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International Journal of Quantum Information

ENTANGLEMENT IN THE DYNAMICAL EVOLUTION OF COMPOSITE FERMIONIC SYSTEMS

V. C. G. OLIVEIRA, H. A. B. SANTOS, L. A. M. TORRES, André M. C. Souza

Published Apr 1, 2008• 27 citations
Publisher

Abstract

The time that a given initial quantum state requires to evolve to an orthogonal state is used to explore the role played by entanglement in the dynamical evolution of composite fermionic systems. We study two different applications of free spin fermions in a four site cluster. It is shown that through selection of the hopping matrix elements, the relationship between entanglement and speed of quantum evolution can be controlled. Consequently, the result recently pointed out [J. Batle et al., Phys. Rev. A72 (2005) 032337] represents only a particular choice of the energy spectrum of the system.

DOI
10.1142/s0219749908003499

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Comment on "Connection between entanglement and the speed of quantum evolution"

H. F. Chau

Batle et al. [Phys. Rev. A {\bf 72}, 032337 (2005)] and Borrás et al. [Phys. Rev. A {\bf 74}, 022326 (2006)] studied the connection between entanglement and speed of quantum evolution for certain low-dimensional bipartite quantum states. However, their studies did not cover all possible cases. And the relation between entanglement and the maximum possible quantum evolution speed for these uncovered cases can very different from the ones that they have studied.

arXiv:1011.3560•Nov 16, 2010•0 citations
The time that a given initial quantum state requires to evolve to an orthogonal state is used to explore the role played by entanglement in the dynamical evolution of composite fermionic systems. We study two different applications of free spin fermions in a four site cluster. It is shown that through selection of the hopping matrix elements, the relationship between entanglement and speed of quantum evolution can be controlled. Consequently, the result recently pointed out [J. Batle et al., Phys. Rev. A72 (2005) 032337] represents only a particular choice of the energy spectrum of the system.
It has been recently pointed out [V. Giovannetti, S. Lloyd, and L. Maccone, Europhys. Lett. 62, 615 (2003)] that, for certain classes of states, quantum entanglement enhances the ``speed'' of evolution of composite quantum systems, as measured by the time a given initial state requires to evolve to an orthogonal state. We provide here a systematic study of this effect for pure states of bipartite systems of low dimensionality, considering both distinguishable (two-qubits) subsystems, and systems constituted of two indistinguishable particles.
The concept of quantum speed limit-time (QSL) was initially introduced as a lower bound to the time interval that a given initial state ${\ensuremath{\psi}}_{I}$ may need so as to evolve into a state orthogonal to itself. Recently [V. Giovanetti, S. Lloyd, and L. Maccone, Phys. Rev. A 67, 052109 (2003)] this bound has been generalized to the case where ${\ensuremath{\psi}}_{I}$ does not necessarily evolve into an orthogonal state, but into any other ${\ensuremath{\psi}}_{F}$. It was pointed out that, for certain classes of states, quantum entanglement enhances the evolution ``speed'' of composite quantum systems. In this work we provide an exhaustive and systematic QSL study for pure and mixed states belonging to the whole 15-dimensional space of two qubits, with ${\ensuremath{\psi}}_{F}$ a not necessarily orthogonal state to ${\ensuremath{\psi}}_{I}$. We display convincing evidence for a clear correlation between concurrence, on the one hand, and the speed of quantum evolution determined by the action of a rather general local Hamiltonian, on the other one.
The time needed by a quantum system to reach a state fully distinguishable from the original one provides a natural way of determining how fast the corresponding dynamical evolution is. This orthogonality time admits a lower bound, expressible in terms of the energy's expectation value and the energy's standard deviation, that yields a 'quantum speed limit'. Composite quantum systems need entanglement in order to achieve this limit. So far, most studies on the connection between entanglement and the quantum speed limit have focused on the case of non-interacting systems. The connection between quantum speed and entanglement is systematically investigated here for the case of a system of two interacting qubits, taking into consideration all possible initial states that evolve into an orthogonal one.
Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum brachistochrone problem. Given an initial and a final state of a composite system we consider the amount of entanglement associated with the brachistochrone evolution between those states, showing that entanglement is an essential resource to achieve the alluded time-optimal quantum evolution.
Abstract We consider the simplest identical-fermion system that exhibits the phenomenon of entanglement (beyond exchange correlations) to analyze its speed of evolution toward an orthogonal state, and revisit the relation between this latter and the amount of fermionic entanglement. A characterization of the quantum speed limit and the orthogonality times is performed, throwing light into the general structure of the faster and the slower states. Such characterization holds not only for fermionic composites, but apply more generally to a wide family of six-dimensional states, irrespective of the specific nature of the system. Further, it is shown that the connection between speed of evolution and entanglement in the fermionic system, though more subtle than in composites of distinguishable parties, may indeed manifest for certain classes of states.