In this job, we will present a theory called Quantum Tomography that is the natural extension of the theory of detection of signals in classical telecommunications to Quantum Mechanics. This theory mainly consists in the reconstruction of a quantum state of a system through a probability distribution measured directly in the laboratory, usually called Tomogram. This Thesis contains five chapters. In the first one, we will show the birth of this theory as an adaptation of homodyne and heterodyne classical detection to Quantum Mechanics. In the second, we will describe a tomographic description of Quantum Mechanics on C*-algebras by splitting the theory in two parts: a Generalized Sampling Theory and a Generalized Positive Transform. In the third, we will present the first numerical algorithm that solves the Clebsh-Gordan decomposition problem for any finite-dimensional unitary representation of any finite or compact Lie group. This algorithm receives the name of SMILY algorithm. In the fourth and fifth chapters, we will discuss tomography of systems parametrized with fields. In the fourth, we will reconstruct states of classical systems and in the fifth, we will do the same with quantum ones, but here in two different ways: the first one will be done by adapting the reconstruction formula of chapter four, and the second one by means of the axiomatic description of Wightamn-Streater of a Quantum Field Theory.
In this job, we will present a theory called Quantum Tomography that is the natural extension of the theory of detection of signals in classical telecommunications to Quantum Mechanics. This theory mainly consists in the reconstruction of a quantum state of a system through a probability distribution measured directly in the laboratory, usually called Tomogram. This Thesis contains five chapters. In the first one, we will show the birth of this theory as an adaptation of homodyne and heterodyne classical detection to Quantum Mechanics. In the second, we will describe a tomographic description of Quantum Mechanics on C*-algebras by splitting the theory in two parts: a Generalized Sampling Theory and a Generalized Positive Transform. In the third, we will present the first numerical algorithm that solves the Clebsh-Gordan decomposition problem for any finite-dimensional unitary representation of any finite or compact Lie group. This algorithm receives the name of SMILY algorithm. In the fourth and fifth chapters, we will discuss tomography of systems parametrized with fields. In the fourth, we will reconstruct states of classical systems and in the fifth, we will do the same with quantum ones, but here in two different ways: the first one will be done by adapting the reconstruction formula of chapter four, and the second one by means of the axiomatic description of Wightamn-Streater of a Quantum Field Theory.
This thesis investigates three distinct facets of the theory of quantum information. The first two, quantum state estimation and quantum process estimation, are closely related and deal with the question of how to estimate the classical parameters in a quantum mechanical model. The third attempts to bring quantum theory as close as possible to classical theory through the formalism of quasi-probability. \n \nBuilding a large scale quantum information processor is a significant challenge. First, we require an accurate characterization of the dynamics experienced by the device to allow for the application of error correcting codes and other tools for implementing useful quantum algorithms. The necessary scaling of computational resources needed to characterize a quantum system as a function of the number of subsystems is by now a well studied problem (the scaling is generally exponential). However, irrespective of the computational resources necessary to just write-down a classical description of a quantum state, we can ask about the experimental resources necessary to obtain data (measurement complexity) and the computational resources necessary to generate such a characterization (estimation complexity). These problems are studied here and approached from two directions. \n \nThe first problem we address is that of quantum state estimation. We apply high-level decision theoretic principles (applied in classical problems such as, for example, universal data compression) to the estimation of a qubit state. We prove that quantum states are more difficult to estimate than their classical counterparts by finding optimal estimation strategies. These strategies, requiring the solution to a difficult optimization problem, are difficult to implement in practise. Fortunately, we find estimation algorithms which come close to optimal but require far fewer resources to compute. Finally, we provide a classical analog of this quantum mechanical problem which reproduces, and gives intuitive explanations for, many of its features, such as why adaptive tomography can quadratically reduce its difficulty. \n \nThe second method for practical characterization of quantum devices takes is applied to the problem of quantum process estimation. This differs from the above analysis in two ways: (1) we apply strong restrictions on knowledge of various estimation and control parameters (the former making the problem easier, the latter making it harder); and (2) we consider the problem of designing future experiments based on the outcomes of past experiments. We show in test cases that adaptive protocols can exponentially outperform their off-line counterparts. Moreover, we adapt machine learning algorithms to the problem which bring these experimental design methodologies to realm of experimental feasibility. \n \nIn the final chapter we move away from estimation problems to show formally that a classical representation of quantum theory is not tenable. This intuitive conclusion is formally borne out through the connection to quasi-probability -- where it is equivalent to the necessity of negative probability in all such representations of quantum theory. In particular, we generalize previous no-go theorems to arbitrary classical representations of quantum systems of arbitrary dimension. We also discuss recent progress in the program to identify quantum resources for subtheories of quantum theory and operational restrictions motivated by quantum computation.
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrödinger–Dirac picture of quantum mechanics on Hilbert spaces. In this picture, states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture, which has evolved in the C ⋆ -algebraic approach to quantum mechanics, starts with the algebra of observables and introduces states as a derived concept. The equivalence between these two pictures amounts, essentially, to the Gelfand–Naimark–Segal construction. In this construction, the abstract C ⋆ -algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation that is defined may be reducible or irreducible, but in either case it allows us to identify a unitary group associated with the C ⋆ -algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group; it also follows that quantum tomograms are strictly related with appropriate functions (positive-type) on the group. In this paper we present, using very simple examples, a tomographic description emerging from the set of ideas connected with the C ⋆ -algebra picture of quantum mechanics. In particular, we introduce the tomographic probability distributions for finite and compact groups, and formulate an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state.
Abstract In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger’s picture of quantum mechanics in which a groupoid is associated with every quantum system. The attention is focussed on spin tomography: in this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a quorum in close analogy with the continuous case.
[eng] This thesis can be classified into the research field of quantum algorithms. A typical paper in this field usually focuses on what kind of things we can do with a quantum computer, whether they are useful or advantageous compared to classical computers, and how we can do it more efficiently. Since the ideal fault-tolerant large-scale quantum computer will not be available soon, many papers focus on noisy intermediate-scale quantum (NISQ) devices, how to characterize them through various benchmarking protocols, and how to make them useful through error mitigation or variational algorithms. Most probably, anyone who ends up opening this thesis is already an expert in certain topics, but they can still find some “out of the box" elements in one chapter or another. The results presented in the thesis might not be directly useful, but hopefully, these elements can provide insights and inspiration. The main body of this thesis is organized into six independent chapters. Each chapter is complete and self-consistent. All these chapters share a similarity: They provide some mathematical ideas of quantum computing in one aspect or another. Since part of the research is conducted in collaboration, the pronoun “we" is used throughout the work. In the first chapter, we focus on actions that transform quantum states into quantum states. We begin with a more theoretical approach. These actions naturally process a group structure. And we ask, under what conditions is this group finite? Given an arbitrary finite group, how can we construct these actions? How can these actions be useful to quantum computing? The set of these actions is named "Quantum representation of finite group," and answers to these questions are provided in the main text. Then, in the second chapter, we switch to a more practical scenario. The inspiration came from a problem that appeared in post-quantum cryptography, the dihedral coset problem, which uses quantum algorithms to recover a secret of the group hidden in quantum states. We simplified this problem and designed an interactive protocol for verifying the computational capacity of a quantum device. Holding a small quantum computer, the verifier prepares quantum state samples that encode a secret and sends them one by one to the prover via a one-way quantum channel. The prover then uses his quantum device to recover the secret with a certain probability. The device’s capacity is then reflected in the accuracy of the solution. This interactive protocol can be used locally on a single device as a benchmarking protocol. In the third chapter, we explore quantum entanglement. We present a new method for visualizing entanglement during a quantum evolution, such as the execution of a quantum algorithm. We name this method "Entanglement trajectory" and show that it can provide a "fingerprint" for each quantum process. We determine the boundaries of these trajectories with Random Matrix Theory and use different examples and entanglement measurements to show that this trajectory has an intrinsic quantum meaning and provides more information about the system. In the fourth chapter, we elaborate on the entanglement presented in a succession of numbers. First, we give a numerical and analytical description of the average entropy of succession, which can provide a standard for studying any successions. Then, we simulate the example of k-almost primes and its unions for different system sizes and explore their particular feature. In the fifth chapter, we will show three different methods of using a quantum annealer to solve Multivariate Quadratic (MQ) problems. In the sixth chapter, we present a data encoder in the fixed Hamming weight subspace, which is parameter-optimal and can be extended to a sparse and binary data encoder. We will give a small summary for each chapter. At the end of this thesis, we will provide an overall conclusion for our readers as “take-home messages".
We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU (3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.