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arXiv:2309.14333

Quantum-Enhanced Parameter Estimation Without Entanglement

Pragati Gupta

Published Sep 25, 2023• 0 citations
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Abstract

Entanglement is generally considered necessary for achieving the Heisenberg limit in quantum metrology. We construct analogues of Dicke and GHZ states on a single $N+1$ dimensional qudit that achieve precision equivalent to symmetrically entangled states on $N$ qubits, showing that entanglement is not necessary for going beyond the standard quantum limit. We define a measure of non-classicality based on quantum Fisher information and estimate the achievable precision, suggesting a close relationship between non-classical states and metrological power of qudits. Our work offers an exponential reduction in the physical resources required for quantum-enhanced parameter estimation, making it accessible on any quantum system with a high-dimensional Hilbert space.

DOI
10.48550/arxiv.2309.14333

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Entanglement is generally considered necessary for achieving the Heisenberg limit in quantum metrology. We construct analogues of Dicke and GHZ states on a single N+1N+1N+1 dimensional qudit that achieve precision equivalent to symmetrically entangled states on NNN qubits, showing that entanglement is not necessary for going beyond the standard quantum limit. We define a measure of non-classicality based on quantum Fisher information and estimate the achievable precision, suggesting a close relationship between non-classical states and metrological power of qudits. Our work offers an exponential reduction in the physical resources required for quantum-enhanced parameter estimation, making it accessible on any quantum system with a high-dimensional Hilbert space.
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We investigate high-precision measurements beyond the standard quantum limit, utilizing nonclassical states. Although entanglement was considered a resource for achieving the Heisenberg limit in measurements, we show that any advantage expected from using entanglement is dependent on the measurement in question. We investigate several measurement scenarios and illustrate the role of entanglement as a resource for quantum high-precision measurement. In particular, we demonstrate that there is a regime wherein entanglement not only does not help, but prevents the achievement of the fundamental limit.
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We introduce a new class of quantum many-particle entangled states, called the Dicke squeezed (or DS) states, which can be used to improve the precision in quantum metrology beyond the standard quantum limit. We show that the enhancement in measurement precision is characterized by a single experimentally detectable parameter, called the Dicke squeezing , which also bounds the entanglement depth for this class of states. The measurement precision approaches the ultimate Heisenberg limit as attains the minimum in an ideal Dicke state. Compared with other entangled states, we show that the DS states are more robust to decoherence and give better measurement precision under typical experimental noise.
Quantum systems allow one to sense physical parameters beyond the reach of classical statistics---with resolutions greater than 1/N1/N1/N, where NNN is the number of constituent particles independently probing the parameter. In principle, the Heisenberg Limit 1/N21/N^{2}1/N2 may even be reached, which requires both the relative size of the largest entangled block and the geometric measure of entanglement to be non-vanishing as N ⁣→ ⁣∞N\!\to\!\inftyN→∞. Yet, we demonstrate that in the asymptotic NNN limit any precision scaling arbitrarily close to the Heisenberg Limit (1/N2−ε1/N^{2-\varepsilon}1/N2−ε with any ε ⁣> ⁣0\varepsilon\!>\!0ε>0) may be attained, even though the system gradually becomes noisier and separable, and both the above entanglement quantifiers asymptotically vanish. Our results thus show that quantum systems achieve nearly optimal resolution scaling despite arbitrarily small amount of entanglement present. In deriving these results, we also establish a continuity relation for the Quantum Fisher Information of independent interest.
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.