merit under special settings, such as pure state or single parameter models, which admit no easy generalization to the more challenging and exciting multiparameter estimation problems with general figures of merit. In particular, the tomographic precision limit in the multiparameter setting is still poorly understood; experimental studies are especially rare. To fill this gap, in this work we report the first experimental verification of the quantum precision limit in adaptive quantum state tomography on optical polarization qubits.
Quantum Precision limit. In practice, the tomographic precision limit is determined by experimental settings. As technology advances, it is ultimately limited by ba-sic principles of quantum mechanics. One fundamental limit is known as the quantum Cramér-Rao (CR) bound [16][17][18][19]; see the supplementary material. In the oneparameter setting, this bound can be saturated locally by measuring a suitable observable, which may depend on the parameter point. To saturate the bound globally, it is usually necessary to employ adaptive measurements. A simple and effective choice is known as the two-step adaptive strategy [20,21], whose basic idea can be sketched as follows. First, we perform a generic informationally complete measurement on c √ N copies of the true state for some positive constant c and compute the maximum likelihood estimator (MLE) [1] according to the measurement statistics. Then we perform the optimal measurement with respect to the estimator on the remaining N − c √ N copies and compute the MLE again.
In the multi-parameter setting, however, the quantum CR bound generally cannot be saturated except when the optimal observables corresponding to different parameters can be chosen to be compatible. The existence of incompatible observables underlies the main distinction between quantum state estimation and classical state estimation and is the main reason why multiparameter estimation problems are difficult and poorly understood. Up to now, the optimal solutions have been found only for a few special cases [2,18], of which the qubit model is the most prominent [9,21,22].
To devise a good measurement scheme in the multiparameter setting, it is indispensable to take into account the subtle information trade-off among incompatible observables. A vivid manifestation of such tradeoff is the wave-particle duality relation between fringe visibility V and path distinguishability D [23,24], This phenomenon is not limited to the double-slit experiment but presents itself whenever we are trying to extract information about incompatible observables. It is especially important in understanding multiparameter estimation problems. Suppose the state of the quantum system is parametrized by a set of parameters denoted collectively by θ, then such trade-off can be succinctly summarized by the following inequality derived by Gill and Massar [21], which is applicable to any measurement on a d-level system. Here I(θ) and J(θ) are the Fisher and quantum Fisher information matrices, respectively (see the supplementary material). The Gill-Massar (GM) inequality may be seen as a generalization of the wave-particle duality relation in the language of quantum estimation theory. To appreciate its significance, it is instructive to point out that the upper bound would be d 2 − 1 if all observables in quantum theory were compatible or, equivalently, if the quantum CR bound could always be saturated.
The GM inequality imposes a fundamental precision limit on quantum state tomography based on individual (non-collective) measurements. For example, it sets a lower bound for the scaled weighted mean square error (WMSE) of any unbiased estimator [9,21], where W is the weighting matrix. Note that the GM bound for the WMSE is E GM W /N if the sample size is N . In the case of a qubit, the GM bound agrees with the bound derived by Hayashi [22] and can always be saturated locally by mutually unbiased measurements [9,21] (see the supplementary material). Recently, the GM inequality and GM bound have been turned into a powerful tool for studying a number of foundational issues entangled with incompatible observables [11], such as the complementarity principle [25] and uncertainty relations [26]. The cross fertilization of quantum estimation theory and foundational studies is due to lead to deeper understanding of both subjects [9][10][11][12]. Determination of the quantum precision limit in the multiparameter setting is thus of primary interest from both practical and foundational perspectives.
Attaining quantum precision limit with two-step adaptive strategy. Here we report the first experimental verification of the GM bound in adaptive quantum state tomography on optical polarization qubits. Our tomographic protocol consists of two-step adaptive measurements and MLE, as illustrated in Fig. 1. In each step, we x , σ ′ y , σ ′ z depend on the estimatorρ1 obtained in the first step and are related to σx, σy, σz via the unitary transformation U (ŝ1) described in the main text. The probabilities p1, p2, and p3 depend on both the estimatorρ1 and the figure of merit. In the large-N limit, it suffices to use the measurement statistics of step 2 to construct the second MLE. When N is finite, it is preferable to retain the measurement statistics of step 1.
need to implement only three projective measurements that are mutually unbiased. Despite the simplicity of this approach, it is capable of attaining the quantum precision limit with respect to most figures of merit of interest. To facilitate applications of our approach, we have determined local optimal measurements with respect to a large family of figures of merit in Sec. II of the supplementary material. For concreteness, the two-step adaptive strategy for minimizing the MSE can be sketched as follows: 1. Measure σ x , σ y , σ z on N 1 /3 copies of the qubit, respectively, and compute the MLEρ 1 based on the measurement statistics. Denote the Bloch vector of ρ 1 byŝ 1 .
2. Choose a unitary transformation U (ŝ 1 ) that rotatesŝ 1 to the z direction. Measure σ x , σ y , σ z with probabilities p 1 , p 2 , p 3 , respectively, according to Eq. (9) in the supplementary material after applying this unitary transformation to the remaining N 2 = N − N 1 copies of the qubit state. Construct the MLEρ 2 again based on the measurement statistics.
Experimental setup. The experimental setup is shown in Fig. 2. A 2-mm-long BBO crystal cut for type-I phase-matched spontaneous parametric down-conversion (SPDC) process is pumped at 404 nm by a 40-mW Vpolarized beam. A pair of H-polarized photons with wave length λ = 808 nm is created via SPDC process. One photon passes through an interference filter whose FWHM is 3 nm, resulting in a coherence length of 270λ, and is detected by a single photon detector acting as a trigger. The polarization state of the other photon is prepared by HWP1, a 770λ quartz crystal, and QWP1. HWP1 and the quartz crystal are used to change the purity or the length of the Bloch vector of the state and QWP1 to change the direction of the Bloch vector. QWP2 and HWP2 followed by a PBS are used to perform arbitrary projective measurements on the qubit. The rotation angles of HWP2 and QWP2 are controlled by a To achieve high tomographic precision in experiments, it is crucial to reduce the systematic error to a very low level besides adopting the correct adaptive strategy. To this end, we have implemented error-compensation measurements proposed in Ref. [27], which are capable of reducing the systematic error from 5 × 10 −5 to 8 × 10 −6 , which is about 100 times smaller than the statistical error when N = 9000 (see the supplementary file).
Quantum precision limit with respect to various figures of merit. To illustrate the power of the two-step adaptive strategy described above, here we investigate quantum precision limits with respect to a variety of figures of merit. In the first experiment, we verify this limit concerning the MSE, with N 1 = 3000 and N 2 = 6000. The MSE is determined by averaging over 4000 repetitions. As comparison, we also implement two other tomographic strategies. In the first one, σ x , σ y , σ z are measured on N/3 copies of the qubit, respectively, and the MLE is computed as before. This standard tomography is widely used in practice because of its simplicity. In the second one, σ x , σ y , σ z are measured with probabilities p 1 , p 2 , p 3 as specified in Eq. (9) in the supplementary material after rotating the Bloch vector of each of the N copies of the qubit state to the z-axis. This "knownstate tomography" assumes knowledge of the true state in designing the optimal measurement though not in reconstructing the state. It is not feasible in most practical situations, where the state is in fact unknown, but it is useful as a benchmark. Figure 3 shows the MSEs associated with the three measurement strategies mentioned above along with the theoretical MSE of the standard tomography and the GM bound. The experimental data agree very well with the theoretical prediction. In contrast with the adaptive strategy proposed in Ref. [14], which offers no advantage over standard tomography with respect to the MSE, our two-step adaptive strategy is significantly more efficient than standard tomography. What is remarkable, the MSE achievable by the two-step adaptive scheme is quite close to that of known-state tomography and saturates the GM bound approximately.
Next, we investigate the quantum precision limit with respect to the mean square Bures distance (MSB). Incidentally, the GM bound for the MSB is equal to 9/(4N ) (see the supplementary material), independent of the unknown qubit state. The two-step adaptive strategy for minimizing the MSB is quite similar to that for minimizing the MSE except that in the second step σ x , σ y , σ z are measured with probability 1/3 each. Similar modification applies to the "known-state tomography". Unlike the previous case, it is not necessary to implement error-compensation measurements [27] because a simple measurement strategy can already achieve a sufficiently high precision. Here our two-step adaptive strategy is similar to the one proposed in Ref. [14], which is tailored to minimize the infidelity. This is not surprising if we no- tice that the infinitesimal square Bures distance is equal to the infidelity. Our study provides further justification of the approach in Ref. [14] from a wider context.
The upper plot of Fig. 4 shows the MSBs associated with three measurement strategies along with the theoretical precision limit set by the GM bound. In standard tomography, the MSB increases rapidly as the purity of the true state increases. In sharp contrast, the MSB achievable by the two-step adaptive scheme is almost independent of the purity and is much smaller than that of standard tomography, the more so the higher purity of the true state. Moreover, it saturates the GM bound approximately, except when the true state is nearly pure.
The small gap from the bound is mainly due to the fact that the experiment under investigation is not close enough to the asymptotic regime as the ratio N 1 /N is nonnegligible. Meanwhile, the efficiency gap of the standard measurement from the local optimal measurement is significant for states with high purities.
A prominent merit of our approach is its versatility in dealing with various figures of merit as emphasized before. To further demonstrate this point, we now turn to verifying the precision limit with respect to WMSEs based on monotone Riemannian metrics [28][29][30]. For concreteness, we shall focus on the family of metrics characterized by Eqs. (10) and (13) in the supplementary material, which includes the Bures metric and the quantum Chernoff metric [31]. Other metrics can be treated on the same footing. The two-step adaptive strategy is quite similar to the one in the precious case, except that in the second step σ x , σ y , σ z should be measured with probabilities p 1 , p 2 , p 3 as determined by Eq. (14) in the supplementary material. The lower plot of Fig. 4 shows the experimental results for a true state whose Bloch vector has length 0.9. The WMSEs achieved by the two-step adaptive strategy are much smaller than that of standard tomography as in the previous case. Moreover, they agree very well with GM bounds.

Conclusion.
We have implemented optimal adaptive qubit state tomography of mixed states in the multiparameter setting. The two-step adaptive strategy employed in our experiment is very easy to realize in practice. Yet it is surprisingly powerful: It is applicable to optimizing most figures of merit of practical interest, such as the WMSE based on the Bures metric, quantum Chernoff metric, or any other metric commonly used. Moreover, it is capable of attaining the precision limit set by the GM bound approximately. Our experiment represents a significant step towards optimal quantum state tomography in the multiparameter setting, which may have profound implications for multiparameter quantum metrology. Furthermore, our study exemplifies the subtle connection between the complementarity principle and quantum precision limit, thereby promoting the cross fertilization of quantum estimation theory and foundational studies.

QUANTUM CRAMÉR-RAO BOUND
Suppose the state ρ(θ) of a given quantum system is characterized by a set of parameters θ 1 , θ 2 , . . . , θ g . To determine the values of these parameters, we may perform measurements and construct an estimator based on the measurement statistics. Once a measurement with outcomes Π ξ (Π ξ are positive operators that sum up to the identity) is chosen, the probability of obtaining each outcome is determined by the Born rule p(ξ|θ) = tr(ρΠ ξ ). Given an estimatorθ(ξ), the accuracy is usually quantified by the mean-square-error (MSE) matrix, with entries given by (1) The famous Cramér-Rao (CR) bound in statistical inference states that the MSE matrix C(θ) of any unbiased estimator is lower bounded by the inverse of the Fisher information matrix I(θ); that is, Meanwhile, I(θ) is upper bounded by the quantum Fisher information matrix J(θ), where and L j (θ) is the Hermitian operator that satisfies the equation and is known as the symmetric logarithmic derivative (SLD) of ρ(θ) with respect to θ j . Together with the classical CR bound, the inequality I(θ) ≤ J(θ) implies the quantum CR bound C(θ) ≥ J −1 (θ) [17][18][19]. In the oneparameter setting, this bound can be saturated locally by measuring the observable L(θ). In the multiparameter setting, however, it generally cannot be saturated except when the L j (θ) commute with each other.

GILL-MASSAR BOUND FOR THE QUBIT
Here we show that the GM bound for the WMSE in the qubit case can always be saturated, and the local optimal measurement can be realized by a complete set of mutually unbiased measurements [9,11,21]. Note that the GM bound in Eq. (3) of the main text can be saturated if and only if there exists a measurement that yields the Fisher information matrix When W and J commute, Eq. (3) in the main text and Eq. (5) reduce to In the case of a qubit, it is convenient to parametrize the state space by the components of the Bloch vector s = (s x , s y , s z ). Then the inverse quantum Fisher information matrix takes on the form J −1 (s) = 1 − ss. Suppose that I W in Eq. (5) has eigenvalues a 1 , a 2 , a 3 along with orthonormal eigenvectors r 1 , r 2 , r 3 . Denote by s 1 , s 2 , s 3 the three components of the Bloch vector in this basis. Then the GM bound can be saturated by measuring each observable σ j := r j · σ with probability a j (1 − s 2 j ). The equality j a j (1 − s 2 j ) = tr(J −1 I W ) = 1 [9,11] guarantees that the set of probabilities is indeed normalized. This observation confirms our claim that the optimal measurement scheme can be realized with a complete set of mutually unbiased measurements.
Next, we derive the GM bound when the weighting matrix is the identity or determined by a monotone Riemannian metric [28][29][30]. Without loss of generality, we may assume that the Bloch vector is aligned with the z-axis, so that J = diag(1, 1, 1/(1 − s 2 )). If the weighting matrix is diagonal, say W = diag(w 1 , w 2 , w 3 ), then Eq. (6) reduces to The optimal measurement scheme can be realized by measuring σ j for j = 1, 2, 3 (σ x , σ y , σ z in this case) with probability The GM bound for the MSE is obtained when w 1 = w 2 = w 3 = 1, in which case we have Every monotone Riemannian metric for a qubit has the following form up to scaling, where dΩ 2 is the metric on the unit sphere, and f is a Morozova-Chentsov function [28][29][30]. If the weighting matrix is determined by this metric, that is, where t = (1 − s)/(1 + s), then , An important family of Morozova-Chentsov functions has the form [28] f n (t) = 1 + t 1/n 2 n .
The metric turns out to be the Bures metric when n = 1, in which case p 1 = p 2 = p 3 = 1/3 and E GM B = 9/4, and the quantum Chernoff metric [31] when n = 2.

DATA COLLECTION.
To stabilize the collecting efficiency, multimode fibers are used to direct photons from free space to singlephoton detectors. The whole experimental setup is covered by a black hood to keep away stray light from outside; the multimode fibers are wrapped up by black plastic films to suppress laser scattering. To balance the data-collection time and random coincidences, the power of the continuous pumping laser is adjusted to 40 mW, and the coincidence window is set at 2 ns. The resulting coincidence rate is 8000 per second, and the coincidence efficiency is about 8%.

SYSTEMATIC ERROR
In the ideal scenario, the error between the calibrated state and the true state vanishes in the large sample limit, so the calibrated state can serve as the true state. In real experiments, however, the error does not vanish due to various experimental imperfections. By the same token, different calibration procedures may result in different calibrated states. To realize the goal of our experiment, it is crucial that the total systematic error is much smaller than the error due to statistical fluctuation. This is especially the case in the experiment concerning the MSE, in which the statistical fluctuation is suppressed by a large sample and large number of repetitions. In this supplementary material, we estimate the magnitude of the systematic error and show that it meets the requirement of our experiment.
The systematic errors in our experiment mainly come from six aspects: uncertainties in the phases and rotation angles of the optical axes of QWP2 and HWP2, the extinction ratio of PBS, and unbalanced collecting efficiencies between the two branches from PBS. In the analysis of systematic errors, we neglect the statistical fluctuation of measurement results. When the Pauli operator r · σ with unit vector r is measured on N copies of the qubit state ρ = (1 + s · σ)/2, the expected photon counts of the outcomes ±1 are where p ± are probabilities given by the Born rule and η ± are collecting efficiencies determined by channel losses as well as coupling and detection efficiencies of the two branches. Consider a PBS with an extinction ratio of 1/β, the measurement operators corresponding to the two outcomes ±1 actually realized are Π ± = [1 ± (1 − 2β)r ·σ]/2. Accordingly, p ± = [1 ± (1 − 2β)s · r]/2. The unbalance between collecting efficiencies of the two branches is defined as η = (η + /η − ) − 1. To the first order in η, the frequencies of measurement outcomes are The expectation value of r · σ iŝ where m = s · r is the expectation value if the extinction ratio is infinity and the collecting efficiencies are balanced. In our experiment, the measurement of r · σ is realized by a configuration of QWP2 and HWP2 as specified by the phases δ 1 , δ 2 and rotation angles θ 1 , θ 2 of QWP2 and HWP2; see Table I for examples. According to Eqs. (17) and (18), the systematic error ofm can be estimated as follows, wherem ζ is the partial derivative ofm with respect to ζ with ζ = β, η, δ 1 , δ 2 , θ 1 , θ 2 taken at ideal values of these parameters, that is, β = η = 0, δ 1 = π/2, δ 2 = π (the ideal values of θ 1 and θ 2 depend on r). These derivatives are (m β ) 2 = 4m 2 , (m η ) 2 = 4(p + p − ) 2 ≤ 1 4 ; if ζ = δ 1 , δ 2 , θ 1 , θ 2 , then (m ζ ) 2 = (m ζ ) 2 = (s · r ζ ) 2 ≤ |r ζ | 2 , where r ζ = ∂r/∂ζ. Although the dependence of r on δ 1 , δ 2 , θ 1 , θ 2 is complicated [27], the magnitudes of its partial derivatives have simple forms: |r θ1 | 2 = 4 + 4 cos 2 (2θ 1 − 4θ 2 ), When r is one of the basis vectors, the magnitudes of these partial derivatives are listed in Table I. In our calibration process, three Pauli operators r j · σ are measured, where r j for j = 1, 2, 3 form an orthonormal basis, so that the eigenbases of r j · σ are mutually unbiased. The calibrated state is determined by the formulaρ The systematic error between the calibrated state and the true state can be calculated as follows, where (M ζ ) 2 = 3 j=1 (m j ζ ) 2 and ζ = β, η, δ 1 , δ 2 , θ j 1 , θ j 2 . In order to decrease the systematic error caused by the PBS, we use a beam displacer (BD) acting as a PBS, whose extinction ratio is about 8000. The resulting contribution to the systematic error is (M β ) 2 (∆β) 2 ≤ 6 × 10 −8 . Notice that since r j for j = 1, 2, 3 are orthonormal. The unbalance between the collecting efficiencies of the two branches is mainly due to the fluctuation caused by the disturbance of the air flow as well as the vibration and shift of the laser propagation direction. With a multimode fiber collecting system, this fluctuation can be reduced to below 0.002 within 16 hours, the measurement duration of 4000 repetitions for one quantum state. This fluctuation results in (∆η) 2 = 4 × 10 −6 and (M η ) 2 (∆η) 2 ≤ 3 × 10 −6 . Notice that (m j η ) 2 ≤ 1 4 and (M η ) 2 = 3 j=1 (m j η ) 2 ≤ 3 4 .
The rotation stages used in our experiments have a precision of 0.01 • ; the uncertainties in rotation angles θ 1 and θ 2 of QWP2 and HWP2 are mainly determined by the uncertainties in the calibration angles of their optical axes, which are about ∆θ = 0.1 • . The resulting systematic error is [(M θ1 ) 2 + (M θ2 ) 2 ](∆θ) 2 ≤ 2 × 10 −4 .
The total systematic errors due to the six aspects of experimental imperfections mentioned above is no larger than 2.6 × 10 −4 . Incidentally, the error between states calibrated by standard tomography and known-state tomography with about 10 7 photons in the experiment is around 5×10 −5 . In our experiments concerning the MSB and WMSEs with respect to a family monotone Riemannian metrics, this systematic is low enough to guarantee that the experimental result agrees well with the theoretical prediction and that the advantage of adaptive state tomography over standard tomography is clearly manifested.
In the experiment concerning the MSE, however, the systematic error is still not small enough since the advantage of adaptive strategy over standard strategy is quite limited and we need to adopt a large sample size and a large number of repetitions to overcome this issue. To solve this problem, we developed and implemented error-compensation measurements [27] so as to alleviate the influences of uncertainties in the phases and the calibration angles of QWP2 and HWP2. In this way, the total systematic error can be reduced to 7 × 10 −6 . Accordingly, the error between the two calibrated states by standard tomography and known-state tomography with about 10 7 photons is reduced to 8 × 10 −6 [27]. Now the systematic error is much smaller than the MSE gap between adaptive tomography and standard tomography. Consequently, the experimental result agrees well with the theoretical prediction again.