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

@article{Kimura2005TheBS, title={The Bloch-Vector Space for N-Level Systems: the Spherical-Coordinate Point of View}, author={Gen Kimura and Andrzej Kossakowski}, journal={Open Systems \& Information Dynamics}, year={2005}, volume={12}, pages={207-229} }

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.

## 56 Citations

### Numerical shadow and geometry of quantum states

- Mathematics
- 2011

The totality of normalized density matrices of dimension N forms a convex set in . Working with the flat geometry induced by the Hilbert–Schmidt distance, we consider images of orthogonal projections…

### Do spins have directions?

- PhysicsSoft Comput.
- 2017

The extended Bloch model is presented and used to investigate the nature of the quantum spin entities and of their relation to the three-dimensional Euclidean theater, and a new view of realism is put forward, that is multiplex realism, providing a specific framework with which to interpret the human observations and understanding of the component parts of the world.

### The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

- PhysicsQuantum
- 2021

This work demonstrates that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter.

### Eigenprojectors, Bloch vectors and quantum geometry of N-band systems

- Physics
- 2021

The eigenvalues of a parameter-dependent N × N Hamiltonian matrix form a band structure in parameter space. Quantum geometric properties (Berry curvature, quantum metric, etc.) of such N-band systems…

### The Bloch vectors formalism for a finite-dimensional quantum system

- Mathematics, PhysicsJournal of Physics A: Mathematical and Theoretical
- 2021

In the present article, we consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. In the frame of this formalism, qudit states and…

### Geometry of Pure States of N Spin-J System

- MathematicsOpen Syst. Inf. Dyn.
- 2010

We present the geometry of pure states of an ensemble of N spin-J systems using a generalisation of the Majorana representation. The approach is based on Schur-Weyl duality that allows for simple…

### Berry curvature and quantum metric in N -band systems: An eigenprojector approach

- PhysicsPhysical Review B
- 2021

The eigenvalues of a parameter-dependent Hamiltonian matrix form a band structure in parameter space. In such N -band systems, the quantum geometric tensor (QGT), consisting of the Berry curvature…

### Geometry of the generalized Bloch sphere for qutrits

- Mathematics
- 2016

The geometry of the generalized Bloch sphere Ω3, the state space of a qutrit, is studied. Closed form expressions for Ω3, its boundary ∂Ω3, and the set of extremals &OHgr; 3 ext ?> are obtained by…

### Four-dimensional Bloch sphere representation of qutrits using Heisenberg-Weyl Operators

- Physics
- 2021

In the Bloch sphere based representation of qudits with dimensions greater than two, the Heisenberg-Weyl operator basis is not preferred because of presence of complex Bloch vector components. We try…

### Dissecting the qutrit

- Physics
- 2012

To visualize a higher dimensional object it is convenient to consider its two-dimensional cross-sections. The set of quantum states for a three level system has eight dimensions. We supplement a…

## References

SHOWING 1-10 OF 33 REFERENCES

### Orbits of quantum states and geometry of Bloch vectors for N-level systems

- Physics, Mathematics
- 2004

Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than 2. To shed some light on the complicated structure of the set of…

### Hilbert–Schmidt volume of the set of mixed quantum states

- Mathematics
- 2003

We compute the volume of the convex (N2 − 1)-dimensional set N of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyper-area of the boundary of this set is also found and…

### Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation

- Mathematics
- 2003

A parametrization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parametrization we find the…

### Entropy production in coherence-vector formulation for N-level systems.

- PhysicsPhysical review. A, General physics
- 1986

General exact formulas for two-level systems are given and, in arbitrary dimensions, for a weakly irreversible process close to the central state, a first-principles derivation of the phenomenological Onsager coefficients is outlined.

### Quantum computation and quantum information

- PhysicsMathematical Structures in Computer Science
- 2007

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal…

### Quantum Dynamical Semigroups and Applications

- Physics
- 1987

In this text the authors develop quantum dynamics of open systems for a wide class of irreversible processes starting from the concept of completely positive semigroups. This unified approach makes…

### A Class of Linear Positive Maps in Matrix Algebras II

- MathematicsOpen Syst. Inf. Dyn.
- 2004

A systematic construction by means of spectra of generators of SU(n) is discussed and a class of linear trace preserving positive maps on matrix algebras which is a generalization of that in [7].

### Separability of mixed states: necessary and sufficient conditions

- Mathematics, Economics
- 1996

### Phys

- Lett. A 286
- 2001