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arXiv:2304.14370

Heisenberg Limit beyond Quantum Fisher Information

Wojciech Górecki

Published Apr 27, 2023• 1 citation
arXivHTMLPDF

Abstract

The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of $N$ resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with $N$ better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources $N$ is strictly constrained, I show that the final measurement uncertainty is $π$ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.

DOI
10.48550/arxiv.2304.14370

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The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter.The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla.In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N → ∞, which is called the asymptotic QFI.Over the years, researchers found various types of upper bounds of the asymptotic QFI, but they were proven achievable only in several specific situations.It was known that the asymptotic QFI of an arbitrary quantum channel grows either linearly or quadratically with N .Here we show that a simple criterion can determine whether the scaling is linear or quadratic.In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are computable via a semidefinite program.When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered.When the scaling is linear, we show that the asymptotic QFI is still in general larger than N times the single-channel QFI and, furthermore, that sequential estimation strategies provide no advantage over parallel ones.
We discuss the Heisenberg limit in the multiparameter metrology within two different paradigms: one where the measurement is repeated many times (so the Cram\'er-Rao bound is guaranteed to be asymptotically saturable). and the second one where all the resources are allocated into one experimental realization (analyzed with the mimimax approach). We investigate the potential advantage of measuring all the parameters simultaneously compared to estimating them individually, while spending the same total amount of resources. We show that in general the existence of such an advantage, its magnitude, and conditions under which it occurs depends on which of the two paradigms has been chosen. In particular, for the problem of magnetic field sensing using NNN entangled spin 12\frac{1}{2}21​, we show that the predictions based purely on the Cram\'er-Rao formalism may be overly pessimistic in this matter: the minimax approach reveals the superiority of measuring all the parameters jointly whereas the Cram\'er-Rao approach indicates lack of such an advantage.
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.
Quantum metrology with indefinite causal order (ICO) has attracted intense interest due to its potential to surpass the limitations of conventional fixed-order strategies. A key open question is whether ICO can fundamentally enhance asymptotic precision scaling. In this work, we bridge this gap for the estimation of a single parameter encoded in NNN identical uses of a finite-dimensional quantum channel. We first establish a universal Heisenberg-scaling upper bound for the full general ICO process-matrix class and show that for unitary channels its optimal quantum Fisher information (QFI) coincides exactly with that of parallel strategies. For noisy channels, a structurally refined bound shows that channels restricted to the standard quantum limit (SQL) under parallel strategies remain SQL-limited under general ICO strategies. Most significantly, an asymptotically tight (AT) bound is derived to close the remaining possibility of an asymptotic ICO advantage by showing that general ICO and optimal parallel strategies have exactly the same leading QFI coefficient in both the SQL and Heisenberg regimes. For the operationally motivated class of quantum circuits with quantum control of causal order, we further obtain an iterative constraint on finite-query precision whose asymptotic limit agrees with that of the AT bound. Our results clarify the ultimate role of indefinite causality as a metrological resource for quantum channel estimation.
Non-Hermitian quantum metrology, an emerging field at the intersection of quantum estimation and non-Hermitian physics, holds promise for revolutionizing precision measurement. Here, we present a comprehensive investigation of non-Hermitian quantum parameter estimation in the quantum regime, with a special focus on achieving Heisenberg scaling. We introduce a concise expression for the quantum Fisher information (QFI) that applies to general non-Hermitian Hamiltonians, enabling the analysis of estimation precision in these systems. Our findings unveil the remarkable potential of non-Hermitian systems to attain the Heisenberg scaling of 1/ t , where t represents time. Moreover, we derive optimal measurement conditions based on the proposed QFI expression, demonstrating the attainment of the quantum Cramér-Rao bound. By constructing non-unitary evolutions governed by two non-Hermitian Hamiltonians, one with parity-time symmetry and the other without specific symmetries, we experimentally validate our theoretical analysis. The experimental results affirm the realization of Heisenberg scaling in estimation precision, marking a substantial milestone in non-Hermitian quantum metrology.
Abstract Heisenberg scaling characterizes the ultimate precision of parameter estimation enabled by quantum mechanics, which represents an important quantum advantage of both theoretical and technological interest. Here, we present a comprehensive and rigorous study of the attainability of strong, global notions of Heisenberg scaling (in contrast to the commonly studied local estimation based on e.g. quantum Fisher information) in the fundamental problem of quantum metrology, in noisy environments. As our first contribution, we formally define two useful notions of Heisenberg scaling in global estimation respectively based on the average estimation error and the limiting distribution of estimation error (which we highlight as a practically important figure of merit). A main result of this work is that for the standard phase damping noise, an O ( n −1 ) noise rate is a necessary and sufficient condition for attaining global Heisenberg scaling. We first prove that O ( n −1 ) is an upper bound on the noise rate for Heisenberg scaling to be possible, and then show by constructing a ‘robust’ estimation procedure that global Heisenberg scaling in both senses can indeed be achieved under Θ( n −1 ) noise. In addition, we provide a practically more friendly adaptive protocol using only an one-qubit memory, which achieves global Heisenberg scaling in terms of limiting distribution as well under O ( n −1 ) noise.