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arXiv:2008.08682 · Physical Review A

Beyond density matrices: Geometric quantum states

Fabio Anzà, James P. Crutchfield

Published Jun 22, 2021• 18 citations
arXivHTMLPDF

Abstract

In quantum mechanics, states are described by density matrices. Though their probabilistic interpretation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express the physical realization of an ensemble. Conveniently, the outcome statistics of projective and positive operator-valued measurements do not depend on the ensemble realization, only on the density matrix. Here, we show how the geometric approach to quantum mechanics tracks ensemble realizations. We do so in two concrete cases of a finite-dimensional quantum system interacting with another one with (i) finite-dimensional Hilbert space, relevant for quantum thermodynamics, and (ii) infinite-dimensional Hilbert space, relevant for state-manipulation protocols.

DOI
10.1103/physreva.103.062218

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In quantum mechanics, states are described by density matrices. Though their probabilistic interpretation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express the physical realization of an ensemble. Conveniently, the outcome statistics of projective and positive operator-valued measurements do not depend on the ensemble realization, only on the density matrix. Here, we show how the geometric approach to quantum mechanics tracks ensemble realizations. We do so in two concrete cases of a finite-dimensional quantum system interacting with another one with (i) finite-dimensional Hilbert space, relevant for quantum thermodynamics, and (ii) infinite-dimensional Hilbert space, relevant for state-manipulation protocols.
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.
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.
Abstract The conventional variational method is reformulated within a geometrical approach to quantum mechanics, both for finite and infinite dimensional Hilbert spaces. This geometrical setting allows us to deal not only with approximated eigenvalues and eigenstates, but also with approximated Schrödinger equations by means of Hamiltonian evolution equations associated with expectation value functions. We shall consider some specific examples where we extend our approach also to density matrices.
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension ddd and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.
Abstract This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out on the ensemble, it will split into a number of sub-ensembles, each sub-ensemble being conditioned on a particular outcome of the measurements. The mathematical description of the collection of sub-ensembles thus created leads to probability distributions on projective Hilbert space. The chapter develops an appropriate mathematical framework which enables the general formulation of such a distribution, and leads to the concepts of stochastic state vectors and stochastic density matrices. These concepts are required in later chapters to construct appropriate stochastic differential equations describing the continuous monitoring of open quantum systems.
Abstract This chapter develops the fundamental notions and concepts of the probabilistic and statistical interpretation of quantum mechanics. It begins with a brief review of the mathematical structure given by a Hilbert space of state vectors and an algebra of observables represented by selfadjoint operators. Special emphasis is laid on the connection between the spectral representation of a selfadjoint operator and a corresponding random variable that describes the possible measurement outcomes. The chapter further introduces the concepts of a composite quantum system, of the reduced density matrix, and of entangled quantum states. A section on quantum entropies is included, as well as the mathematical formulation of the generalized theory of quantum measurements in terms of completely positive quantum operations and effects.