The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.
The problems of optimally estimating a phase, a direction, and the orientation of a Cartesian frame (or trihedron) with general pure states are addressed. Special emphasis is put on estimation schemes that allow for inconclusive answers or abstention. It is shown that such schemes enable drastic improvements, up to the extent of attaining the Heisenberg limit in some cases, and the required amount of abstention is quantified. A general mathematical framework to deal with the asymptotic limit of many qubits or large angular momentum is introduced and used to obtain analytical results for all the relevant cases under consideration. Parameter estimation with abstention is also formulated as a semidefinite programming problem, for which very efficient numerical optimization techniques exist.
The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an inconclusive answer at readout, can drastically improve the measurement precision and even lead to a change in its asymptotic behavior, from the shot-noise to the Heisenberg scaling. We focus on phase estimation and quantify the required amount of abstention for a given precision. We also develop analytical tools to obtain the asymptotic behavior of the precision and required rate of abstention for arbitrary pure states.
We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consist of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are illustrated for small values of N . For asymptotically large N , we derive analytical expressions of the fidelity and the probability of abstention and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback–Leibler (relative) entropy. As a byproduct, we give an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides the most significant improvement as compared to the standard approach.
Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy -- First, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j, then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N=2,3, or approximately when the qubits are nearly pure for other N values. Our results point to the conjecture that using a Bell multiport setup, one can distinguish between projection onto the j=N/2 and j=N/2−1 subspaces from their respective interference signatures at the output.
Abstract Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of two parameters that are respectively associate with the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy—first, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j , then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show that when the state is nearly pure, for sufficiently but not arbitrarily large N , most information will be captured in the largest three j -subspaces. Then, we study how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N = 2, 3, or approximately when the qubits are nearly pure for other N values. We also obtain numerical results that suggest that using a Bell multiport setup, one can distinguish between projection onto the j = N /2 and j = N /2 − 1 subspaces from their respective interference signatures at the output.
Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances of the problem, but in general even numerical methods are not known. Here, we focus on the single-shot Bayesian setting, where the goal is to find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024
We investigate the ultimate precision limits for quantum phase estimation in terms of the coherence, C, of the probe. For pure states, we give the minimum estimation variance attainable, V(C), and the optimal state, in the asymptotic limit when the probe system size, n, is large. We prove that pure states are optimal only if C scales as n with a sufficiently large proportionality factor, and that the rank of the optimal state increases with decreasing C, eventually becoming full-rank. We show that the variance exhibits a Heisenberg-like scaling, V(C)∼an/C2, where an decreases to π2/3 as n increases, leading to a dimension-independent relation.