A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources.
A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources.
Open questions from Sarovar and Milburn (2006 J.Phys. A: Math. Gen. 39 8487) are answered. Sarovar and Milburn derived a convenient upper bound for the Fisher information of a one-parameter quantum channel. They showed that for quasi-classical models their bound is achievable and they gave a necessary and sufficient condition for positive operator-valued measures (POVMs) attaining this bound. They asked (i) whether their bound is attainable more generally, (ii) whether explicit expressions for optimal POVMs can be derived from the attainability condition. We show that the symmetric logarithmic derivative (SLD) quantum information is less than or equal to the SM bound, i.e.\ H(θ)≤CΥ(θ) and we find conditions for equality. As the Fisher information is less than or equal to the SLD quantum information, i.e. FM(θ)≤H(θ), we can deduce when equality holds in FM(θ)≤CΥ(θ). Equality does not hold for all channels. As a consequence, the attainability condition cannot be used to test for optimal POVMs for all channels. These results are extended to multi-parameter 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
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it.
A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties.
Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement.
Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output.
In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
Morozova and Chentsov (Morozova and Chentsov 90) studied Riemannian metrics on the set of probability measures. They showed that, up to a constant factor, the Fisher information is the only Riemannian metric which is monotone under stochastic transformation. Sarovar and Milburn (Sarovar and Milburn 06) computed an upper bound on the Fisher information for one-parameter channels. In (O'Loan 07) we extended their bound to an upper bound on the Fisher information of multi-parameter families of states; we call this the SM quantum information. Petz and Sudár (Petz 95) characterized fully the set of monotone metrics on the space of all density matrices. We analyse the SM quantum information in light of their work. We show that the SM quantum information is not a well-defined metric on the space of density matrices: different choices of phase of the eigenvectors lead to different metrics. We define a new metric C_L as a lower bound among the SM quantum informations. We look at properties of C_L and show that it is invariant but not monotone.
In this thesis I study optimal strategies to estimate or characterize quantum states and processes (quantum channels). I consider the problem of estimating a phase encoded on a gaussian state. Phase estimation is at the heart of many quantum metrology applications, such as gravitational wave detectors or clock synchronization. I address the problem to the phase estimation fidelity for pure and mixed gaussian states and study how temperature affects the precision of estimation. Although the case of pure states has been studied before, very little is known about estimation in mixed states. The interest in mixed gaussian states relies on the fact that these are the states one has in real experiments with light due to the unavoidable presence of noise. Displaced and squeezed thermal states are studied. For the case of displaced states, I find that fidelity estimation gets worse when the temperature increases. On the contrary, for squeezed thermal states I find the surprising result that a larger temperature yields better estimation. This apparently paradoxical feature, also appears when I derive the estimation accuracy that can be obtained when an asymptotically large number of copies of the state is available. I also study quantum benchmarks that can be applied to a wide variety of protocols, such us quantum storage or quantum teleportation. A very important question in QIT is to know whether a particular quantum protocol can be realized by classical means with the same efficiency. After all, QIT lives from the promise of this quantum advantage. This question becomes specially relevant in experimental (real) implementations of quantum schemes. In real experiments there are always losses and the ideal quantum protocols become imperfect. It is then necessary to assess whether the same experimental result could be obtained by classical (less costly) means. So, for example, in a particular teleportation experiment -which involves generation of entanglement, complicated Bell measurements, fighting decoherence, etc.- one might ask whether one could achieve the same (i.e. mapping the state of a system to a second system located in a different place or time) by measuring the system, transmitting the collected information to the other location, and finally preparing the system accordingly. It is hence of paramount importance to give bounds on the efficiency of these measure and prepare (or entanglement breaking) channels. For this purpose we propose a threshold based on a phase-covariant family of states. Besides of being a family with a straightforward group-theoretical structure, which gives obvious computational advantages, this family is interesting because it can be tested experimentally also in a straightforward way. A crucial feature for the practical use of our benchmark is that it does not require pure (ideal) input states. I finally consider the problem of estimating the parameter of the amplification channel. The amplification channel is interesting by itself but also by the fact that the Hawking-Unruh effect can be formally modeled by it. The Unruh-Hawking effect is a relativistic effect and as such it is hard to measure and detect in current experiments. I investigate whether quantum strategies can provide an improvement over classical protocols in the estimation of the amplification parameter. If the field is inertially prepared in the vacuum one could measure its temperature as detected by a noninertial observer and extract information about the parameter. But I would like to know if the vacuum state is the optimal probe state in an inertial frame for such a task or a better sensitivity could be achieved if the field were in a state of nonzero energy (mean photon number) from an inertial perspective. The aim is to determine the optimal input states, so that, by optimally measuring the output, one can construct an estimator for amplification channel parameter which is unbiased and has the minimum possible variance.
One of the fundamental tasks in quantum information processing is to measure the quantum channels. Similar to measurements of quantum states, measurements of quantum channels are inherently stochastic, that is, quantum theory provides a formula to calculate the probability of obtaining an outcome. The upper bound on each probability associated with the measurement outcome of the quantum channels is a fundamental and important quantity. In this study, we derived the upper bounds of the probability in a channel measurement for specific classes of quantum channels. We also present two applications for the upper bounds. The first is the notion of convertibility considered by Alberti and Uhlmann and the second is the detection problem of a quantum channel. These applications demonstrate the significance of the obtained upper bounds.