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arXiv:quant-ph/0507030 · Physical Review A

Connection between entanglement and the speed of quantum evolution

Josep Amengual i Batle, M. Casas, Angel Ricardo Plastino, Angel Ricardo Plastino

Published Sep 29, 2005• 75 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.72.032337

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Reply to “Comment on ‘Connection between entanglement and the speed of quantum evolution’ and on ‘Entanglement and the lower bounds on the speed of quantum evolution’ ”

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Chau's Comment deals with the properties exhibited by a particular set of two-qubit states in connection with entanglement and the quantum evolution of some low-dimensional composite systems. However, there is an important aspect of the previously mentioned two-qubit states and of the role they play in relation with the ``speed'' of quantum evolution that was not mentioned by Chau and deserves to be pointed out: for the two-qubit system under consideration, these states require the longest possible absolute time to evolve to an orthogonal state.

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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.
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.
Chau's Comment deals with the properties exhibited by a particular set of two-qubit states in connection with entanglement and the quantum evolution of some low-dimensional composite systems. However, there is an important aspect of the previously mentioned two-qubit states and of the role they play in relation with the ``speed'' of quantum evolution that was not mentioned by Chau and deserves to be pointed out: for the two-qubit system under consideration, these states require the longest possible absolute time to evolve to an orthogonal state.
There is a lower bound for the 'speed' of quantum evolution as measured by the time needed to reach an orthogonal state. We show that, for two-qubits systems, states saturating the quantum speed limit tend to exhibit a small amount of local evolution, as measured by the fidelity between the initial and final single qubit states after the time τ required by the composite system to reach an orthogonal state. Consequently, a trade-off between the speed of global evolution and the amount of local evolution seems to be at work.