The Bloch vectors formalism for a finite-dimensional quantum system

• URL: http://arxiv.org/abs/2102.11829v3
• Date: Sat, 27 Jan 2024 13:58:09 GMT
• Title: The Bloch vectors formalism for a finite-dimensional quantum system
• Authors: Elena R. Loubenets and Maxim S. Kulakov
• Abstract summary: We consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated. For a pure bipartite state of a dimension $d_1times d_2$, we quantify its entanglement in terms of the Bloch vectors for its reduced states.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $d\geq2$ 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 $d\geq2$. 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.

Related papers

• Bloch Sphere of the Qutrit System [0.0]
  We use the SU(3) group as the basis group to obtain the Bloch vectors of different configurations of the qutrit system. The Bloch trajectories of the two sectors across different configurations exhibit a range of simple to complex curves.
  arXiv  Detail & Related papers  (2024-11-25T15:21:05Z)
• The Schmidt rank for the commuting operator framework [58.720142291102135]
  The Schmidt rank is a measure for the entanglement dimension of a pure bipartite state. We generalize the Schmidt rank to the commuting operator framework. We analyze bipartite states and compute the Schmidt rank in several examples.
  arXiv  Detail & Related papers  (2023-07-21T14:37:33Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• Trace class operators and states in p-adic quantum mechanics [0.0]
  We show that one can define a suitable space of trace class operators in the non-Archimedean setting. The analogies, but also the several (highly non-trivial) differences, with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.
  arXiv  Detail & Related papers  (2022-10-04T12:44:22Z)
• Entanglement dynamics of spins using a few complex trajectories [77.34726150561087]
  We consider two spins initially prepared in a product of coherent states and study their entanglement dynamics. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator.
  arXiv  Detail & Related papers  (2021-08-13T01:44:24Z)
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states [11.04121146441257]
  We prove that the entanglement entropy of any pure initial state of a bosonic quantum system grows linearly in time. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems.
  arXiv  Detail & Related papers  (2021-07-23T07:55:38Z)
• The classical limit of Schr\"{o}dinger operators in the framework of Berezin quantization and spontaneous symmetry breaking as emergent phenomenon [0.0]
  A strict deformation quantization is analysed on the classical phase space $bR2n$. The existence of this classical limit is in particular proved for ground states of a wide class of Schr"odinger operators. The support of the classical state is included in certain orbits in $bR2n$ depending on the symmetry of the potential.
  arXiv  Detail & Related papers  (2021-03-22T14:55:57Z)
• Geometric quantification of multiparty entanglement through orthogonality of vectors [0.0]
  We show that post-measurement vectors can yield non-identical set of maximally entangled states. We discuss the trade-off between the local properties namely predictability and coherence with the global property.
  arXiv  Detail & Related papers  (2021-03-06T08:28:07Z)
• Complete entropic inequalities for quantum Markov chains [17.21921346541951]
  We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional algebra satisfies a modified log-Sobolev inequality. We also establish the first general approximateization property of relative entropy.
  arXiv  Detail & Related papers  (2021-02-08T11:47:37Z)
• Bloch sphere analog of qudits using Heisenberg-Weyl Operators [0.0]
  We study an analogous Bloch sphere of higher-level quantum systems using the Heisenberg-Weyl operator basis. It is well known that we need eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits)
  arXiv  Detail & Related papers  (2021-01-16T08:59:45Z)
• Quantum particle across Grushin singularity [77.34726150561087]
  We study the phenomenon of transmission across the singularity that separates the two half-cylinders. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection. This allows to comprehend the distinguished status of the so-called bridging' transmission protocol previously identified in the literature.
  arXiv  Detail & Related papers  (2020-11-27T12:53:23Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.