This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and define dcq-computable states as those that can be approximated by a dcq-TM. Moreover, we introduce (conditional) Kolmogorov complexity of quantum states and use it to study three particular aspects of the algorithmic information contained in a quantum state: a comparison of the information in a quantum state with that of its classical representation as an array of real numbers, an exploration of the limits of quantum state copying in the context of algorithmic complexity, and study of the complexity of correlations in quantum systems, resulting in a correlation-aware definition for algorithmic mutual information that satisfies symmetry of information property.This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and define dcq-computable states as those that can be approximated by a dcq-TM. Moreover, we introduce (conditional) Kolmogorov complexity of quantum states and use it to study three particular aspects of the algorithmic information contained in a quantum state: a comparison of the information in a quantum state with that of its classical representation as an array of real numbers, an exploration of the limits of quantum state copying in the context of algorithmic complexity, and study of the complexity of correlations in quantum systems, resulting in a correlation-aware definition for algorithmic mutual information that satisfies symmetry of information property.

arXiv:2305.14252QuantumJan 18, 20243 citations

Quantum Kolmogorov Complexity

André Berthiaume, Wim van Dam, Sophie Laplante

arXiv:quant-ph/0005018Journal of Computer and System SciencesSep 1, 2001108 citations

Hydrodynamic theory of scrambling in chaotic long-range interacting systems

Tianci Zhou, Andrew Y. Guo, Shenglong Xu +2 more

The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a mean-field theory of out-of-time-ordered commutators in locally interacting quantum chaotic systems at high energy density. In systems with power-law interactions, the corresponding fractional-derivative FKPP equation provides an analogous mean-field theory. However, the fractional FKPP description is potentially subject to strong quantum fluctuation effects, so it is not clear a priori if it provides a suitable effective description for generic chaotic systems with power-law interactions. Here we study this problem using a model of coupled quantum dots with interactions decaying as $1/{r}^{\ensuremath{\alpha}}$, where each dot hosts $N$ degrees of freedom. The large- $N$ limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by $1/N$ corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for $N=1$ and generic finite $N$ . In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.The Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation provides a mean-field theory of out-of-time-ordered commutators in locally interacting quantum chaotic systems at high energy density. In systems with power-law interactions, the corresponding fractional-derivative FKPP equation provides an analogous mean-field theory. However, the fractional FKPP description is potentially subject to strong quantum fluctuation effects, so it is not clear a priori if it provides a suitable effective description for generic chaotic systems with power-law interactions. Here we study this problem using a model of coupled quantum dots with interactions decaying as $1/{r}^{\ensuremath{\alpha}}$, where each dot hosts $N$ degrees of freedom. The large-$N$ limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by $1/N$ corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for $N=1$ and generic finite $N$. In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.

arXiv:2208.01649Physical review. B./Physical review. BJan 3, 202310 citations

Quantum Stochastics

Günter Mahler A, Volker A. Weberruß

Springer eBooksJan 1, 19950 citations

Stochastic dynamics of quantum jumps

Heinz‐Peter Breuer, Francesco Petruccione

The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form ${\mathit{tsum}}_{\mathit{i}}$ ${\mathit{A}}_{\mathit{i}}$ \ensuremath{\bigotimes} ${\mathit{B}}_{\mathit{i}}$ , where ${\mathit{A}}_{\mathit{i}}$ and ${\mathit{B}}_{\mathit{i}}$ are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.The dynamics of an open quantum system coupled to an external reservoir is studied on the basis of a recently proposed formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. The previous result is generalized to include interaction Hamiltonians of the form ${\mathit{tsum}}_{\mathit{i}}$${\mathit{A}}_{\mathit{i}}$\ensuremath{\bigotimes}${\mathit{B}}_{\mathit{i}}$, where ${\mathit{A}}_{\mathit{i}}$ and ${\mathit{B}}_{\mathit{i}}$ are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 1, 199556 citations

Quantum Stochastics

Günter Mahler A, Volker A. Weberruß

Springer eBooksJan 1, 19980 citations

N

degrees of freedom. The large-

N

limit corresponds to the mean-field description, while quantum fluctuations contributing to the OTOC can be modeled by

1/N

corrections consisting of a cutoff function and noise. Within this framework, we show that the parameters of the effective theory can be chosen to reproduce the butterfly light cone scalings previously found for

N=1

and generic finite

N

. In order to reproduce these scalings, the fractional index $\ensuremath{\mu}$ in the FKPP equation needs to be shifted from the na\"{\i}ve value of $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}1$ to a renormalized value $\ensuremath{\mu}=2\ensuremath{\alpha}\ensuremath{-}2$. We provide supporting analytic evidence for the cutoff model and numerical confirmation for the full fractional FKPP equation with cutoff and noise.

{\mathit{tsum}}_{\mathit{i}}

{\mathit{A}}_{\mathit{i}}

\ensuremath{\bigotimes}

{\mathit{B}}_{\mathit{i}}

, where

{\mathit{A}}_{\mathit{i}}

and

{\mathit{B}}_{\mathit{i}}

are operators acting on the Hilbert space of the reduced system and of the reservoir, respectively. The differential Chapman-Kolmogorov equation governing the dynamics of the conditional transition probability of the reduced system is derived from the underlying microscopic theory based on the Schr\"odinger equation for the total system. The stochastic process turns out to be a piecewise deterministic Markovian jump process in the projective Hilbert space of the reduced system. The sample paths are derived and shown to be similar to those of the Monte Carlo wave function simulation methods proposed in the literature. Finally, a diffusion-noise expansion of the Liouville master equation is performed and demonstrated to yield a stochastic differential equation for the state vector of the open system.