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The Metrological Power of Nonclassical Single-Mode States

Wenchao Ge, Kurt A. Jacobs, Muhammad Suhail Zubairy

Published Mar 31, 2021• 0 citations
OpenAlex

Abstract

Nonclassical states enable metrology with a precision beyond that possible with classical physics. Both for practical applications and to understand non-classicality as a resource, it is useful to know the maximum quantum advantage that can be provided by a nonclassical state when it is combined with arbitrary classical states. This advantage has been termed the power of a quantum state. A key open question is whether the metrological powers for the metrology of different quantities are related, especially metrology of force (acceleration) and phase shifts (time). Here we answer this question for all single-mode states, both for local and distributed metrology, by obtaining complete expressions for the metrological powers for the metrology of essentialy any quantity (any single-mode unitary transformation), as well as the linear networks that achieve this maximal precision. This shows that the metrological powers for all quantities are proportional to a single property of the state, which for pure states is the quadrature variance, maximized over all quadratures.

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arXiv:1406.5857•Physical Review A•Aug 20, 2014•53 citations
Nonclassical states enable metrology with a precision beyond that possible with classical physics. Both for practical applications and to understand non-classicality as a resource, it is useful to know the maximum quantum advantage that can be provided by a nonclassical state when it is combined with arbitrary classical states. This advantage has been termed the power of a quantum state. A key open question is whether the metrological powers for the metrology of different quantities are related, especially metrology of force (acceleration) and phase shifts (time). Here we answer this question for all single-mode states, both for local and distributed metrology, by obtaining complete expressions for the metrological powers for the metrology of essentialy any quantity (any single-mode unitary transformation), as well as the linear networks that achieve this maximal precision. This shows that the metrological powers for all quantities are proportional to a single property of the state, which for pure states is the quadrature variance, maximized over all quadratures.
Abstract It is well-known that the precision of a phase measurement with a Mach-Zehnder interferometer employing strong classic light can be greatly enhanced with the addition of weak nonclassical light. In the context of quantifying nonclassicality, the amount by which a nonclassical state can enhance precision in this way has been termed its ’metrological power’. To-date, the enhancement provided by weak nonclassical states has been calculated only for specific measurement configurations. Here we are able to optimize over all measurement configurations to obtain the maximum enhancement that can be achieved by any single or multi-mode nonclassical state together with strong classical states, for local and distributed quantum metrology employing any linear or nonlinear single-mode unitary transformation. Our analysis reveals that the quantum Fisher information for quadrature-displacement sensing is the sole property that determines the maximum achievable enhancement in all of these different scenarios, providing a unified quantification of the metrological power.
We establish the nonclassicality of continuous-variable states as a resource for quantum metrology. Based on the quantum Fisher information of multimode quadratures, we introduce the metrological power as a measure of nonclassicality with a concrete operational meaning of displacement sensitivity beyond the classical limit. This measure belongs to the resource theory of nonclassicality, which is nonincreasing under linear optical elements. Our Letter reveals that a single copy, highly nonclassical quantum state is intrinsically advantageous when compared to multiple copies of a quantum state with moderate nonclassicality. This suggests that metrological power is related to the degree of quantum macroscopicity. Finally, we demonstrate that metrological resources useful for nonclassical displacement sensing tasks can be always converted into a useful resource state for phase sensitivity beyond the classical limit.
Quantum metrology exploits quantum mechanical effects to increase the precision of measurements of physical quantities. A wide variety of applications are currently being developed for scientific and technological purposes, however, most research relies on the use of highly entangled resource states that are challenging to generate and control in a given physical system. Here, we study the use of weighted graph states as more accessible resources for quantum metrology, which yield a favorable precision beyond the classical limit, approaching the Heisenberg limit. We find a notable robustness to variation in weights and less challenging weight requirements compared to standard graph states, which require a maximal weight at all edges. Both of these aspects reduce the practical demands in a physical setup, with the latter implying significantly less entanglement is required to gain a quantum advantage in metrology. We study the quantum Fisher information and optimized estimator variance of two identified sub classes of weighted graph states for an arbitrary number of N qubits, providing analytical forms and investigating their scaling. Our work opens up opportunities for using weakly entangled states in quantum-enhanced metrology.
Quantifying nonclassicality of a bosonic mode is an important but challenging task in quantum optics. Recently, a nonclassicality measure based on the concept of operational resource theory has been proposed [W. Ge, K. Jacobs, S. Asiri, M. Foss-Feig, and M. S. Zubairy, Phys. Rev. Res. 2, 023400 (2020)], which shows several crucial properties as a resource for quantum metrology. Here we apply the measure to evaluate and categorize different classes of nonclassical states. We discover a class of states that can achieve the maximum nonclassicality in the asymptotic limit of a large mean number of excitations. These states can provide the same sensitivity in place of a squeezed vacuum in a sensing scheme, e.g., the laser interferometer gravitational-wave observatory (LIGO) experiments. We also discover that the nonclassicality of certain states can be greatly improved with a single-photon addition. In addition, we explore some examples on how to evaluate analytically the nonclassicality of mixed states, which are in general difficult to calculate due to the convex roof nature of the measure. Our results will be useful for preparing and utilizing nonclassical states for applications in precision-sensing tasks, such as quadrature sensing and phase sensing in a Mach-Zehnder interferometer.
What is the most effective metric for quantifying the nonclassical characteristics of an individual quantum state? Our answer to this question is that it depends on what will be used for. Here we present some of the most widely used measures, as well as some promising ones. In this discussion, we analyze the question and ultimately determine that there is no singular superior metric. However, a mix of measures can provide a comprehensive understanding of the situation. Each measure has a potential application, some can be classified as a general nonclassicality measure, while others are dedicated measures. Some measures have a practical physical meaning, while others are justified by the results. We show that the negativity of the Wigner function, Mandel’s Q parameter and Hilbert–Schmidt distance are good dedicated measures, while Roughness is a general one, like Skewness in this case for pure states.
The interferometric power of a bipartite quantum state quantifies the precision, measured by quantum Fisher information, that such a state enables for the estimation of a parameter embedded in a unitary dynamics applied to one subsystem only, in the worst-case scenario where a full knowledge of the generator of the dynamics is not available a priori. For finite-dimensional systems, this quantity was proven to be a faithful measure of quantum correlations beyond entanglement. Here we extend the notion of interferometric power to the technologically relevant setting of optical interferometry with continuous-variable probes. By restricting to Gaussian local dynamics, we obtain a closed formula for the interferometric power of all two-mode Gaussian states. We identify separable and entangled Gaussian states which maximize the interferometric power at fixed mean photon number of the probes and discuss the associated metrological scaling. At fixed entanglement of the probes, highly thermalized states can guarantee considerably larger precision than pure two-mode squeezed states.