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arXiv:2507.03512

Entanglement-Constrained Quantum Metrology: Rapid Low-Entanglement Gains, Tapered High-Level Growth

Debarupa Saha, Ujjwal Sen

Published Jul 4, 2025• 0 citations
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Abstract

It is a specific type of quantum correlated state that achieves optimal precision in parameterestimation under unitary encoding. We consider the potential experimental limitation on probe entanglement, and find a relation between achievable precision and initial probe entanglement, in both bipartite and multipartite scenarios. For two-qubit probes, we analytically derive an exact relationship between the entanglement-constrained optimal quantum Fisher information and the limited initial entanglement, measured via both generalized geometric measure and entanglement entropy. We demonstrate that this fundamental relationship persists across the same range of the entanglement measures even when higher-dimensional bipartite probes are considered. Furthermore, we identify the specific states that realize maximum precision in these scenarios. Additionally, by considering the geometric measure of entanglement, we extend our approach to multiqubit probes. We find that in every case, the optimal quantum Fisher information exhibits a universal behavior:a steep increase in the low-entanglement regime, followed by a gradual and nearly-saturated improvement as the probe entanglement approaches values close to those required for achieving the Heisenberg limit.

DOI
10.48550/arxiv.2507.03512

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It is a specific type of quantum correlated state that achieves optimal precision in parameterestimation under unitary encoding. We consider the potential experimental limitation on probe entanglement, and find a relation between achievable precision and initial probe entanglement, in both bipartite and multipartite scenarios. For two-qubit probes, we analytically derive an exact relationship between the entanglement-constrained optimal quantum Fisher information and the limited initial entanglement, measured via both generalized geometric measure and entanglement entropy. We demonstrate that this fundamental relationship persists across the same range of the entanglement measures even when higher-dimensional bipartite probes are considered. Furthermore, we identify the specific states that realize maximum precision in these scenarios. Additionally, by considering the geometric measure of entanglement, we extend our approach to multiqubit probes. We find that in every case, the optimal quantum Fisher information exhibits a universal behavior:a steep increase in the low-entanglement regime, followed by a gradual and nearly-saturated improvement as the probe entanglement approaches values close to those required for achieving the Heisenberg limit.
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 a parameter. In the canonical phase-sensing scenario the Heisenberg limit 1/N21/{N}^{2}1/N2 may be reached, which requires, as we show, both the relative size of the largest entangled block and the geometric measure of entanglement to be nonvanishing as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Yet, we also demonstrate that in the asymptotic NNN limit any precision scaling arbitrarily close to the Heisenberg limit ($1/{N}^{2\ensuremath{-}\ensuremath{\varepsilon}}$ with any $\ensuremath{\varepsilon}>0$) may be attained, even though the system gradually becomes noisier and separable, so that both the above entanglement quantifiers asymptotically vanish. Our work shows that sufficiently large quantum systems achieve nearly optimal resolutions despite their relative amount of entanglement being arbitrarily small. In deriving our results, we establish the continuity relation of the quantum Fisher information evaluated for a phaselike parameter, which lets us link it directly to the geometry of quantum states, and hence naturally to the geometric measure of entanglement.
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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.
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.
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.
We have introduced a measure of Gaussian quantum correlations based on quantum Fisher information. For bipartite Gaussian states the minimum quantum Fisher information due to local unitary evolution on one of the parties reliably quantifies quantum correlation. In quantum metrology the proposed measure becomes the tool to investigate the role of quantum orrelation in setting metrological precision. In particular, a deeper insights can be gained on how quantum correlations are instrumental to enhance metrological precision. Our analysis demonstrates that not only entanglement but also quantum correlation plays an important role to enhance metrological precision. Clearly unraveling the underlaying mechanism we show that quantum correlations, even in the absence of entanglement, can be exploited as the resource to beat standard quantum limit and attain Heisenberg limit in quantum metrology.