Brachistochrone of Entanglement for Spin Chains

TL;DR

This paper analytically explores how entanglement influences the minimal time evolution of a three-qubit system, revealing optimal control laws and entanglement dynamics in quantum information processing.

Contribution

It introduces an analytical framework for the quantum brachistochrone in a three-qubit system, linking entanglement measures to time-optimal state evolution.

Findings

Entanglement affects the minimal evolution time for certain initial states.

The optimal Hamiltonian law is derived for the three-qubit model.

Residual entanglement is maximized at the minimal evolution time.

Abstract

We analytically investigate the role of entanglement in time-optimal state evolution as an appli- cation of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly cou- pled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entan- glement plays a role for W and GHZ initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.