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arXiv:1701.05518 · Physical Review A

Bounding the quantum limits of precision for phase estimation with loss and thermal noise

Christos N. Gagatsos, Boulat A. Bash, Saikat Guha, Animesh Datta

Published Dec 4, 2017• 22 citations
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Abstract

We consider the problem of estimating an unknown but constant carrier phase modulation $\ensuremath{\theta}$ using a general, possibly entangled, $n$-mode optical probe through $n$ independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\ensuremath{\theta}$ as a function of $n$, the mean and variance of the total number of photons ${N}_{\mathrm{S}}$ in the $n$-mode probe, the transmissivity $\ensuremath{\eta}$, and mean thermal photon number per mode ${\overline{n}}_{\mathrm{B}}$ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-square error (MSE) of an unbiased estimator $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\theta}}$ of $\ensuremath{\theta}$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.

DOI
10.1103/physreva.96.062306

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We present a square root law for active sensing of phase θθθ of a single pixel using optical probes that pass through a single-mode lossy thermal-noise bosonic channel. Specifically, we show that, when the sensor uses an nnn-mode covert optical probe, the mean squared error (MSE) of the resulting estimator θ^n\hatθ_nθ^n​ scales as ⟨(θ−θ^n)2⟩=O(1/n)\langle (θ-\hatθ_n)^2\rangle=\mathcal{O}(1/\sqrt{n})⟨(θ−θ^n​)2⟩=O(1/n​); improving the scaling necessarily leads to detection by the adversary with high probability. We fully characterize this limit and show that it is achievable using laser light illumination and a heterodyne receiver, even when the adversary captures every photon that does not return to the sensor and performs arbitrarily complex measurement as permitted by the laws of quantum mechanics.
We find an analytical upper bound for the quantum Fisher information (QFI) for estimating the phase shift θ\thetaθ when an arbitrary probe state suffers thermal losses, i.e., when the probe state interacts with a thermal state via a beam splitter. We consider the single-mode and nnn-mode problems. The strength of our result is that the bound is calculated analytically, it decreases with the environment's mean thermal photons, and the initial state is arbitrary. We also provide a proof that the position of the phase shifting, i.e., before or after the thermal loss channel, plays a role in the calculation of the aforementioned bound but it is irrelevant in the calculation of the Fisher information to be bounded.
We present a square root law for active sensing of phase θ of a single pixel using optical probes that pass through a single-mode lossy thermal-noise bosonic channel. Specifically, we show that, when the sensor uses an n-mode covert optical probe, the mean squared error (MSE) of the resulting estimator θnscales as 〈(θ-θ̂n)2〉 = O(1/√n) improving the scaling necessarily leads to detection by the adversary with high probability. We fully characterize this limit and show that it is achievable using laser light illumination and a heterodyne receiver, even when the adversary captures every photon that does not return to the sensor and performs arbitrarily complex measurement as permitted by the laws of quantum mechanics.
We obtain universal (i.e., probe and measurement-independent) performance bounds on ancilla-assisted quantum sensing of multiple parameters of phase-covariant optical channels under energy and mode-number constraints. We first show that for any such constrained problem, an optimal ancilla-entangled probe can always be found whose reduced state on the modes probing the channel is diagonal in the photon-number basis. For parameters that are encoded in single-mode Gaussian channels, we derive a universal upper bound on the quantum Fisher information matrix that delineates the roles played by the energy and mode constraints. We illustrate our results for sensing of the transmittance of a thermal loss channel under both the no-passive-signature and passive-signature paradigms, and in the problem of sensing the noise variance of an additive-noise channel. In both cases, we show that two-mode squeezed vacuum probes are near-optimal under the constraints in the regime of low signal brightness, i.e., per-mode average photon number. More generally, our work sets down a uniform framework for readily evaluating universal limits for any sensing problem involving Gaussian channels.
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2kQ_{\mathrm{coh}}(T)\sim T^{2k}Qcoh​(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2kQ(T)\sim T^{2k}Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1Q(T)\sim T^{2k-1}Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4T^{4}T4 and T3T^{3}T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.
We consider the problem of estimating unknown loss η overnnnuses of single-mode lossy thermal noise bosonic channel under an average photon number constraint per mode. We prove that a product ofnnntwo-mode squeezed vacuum (TMSV) states achieves minimal quantum Cramér-Rao bound (QCRB) over Gaussian quantum states in this scenario, and characterize the optimal receiver structure. We show that TMSV minimizes QCRB over all quantum states in the limit of low input photon number. Finally, we compare the performance of our optimal receiver for TMSV to other receivers.