The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.
The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.
Properties and structure of the 15-dimensional parameter space of four-state density matrices are examined using the SU(4) generator expansion. Appropriate classification of one-, two- and three-parameter density matrices is obtained, based on the sameness of the characteristic polynomial of density matrices belonging to a given type. It is found that in the one parameter case of 15 different density matrices only three distinct types exist, while in the two parameter case 105 different density matrices group into 11 distinct types. In the three parameter case appropriate classification of 455 different density matrices into 44 types is determined. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are illustrated by randomly drawing matrices for several types of density matrices, providing some insight into the intricate and complex structure of the space of density matrices for a four-state system. Positions of the representative points corresponding to the pure states are found for all types. Global properties of observables are determined by generating, by the Monte Carlo sampling method, and averaging over nearly all density matrices pertaining to a given type.
A possible method of classifying all density matrices for a N-state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the SU(N) generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined time-dependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a N-state system. The general discussion is illustrated throughout with the special case of a N=4-state quantum system. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.
Abstract The geometry of the generalized Bloch sphere Ω 3 , the state space of a qutrit, is studied. Closed form expressions for Ω 3 , its boundary ∂Ω 3 , and the set of extremals are obtained by use of an elementary observation. These expressions and analytic methods are used to classify the 28 two-sections and the 56 three-sections of Ω 3 into unitary equivalence classes, completing the works of earlier authors. It is shown, in particular, that there are families of two-sections and of three-sections which are equivalent geometrically but not unitarily, a feature that does not appear to have been appreciated earlier. A family of three-sections of obese-tetrahedral shape whose symmetry corresponds to the 24-element tetrahedral point group T d is examined in detail. This symmetry is traced to the natural reduction of the adjoint representation of SU (3), the symmetry underlying Ω 3 , into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of T d .
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.
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 present a novel method to study the Bloch space of the qutrit system by examining the Bloch trajectories in it. Since such system is inherently a three-level quantum system, therefore we use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of it. The norm of the Bloch space is evaluated from the geometric consideration and also from the dynamics of the Bloch vectors and both results are found to be identical. The analysis of the dynamical evolution of the Bloch vectors reveals an additional feature that, under resonant conditions, the Bloch sphere S7 splits into two parts, a four-sphere S4 and a two-sphere S2. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves, highlighting the non-trivial structure of the Bloch space of the qutrit system.