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Schematic diagram of adaptive quantum state estimation for single photonic qubits. Single photons with a fixed unknown polarization are emitted from a photon generator. The polarization is analyzed by QWP1, HWP1, and a polarizing beam splitter (PBS). The controller sets QWP1 and HWP1 to an angle calculated based on the photon measurement results.
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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 knowledg...
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... paper, we report the first experimen- tal demonstration of a multi-parameter AQSE for totally unknown photonic qubits. Similar to the one-parameter AQSE presented in [14], the measurement configuration is updated using the results of each photon detection event so that our method does not require prior knowl- edge of the total number of samples (Fig. 1). The exper- imental results obtained herein demonstrate both strong consistency and asymptotic efficiency through several rig- orous statistical tests. Furthermore, we show that the experimentally obtained distribution of the states esti- mated using AQSE is significantly different from that obtained by conventional state tomography ...
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... superiority of AQSE compared to tomography becomes much clearer if the true state is taken in the vicinity of the surface of the Bloch ball, that is, if the true state is (almost) pure. Fig. 10 shows the 200 esti- mated states at n = 798 when the true state is set to be θ * = (0.577, 0.577, 0.577). For this true state, the esti- mated states of AQSE (Fig. 10(a)) have a much thinner pancake-like distribution, as compared with Fig. 8(a). Simple tomography (Fig. 10(b)) still has a spherical dis- tribution around θ * and, since ...
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... of AQSE compared to tomography becomes much clearer if the true state is taken in the vicinity of the surface of the Bloch ball, that is, if the true state is (almost) pure. Fig. 10 shows the 200 esti- mated states at n = 798 when the true state is set to be θ * = (0.577, 0.577, 0.577). For this true state, the esti- mated states of AQSE (Fig. 10(a)) have a much thinner pancake-like distribution, as compared with Fig. 8(a). Simple tomography (Fig. 10(b)) still has a spherical dis- tribution around θ * and, since the center point is near the surface now, approximately half of the estimated states are unphysical and are to be projected onto the Bloch ball by ML tomography (Fig. ...
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... the surface of the Bloch ball, that is, if the true state is (almost) pure. Fig. 10 shows the 200 esti- mated states at n = 798 when the true state is set to be θ * = (0.577, 0.577, 0.577). For this true state, the esti- mated states of AQSE (Fig. 10(a)) have a much thinner pancake-like distribution, as compared with Fig. 8(a). Simple tomography (Fig. 10(b)) still has a spherical dis- tribution around θ * and, since the center point is near the surface now, approximately half of the estimated states are unphysical and are to be projected onto the Bloch ball by ML tomography (Fig. 10(c)). In this way, the number of unphysical estimates increases as the purity approaches 1, and thus the ...
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... of AQSE (Fig. 10(a)) have a much thinner pancake-like distribution, as compared with Fig. 8(a). Simple tomography (Fig. 10(b)) still has a spherical dis- tribution around θ * and, since the center point is near the surface now, approximately half of the estimated states are unphysical and are to be projected onto the Bloch ball by ML tomography (Fig. 10(c)). In this way, the number of unphysical estimates increases as the purity approaches 1, and thus the effect of the maximum likeli- hood data processing is sensitive to the ...
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... us observe this tendency from a different view- point. Fig. 11 shows the evolution of the weighted trace of sample covariance for θ * = (0.577, 0.577, 0.577). The purple, blue, and green dots indicate the data obtained by AQSE, simple tomography, and ML tomography, re- spectively. Furthermore, the purple and blue dashed horizontal lines indicate the corresponding theoretical values for AQSE and ...
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Citations
... In particular, various forms of phase transitions in such systems have been used for achieving quantum-enhanced sensitivity, including first-order, [36][37][38][39] second-order, [25,[40][41][42][43][44][45][46][47][48][49][50][51][52][53] Floquet, [54,55] dissipative, [56][57][58][59][60][61][62] time crystals, [63,64] topological, [65][66][67][68] many-body [69] and Stark localization [70] phase transitions. Other types of manybody probes profit from diverse measurement methods including adaptive, [71][72][73][74][75][76][77][78][79][80] continuous, [29,62,81,84] and sequential [85,86] measurements. Since most of the sensing proposals in many-body probes have been dedicated to short-range interactions, a key open problem is whether long-range interactions can provide more benefits for sensing tasks? ...
In contrast to interferometry-based quantum sensing, where interparticle interaction is detrimental, quantum many-body probes exploit such interactions to achieve quantum-enhanced sensitivity. In most of the studied quantum many-body probes, the interaction is considered to be short-ranged. Here, we investigate the impact of long-range interaction at various filling factors on the performance of Stark quantum probes for measuring a small gradient field. These probes harness the ground state Stark localization phase transition which happens at an infinitesimal gradient field as the system size increases. Our results show that while superHeisenberg precision is always achievable in all ranges of interaction, the longrange interacting Stark probe reveals two distinct behaviors. First, by algebraically increasing the range of interaction, the localization power enhances and thus the sensitivity of the probe decreases. Second, as the interaction range becomes close to a fully connected graph its effective localization power disappears and thus the sensitivity of the probe starts to enhance again. The super-Heisenberg precision is achievable throughout the extended phase until the transition point and remains valid even when the state preparation time is incorporated in the resource analysis. As the probe enters the localized phase, the sensitivity decreases and its performance becomes size-independent, following a universal behavior. In addition, our analysis shows that lower filling factors lead to better precision for measuring weak gradient fields.
... Thus, we present simulations of two adaptive methods in this section to demonstrate the potential usefulness of the ST-POVM in the scenario where we have some limited amount of prior knowledge. Adaptive approaches to tomography have been considered before both theoretically [40] and experimentally [42][43][44]. In this section, we will first illustrate the simulations of an adaptive measurement scheme on states assumed to be completely unknown. ...
Quantum bits, or qubits, are the fundamental building blocks of present quantum computers. Hence, it is important to be able to characterize the state of a qubit as accurately as possible. By evaluating the qubit characterization problem from the viewpoint of quantum metrology, we are able to find optimal measurements under the assumption of good prior knowledge. We implement these measurements on a superconducting quantum computer. Our demonstration produces sufficiently low error to allow the saturation of the theoretical limits, given by the Nagaoka-Hayashi bound. We also present simulations of adaptive measurement schemes utilizing the proposed method. The results of the simulations show the robustness of the method in characterizing arbitrary qubit states with different amounts of prior knowledge.
... An enormous ensemble of the unknown state is used to extract the quantum state using the set of known bases. These bases could either be fixed or adaptive depending on the feasibility of the method according to the available resources and required metrics [1][2][3] . A renowned conundrum in quantum state tomography is the exponential growth of associated parameters with an increase in the dimensionality of the quantum state 4 . ...
Quantum state tomography is a process for estimating an unknown quantum state; which is innately probabilistic. The exponential growth of unknown parameters to be estimated is a fundamental difficulty in realizing quantum state tomography for higher dimensions. Iterative optimization algorithms like self-guided quantum tomography have been effective in robust and accurate ascertaining a quantum state even with exponential growth in Hilbert space. We propose a faster convergent simultaneous perturbation stochastic approximation algorithm which is more practical in a resource-deprived situation for determining the underlying quantum states by incorporating the Barzilai–Borwein two-point step size gradient method with minimal loss of accuracy.
... terval. Hence, tuning the system to operate optimally at its quantum phase transition point can be very elusive and practically demanding, e.g., adaptive sensing strategies with complex measurement types have to be employed [2,[59][60][61][62][63][64][65][66]. A key open question is whether one can employ such sensors for global sensing, where the unknown parameter varies over a wide range. ...
Quantum sensing is one of the key areas which exemplifies the superiority of quantum technologies. Nonetheless, most quantum sensing protocols operate efficiently only when the unknown parameters vary within a very narrow region, i.e., local sensing. Here, we provide a systematic formulation for quantifying the precision of a probe for multi-parameter global sensing when there is no prior information about the parameters. In many-body probes, in which extra tunable parameters exist, our protocol can tune the performance for harnessing the quantum criticality over arbitrarily large sensing intervals. For the single-parameter sensing, our protocol optimizes a control field such that an Ising probe is tuned to always operate around its criticality. This significantly enhances the performance of the probe even when the interval of interest is so large that the precision is bounded by the standard limit. For the multi-parameter case, our protocol optimizes the control fields such that the probe operates at the most efficient point along its critical line. Interestingly, for an Ising probe, it is predominantly determined by the longitudinal field. Finally, we show that even a simple magnetization measurement significantly benefits from our optimization and moderately delivers the theoretical precision.
... One of the complex problems in quantum metrology is that the optimal measurement basis, computed from the eigenvetors of the SLD operator L T , depend on the unknown 043338-5 parameter, heren. The typical recipe for this problem is to follow complex adaptive approaches [69][70][71][72][73][74] to update the measurement basis iteratively by extracting information about the exact value of the unknown parameter. In practice, to avoid such complexity, it is of significant importance if one can determine a fixed measurement basis which is independent of the unknown parameter and maximizes the quantum Fisher information. ...
Optomechanical systems are promising platforms for controlled light-matter interactions. They are capable of providing several fundamental and practical novel features when the mechanical oscillator is cooled down to nearly reach its ground state. In this framework, measuring the effective temperature of the oscillator is perhaps the most relevant step in the characterization of those systems. In conventional schemes, the cavity is driven strongly, and the overall system is well-described by a linear (Gaussian preserving) Hamiltonian. Here, we depart from this regime by considering an undriven optomechanical system via non-Gaussian radiation-pressure interaction. To measure the temperature of the mechanical oscillator, initially in a thermal state, we use light as a probe to coherently interact with it and create an entangled state. We show that the optical probe gets a nonlinear phase, resulting from the non-Gaussian interaction, and undergoes an incoherent phase diffusion process. To efficiently infer the temperature from the entangled light-matter state, we propose using a nonlinear Kerr medium before a homodyne detector. Remarkably, placing the Kerr medium enhances the precision to nearly saturate the ultimate quantum bound given by the quantum Fisher information. Furthermore, it also simplifies the thermometry procedure as it makes the choice of the homodyne local phase independent of the temperature.
... One of the complex problems in quantum metrology is that the optimal measurement basis, computed from the eigenvetors of the SLD operator L T , depend on the unknown parameter, heren. The typical recipe for this problem is to follow complex adaptive approaches [69][70][71][72][73][74] to update the measurement basis iteratively by extracting information about the exact value of the unknown parameter. In practice, to avoid such complexity, it is of significant importance if one can determine a fixed measurement basis which is independent of the unknown parameter and maximizes the quantum Fisher information. ...
Optomechanical systems are promising platforms for controlled light-matter interactions. They are capable of providing several fundamental and practical novel features when the mechanical oscillator is cooled down to nearly reach its ground state. In this framework, measuring the effective temperature of the oscillator is perhaps the most relevant step in the characterization of those systems. In conventional schemes, the cavity is driven strongly, and the overall system is well-described by a linear (Gaussian preserving) Hamiltonian. Here, we depart from this regime by considering an undriven optomechanical system via non-Gaussian radiation-pressure interaction. To measure the temperature of the mechanical oscillator, initially in a thermal state, we use light as a probe to coherently interact with it and create an entangled state. We show that the optical probe gets a nonlinear phase, resulting from the non-Gaussian interaction, and undergoes an incoherent phase diffusion process. To efficiently infer the temperature from the entangled light-matter state, we propose using a nonlinear Kerr medium before a homodyne detector. Remarkably, placing the Kerr medium enhances the precision to nearly saturate the ultimate quantum bound given by the quantum Fisher information. Furthermore, it also simplifies the thermometry procedure as it makes the choice of the homodyne local phase independent of the temperature, which avoids the need for adaptive sensing protocols.
... Other various types of adaptive schemes have been intensively studied recently both in theory and experiment. See, for example, [47][48][49][50][51][52] and a review paper [53] and references therein. ...
In this paper, we analyze quantum-state estimation problems when some of the parameters are of no interest to be estimated. In classical statistics, these irrelevant parameters are called nuisance parameters, and this problem is of great importance in many practical applications of statistics. However, little is known regarding the effects of nuisance parameters in quantum-state estimation problems. The main contribution of this paper is first to formulate the nuisance parameter problem in the quantum-state estimation theory for the finite sample size case. We then derive the precision limit for the parameters of interest based on the concept of locally unbiasedness for the parameters of interest. We also propose a method of how to eliminate the nuisance parameters to obtain an upper bound for estimation error of the parameters of interest. Based on the proposed method, we clarify an intrinsic tradeoff relation to estimate the nuisance parameters and parameters of interest. The general qubit models are examined in detail to emphasize that we cannot ignore the effects of the nuisance parameters in general. Several examples are worked out to illustrate our findings.
... [737][738][739][740] This has been experimentally demonstrated in the estimation of the photon polarization. [741][742][743][744][745] A detailed review on implementations of feedback controls in quantum systems can be found in Ref. 713. ...
Quantum metrology is one of the most promising applications of quantum technologies. The aim of this research field is the estimation of unknown parameters exploiting quantum resources, whose application can lead to enhanced performances with respect to classical strategies. Several physical quantum systems can be employed to develop quantum sensors, and photonic systems represent ideal probes for a large number of metrological tasks. Here, the authors review the basic concepts behind quantum metrology and then focus on the application of photonic technology for this task, with particular attention to phase estimation. The authors describe the current state of the art in the field in terms of platforms and quantum resources. Furthermore, the authors present the research area of multiparameter quantum metrology, where multiple parameters have to be estimated at the same time. The authors conclude by discussing the current experimental and theoretical challenges and the open questions toward implementation of photonic quantum sensors with quantum-enhanced performances in the presence of noise.
... Finally, adaptive protocols can be also exploited to enhance state discrimination and more general in quantum tomography [708][709][710][711] . This has been experimentally demonstrated in the estimation of the photons polarization [712][713][714][715][716] . A detailed review on implementations of feedback controls in quantum systems can be found in Ref. 687. ...
Quantum Metrology is one of the most promising application of quantum technologies. The aim of this research field is the estimation of unknown parameters exploiting quantum resources, whose application can lead to enhanced performances with respect to classical strategies. Several physical quantum systems can be employed to develop quantum sensors, and photonic systems represent ideal probes for a large number of metrological tasks. Here we review the basic concepts behind quantum metrology and then focus on the application of photonic technology for this task, with particular attention to phase estimation. We describe the current state of the art in the field in terms of platforms and quantum resources. Furthermore, we present the research area of multiparameter quantum metrology, where multiple parameters have to be estimated at the same time. We conclude by discussing the current experimental and theoretical challenges, and the open questions towards implementation of photonic quantum sensors with quantum-enhanced performances in the presence of noise.
... As far as this demonstration is concerned, our method works very efficiently. We note that the present method has been successfully applied to an experimental study using photonic qubits [16]. ...
A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space $(-1,1)^3$ of the Stokes parameters $(\xi_1, \xi_2,\xi_3)$ that specify qubit states should be regarded as a Riemannian manifold endowed with a metric $g_{ij}:=\delta_{ij}/(1-(\xi_i)^2)$, and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric $g_{ij}$. An efficient algorithm for computing the maximum likelihood estimate is also proposed.
