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arXiv:chao-dyn/9508003 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Chaotic evolution in quantum mechanics

Asher Peres

Published May 1, 1996• 8 citations
arXivHTMLPDF

Abstract

A quantum system is described whose wave function has a complexity that increases exponentially with time. Namely, for any fixed orthonormal basis, the number of components required for an accurate representation of the wave function increases exponentially.
DOI
10.1103/physreve.53.4524

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The evolution of a classically chaotic Hamiltonian system is characterised by a computational complexity that increases exponentially with the time elapsed. Quantum systems do not exhibit such behaviour. It is therefore tempting to solve a classical problem by quantising it and following the motion of a small wavepacket. This paper shows that nothing can be gained by this subterfuge: an adequate quantum simulation requires a Hilbert space whose dimension increases exponentially with the planned duration of the simulation.
Highly excited many-particle states in quantum systems (nuclei, atoms, quantum dots, spin systems, quantum computers) can be ‘chaotic’ superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is a result of the very high energy level density of many-body states which can be easily mixed by a residual interaction between particles. We consider the time dynamics of wave functions and increase of entropy in such chaotic systems. As an example, we present the time evolution in a closed quantum computer. A time scale for the entropy S ( t ) increase is t c ~&tgr; 0 /( n log 2 n ), where &tgr; 0 is the qubit ‘lifetime’, n is the number of qubits, S (0) = 0 and S ( t c )=1. At t _ t c the entropy is small: S ~ nt 2 J 2 log 2 (1/ t 2 J 2 ), where J is the inter-qubit interaction strength. At t > t c the number of ‘wrong’ states increases exponentially as 2 S ( t ) . Therefore, t c may be interpreted as a maximal time for operation of a quantum computer. At t >> t c the system entropy approaches that for chaotic eigenstates.
A number of quantum states form an orthonormal and complete basis set if they are normalized and mutually orthogonal and any quantum state can be expressed as a linear combination. In classical mechanics, any two physical variables are commutable. However, this property is no longer valid for physical operators. Commutation of operators is an important property of quantum mechanics. Although there are many representations of wave functions and observables connected by unitary transformations, it is useful to distinguish certain classes of representations, called pictures, which differ in the way the time evolution of the system is treated. The Schrodinger picture is one in which the operators are time-independent but the wave function is time dependent. The virial theorem relates the kinetic energy to its potential in an energy eigenstate.
A bstract Given a choice of an ordered, orthonormal basis for a D -dimensional Hilbert space, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents any quantum state as a real, normalized function on a discrete phase space. The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. Further, Wigner negativity also gives a lower bound on an entropic measure of spread complexity. In this paper, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. In [1], a perturbative argument was given to show that the Krylov basis minimizes the early time growth of Wigner negativity in the large- D limit. Using tools from random matrix theory, here we show that for a generic choice of basis, the Wigner negativity for a classical initial state becomes exponentially large in an O (1) amount of time evolution. On the other hand, we show that in the Krylov basis the negativity grows at most as a power law, and becomes exponentially large only at exponential times. We take this as evidence that the Krylov basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large- D at sub-exponential times. For the Gaussian unitary ensemble, this effective description is the q → 0 limit of q -deformed JT gravity.