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Physical Review A

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

H. F. Chau

Published Nov 30, 2010• 6 citations
Publisher

Abstract

Batle et al. [Phys. Rev. A 72, 032337 (2005)] and Borr\'as et al. [Phys. Rev. A 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 be very different from the ones that they have studied.

DOI
10.1103/physreva.82.056301

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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.

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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.

Physical Review A•Nov 30, 2010•4 citations

Connection between entanglement and the speed of quantum evolution

Josep Amengual i Batle, M. Casas, Angel Ricardo Plastino +1 more

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.

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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.

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Entanglement and the speed of evolution of two interacting qubits

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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.

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Two particles in a double well: illustrating the connection between entanglement and the speed of quantum evolution

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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.

European Journal of Physics•Jul 31, 2006•10 citations
Batle et al. [Phys. Rev. A 72, 032337 (2005)] and Borr'as et al. [Phys. Rev. A 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 be very different from the ones that they have studied.
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.
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.