The connection between entanglement and the speed of quantum evolution (as measured by the time needed to reach an orthogonal state) is discussed in the case of two quantum particles moving in a one-dimensional double well. The aforementioned connection offers an interesting opportunity of discussing the basic features of quantum entanglement within an elementary context, using concepts and methods usually included in university courses of quantum mechanics.
The connection between entanglement and the speed of quantum evolution (as measured by the time needed to reach an orthogonal state) is discussed in the case of two quantum particles moving in a one-dimensional double well. The aforementioned connection offers an interesting opportunity of discussing the basic features of quantum entanglement within an elementary context, using concepts and methods usually included in university courses of quantum mechanics.
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.
Due to its great importance, both from the fundamental and from the practical points of view, it is imperative that the concept of entanglement is explored. In this thesis I investigate the connection between information measures, entanglement and the “speed” of quantum evolution. In Chapter 1 a brief review of the different information and entanglement measures as well as of the concept of “speed” of quantum evolution is given. An illustration of the quantum no-cloning theorem in connection with closed timelike curves is also provided. The work leading up to this thesis has resulted in the following three publications and in one conference proceeding: (A) C. Zander and A.R. Plastino, “Composite systems with extensive Sq (power-law) entropies”, Physica A 364, (2006) pp. 145-156 (B) S. Curilef, C. Zander and A.R. Plastino, “Two particles in a double well: illustrating the connection between entanglement and the speed of quantum evolution”, Eur. J. Phys. 27, (2006) pp. 1193-1203 (C) C. Zander, A.R. Plastino, A. Plastino and M. Casas, “Entanglement and the speed of evolution of multi-partite quantum systems”, J. Phys. A: Math. Theor. 40 (11), (2007) pp. 2861-2872 (D) A.R. Plastino and C. Zander, “Would Closed Timelike Curves Help to Do Quantum Cloning?”, AIP Conference Proceedings: A century of relativity physics, ERE 841, (2005) pp. 570-573. Chapter 2 is based on (A) and is an application of the Sq (powerlaw) entropy. A family of models for the probability occupancy of phase space exhibiting an extensive behaviour of Sq and allowing for an explicit analysis of the thermodynamic limit is proposed. Chapter 3 is based on (B). The connection between entanglement and the speed of quantum evolution as measured by the time needed to reach an orthogonal state is discussed in the case of two quantum particles moving in a one-dimensional double well. This illustration is meant to be incorporated into the teaching of quantum entanglement. Chapter 4 is based on (C). The role of entanglement in time evolution is investigated in the cases of two-, three- and N-qubit systems. A clear correlation is seen between entanglement and the speed of evolution. States saturating the speed bound are explored in detail. Chapter 5 summarizes the conclusions drawn in the previous chapters.
There exists an interesting relationship between entanglement and the time evolution of composite quantum systems: quantum entanglement enhances the 'speed' of evolution of certain quantum states, as measured by the time needed to reach an orthogonal state. Previous research done on this subject has been focused upon comparing extreme cases (highly entangled states versus separable states) or upon bi-partite systems. In the present contribution we explore the aforementioned connection (between entanglement and time evolution) in the cases of two-qubits and N -qubits systems, taking into account states of intermediate entanglement. In particular, we investigate a family of energetically symmetric states of low entanglement that saturate the quantum speed bound. We show that, as the number of qubits increases, very little entanglement is needed to reach the quantum speed limit.
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.
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.