Pure coherent states are known as the most classical states in quantum mechanics. Similarly, pure spin coherent states based on the rotation group also exist in finite dimensions. We call these states pure classical states, and the mixed classical states are the statistical mixtures of these states. The set of all classical states is a subset of the set of all pure & mixed states. We present an essentially complete description of the geometry of this set for 3-dimensional Hilbert space. While studying its geometry we find a remarkable similarity between the geometry of this set and the geometry of the set of all classical probability distributions. This study also improves a recent work on this set done by Giraud at el [4] and relates it to an earlier work by Wootters [14].
Pure coherent states are known as the most classical states in quantum mechanics. Similarly, pure spin coherent states based on the rotation group also exist in finite dimensions. We call these states pure classical states, and the mixed classical states are the statistical mixtures of these states. The set of all classical states is a subset of the set of all pure & mixed states. We present an essentially complete description of the geometry of this set for 3-dimensional Hilbert space. While studying its geometry we find a remarkable similarity between the geometry of this set and the geometry of the set of all classical probability distributions. This study also improves a recent work on this set done by Giraud at el [4] and relates it to an earlier work by Wootters [14].
We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, ``most classical'' can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons, as well as mixed spin states decomposable into probabilistic mixtures of pure coherent states. The latter was the subject of the recent paper by Giraud et al. [Phys. Rev. A 78, 042112 (2008)], who showed that in the lowest-dimensional nontrivial case of spin 1, each such state can be decomposed into a mixture of eight pure states. Using our method, we prove that in fact four pure states always suffice.
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.
Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.
We investigate quantumness of spin-1 states, defined as the Hilbert-Schmidt distance to the convex hull of spin coherent states. We derive its analytic expression in the case of pure states as a function of the smallest eigenvalue of the Bloch matrix and give explicitly the closest classical state for an arbitrary pure state. Numerical evidence is given that the exact formula for pure states provides an upper bound on the quantumness of mixed states. Due to the connection between quantumness and entanglement we obtain new insights into the geometry of symmetric entangled states.
A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schrödinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15—to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS , might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms—the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and applications of the geometric approach. The first four chapters contain the standard mathematics required to understand the rest of the material presented: specific areas in colour theory, set theory, probability theory, differential geometry and projective geometry are all covered with an eye to the material that follows. Chapter 5 starts the first real discussion of quantum theory in GQS and serves as an elegant, succinct introduction to the geometry which underlies quantum theory. This may be the most worthwhile chapter for the casual reader who wants to understand the key ideas in this field. Chapter 6 builds on the discussion in Chapter 5, introducing a group theoretic approach to understand coherent states and Chapter 7 describes a geometric tool in the form of an approach to complex projective geometry called 'the stellar representation'. Chapter 8 returns to a more purely quantum mechanical discussion as the authors turn to study the space of density matrices. This chapter completes the discussion which started in Chapter 5. Chapter 9 begins the part of the book concerned with applications of the geometric approach. From this point on the book aims, specifically, to prepare the reader for the material in Chapter 15 beginning with a discussion on the purification of mixed quantum states. In the succeeding chapters a definite choice has been made to present a geometric approach to certain quantum information problems. For example, Chapter 10 contains an extremely well formulated discussion of measurement and positive operator-valued measures with several well illustrated examples and Chapter 11 reopens the discussion of density matrices. Entropy and majorization are again revisited in Chapter 12 in much greater detail than in previous chapters. Chapters 13 and 14 concern themselves with a discussion of various metrics and their relation to the problem of distinguishing between probability distributions and their suitability as probability measures.
Topological and geometrical properties of the set of mixed quantum states in the N -dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU (2) coherent states and the generalized Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ * , proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized `chaotic' states are close to ρ * . This contrasts the geometry induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, for which the distance from ρ * is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states and more suitable for numerical computation.