Directly Characterizing the Coherence of Quantum Detectors by Sequential Measurement

The quantum properties of quantum measurements are indispensable resources in quantum information processing and have drawn extensive research interest. The conventional approach to reveal the quantum properties relies on the reconstruction of the entire measurement operators by quantum detector tomography. However, many specific properties can be determined by a part of matrix entries of the measurement operators, which provides us the possibility to simplify the process of property characterization. Here, we propose a general framework to directly obtain individual matrix entries of the measurement operators by sequentially measuring two non-compatible observables. This method allows us to circumvent the complete tomography of the quantum measurement and extract the useful information for our purpose. We experimentally implement this scheme to monitor the coherent evolution of a general quantum measurement by determining the off-diagonal matrix entries. The investigation of the measurement precision indicates the good feasibility of our protocol to the arbitrary quantum measurements. Our results pave the way for revealing the quantum properties of quantum measurements by selectively determining the matrix entries of the measurement operators.


Introduction
The quantum properties of quantum measurements have been widely regarded as essential quantum resources for the preparation of quantum states, 1-3 achieving the advantages of quantum technologies 4-7 as well as the study of fundamental quantum theories. [8][9][10][11][12][13][14][15] The time-reversal approach allows to investigate the properties of quantum measurements qualitatively from the perspective of quantum states. [16][17][18] In addition, the quantum resource theories to properly quantify the quantum properties of quantum measurements has been developed very recently, [19][20][21][22] and has been applied to investigate an important quantum property, the coherence, of quantum-optical detectors. 23 Thus, developing efficient approaches to characterize the quantum properties of quantum measurements is important for both the fundamental investigations and practical applications.
A general quantum measurement, and all its properties, can be completely determined by the positive operator-valued measure (POVM) {Π l }, in which the elementΠ l denotes the measurement operator corresponding to the outcome l. Several approaches have been developed to determine the unknown POVM, [24][25][26][27] of which the most representative one is quantum detector tomography (QDT). 24 In QDT, a set of probe states {ρ (m) } are prepared to input the unknown measurement apparatus and the probability of obtaining the outcome l is given by p can be reconstructed by minimizing the gap between the theoretical calculation and the experimental results. To date, QDT has achieved great success in characterizing a variety of quantum detectors, including avalanche photodiodes, 28 time-multiplexed photon-number-resolving detectors, 24,29,30 transition edge sensors, 31 and superconducting nanowire detectors. 32 As the quantum detectors become increasingly complicated, the standard QDT is confronted with the experimental and computational challenges, which prompts the exploration of some helpful shortcuts. For example, the determination of a few key parameters that describe the quantum detectors allows to largely reduce the characterization complexity. 33 The quantum detectors can also be self-tested with certain quantum states in the absence of the prior knowledge of the apparatus. [34][35][36][37] The emerging data-pattern approach realizes the operational tomography of quantum states through fitting the detector response, which is robust to imperfections of the experimental setup. 38,39 Though QDT is a generic protocol to acquire the entire measurement operators, it does not have the direct access to the single matrix entries of the measurement operator. The complexity of the reconstruction algorithm in QDT increases dramatically with the increase of the dimensional of the quantum system. Typically, tomography of the full measurement operators is considered as the prerequisite for characterizing the properties of quantum measurements. 23 However, in some situ-ations, the complete determination of the measurement operators is not necessary to fulfil specific tasks, which makes it possible to simplify the characterization process. For example, if the input state is known to lie in the subspace of the quantum system, it only requires the corresponding matrix entries of the measurement operators to predict the probability of outcomes. 29,40,41 In particular, the coherence of a quantum measurement is largely determined by the off-diagonal matrix entries of its measurement operators in certain basis. 23 Recently, Lundeen et.al proposed a method to directly measure the probability amplitudes of the wavefunction using the formalism of the weak measurement and weak values. 42 This method, known as the direct quantum state tomography, opens up a new avenue for the quantum tomography technique. The direct tomography (DT) protocol has been extensively studied and the scope of its application is expanded to high-dimensional states, 43-51 mixed states [52][53][54][55][56] and entangled states, 57, 58 quantum processes 59 and quantum measurements. 60 The development of the DT theory from the original weak-coupling approximation to the rigorous validation with the arbitrary coupling strength ensures the accuracy and simultaneously improves the precision. [61][62][63][64][65][66][67][68] Moreover, the direct state tomography allows to directly measure any single matrix entry of the density matrix, which has provided an exponential reduction of measurement complexity compared to the standard quantum state tomography in determining a sparse multiparticle state. [53][54][55][56]69 Recent work has extended the idea to realize the direct characterization of the full measurement operators based on weak values, showing the potential advantages over QDT in the operational and computational complexity. ? In view of the unique advantages of the DT, it is expected that the generalization of the DT scheme to directly characterizing the matrix entries of the measurement operators allows to extract the properties of the quantum measurement in a more efficient way.
In this paper, we propose a framework to directly characterize the individual matrix entries of the measurement operators by sequentially measuring two non-compatible observables with two independent meter states. In the following, the unknown quantum detector performs measurement on the quantum system. The specific matrix entry of the measurement operator can be extracted from the collective measurements on the meter states when the corresponding outcomes of the quantum detector are obtained. Our procedure is rigorously valid with the arbitrary non-zero coupling strength. The investigations of the measurement precision indicate the good feasibility of our scheme to characterize arbitrary quantum measurement. We experimentally demonstrate our protocol to monitor the evolution of coherence of the quantum measurement in two different situations, the dephasing and the phase rotation, by characterizing the associated off-diagonal matrix entries. Our results show the great potential of the DT in capturing the quantum properties of the quantum measurement through partial determination of the measurement operators.
2 Theoretical Framework

Directly Determining the Matrix Entries of the Measurement Operators
The schematic diagram of directly measuring the matrix entries of the POVM is shown in Fig.   1. We represent the POVM {Π l } acting on the d-dimensional quantum system (QS) with the orthogonal basis {|a j } (A) and the matrix entry of the measurement operatorΠ l is given by a j a k corresponds to the diagonal matrix entry which can be easily determined by inputting pre-selected QS state ρ (j) s = |a j a j | to the quantum detector and collecting the probability p l = a j |Π l |a j of obtaining the outcome l. By contrast, the off-diagonal matrix entry E A do not commute, the measurement has to be performed in a particular order.
The sequential measurement process can be described by the unitary evolution of the system- and the second transformationÛ leading to the joint state Then, the unknown quantum detector to be characterized performs the post-selection measurement  A by Both the observablesÔ B andÔ a j a k : Here, the subscripts of the coupling strength g and the Pauli operators coincide with those of the operatorsP andQ. For the sake of convenience, we take g A = g B = g in the rest of this article.

Precision analysis on directly characterizing the matrix entries of the measurement operators
The accuracy and the precision are two essential indicators to evaluate a measurement scheme.
There is no systematic errors in our protocol, since the derivation is rigorous for the arbitrary coupling strength g. According to the previous studies, the precision of the DT applied to the quantum states is sensitive to both the coupling strength and the unknown states. 70 The increase of the coupling strength is beneficial to improve the precision. [62][63][64][65][66][67][68] When the unknown state approaches being orthogonal to the post-selected state, the DT protocol is prone to large statistical errors and therefore highly inefficient. 70,71 Here, we theoretically investigate the precision of the DT protocol applied to the quantum measurement to verify the feasibility of our protocol.
Given that the real and the imaginary parts of the matrix entries are independently measured, we quantify the measurement precision with the total variance ∆ 2 E (l) According to the Eq. (6), the variance can be derived by where Consider N particles are used for one measurement of W mn . The variance of the probability is approximated as δ 2 W mn ≈ W mn /N in the large N limit due to the Poissonian statistic.
As a demonstration, we theoretically derive the precision of directly measuring the off-diagonal matrix entry E 1,0 (θ) of a general measurement operator for a two-dimensional QŜ with different coupling strength g. According to the Eq. (8), the variance of the off-diagonal matrix entry E 1,0 (θ) is given by In Fig. 2 (A), we show how the variance of E 1,0 (θ) changes with different g for four values of θ.
We find that the statistical errors ∆ 2 E 1,0 (θ) become large with a small coupling strength (g → 0 or g → π/2), while the strong coupling strength (g → π/4) significantly decreases the variance to We also compare the characterization precision of E 1,0 (θ) associated with different POVM parameter θ in Fig. 2 (B). The statistical errors ∆ 2 E 1,0 (θ) remain finite over all θ indicating that our protocol is applicable to characterize the arbitrary POVM of The variance of E 1,0 (θ) changes with different parameter θ for the coupling strength g = π/16, π/8, π/4, 3π/8. Here, we take η = 1/2 and N = 12790 to coincide with our experimental conditions. The points X and Y refer to the precision of directly measuring the off-diagonal matrix entry of the two-dimensional symmetric informationally complete positive operator-valued measure with the coupling strength g = π/4. two-dimensional quantum system. In addition, the variance ∆ 2 E 1,0 (θ) is related to the parameter θ but does not depend on the value of E 1,0 (θ). This implies that the change of the off-diagonal matrix entries of the measurement operator, such as the dephasing and the phase rotation process will not affect the characterization precision. We note that the choice of the sequential observablesÔ B and A is indeed not unique. How to choose the optimal observables of the quantum system to achieve the best characterization precision remains an open question in the field of direct tomography. If the sequential observables of the quantum system are changed, the collective observablesR B,A and T B,A of the meter states should also be changed correspondingly to reveal the matrix entries E It has been shown that the completeness condition of the POVM {Π l }, i.e., lΠ l =Î, can be used to improve the precision of direct quantum detector tomography. ? In the following, we prove that the same condition is also helpful to improve the precision in the direct characterization of E (l) a j a k (j = k). Since the real part of the entries E leading to the optimal precision

Experiment
In the experiment, we apply the DT protocol to characterize the symmetric informationally complete (SIC) POVM in the polarization degree of freedom (DOF) of photons. Since the coherence between two polarization base states only changes the off-diagonal entries of the measurement operators, we demonstrate that the dephasing and the phase rotation of the SIC POVM can be monitored by only characterizing the corresponding matrix entries.
The experimental setup is shown in Fig. 3.  Finally, the photons input the unknown detector for the post-selection. By collecting the photons that arrive the outputs, we obtain the measurement results.

Results
In Fig. 4, we compare the experimental results of DT with those of the conventional tomography (CT) as well as the ideal SIC POVM during the dephasing and phase rotation process. The detailed information of characterizing the experimental SIC POVM by CT is provided in the Supplementary Materials. The results of CT shown in Fig. 4 are inferred from the experimental SIC POVM and the calibrated coefficient ξ (during the dephasing process) or the phase φ lc (during the phase rotation process). As shown in Fig. 4 (A), the points in each connecting solid line along the direction of arrows correspond to the relative time delay = 0, 20, 40, 60, 80, 120, 160, 200, 240 (λ). The increase of the relative time delay between the separated wave packets reduces the overlap of the temporal wavefunction ξ = ϕ(t H )|ϕ(t V ) , which leads to the dephasing of the quantum measurement. The relation between the relative time delay and the coefficient ξ is calibrated in In Fig. 4 (B), we plot E The statistical noise determines the ultimate precision that can be achieved for a specific amount of input resources, which is an important metric to evaluate whether a measurement protocol is efficient or not.
The statistical errors of the experimental results are shown in Fig. 4 (C). The theoretical precision, represented by dashed lines in (C) and (D), is inferred by assuming that the matrix entries rotation of the quantum measurement. We can also find that the characterization precision after using the completeness condition in Fig. 4 (D) is significantly improved compared to the original precision in Fig. 4 (C).

Discussion and Conclusions
We have proposed a protocol to directly characterize the individual matrix entries of the general POVM, extending the scope of the direct tomography scheme. Our expression is rigorous for the arbitrary coupling strength, which allows to change the coupling strength to improve the precision and simultaneously maintain the accuracy. The statistical errors are finite over all the choice of the POVM parameter demonstrating the feasibility of our protocol for the arbitrary POVM. In particuliar, if the completeness condition of the POVM is appropriately used, the measurement precision can be further improved. Our results indicate that the characterization precision is not affected by the dephasing and phase rotation that only change the off-diagonal matrix entries of the measurement operators. Another typical noise is the phase diffusion meaning that the phase of the quantum measurements randomly jitters. According to the derivations in the paper, 73 the phase diffusion decreases the modulus of the off-diagonal matrix entries in a similar way to the dephasing in our work. Therefore, it is expected that the precision of our protocol is immune to the incoherent noise, such as phase diffusion.
Since some properties of quantum measurements may depend on a part of matrix entries of the measurement operators, this protocol allows us to reveal these properties without the full tomography. We experimentally demonstrate that the evolution of the coherence of a quantum measurement can be monitored through determining the off-diagonal matrix entries of the measurement operators. Our scheme makes no assumptions about the basis to represent the measurement operators.
The choice of the basis depends on the specific conditions or can be optimized according to the research goals. For example, the quantum properties can be basis-dependent (e.g., coherence), or are better revealed with proper choice of the basis (e.g., entanglement). Our scheme provides the flexibility to characterize the matrix entries of the measurement operators in any basis of interest by adjusting the initial quantum state as well as the sequential observables while other parts of the theoretical framework remain unchanged. This feature is an advantage for us to explore the coherence properties or to seek the optimal entanglement witness. 74 Our protocol can be extended to high-dimensional quantum system, in which the coherence information of the quantum measurement among specified base states is of interest.
where Tr W {·} denotes the partial trace in the walker position DOF. We implement the dephasing of the POVM {Π l } by coupling the QS to the environment state ρ E under the Hamilto-nianĤ SE = 2 δ(t − t 0 )ĈΩ, in whichĈ = |a j a j | − |a k a k | andΩ are the observables of the QS and the environment, respectively. By reducing the environment DOF, the measurement operatorΠ l is transformed toΠ D l = Tr E Û † SEΠ l ⊗ ρ EÛSE . We can infer that the dephasing process only changes the related matrix entries E (l) a j a k ξ with the coefficient

A.2 Experimental Calibration
To calibrate the relation between the coefficient ξ and the relative time delay = |t H − t V |, we construct the setup shown in Fig. 5 (a), in which both the half-wave plates (HWPs) are set to 22.5 • .
The photons in |H inputs the calibration setup resulting in the final state after the second HWP Then, ρ D is projected to the basis {|H , |V } with a polarizing beam displacer (PBD), obtaining the probabilities P H and P V . The parameter ξ is given by ξ = P H − P V . The relation between ξ and the relative time delay is shown in the Fig. 5 (B), in which we take from 0 to 260 times the wavelength (λ = 830nm) and the red circled points are adopted for the experiment. The liquid crystal imposes a relative phase φ lc between |H and |V controlled by the voltage.
Through the calibration setup in Fig. 5 (A), the phase can be obtained by φ lc = arccos[2(P H − The calibration results of the relation between the phase φ lc and the applied voltage are shown in Fig. 5 (C). Here, we adjust the voltages to 1.32V and 2.01V and the relative phases are approximately 4π/5 and 2π/5.

Disclosures
The authors declare no conflicts of interest. † These authors contribute equally to this work.