Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.
Introduction The aim of information geometry is to introduce a suitable geometrical structure on families of probability distributions or quantum states. For parametrised statistical models, such structure is based on two fundamental notions: the Fisher information and the exponential family with its dual mixed parametrisation, see for example (Amari 1985, Amari and Nagaoka 2000). For the non-parametric situation, the solution was given by Pistone and Sempi (Pistone and Sempi 1995, Pistone and Rogantin 1999), who introduced a Banach manifold structure on the set P of probability distributions, equivalent to a given one. For each µ ∈ ρ, the authors considered the non-parametric exponential family at µ. As it turned out, this provides a C ∞ -atlas on ρ, with the exponential Orlicz spaces Lφ(µ) as the underlying Banach spaces, here φ is the Young function of the form φ(x) = cosh(x) — 1. The present contribution deals with the case of quantum states: we want to introduce a similar manifold structure on the set of faithful normal states of a von Neumann algebra M. Since there is no suitable definition of a non-commutative Orlicz space with respect to a state φ, it is not clear how to choose the Banach space for the manifold. Of course, there is a natural Banach space structure, inherited from the predualM. But, as it was already pointed out in (Streater 2004), this structure is not suitable to define the geometry of states: for example, any neighbourhood of a state φ contains states such that the relative entropy with respect to φ is infinite.
Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of
C∗ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given
C∗ C∗ -algebra
A A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states
S S of a possibly infinite-dimensional, unital
C∗ C∗ -algebra
A A is partitioned into the disjoint union of the orbits of an action of the group
G G of invertible elements of
A A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space
H H are smooth, homogeneous Banach manifolds of
G=GL(H) G=GL(H) , and, when
A A admits a faithful tracial state
τ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through
τ τ is a smooth, homogeneous Banach manifold for
G G .
Motivated by the engineering applications of uncertainty quantification, in this work we draw connections between the notions of random quantum states and operations in quantum information with probability distributions commonly encountered in the field of orientation statistics. This approach identifies natural probability distributions that can be used in the analysis, simulation, and inference of quantum information systems. The theory of exponential families on Stiefel manifolds provides the appropriate generalization to the classical case, and fortunately there are many existing techniques for inference and sampling that exist for these distributions. Furthermore, this viewpoint motivates a number of additional questions into the convex geometry of quantum operations relative to both the differential geometry of Stiefel manifolds as well as the information geometry of exponential families defined upon them. In particular, we draw on results from convex geometry to characterize which quantum operations can be represented as the average of a random quantum operation.
This document is meant as a pedagogical introduction to the modern language used to talk about quantum theory, especially in the field of quantum information. It assumes that the reader has taken a first traditional course on quantum mechanics, and is familiar with the concept of Hilbert space and elementary linear algebra. As in the popular textbook on quantum information by Nielsen and Chuang, we introduce the generalised concept of states (density matrices), observables (POVMs) and transformations (channels), but we also characterise these structures from an algebraic standpoint, which provides many useful technical tools, and clarity as to their generality. This approach also makes it manifest that quantum theory is a direct generalisation of probability theory, and provides a unifying formalism for both fields. The focus on finite-dimensional systems allows for a self-contained presentation which avoids many of the technicalities inherent to the more general
C∗-algebraic approach, while being appropriate for the quantum information literature.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.