The Bloch vectors formalism for a finite-dimensional quantum system

Abstract

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 their evolution in time, qudit observables and their expectations, entanglement and nonlocality, etc. are expressed in terms of the Bloch vectors -- the vectors in the Euclidean space ${\mathbb{R}}^{d^2-1}$ arising under decompositions of observables and states in different operator bases. Within this formalism, we specify for all $d\ge 2$ the set of Bloch vectors of traceless qudit observables and describe its properties; also, find for the sets of the Bloch vectors of qudit states, pure and mixed, the new compact expressions in terms of the operator norms that explicitly reveal the general properties of these sets and have the unified form for all $d\ge 2$. For the sets of the Bloch vectors of qudit states under the generalized Gell-Mann representation, these general properties cannot be analytically extracted from the known equivalent specifications of these sets via the system of algebraic equations. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated and if it is open and find for both cases the main properties of the Bloch vector evolution in time. For a pure bipartite state of a dimension $d_1\times d_2$, we quantify its entanglement in terms of the Bloch vectors for its reduced states. The introduced general formalism is important both for the theoretical analysis of quantum system properties and for quantum applications, in particular, for optimal quantum control, since, for systems where states are described by vectors in the Euclidean space, the methods of optimal control, analytical and numerical, are well developed.

25 pages, typos in indices corrected

References in corpus (3)

• Bloch vectors for qudits
• Orbits of quantum states and geometry of Bloch vectors for $N$-level systems
• Specifying the unitary evolution of a qudit for a general nonstationary Hamiltonian via the generalized Gell-Mann representation

Cited by in corpus (5)

• Quantifying Bell nonlocality of a pure two-qudit state via its entanglement
• Geometric quantum discord of an arbitrary two-qudit state: the exact value and general upper bounds
• Specifying nonlocality of a pure bipartite state and analytical relations between measures for bipartite nonlocality and entanglement
• Uncertainty relation and the constrained quadratic programming
• High-spin measurements in an arbitrary two-qudit state