# Quantum state geometry and entanglement of two spins with anisotropic interaction in evolution

@article{Kuzmak2017QuantumSG, title={Quantum state geometry and entanglement of two spins with anisotropic interaction in evolution}, author={Andrij Kuzmak}, journal={Journal of Geometry and Physics}, year={2017}, volume={116}, pages={81-89} }

Abstract Quantum evolution of a two-spin system with anisotropic Heisenberg Hamiltonian in the magnetic field is considered. We show that this evolution happens on some manifold with geometry depending on the ratio between the interaction couplings and on the initial state. The Fubini-Study metric of this manifold is calculated. The entanglement of the states belonging to this manifold is examined. Also we investigate similar problem for a two-spin system described by the Dzyaloshinsky–Moria… Expand

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#### References

SHOWING 1-10 OF 59 REFERENCES

Geometry of a two-spin quantum state in evolution

- Physics
- 2016

We study the quantum evolution of a two-spin system described by the isotropic Heisenberg Hamiltonian in the external magnetic field. It is shown that this evolution happens on a two-parametric… Expand

Geometric quantum mechanics

- Physics
- 2001

Abstract The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Fubini–Study metric. According to the principles of geometric quantum… Expand

Quantum-state space metric and correlations.

- Physics, Medicine
- Physical review. A, Atomic, molecular, and optical physics
- 1992

It is shown that if the operators are Hermitian, then the off-diagonal components of the metric describe the correlations between them and not only the uncertainty but also the correlation acquires its geometrical interpretation. Expand

Classifying and measuring geometry of a quantum ground state manifold

- Physics
- 2013

From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent… Expand

Riemannian structure on manifolds of quantum states

- Mathematics
- 1980

A metric tensor is defined from the underlying Hilbert space structure for any submanifold of quantum states. The case where the manifold is generated by the action of a Lie group on a fixed state… Expand

The quantum brachistochrone problem for an arbitrary spin in a magnetic field

- Physics, Mathematics
- 2015

We consider quantum brachistochrone evolution for a spin-s system on rotational manifolds. Such manifolds are determined by the rotation of the eigenstates of the operator of projection of spin-s on… Expand

Entanglement of Formation of an Arbitrary State of Two Qubits

- Physics
- 1998

The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state is defined as the minimum average… Expand

Geometry of Quantum Computation with Qutrits

- Physics, Medicine
- Scientific reports
- 2013

It is shown that the optimal quantum circuits are essentially equivalent to the shortest path between two points in a certain curved geometry of SU(3n) so that three-qutrit systems are investigated in detail. Expand

Quantum Gate Design Metric

- Mathematics, Physics
- 2006

What is the time-optimal way of realizing quantum operations? Here, we show how important instances of this problem can be related to the study of shortest paths on the surface of a sphere under a… Expand

Elementary solution to the time-independent quantum navigation problem

- Mathematics, Physics
- 2014

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realizes a required quantum process or task, under the influence of a prevailing 'background' Hamiltonian… Expand