Typicality of Heisenberg scaling precision in multimode quantum metrology

Open Access

Typicality of Heisenberg scaling precision in multimode quantum metrology

Giovanni Gramegna, Danilo Triggiani, Paolo Facchi, Frank A. Narducci, and Vincenzo Tamma

Phys. Rev. Research 3, 013152 – Published 16 February 2021

Abstract

We propose a measurement setup reaching Heisenberg scaling precision for the estimation of any distributed parameter $φ$ (not necessarily a phase) encoded into a generic $M$ -port linear network composed only of passive elements. The scheme proposed can be easily implemented from an experimental point of view since it employs only Gaussian states and Gaussian measurements. Due to the complete generality of the estimation problem considered, it was predicted that one would need to carry out an adaptive procedure which involves both the input states employed and the measurement performed at the output; we show that this is not necessary: Heisenberg scaling precision is still achievable by only adapting a single stage. The nonadapted stage only affects the value of a prefactor multiplying the Heisenberg scaling precision: We show that, for large values of $M$ and a random (unbiased) choice of the nonadapted stage, this prefactor takes a typical value which can be controlled through the encoding of the parameter $φ$ into the linear network.

Received 28 March 2020
Accepted 21 January 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013152

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum measurements Quantum metrology Quantum sensing Squeezing of quantum noise

Homodyne & heterodyne detection Phase space methods Random matrix theory

Quantum Information, Science & TechnologyGeneral PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Giovanni Gramegna ^1,2,*, Danilo Triggiani ^3,†, Paolo Facchi ^1,2, Frank A. Narducci ⁴, and Vincenzo Tamma ^3,5,‡

¹Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy
²INFN, Sezione di Bari, I-70126 Bari, Italy
³School of Mathematics and Physics, University of Portsmouth, Portsmouth PO1 3QL, United Kingdom
⁴Department of Physics, Naval Postgraduate School, Monterey, California 93943, USA
⁵Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, United Kingdom

^*giovanni.gramegna@ba.infn.it
^†danilo.triggiani@port.ac.uk
^‡vincenzo.tamma@port.ac.uk

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Vol. 3, Iss. 1 — February - April 2021

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Images

Figure 1
Block diagram of the investigated setup [34]. A single-mode squeezed vacuum state with real squeezing parameter $r$ is injected into the first preparation stage ${\hat{V}}_{in}$ , which inputs the linear network ${\hat{U}}_{φ}$ encoding the parameter $φ$ to be estimated. After the network, there is a second stage ${\hat{V}}_{out}$ before the measurement. Finally, homodyne detection on the first output port of ${\hat{V}}_{out}$ is performed, and the quadrature field ${\hat{x}}_{θ}$ is measured. In order to reach Heisenberg scaling sensitivity in the estimation of $φ$ , it suffices to optimize only one of the two auxiliary stages, in such a way that one of the conditions (22) or (23) holds.
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Figure 2
Phase space representation of the squeezed vacuum state (with squeezing parameter $r e^{2 i γ_{φ}}$ ) at the first output channel of the whole setup shown in Fig. 1 (blue), and of the Fisher information (10) (four red lobes). We have considered for simplicity the case where all the photons are refocused in the first output channel (when condition (14) reduces to $P_{φ} = 1$ ). Given any axis at an angle $θ$ with respect to the horizontal axis, corresponding to the $x_{0} = x$ quadrature, the distance between its intersections with the ellipse and the origin represents the standard deviation $\sqrt{Δ_{φ}}$ of the quadrature ${\hat{x}}_{θ}$ . In other words, the blue graph is the polar plot of $\sqrt{Δ_{φ}}$ shown in (6) as a function of $θ$ . A polar plot of the Fisher information is overlaid in red. The Fisher information takes vanishing values if the minimum-variance quadratures are measured, namely for $θ_{\min} = γ_{φ} \pm π / 2$ . This happens because the variance $Δ_{φ}$ of the quadrature along $θ_{min}$ is locally insensitive to the variations of $φ$ . Thus, one needs to get far enough from $θ_{min}$ to achieve a suitable high value of the Fisher information. In particular, we have shown that for a large number $N$ of photons it is enough to move from $θ_{min}$ of an additional angle of the order $1 / N$ as in (15) to reach the Heisenberg scaling in the measure of the parameter $φ$ .
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Figure 3
Histograms of the random variable $f (U, G_{φ}) = {(U^{†} G_{φ} U)}_{11}^{2}$ numerically obtained with $10^{5}$ samplings of $U$ from the unitary group with Haar measure and choosing $G_{φ}$ as a diagonal matrix with half entries $3 s$ and half entries made of $1 s$ , with $M = 2$ (top), $M = 20$ (middle), and $M = 200$ (bottom). The normalization is chosen in such a way that the total area under the curve equals 1. For these particular cases an explicit analytic expression of the distribution is achievable, and is given by the orange curves. The derivation is shown in Appendix pp3.
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