Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper, we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation, it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.
Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper, we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation, it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.
A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square–integrable functions. More precisely, by consideration of the square–root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H embodies the ‘state space’ of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher–Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér–Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.
A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square–integrable functions. More precisely, by consideration of the square–root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H embodies the ‘state space’ of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher–Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér–Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.
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.
A statistical model M is specified by a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square-integrable functions. More precisely, by consideration of the square-root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H effectively embodies the `state space' of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), and as a consequence various aspects of classical statistical inference can be formulated in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér-Rao and Bhattacharyya bounds, described in terms of the geometry of the underlying real Hilbert space. The statistical model M can then be specialised to the case of a submanifold of the state space of a quantum mechanical system. This can be pursued by introducing a compatible complex structure on the underlying real Hilbert space, thus allowing the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds to quantum statistical estimation is shown to lead to higher order corrections to the Heisenberg uncertainty relations.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert space formulation of classical statistical thermodynamics, we introduce projective geometry as a basis for analysing probabilistic aspects of statistical physics. In particular, the specification of a canonical polarity on RPn induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, we show that equilibrium thermal states are determined by the Hamiltonian gradient flow with respect to this metric. This flow is concisely characterised by the fact that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is then extended to include the quantum mechanical dynamics of equilibrium thermal states. In this case the relevant state space is complex projective space, here regarded as a real manifold endowed with the natural Fubini-Study metric. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the nonuniqueness of the infinite temperature state. We are then led to formulate a geometric characterisation of the standard KMS-relation often considered in the context of C∗-algebras. The example of a quantum spin one-half particle in heat bath is studied in detail.