Practical quantum metrology in noisy environments

Rosanna Nichols, Thomas R. Bromley, Luis A. Correa, and Gerardo Adesso

Phys. Rev. A 94, 042101 – Published 3 October 2016

Abstract

The problem of estimating an unknown phase $φ$ using two-level probes in the presence of unital phase-covariant noise and using finite resources is investigated. We introduce a simple model in which the phase-imprinting operation on the probes is realized by a unitary transformation with a randomly sampled generator. We determine the optimal phase sensitivity in a sequential estimation protocol and derive a general (tight-fitting) lower bound. The sensitivity grows quadratically with the number of applications $N$ of the phase-imprinting operation, then attains a maximum at some $N_{opt}$ , and eventually decays to zero. We provide an estimate of $N_{opt}$ in terms of accessible geometric properties of the noise and illustrate its usefulness as a guideline for optimizing the estimation protocol. The use of passive ancillas and of entangled probes in parallel to improve the phase sensitivity is also considered. We find that multiprobe entanglement may offer no practical advantage over single-probe coherence if the interrogation at the output is restricted to measuring local observables.

Received 8 April 2016

DOI:https://doi.org/10.1103/PhysRevA.94.042101

Physics Subject Headings (PhySH)

Open quantum systems & decoherence Quantum metrology Quantum parameter estimation

General Physics

Authors & Affiliations

Rosanna Nichols, Thomas R. Bromley, Luis A. Correa, and Gerardo Adesso^*

School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

^*gerardo.adesso@nottingham.ac.uk

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Vol. 94, Iss. 4 — October 2016

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Images

Figure 1
Quantum Fisher information $F_{N} (φ)$ (solid line) and lower bound $f_{N} (φ)$ (dashed line) vs number of rounds $N$ under the channel of Eq. (1) for the von Mises–Fisher distribution $p_{κ} (θ)$ , with $φ = 0.1$ and $κ = 1$ , and probe initialized in the $|+〉$ state. The maximum of $f_{N} (φ)$ is at $N_{opt}^{'} = 85$ , only one round further than the actual maximum of $F_{N} (φ)$ . $N$ is discrete and the continuous lines are a guide to the eye. All of the quantities plotted are dimensionless.
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Figure 2
Quantum Fisher information vs $N$ for the following: the $N$ -round sequential setting with a single-qubit initial state $|+〉$ (solid line), the $N$ -round sequential setting with a passive ancilla and two-qubit initial Bell state (dot-dashed line), and the parallel-entangled setting with $N$ -qubit initial GHZ state (dotted line). The dashed line amounts to the phase sensitivity of ${\hat{σ}}_{x}$ , ${\hat{σ}}_{x}^{\otimes 2}$ , and ${\hat{σ}}_{x}^{\otimes N}$ in each of the three settings, respectively. The model parameters are the same as in Fig. 1. All of the quantities plotted are dimensionless.
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Figure 3
Visualization of the vMF distribution for varying $κ$ . The probability density runs from indigo at its lowest to red at its highest. Note that the setting $κ = 1$ , used to generate the illustrations of the main text, actually corresponds to a very broad distribution for the random generator ${\hat{H}}_{n}$ . All of the quantities plotted are dimensionless.
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Figure 4
Contour plot of the optimal number of iterations $N_{opt}$ as obtained from the maximization of $f_{N} (φ)$ , as a function of the phase $φ$ and the concentration parameter $κ$ . All of the quantities plotted are dimensionless.
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Figure 5
Top: QFI (solid line) and phase sensitivity of ${\hat{σ}}_{x}$ (dashed line) for $φ = 0.01$ and $κ = 0.5$ . Representations of the evolved probe state (purple arrow), optimal measurement basis (red arrow), and suboptimal measurement basis (blue arrow) as Bloch vectors on the equatorial plane are shown in the bottom panels. (A) Initially, the probe state and both measurements are aligned. (B) The phase sensitivity of ${\hat{σ}}_{x}$ meets the QFI when the measurement bases realign. (C) The phase sensitivity of ${\hat{σ}}_{x}$ vanishes whenever the measurement bases become perpendicular. (D) Otherwise, the phase sensitivity of ${\hat{σ}}_{x}$ oscillates between zero and the QFI. Note that as $N$ grows, the optimal basis vectors become perpendicular to the probe state vector as they rotate. Note as well that the probe state vector gradually shortens due to the loss of purity. All of the quantities plotted are dimensionless.
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Figure 6
QFI for the sequential setting when the probe is supplemented with a passive ancilla (solid) as a function of the number of applications of the channel $N$ . The QFI for a single two-level probe (without ancilla) is included for comparison (dot-dashed line). The phase sensitivity of ${\hat{σ}}_{x} \otimes {\hat{σ}}_{x}$ (dotted line) and $\hat{O} \equiv |Ψ_{+}〉 〈Ψ_{+}| - |Ψ_{-}〉 〈Ψ_{-}|$ (dashed line) have also been plotted. The vMF distribution is assumed for the stochastic generator of the phase rotations, with $φ = 0.1$ and $κ = 1$ . All of the quantities plotted are dimensionless.
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