Fidelity Decay Saturation Level for Initial Eigenstates
Yaakov S. Weinstein,
Received November 1, 2002; accepted January 31, 2003
We show that the ﬁdelity decay between an initial eigenstate evolved under a unitary
chaotic operator and the same eigenstate evolved under a perturbed operator
saturates well before the 1=N limit expected for a generic initial state, where N is the
dimension of the Hilbert space. We provide a theoretical argument and numerical
evidence that, for perturbations of intermediate strength, the saturation level
depends quadratically on the perturbation strength.
KEY WORDS: Quantum chaos; ﬁdelity decay; local density of states.
PACS: 05.45.Mt; 03.67.Lx.
Over the past twenty years different phenomena found in quantum systems
that have chaotic classical analogs have been suggested as appropriate
signatures of quantum chaos.
One of these conjectures, that of Peres,
is that the initial rate and behavior of a system’s ﬁdelity decay due to a small
perturbation in the Hamiltonian will differentiate between chaotic and non-
chaotic systems. This signature is analogous to the sensitivity to initial
conditions which characterizes classical chaos but, as a consequence of
strictly unitary evolution, cannot emerge in quantum systems. Recent
have led to a more detailed understanding of this signature.
Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge,
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
To whom correspondence should be addressed. E-mail: firstname.lastname@example.org
Quantum Information Processing, Vol. 1, No. 6, December 2002 (# 2003)
1570-0755/02/1200–0439/0 # 2003 Plenum Publishing Corporation