Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase-space, reminiscent of Gibbs' paradox in statistical mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos and how the unitary time evolution in a closed system directly implies the "quantum death" of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, could it result from an exchange of entropy with the macroscopic meter? This possibility is explored, presenting a model for spin measurement in a unitary setting and some preliminary analytical results based on it.
Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase-space, reminiscent of Gibbs' paradox in statistical mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos and how the unitary time evolution in a closed system directly implies the "quantum death" of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, could it result from an exchange of entropy with the macroscopic meter? This possibility is explored, presenting a model for spin measurement in a unitary setting and some preliminary analytical results based on it.
The notion of Shannon entropy is crucial for the theory of classical information. In quantum information theory, an analogous key role is played by the von Neumann entropy: quantum information processing is closely related to entropy dynamics. This reveals a direct link with the theory of quantum chaotic systems, which can be characterized by a positive entropy production. Furthermore, noise, which inevitably affects any quantum system, can be modeled by a random quantum operation or by coupling to an environment in a generic chaotic state. In this contribution, we emphasize the universality of quantum chaotic dynamics and discuss its implications for quantum information processing.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.
Quantum chaos is the study of quantum systems whose classical description is chaotic. How does chaos manifest itself in the quantum world? In this spirit, we study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled-tops, where chaos and entanglement arise from the same physical mechanism. The long-time entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section - it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudo-random states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. Continuing on our journey to find the footprints of chaos in the quantum world, we explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. We find an increase in the rate of information gain and hence higher fidelities in the process when the Floquet maps employed increase in chaoticity. We make predictions for the information gain using random matrix theory in the fully chaotic regime and show a remarkable agreement between the two. The last part of this thesis is devoted to the study of the nature of quantum correlations themselves. We present an operational interpretation of quantum discord based on the quantum state merging protocol. Quantum discord is the markup in the cost of quantum communication in the process of quantum state merging, if one discards relevant prior information.
A bstract A classical dynamical system can be viewed as a probability space equipped with a measure-preserving time evolution map, admitting a purely algebraic formulation in terms of the algebra of bounded functions on the phase space. Similarly, a quantum dynamical system can be formulated using an algebra of bounded operators in a non-commutative probability space equipped with a time evolution map. Chaos, in either setting, can be characterized by statistical independence between observables at t = 0 and t → ∞, leading to the vanishing of cumulants involving these observables. In the quantum case, the notion of independence is replaced by free independence, which only emerges in the thermodynamic limit (asymptotic freeness). In this work, we propose a definition of quantum chaos based on asymptotic freeness and investigate its emergence in quantum many-body systems including the mixed-field Ising model with a random magnetic field, a higher spin version of the same model, and the SYK model. The hallmark of asymptotic freeness is the emergence of the free convolution prediction for the spectrum of operators of the form A (0) + B ( t ), implying the vanishing of all free cumulants between A (0) and B ( t ) in the thermodynamic limit for an infinite-temperature thermal state. We systematically investigate the spectral properties of A (0) + B ( t ) in the above-mentioned models, show that fluctuations on top of the free convolution prediction follow universal Wigner-Dyson statistics, and discuss the connection with quantum chaos. Finally, we argue that free probability theory provides a rigorous framework for understanding quantum chaos, offering a unifying perspective that connects many different manifestations of it.
In a previous essay I argued that quantum chaos cannot be exhibited in models of quantum systems within von Neumann's mathematical framework for quantum mechanics, and that it can be exhibited in models within Dirac's formal framework. In this essay, the negative thesis concerning von Neumann's framework is elaborated further by extending it to the case of Hamiltonian operators having a continuous spectrum. The positive thesis concerning Dirac's formal framework is also elaborated further by constructing a chaotic model of an open quantum system in which an entropy measure is shown to approach its maximum value as time goes to infinity. Having such an entropy measure is a characteristic that is closely connected to chaotic behavior in phase space models of classical systems.