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arXiv:quant-ph/0303085 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The quantum speed limit

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone

Published May 16, 2003• 21 citations
arXivHTMLPDF

Abstract

How fast can a quantum system evolve? We answer this question focusing on the role of entanglement and interactions among subsystems. In particular, we analyze how the order of the interactions shapes the dynamics.
DOI
10.1117/12.507486

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Let us have a Hamiltonian H and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time t. Now let us fix these two states ; then we may have infinite Hamiltonians (other than H, and one of them is say H') that can time evolve one of them into the other. But the time taken for this evolution varies from t and is say t' (that is, t' is the time Hamiltonian H' takes to time evolve one of the fixed states into the other). From now on, we shall consider only those Hamiltonians that can time evolve one of the states into the other. The question is given any small positive number a, will there exist a Hamiltonian H such that t' = a? Answer: Yes , the Hamiltonian Ht/a (obtained by rescaling the original Hamiltonian) will work. But, there are bounds that show that as long as the average energy and its fluctuations are bounded (thus, no rescaling allowed), you cannot go arbitrarily fast from one state to another orthogonal state. This discussion falls under the study of quantum speed limits. This document looks into this field of active research. We shall study with care how the question of a speed limit arises: from varying the Hamiltonians between two fixed states ; and hence produce an accurate speed limit between any two fixed states. The process and results shall develop as concretely and smoothly as the problem which is produced.

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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.
The quantum speed limit sets a fundamental restriction on the evolution time of quantum systems. We explore the relationship between quantum imaginarity and the quantum speed limit by utilizing measures such as relative entropy, trace distance, and geometric imaginarity. These speed limits define the fundamental constraints on the minimum time necessary for quantum systems to evolve under various dynamical processes. As applications the dephasing dynamics and dissipative dynamics are analyzed in detail. The quantum speed limit in stochastic-approximate transformations is also investigated. Our quantum speed limits provide lower bounds on how fast a physical system evolves to attain or lose certain imaginarity, with potential applications in efficient quantum computation designs, quantum control and quantum sensing.
Let us have a Hamiltonian H and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time t. Now let us fix these two states ; then we may have infinite Hamiltonians (other than H, and one of them is say H') that can time evolve one of them into the other. But the time taken for this evolution varies from t and is say t' (that is, t' is the time Hamiltonian H' takes to time evolve one of the fixed states into the other). From now on, we shall consider only those Hamiltonians that can time evolve one of the states into the other. The question is given any small positive number a, will there exist a Hamiltonian H such that t' = a? Answer: Yes , the Hamiltonian Ht/a (obtained by rescaling the original Hamiltonian) will work. But, there are bounds that show that as long as the average energy and its fluctuations are bounded (thus, no rescaling allowed), you cannot go arbitrarily fast from one state to another orthogonal state. This discussion falls under the study of quantum speed limits. This document looks into this field of active research. We shall study with care how the question of a speed limit arises: from varying the Hamiltonians between two fixed states ; and hence produce an accurate speed limit between any two fixed states. The process and results shall develop as concretely and smoothly as the problem which is produced.