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 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.
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.
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.
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.
The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel's Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when
Abstract We review recent results on the simulation of quantum channels, the reduction of adaptive protocols (teleportation stretching), and the derivation of converse bounds for quantum and private communication, as established in PLOB (Pirandola et al 2017 Nat. Commun. 8 15043). We start by introducing a general weak converse bound for private communication based on the relative entropy of entanglement. We discuss how combining this bound with channel simulation and teleportation stretching, PLOB established the two-way quantum and private capacities of several fundamental channels, including the bosonic lossy channel. We then provide a rigorous proof of the strong converse property of these bounds by adopting a correct use of the Braunstein–Kimble teleportation protocol for the simulation of bosonic Gaussian channels. This analysis provides a full justification of claims presented in the follow-up paper WTB (Wilde et al 2017 IEEE Trans. Inf. Theory 63 1792–817) whose upper bounds for Gaussian channels would be otherwise infinitely large. Besides clarifying contributions in the area of channel simulation and protocol reduction, we also present some generalizations of the tools to other entanglement measures and novel results on the maximum excess noise which is tolerable in quantum key distribution.
Abstract Quantum metrology aims at delivering new quantum-mechanical improvement to technologies of parameter estimations with precision bounded by the quantum Cramér-Rao bound. The currently used quantum Cramér-Rao bound was established with measurements of observables restricted to be Hermitian. This constrains the bound and limits the precision of parameter estimation. In this paper, we lift the constraint and derive a previously unknown quantum Cramér-Rao bound. We find that the new bound can reach arbitrary small value with mixed states and it breaks the Heisenberg limit in some cases. We construct a setup to measure non-Hermitian operators and discuss the saturation of the present bound. Two examples—the phase estimation with Greenberger-Horne-Zeilinger states of trapped ions and the adiabatic quantum parameter estimation with the nuclear magnetic resonance—are employed to demonstrate the theory. The present study might open a new research direction—non-Hermitian quantum metrology.