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arXiv:1609.02160 · Physical Review Letters

Ultimate Precision of Adaptive Noise Estimation

Stefano Pirandola, Cosmo Lupo

Published Mar 7, 2017• 108 citations
arXivHTMLPDF

Abstract

We consider the estimation of noise parameters in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This is based on the exploitation of unlimited entanglement and arbitrary quantum operations, so that the channel inputs may be interactively updated. In this general scenario, we draw a novel connection between quantum metrology and teleportation. In fact, for any teleportation-covariant channel (e.g., Pauli, erasure, or Gaussian channel), we find that adaptive noise estimation cannot beat the standard quantum limit, with the quantum Fisher information being determined by the channel's Choi matrix. As an example, we establish the ultimate precision for estimating excess noise in a thermal-loss channel, which is crucial for quantum cryptography. Because our general methodology applies to any functional that is monotonic under trace-preserving maps, it can be applied to simplify other adaptive protocols, including those for quantum channel discrimination. Setting the ultimate limits for noise estimation and discrimination paves the way for exploring the boundaries of quantum sensing, imaging, and tomography.

DOI
10.1103/physrevlett.118.100502

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We consider the estimation of noise parameters in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This is based on the exploitation of unlimited entanglement and arbitrary quantum operations, so that the channel inputs may be interactively updated. In this general scenario, we draw a novel connection between quantum metrology and teleportation. In fact, for any teleportation-covariant channel (e.g., Pauli, erasure, or Gaussian channel), we find that adaptive noise estimation cannot beat the standard quantum limit, with the quantum Fisher information being determined by the channel's Choi matrix. As an example, we establish the ultimate precision for estimating excess noise in a thermal-loss channel, which is crucial for quantum cryptography. Because our general methodology applies to any functional that is monotonic under trace-preserving maps, it can be applied to simplify other adaptive protocols, including those for quantum channel discrimination. Setting the ultimate limits for noise estimation and discrimination paves the way for exploring the boundaries of quantum sensing, imaging, and tomography.
We consider the problem of estimating a classical parameter encoded in a quantum channel, assuming the most general strategy allowed by quantum mechanics. This strategy is based on the exploitation of an unlimited amount of pre-shared entanglement plus the use of adaptive probings, where the input of the channel is interactively updated during the protocol. We show that, for the wide class of teleportation-stretchable channels in finite dimension, including all Pauli channels and erasure channels, the quantum Fisher information cannot exceed an ultimate bound given by the Choi matrix of the encoding channel. We also extend our methods and results to quantum channel discrimination, finding a corresponding ultimate bound for the minimum error probability. Thus, our findings establish the ultimate precision limits that are achievable in quantum metrology and quantum discrimination for the most basic models of discrete-variable quantum channels.
Abstract In this review we discuss how channel simulation can be used to simplify the most general protocols of quantum parameter estimation, where unlimited entanglement and adaptive joint operations may be employed. Whenever the unknown parameter encoded in a quantum channel is completely transferred in an environmental program state simulating the channel, the optimal adaptive estimation cannot beat the standard quantum limit. In this setting, we elucidate the crucial role of quantum teleportation as a primitive operation which allows one to completely reduce adaptive protocols over suitable teleportation-covariant channels and derive matching upper and lower bounds for parameter estimation. For these channels,wemay express the quantum Cramér Rao bound directly in terms of their Choi matrices. Our review considers both discrete- and continuous-variable systems, also presenting some new results for bosonic Gaussian channels using an alternative sub-optimal simulation. It is an open problem to design simulations for quantum channels that achieve the Heisenberg limit.
Abstract What is the ultimate performance for discriminating two arbitrary quantum channels acting on a finite-dimensional Hilbert space? Here we address this basic question by deriving a general and fundamental lower bound. More precisely, we investigate the symmetric discrimination of two arbitrary qudit channels by means of the most general protocols based on adaptive (feedback-assisted) quantum operations. In this general scenario, we first show how port-based teleportation can be used to simplify these adaptive protocols into a much simpler non-adaptive form, designing a new type of teleportation stretching. Then, we prove that the minimum error probability affecting the channel discrimination cannot beat a bound determined by the Choi matrices of the channels, establishing a general, yet computable formula for quantum hypothesis testing. As a consequence of this bound, we derive ultimate limits and no-go theorems for adaptive quantum illumination and single-photon quantum optical resolution. Finally, we show how the methodology can also be applied to other tasks, such as quantum metrology, quantum communication and secret key generation.
Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time TTT, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, RRR: when the stochastic signal exhibits high-rank correlations across the RRR channels and the probe is entangled, the average variance (per parameter) scales no better than Ω(1/(TR2))Ω(1/(TR^2))Ω(1/(TR2)). For collective kkk-body dissipation, R=Θ(Nk)R=Θ(N^k)R=Θ(Nk), signifying super-Heisenberg scaling with the system size NNN. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.
In this thesis we investigate the fundamental limitations that the laws of the quantum nature impose on the performance of quantum communications, quantum metrology and quantum channel discrimination. In a quantum communication scenario, the typical tasks are represented by the simple transmission of quantum bits, the distribution of entangle- ment and the sharing of quantum secret keys. The ultimate rates for each of these protocols are given by the two-way quantum capacities of the quantum channel which are in turn defined by considering the most general adaptive strategies that can be implemented over the channel. To assess these quantum capacities, we combine the simulation of quantum channels, suitably generalized to systems of arbitrary dimension, with quantum telepor- tation and the relative entropy of entanglement. This procedure is called teleportation stretching. Relying on this, we are able to reduce any adaptive protocols into simpler block ones and to determine the tightest upper bound on the two-way quantum capacities. Re- markably, we also prove the existence of a particular class of quantum channel for which the lower and the upper bounds coincide. By employing a slight modification of the tele- portation scheme, allowing the two parties to share a multi-copy resource state, we apply our technique to simplify adaptive protocols for quantum metrology and quantum channel discrimination. In the first case we show that the modified teleportation stretching implies a quantum Cram ́er-Rao bound that follows asymptotically the Heisenberg scaling. In the second scenario we are able to derive the only known so far fundamental lower bound on the probability of error affecting the discrimination of two arbitrary finite-dimensional quantum channels.
The major problem of multiparameter quantum estimation theory is to find an ultimate measurement scheme to go beyond the standard quantum limits that each quasi-classical estimation measurement is limited by. Although, in some specifics quantum protocols without environmental noise, the ultimate sensitivity of a multiparameter quantum estimation can beat the standard quantum limit. However, the presence of noise imposes limitations on the enhancement of precision due to the inevitable existence of environmental fluctuations. Here, we address the motivation behind the usage of Gaussian quantum resources and their advantages in reaching the standard quantum limits under realistic noise. In this context, our work aims to explore the ultimate limits of precision for the simultaneous estimation of a pair of parameters that characterize the displacement channel acting on Gaussian probes and subjected to open dynamics. More precisely, we focus on a general two-mode mixed squeezed displaced thermal state, after reducing it to various Gaussian probes states, like; a two-mode pure squeezed vacuum, two-mode pure displaced vacuum, two-mode mixed displaced thermal, two-mode mixed squeezed thermal. To study the ultimate estimation precision, we evaluate the upper and bottom bound of HCRB in various cases. We find that when the entangled states, two-mode pure squeezed vacuum and two-mode mixed squeezed thermal, are employed as probes states, the upper and bottom bound of HCRB beats the standard quantum limit in the presence of a noisy environment.