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arXiv:1607.06124 · Physical Review A

Characterization of qubit chains by Feynman probes

Dario Tamascelli, Claudia Benedetti, Stefano Olivares, Matteo G. A. Paris

Published Oct 26, 2016• 41 citations
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Abstract

We address the characterization of qubit chains and assess the performances of local measurements compared to those provided by Feynman probes, i.e., nonlocal measurements realized by coupling a single-qubit register to the chain. We show that local measurements are suitable to estimate small values of the coupling and that a Bayesian strategy may be successfully exploited to achieve optimal precision. For larger values of the coupling Bayesian local strategies do not lead to a consistent estimate. In this regime, Feynman probes may be exploited to build a consistent Bayesian estimator that saturates the Cram\'er-Rao bound, thus providing an effective characterization of the chain. Finally, we show that ultimate bounds to precision, i.e., saturation of the quantum Cram\'er-Rao bound, may be achieved by a two-step scheme employing Feynman probes followed by local measurements.

DOI
10.1103/physreva.94.042129

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We address the characterization of qubit chains and assess the performances of local measurements compared to those provided by Feynman probes, i.e., nonlocal measurements realized by coupling a single-qubit register to the chain. We show that local measurements are suitable to estimate small values of the coupling and that a Bayesian strategy may be successfully exploited to achieve optimal precision. For larger values of the coupling Bayesian local strategies do not lead to a consistent estimate. In this regime, Feynman probes may be exploited to build a consistent Bayesian estimator that saturates the Cram'er-Rao bound, thus providing an effective characterization of the chain. Finally, we show that ultimate bounds to precision, i.e., saturation of the quantum Cram'er-Rao bound, may be achieved by a two-step scheme employing Feynman probes followed by local measurements.
Quantum Cramér--Rao theory is intrinsically local: it bounds precision near a specified parameter value, and its saturating measurement generally depends on that value. Barankin-type bounds use finite parameter displacements, but remain anchored to a chosen reference value. This leaves open a basic global-estimation problem: when the parameter is known only within a broad domain, what precision can be guaranteed by a single estimator and a single measurement strategy fixed before the true value is localized? We answer this question by introducing global score functions tied to a weighted variance over the whole parameter domain. Their correlations generate a hierarchy of precision bounds: global Cramér--Rao and Barankin-type bounds arise as restricted levels, whereas unrestricted score correlations yield a fully global bound for the prescribed weighted variance. The hierarchy recovers local Cramér--Rao theory in the many-repetition limit and reveals genuinely global precision limits for finite data over broad domains. In the quantum setting, the construction identifies when this fully global bound can be realized by a single parameter-independent measurement. The same framework extends to Bayesian estimation, recovering the Van Trees bound in the local limit while yielding stronger finite-width lower bounds on the Bayesian mean-square error beyond this limit.
We address estimation of one-parameter unitary gates for qubit systems and seek optimal probes and measurements. Single- and two-qubit probes are analysed in detail focusing on precision and stability of the estimation procedure. Bayesian inference is employed and compared with the ultimate quantum limits to precision, taking into account the biased nature of the Bayes estimator in the non-asymptotic regime. In addition, through the evaluation of the asymptotic a posteriori distribution for the gate parameter and a comparison with the results of Monte Carlo simulated experiments, we show that asymptotic optimality of the Bayes estimator is actually achieved after a limited number of runs. The robustness of the estimation procedure against fluctuations of the measurement settings is investigated and the use of entanglement to improve the overall stability of the estimation scheme is also analysed in some detail.
We consider the readout process on a probe made of NNN qubits (two-level systems) in a quantum single-parameter estimation scheme. The parameter-dependent evolution of the probe as well as the measurements that are done for readout are assumed to be fixed. We find the optimal initial states of the probe that will saturate the quantum Cram\'er-Rao bound under these conditions. The relation between the optimal state of the NNN-qubit probe and that of the one-qubit probe is obtained for both entangling and nonentangling dynamics for the probe qubits. We study the limitations placed on the optimal initial state of the probe and on the achievable measurement uncertainty by the restrictions on the readout procedure.
Measuring the temperature of a quantum system is an essential task in almost all aspects of quantum technologies. Theoretically, an optimal strategy for thermometry often requires measuring energy, which demands full accessibility over the entire system as well as a complex entangled measurement basis. In this paper, we take a different approach and show that single-qubit sequential measurements in the computational basis not only allow for precise thermometry of a many-body system, but may also achieve precision beyond the thermometry capacity of the probe at equilibrium, given by the Cram\'er-Rao bound. Thus, using consecutive single-qubit measurements of the probe out of equilibrium is, in most cases, very beneficial, as it achieves lower-temperature uncertainties and avoids demanding energy measurements when compared with probes at thermal equilibrium. To obtain such precision, the time between the two subsequent measurements should be smaller than the thermalization time so that the probe never thermalizes. Therefore, the nonequilibrium dynamics of the system continuously imprint information about temperature in the state of the probe. To demonstrate the generality of our findings, we consider thermometry in both spin chains and the Jaynes-Cummings model.
Quantum measurement plays a crucial role in quantum metrology. Due to the limitations of experimental capabilities, collectively measuring multiple copies of probing systems can present significant challenges. Therefore, the concept of locality in quantum measurements must be considered. In this work, we investigate the possibility of achieving the quantum Cramér-Rao bound (QCRB) through local measurements (LMs). We first demonstrate that if there exists a LM to saturate the QCRB for qubit systems, then we can construct another rank-1 local projective measurement to saturate the QCRB. In this sense, rank-1 local projective measurements are sufficient to analyze the problem of saturating the QCRB. For pure qubits, we propose two necessary and sufficient methods to determine whether and how a given parameter estimation model can achieve the QCRB through LMs. The first method, dubbed the iterative matrix partition method and based on unitary transformations that render the diagonal entries of a traceless matrix vanish, elucidates the underlying mathematical structure of LMs as well as the local measurements with classical communications (LMCC), generalizing the result by Zhou . [], which only holds for the later case. We clarify that the saturation of the QCRB through LMs for the Greenberger–Horne–Zeilinger-encoded states is actually due to the self-similar structure in this approach. The second method, dubbed the hierarchy of orthogonality conditions and based on the parametrization of rank-1 measurements for qubit systems, allows us to construct several examples of saturating QCRBs, including the three-qubit W states and the N -qubit W states ( N ≥ 3 ). Our findings offer insights into achieving optimal performance in quantum metrology when measurement resources are limited.
Distributed quantum sensing network has the potential of enhancing the precision in estimating a global function of local parameters by utilizing an entangled probe, compared with that achieved with separable probes. This advantage is often characterized as a quadratic improvement of the quantum Cramér-Rao bound (QCRB). This argument is incomplete in that QCRB assumes a team of all-powerful sensors that can perform arbitrary joint measurements allowed by quantum mechanics. An immediate question arises as to whether such an advantage persists for isolated sensors with physically motivated constraints in their measurement strategies. In this paper, we first consider local operations and classical communication (LOCC) strategies and prove that the QCRB is indeed asymptotically attainable for arbitrary pure probe states, by extending previous work on single-parameter estimation [1]. We further numerically analyze a more restricted scenario where the sensors can only make independent local measurements, and provide evidence that the QCRB is not informative enough for comparing different probe states.