Limits on Parameter Estimation of 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\u27s 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 estimating multiple parameters simultaneously. For the single parameter case, we establish Cramer--Rao bounds for general quantum channels using both the SLD and RLD Fisher information. The task of estimating multiple parameters simultaneously is more involved than the single parameter case. In the multiparameter case, Cramer--Rao bounds take the form of matrix inequalities. We provide a method to obtain scalar Cramer--Rao bounds from the corresponding matrix inequalities. We then establish a scalar Cramer--Rao bound using the RLD Fisher information. Our bounds apply universally and we also show how they are efficiently computable by casting them as optimization problems. In the single parameter case, we recover the so-called Heisenberg scaling\u27\u27 using our SLD-based bound. On the other hand, we provide a no-go condition for Heisenberg scaling using our RLD-based bound for both the single and multiparameter settings. Finally, we apply our bounds to the example of estimating the parameters of a generalized amplitude damping channel