This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.
This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.
We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon-Nikodym density with respect to the trace in the sense of Belavkin-Staszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the trace-norm distance.
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.
In a unital C*-algebra with a faithful trace functional τ, the set Dτ(A) of positive ρ∈A of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space Dτ(A) in terms of the Bures metric. Linear maps on A that map Dτ(A) back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space Dτ(A), (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space Dτ(A), and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra A∗∗.
The notion of fidelity for quantum states is a measure of how much two states overlap. In the matrix formalism of quantum mechanics, states are represented by density operators, i.e., positive semi-definite matrices with trace equal to 1 in a complex Euclidean space Mn(C). Felipe-Sosa and Felipe (2022) introduced the notion of quantum states on certain Krein spaces with indefinite metric induced by a fundamental symmetry J, calling these J-states. We define an analogous notion of measurement for J-states to the regular quantum theory and use it to show that a notion of fidelity holds in the Krein setting. We also show that an analogous result to the Fuchs-Caves measurement holds in this setting. Following the developments of Bag, Rohilla, and Trivedi (2024), we then extend this definition of fidelity to U-quantum states on S-spaces. We demonstrate that the analogous geometric motivation holds in the Krein and S-space setting, as holds for quantum fidelity and geometric means of operators.
Fidelity plays an important role in quantum information theory. In this paper, two pairs of metrics of quantum states are introduced based on the Uhlmann-Jozsa fidelity and super-fidelity. They are proved to satisfy the axioms of metrics and return to the Sine metric and Bures metric for the qubit case. The CP expansive property and convex property of the metrics are studied. These metrics have deep connections with the known important metrics such as trace metric, Sine metric, Bures metric, and spectral metric, and are useful to quantify entanglement. I. Background, problem, and result Suppose one has two quantum states ρ and σ, then the Uhlmann-Jozsa fidelity [1] between ρ and σ is given by F(ρ, σ) = [Tr ρ 1 2 σρ 1 2] 2. Uhlmann-Jozsa fidelity plays an important role in quantum information theory and quantum computation [1], [2]. We know that for the case of qubit, Uhlmann-Jozsa fidelity has a simple form. From the Bloch sphere representation of quantum states, a qubit is described by a density matrix as([3]): ρ(u) = 1 (I + σ · u), 2 where I is the 2 × 2 unit matrix and σ = (σ1, σ2, σ3) are the Pauli matrices. Assume ρ(u) and ρ(v) are two states of one qubit, then they can represented by two vectors u and v in the Bloch sphere. The Uhlmann-Jozsa fidelity for qubits has an elegant form([4] and [5]): F(ρ(u), ρ(v)) = 1 2 [1 + u · v + √ 1 − |u | 2 √ 1 − |v | 2], where u · v is the inner product of two vectors u and v, and |u | is the magnitude of u. We know that for N × N quantum states, the Uhlmann-Jozsa fidelity has no simple form like the case of qubits. To use the simple form of fidelity, we note that in [6], the authors introduce a new fidelity, called super-fidelity, and proved that Uhlmann-Jozsa fidelity can be bounded above by the super-fidelity G(ρ1, ρ2): F(ρ1, ρ2) ≤ G(ρ1, ρ2)
A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras. In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data. Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization. In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.