A geometric Hamiltonian description of composite quantum systems and quantum entanglement

TL;DR

This paper develops a geometric Hamiltonian framework for finite-dimensional quantum systems, especially bipartite systems, providing a new perspective on quantum entanglement and separability criteria.

Contribution

It introduces a geometric Hamiltonian description of composite quantum systems and proposes a novel entanglement measure within this framework.

Findings

Phase space of bipartite systems is the projective space of the tensor product, not the Cartesian product.

A geometric definition and measure of quantum entanglement are proposed.

Two separability criteria are implemented in the Hamiltonian formalism.

Abstract

Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kahler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be given by the projective space of the tensor product of two Hilbert spaces H and K and not simply by the cartesian product P(H)xP(K) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.