Quantum science and technology have seen rapid progress in recent decades, finding applications in computation, information processing, communication, and sensing. Accurate characterization and manipulation of quantum states is essential to the successful implementation of these applications. Moreover, optimal quantum state estimation is closely related to fundamental questions, such as limits on how fast information can be extracted about quantum systems. In this thesis, we present several contributions to efficiently estimating and transforming quantum states. We discuss optimal quantum state estimation, both in the regime of few available quantum systems as well as the asymptotic regime. We numerically investigate how assuming a pure-state can affect the performance of tomographic methods, and where adaptive quantum state tomography is in fact advantageous. While collective measurements have been formally considered as optimal solutions to various problems in quantum information, their implementation has been largely neglected due to experimental challenges. We present a proof-of-concept experiment realizing Massar and Popescu's optimal collective measurement for quantum state estimation, utilizing two-photon interference and polarization-dependent loss to achieve non-maximally entangled state projections. We demonstrate that collective strategies can outperform the best separable measurements. Furthermore, we introduce and experimentally implement a fast and automated optical polarization compensation scheme using liquid crystal variable retarders and rotating quarter-wave plate tomography. Desired accuracy thresholds can be achieved within a few compensation steps. Finally, we use a matrix-model approach to investigate the realization of spatial unitary transformations via sequences of thin phase gratings and free-space propagation. We numerically demonstrate that the grating parameters and propagation distances can be optimized to achieve various two-beam and three-beam unitary transformations, such as Pauli gates and the discrete Fourier transform. Through experimental and numerical approaches, this thesis offers new perspectives on methods for quantum state estimation and unitary transformation.
Quantum science and technology have seen rapid progress in recent decades, finding applications in computation, information processing, communication, and sensing. Accurate characterization and manipulation of quantum states is essential to the successful implementation of these applications. Moreover, optimal quantum state estimation is closely related to fundamental questions, such as limits on how fast information can be extracted about quantum systems. In this thesis, we present several contributions to efficiently estimating and transforming quantum states. We discuss optimal quantum state estimation, both in the regime of few available quantum systems as well as the asymptotic regime. We numerically investigate how assuming a pure-state can affect the performance of tomographic methods, and where adaptive quantum state tomography is in fact advantageous. While collective measurements have been formally considered as optimal solutions to various problems in quantum information, their implementation has been largely neglected due to experimental challenges. We present a proof-of-concept experiment realizing Massar and Popescu's optimal collective measurement for quantum state estimation, utilizing two-photon interference and polarization-dependent loss to achieve non-maximally entangled state projections. We demonstrate that collective strategies can outperform the best separable measurements. Furthermore, we introduce and experimentally implement a fast and automated optical polarization compensation scheme using liquid crystal variable retarders and rotating quarter-wave plate tomography. Desired accuracy thresholds can be achieved within a few compensation steps. Finally, we use a matrix-model approach to investigate the realization of spatial unitary transformations via sequences of thin phase gratings and free-space propagation. We numerically demonstrate that the grating parameters and propagation distances can be optimized to achieve various two-beam and three-beam unitary transformations, such as Pauli gates and the discrete Fourier transform. Through experimental and numerical approaches, this thesis offers new perspectives on methods for quantum state estimation and unitary transformation.
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.
In this thesis, I present experimental and theoretical work which assesses and develops a range of tools that are required for performing quantum information processing, particularly in photonic systems. I investigate the three degrees of freedom of a single photon—its polarisation, spatialmomentum and time-frequency distributions. For polarisation, I show that using wave plates to implement arbitrary, single-qubit rotations is more complicated than is commonly appreciated. In the spatial-momentum and time-frequency domains, I develop ways to perform tomographic analysis of quantum states, and I report the first demonstrations of these techniques. In the time-frequency domain, the tomography technique utilises entanglement in the photon polarisation as a resource to store and provide access to the time-frequency information. In my first two experiments, I use spontaneous parametric down-conversion to produce entanglement between pairs of single photons in all three degrees of freedom. I demonstrate the first characterisation of entanglement in spatial modes and the time-frequency domain, the first quantitative measurement of entangled qutrit states, and the highest quality entangled states yet measured in both polarisation and spatial modes. I also report the first realisation of complete hyperentanglement, and a full, black-box tomography of a 36-dimensional two-photon state—the largest system to be characterised in this way to date. In my final experiment, I model and implement a new architecture for a controlled-Z gate which is much simpler to align than previous implementations. This gate requires only one non-classical interference condition, the visibility of which is the main limitation to its performance. I show that the gate operates effectively as a means of both creating entanglement and discriminating between the four elements in a basis of maximally entangled, Bell-type states. Indeed, its observed performance as a Bell analyser would be sufficient to build a quantum state teleporter which would guarantee that the recipient would be left with a better copy of an unknown input state than any eavesdropper. In a series of numerical simulations, I investigate some of the practicalities that arise when using tomographic reconstruction techniques, including how to estimate errors, which measurements to make and what is actually the optimal reconstruction. In particular, I show that tomographies perform better when based on the results from over-complete sets of measurements. Finally, I discuss the important issue of how to compare two processes, particularly when trying to assess the quality of a measured process by comparing it to some expected ideal. Judging possible candidate measures against a set of experimentally and theoretically motivated
Accurately estimating the overlap between quantum states is a fundamental task in quantum information processing. While various strategies using distinct quantum measurements have been proposed for overlap estimation, the lack of experimental benchmarks on estimation precision limits strategy selection in different situations. Here we compare the performance of four practical strategies for overlap estimation, including tomography-tomography, tomography-projection, Schur collective measurement and optical swap test using photonic quantum systems. We encode the quantum states on the polarization and path degrees of freedom of single photons. The corresponding measurements are performed by photon detection on certain modes following single-photon mode transformation or two-photon interference. We further propose an adaptive strategy with optimized precision in full-range overlap estimation. Our results shed new light on extracting the parameter of interest from quantum systems, prompting the design of efficient quantum protocols.
Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.
Quantum information science and technology is experiencing rapid development. The process raises an emerging demand for experiments to implement and demonstrate novel ideas to overcome the challenges towards the further application of quantum information. Linear optical systems have long been the most available and efficient platform for the demonstration of quantum information methods and concepts due to its robustness against the influence of environment. However, the large-scale multi-qubit linear optical information systems have long been suffering the low generated efficiency of the single photon sources, leading to a limited availability and scalability. In recent years, novel techniques such as beam-like SPDC open up new possibilities to break this bottleneck and further develop linear optical quantum information systems. Driven by the increasing demand and inspired by the technical progress, in my degree candidature, I carried out a project to build an advanced linear optical multi-qubit experimental platform to implement quantum information experiment and to discover novel methodologies of quantum optics research. In this thesis, I will present our progress and future perspectives. In Chapters 1-2, I will introduce the background and the motivation of this project first, and then present the basic concepts and principles of linear optical quantum information that are involved in my research. Chapter 3 describes my work on the generation and measurement of linear optical multi-qubit entanglement based on beam-like SPDC sources. By analyzing and containing the entanglement destruction from time-space correlation and group delay dispersion, the specifications of linear optical multi-qubit system can reach a higher level that allows for more applications. In Chapter 4, we discuss the methodology we developed to statistically estimate quantum states based on measurement results. In this project, we developed and applied a new algorithm named Bayesian mean estimation based on Gibbs sampling to estimate quantum states. I will describe the background, principle, and advantages compared to other widely accepted methods of this algorithm. In Chapter 5, I report our progress in experimentally demonstrating quantum overlapping tomography, a scheme focusing on characterizing critical information of a many-body quantum system in logarithmic time complexity. We compare the estimate results of full-state tomography and overlapping tomography to confirm that overlapping tomography gives accurate information of the system with much fewer state measurements than full-state tomography. Furthermore, under the condition that the system is too large to perform a tomography, we confirm that the state is still able to be measured with overlapping tomography. Chapter 6 begins with the conclusion of the thesis. Furthermore, I present outlooks on some potential directions to further increase the linear optical quantum information system.
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.