Accurate estimation of unknown quantum states using limited number of samples is very important but difficult task using fixed measurement basis states since the best local unbiased estimator (LUE) for estimating the parameter depends, in general, on the unknown parameter itself. In order to solve this problem, Nagaoka advocated an adaptive quantum state estimation (AQSE) procedure [1], where the measurement basis states are adaptively selected for each measurement result. Fujiwara proved the strong consistency and asymptotic efficiency for AQSE [2]. Recently, AQSE has been demonstrated by us for the linear polarizations of photons [3]. However, AQSE was designed for a static quantum state, and was not applicable to dynamic states where quantum states change in time. Here, we propose a new protocol, sequential adaptive quantum state estimation (SAQSE) for dynamic quantum states. Our numerical simulation (shown below) clearly show that the estimated value given by SAQSE can follow the true value changing in time whereas AQSE fails. We believe this new protocol will open up the use of quantum state estimation for various applications including biology, e.g. monitoring the motion of fluorescent molecules.
Accurate estimation of unknown quantum states using limited number of samples is very important but difficult task using fixed measurement basis states since the best local unbiased estimator (LUE) for estimating the parameter depends, in general, on the unknown parameter itself. In order to solve this problem, Nagaoka advocated an adaptive quantum state estimation (AQSE) procedure [1], where the measurement basis states are adaptively selected for each measurement result. Fujiwara proved the strong consistency and asymptotic efficiency for AQSE [2]. Recently, AQSE has been demonstrated by us for the linear polarizations of photons [3]. However, AQSE was designed for a static quantum state, and was not applicable to dynamic states where quantum states change in time. Here, we propose a new protocol, sequential adaptive quantum state estimation (SAQSE) for dynamic quantum states. Our numerical simulation (shown below) clearly show that the estimated value given by SAQSE can follow the true value changing in time whereas AQSE fails. We believe this new protocol will open up the use of quantum state estimation for various applications including biology, e.g. monitoring the motion of fluorescent molecules.
Accurate estimation of unknown quantum states using limited number of samples is very important but difficult task using fixed measurement basis states since the best local unbiased estimator (LUE) for estimating the parameter depends, in general, on the unknown parameter itself. In order to solve this problem, Nagaoka advocated an adaptive quantum state estimation (AQSE) procedure. Later, Fujiwara proved the strong consistency and asymptotic efficiency for AQSE. AQSE has been demonstrated by us for the linear polarizations of photons, and recently for a single polarization qubit. However, in order to apply AQSE to actual measurement including application to life science, we need to improve the speed of the system which is limited by the response time of the mechanical rotation stage used in the previous experiment [Phys. Rev. Lett. 109, 130404 (2012)]. To solve this problem, we realized high-speed AQSE experiment using liquid crystal retarder (LCR). We demonstrate the adaptive quantum state estimation using a LCR instead of a half wave plate on a mechanical rotator in previous research. We realized a substantial speed-up of estimation time, keeping the accuracy of estimation at the same level. In fact, it took 0.40 second per photon in the previous study, whereas 0.073 second per photon in the present. Specifically, the time to determine the polarization measurement basis was greatly decreased, from 0.34 second per photon in the previous study to 0.010 second per photon in the present.
We propose an adaptive quantum state estimation method for nonstationary quantum states and verify the method by numerical simulations and experimental investigations. Adaptive quantum state estimation provides an asymptotically optimal scheme for estimating an unknown input quantum state by updating the measurement configuration upon the detection of each quantum. However, previous methods are only valid for quanta with the same quantum state (a stationary state). By adopting the likelihood function for a fixed number of recent detection results, our sequential adaptive quantum state estimation allows quantum states changing in time to be estimated. The numerical simulation results and experimental demonstration using photons agree well with the theoretical predictions. This method will find applications in various fields where dynamically changing quantum states need to be estimated.
Summary form only given. Quantum theory is inherently statistical. This entails repetition of experiments over a number of identically prepared quantum objects, for example, quantum states, if one wants to know the “true state” or the “true value” of the parameter that specifies the quantum state. Such an estimation procedure is particularly important for quantum communication and quantum computation, and is also indispensable to quantum metrology [1,2]. In applications, one needs to design the estimation procedure in such a way that the estimated value of the parameter should be close to the true value (consistency), and that the uncertainty of the estimated value should be as small as possible (efficiency) for a given limited number of samples. In order to realize these requirements, Nagaoka advocated an adaptive quantum state estimation (AQSE) procedure [3], and recently Fujiwara proved the strong consistency and asymptotic efficiency for AQSE [4]. In this paper, we report the first experimental demonstration of AQSE using photons[5]. The angle of a half wave plate (HWP) that initializes the linear polarization of input photons is estimated using AQSE (Fig. 1). sequence of AQSE is carried out with 300 input photons, and the sequence is repeated 500 times for four different settings of HWP. The statistical analysis of these results verifies the strong consistency and asymptotic efficiency of AQSE. Recently, it has been mathematically proven that the precision of AQSE outperforms the conventional state tomography [6]. It is thus expected that AQSE will provide a useful methodology in the broad area of quantum information processing, communication, and metrology. This work was supported in part by JSPS Quantum Cybernetics, JSPS-Kakenhi, JST-CREST, FIRST Program, Special Coordination Funds for Promoting Science and Technology, Research Foundation for Opto-Science and Technology.
The first experimental demonstration of an adaptive quantum state estimation (AQSE) is reported. The strong consistency and asymptotic efficiency of AQSE have been mathematically proven [A. Fujiwara, J. Phys. A 39, 12489 (2006)]. In this Letter, the angle of linear polarization of single photons, the phase parameter between the right and the left circularly polarization, is estimated using AQSE, and the strong consistency and asymptotic efficiency are experimentally verified. AQSE will provide a general useful method in both quantum information processing and metrology.
Abstract Estimation of physical observables for unknown quantum states is an important problem that underlies a wide range of fields, including quantum information processing, quantum physics, and quantum chemistry. In the context of quantum computation, in particular, existing studies have mainly focused on holistic state tomography or estimation on specific observables with known classical descriptions, while this lacks the important class of problems where the estimation target itself relies on the measurement outcome. In this work, we propose an adaptive measurement optimization method that is useful for the quantum subspace methods, namely the variational simulation methods that utilize classical postprocessing on measurement outcomes. The proposed method first determines the measurement protocol for classically simulatable states, and then adaptively updates the protocol of quantum subspace expansion (QSE) according to the quantum measurement result. As a numerical demonstration, we have shown for excited-state simulation of molecules that (i) we are able to reduce the number of measurements by an order of magnitude by constructing an appropriate measurement strategy (ii) the adaptive iteration converges successfully even for a strongly correlated molecule of H4. Our work reveals that the potential of the QSE method can be empowered by elaborated measurement protocols, and opens a path to further pursue efficient quantum measurement techniques in practical computations.
We herein report the experimental demonstration of adaptive quantum state estimation for totally unknown photonic qubits. Similar to our previous study [R. Okamoto et al., Phys. Rev. Lett. 109, 130404 (2012)], the measurement configuration is updated using the results of each photon detection event so that our method does not require prior knowledge of the total number of samples. The experimental results obtained herein demonstrate both strong consistency and asymptotic efficiency through several rigorous statistical tests. Furthermore, we demonstrate that the experimentally obtained distribution of the states estimated using an adaptive quantum state estimation is significantly different from that obtained by conventional state tomography and agrees well with theoretical predictions.