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arXiv:2207.05656

Phase estimation with limited coherence

D. Munoz-Lahoz, John Calsamiglia, János A. Bergou, E. Bagán

Published Jul 12, 2022• 0 citations
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Abstract

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, $V(C) \sim a_n/C^2$, where $a_n$ decreases to $π^2/3$ as $n$ increases, leading to a dimension-independent relation.

DOI
10.48550/arxiv.2207.05656

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The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it allows to derive the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In Quantum Phase Estimation protocols, this leads to identify coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e., the Symmetric Logarithmic Derivative, in many cases allows to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows: to discuss the role of coherence vs correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behaviour. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behaviour proper of a large class of quantum phase transitions.
We study the optimization of input states for phase estimation under covariant measurements. Building on Holevo's framework, which provides the optimal covariant measurement for a fixed input state, we further optimize over the input state itself. For a general even 2π2π2π-periodic cost function with non-negative Fourier coefficients, we derive a necessary and sufficient condition for the optimal input state: Its Fock coefficients are determined, up to arbitrary phases, by the eigenvector corresponding to the largest eigenvalue of a Toeplitz matrix defined by the cost function. This characterization yields an explicit expression for the attainable lower bound of the average cost under optimal covariant measurements and shows that this bound asymptotically approaches zero in the infinite-energy limit. For the specific cost function W(θ,θ~)=4sin⁡2[(θ−θ~)/2]W(θ,\tildeθ)=4\sin^2[(θ-\tildeθ)/2]W(θ,θ~)=4sin2[(θ−θ~)/2], we obtain the optimal input state and the corresponding minimum average cost in closed form, demonstrating Heisenberg scaling with respect to the mean photon number.
Phase estimation is a major mission in quantum metrology, especially in quantum interferometry. A full phase estimation scheme usually includes the optimal probe state and measurement. For the finite-dimensional states in Fock basis, the NOON state ceases to be optimal when the average particle number is fixed yet not equal to the Fock dimension (Fock number of the highest occupied Fock state of one mode), and what is the true optimal finite-dimensional probe state in this case is still undiscovered. Hereby we present several theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice. These optimal finite-dimensional probe states reveal an important fact that the Fock dimension could be treated as a metrological resource, and the given scheme is particularly useful in scenarios where weak light or limited particle number is demanded.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
The laws of quantum mechanics place fundamental limits on the accuracy of measurements and therefore on the estimation of physical parameters by a quantum system. In this work, we prove lower bounds on the size of confidence regions reported by any region estimator for a given ensemble of probe states and probability of success. Our bounds are derived from a previously unnoticed connection between the size of confidence regions and the error probabilities of a corresponding binary hypothesis test. In group-covariant scenarios, we find that there is an ultimate bound for any estimation scheme which depends only on the representation-theoretic data of the probe system, and we evaluate its asymptotics in the limit of many systems, establishing a general “Heisenberg limit” for region estimation. We apply our results to several scenarios, in particular to phase estimation, where our bounds strengthen the well-known Heisenberg and shot-noise scaling.