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The role of entanglement in dynamical evolution

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone

Published May 31, 2002• 39 citations
OpenAlex

Abstract

Entanglement is a fundamental resource to achieve the maximum allowed speed in the dynamical evolution of a system. In general, energy entangled states evolve to an orthogonal state faster than their unentangled counterparts.

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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.
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.
Quantum speed limits, establishing fundamental thresholds for the speed of quantum processes, are central to a wide spectrum of applications of quantum science, ranging from speeding up gate operations in quantum computations and optimizing communication channels to unraveling the rate at which correlations can propagate within many-body systems. In the quest to tailor optimal quantum dynamical speeds, the role of quantum entanglement has been showcased in specific instances. However, a robust universal link between system entanglement and its dynamical speed remains elusive. Here, we formulate the genuine contribution of quantum entanglement by developing universal criteria determining this speed enhancement within a quantum system. Specifically, we establish an upper limit that rigorously bounds the speed of the dynamical evolution of separable states, providing an unattainable speed threshold for these states. Our findings underscore the capacity of quantum entanglement to surpass this limit, demonstrating its capability as a resource for enhancing dynamical speed 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.
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.