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arXiv:chao-dyn/9303002 · Physical Review Letters

Hypersensitivity to perturbations in the quantum baker’s map

Rüdiger Schack, Carlton M. Caves

Published Jul 26, 1993• 64 citations
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Abstract

We analyze a randomly perturbed quantum version of the baker's transformation, a prototype of an area-conserving chaotic map. By simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this grows very rapidly and becomes larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker's map displays a hypersensitivity to perturbations analogous to behavior found in the classical case. This hypersensitivity characterizes ``quantum chaos'' in a way that is relevant to statistical physics.

DOI
10.1103/physrevlett.71.525

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We analyze a randomly perturbed quantum version of the baker's transformation, a prototype of an area-conserving chaotic map. By simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this grows very rapidly and becomes larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker's map displays a hypersensitivity to perturbations analogous to behavior found in the classical case. This hypersensitivity characterizes ``quantum chaos'' in a way that is relevant to statistical physics.
Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has lead to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the baker's map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.
For the quantum kicked top we study numerically the distribution of Hilbert-space vectors evolving in the presence of a small random perturbation. For an initial coherent state centered in a chaotic region of the classical dynamics, the evolved perturbed vectors are distributed essentially like random vectors in Hilbert space. In contrast, for an initial coherent state centered near an elliptic (regular) fixed point of the classical dynamics, the evolved perturbed vectors remain close together, explore only a few dimensions of Hilbert space, and do not explore them randomly. These results support and extend the results of earlier studies, thereby providing additional support for a characterization of quantum chaos that uses concepts from information theory.
Randomness generation through quantum-chaotic evolution underpins foundational questions in statistical mechanics and applications across quantum information science, including benchmarking, tomography, metrology, and demonstrations of quantum computational advantage. While statistical mechanics successfully captures the temporal averages of local observables, understanding randomness at the level of higher statistical moments remains a daunting challenge, with analytic progress largely confined to random quantum circuit models or fine-tuned systems exhibiting space-time duality. Here we study how much randomness can be dynamically generated by generic quantum-chaotic evolution under physical, non-random Hamiltonians. Combining theoretical insights with numerical simulations, we show that for broad classes of initially unentangled states, the dynamics become effectively Haar-random well before the system can ergodically explore the physically accessible Hilbert space. Both local and highly nonlocal observables, including entanglement measures, equilibrate to their Haar expectation values and fluctuations on polynomial timescales with remarkably high numerical precision, and with the fastest randomization occurring in regions of parameter space previously identified as maximally chaotic. Interestingly, this effective randomization can occur on timescales linear in system size, suggesting that the sub-ballistic growth of Renyi entropies typically observed in systems with conservation laws can be bypassed in non-random Hamiltonians with an appropriate choice of initial conditions.