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arXiv:quant-ph/0408014 · Open Systems & Information Dynamics

The Bloch-Vector Space for N-Level Systems: the Spherical-Coordinate Point of View

Gen Kimura, Andrzej Kossakowski

Published Aug 24, 2005• 153 citations
arXivHTMLPDF

Abstract

Bloch-vector spaces for N-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU (N). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.

DOI
10.1007/s11080-005-0919-y

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Bloch-vector spaces for N-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU (N). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.
The properties of the geometric phases between three quantum states are investigated in a high-dimensional Hilbert space using the Majorana representation of symmetric quantum states. We found that the geometric phases between the three quantum states in an NNN-state quantum system can be represented by $N\ensuremath{-}1$ spherical triangles on the Bloch sphere. The parameter dependence of the geometric phase was analyzed based on this picture. We found that the geometric phase exhibits rich nonlinear behavior in a high-dimensional Hilbert space.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects.
The geometric interpretation of (pseudo)spin-1/21/21/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high-energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general NNN-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N)\mathrm{SU}(N)SU(N) algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising $N\ensuremath{-}1$ two-spheres. We demonstrate that for the NNN-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up another avenue for the interpretation of the topological classification and the dynamical illustration of multilevel systems, which in turn is anticipated to help in the design of new experimental probes.
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.