Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

Benatti, Fabio; Krüger, Tyll; Müller, Markus P.; Siegmund‐Schultze, Rainer; Szkola, Arleta

doi:10.1007/s00220-006-0027-z

Cited by 32 publications

(27 citation statements)

References 48 publications

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“…Nevertheless, it has to be stressed that quantum dynamical entropies can be related to the notion of quantum algorithmic complexity, mentioned in section 5, by a quantum version of Brudno theorem established in Ref. [81].…”

Section: Dynamical Entropiesmentioning

confidence: 99%

Chaos and complexity of quantum motion

Prosen¹

2007

J. Phys. A: Math. Theor.

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Abstract. The problem of characterizing complexity of quantum dynamics -in particular of locally interacting chains of quantum particles -will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested.

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Section: Dynamical Entropiesmentioning

confidence: 99%

Chaos and complexity of quantum motion

Prosen¹

2007

J. Phys. A: Math. Theor.

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“…[13] for the classical case and [19] for the quantum case. Also, using f (k) := 1/k as accuracy required on input k is not important; any other computable and strictly decreasing function f that tends to zero for k → ∞ such that f −1 is also computable will give the same result up to an additive constant.…”

Section: Quantum Kolmogorov Complexitymentioning

confidence: 99%

“…In [19], we have shown that for ergodic quantum information sources, emitted states |ψ ∈ C 2 ⊗n have a complexity rate…”

Section: Quantum Kolmogorov Complexitymentioning

confidence: 99%

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

Müller

2008

IEEE Trans. Inform. Theory

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Abstract-We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for an arbitrary, but preassigned number of time steps.As a corollary to this result, we give a rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant.Our proof is based on a new mathematical framework for QTMs, including a thorough analysis of their halting behaviour. We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string can always be effectively decomposed into a classical and a quantum part.

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“…Quantum information theory is a nontrivial generalization of the classical one; hence, one expects that the above arguments can be also generalized that way. Indeed, there are various approaches of quantum information to form the concept of quantum entropy rate [2][3][4][5]. In the present paper, however, we will focus on a description in terms of classical information theory.…”

Section: Introductionmentioning

confidence: 99%

Entropy rate of message sources driven by quantum walks

Kollár

Koniorczyk

2014

Phys. Rev. A

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The amount of information generated by a discrete time stochastic processes in a single step can be quantified by the entropy rate. We investigate the differences between two discrete time walk models, the discrete time quantum walk and the classical random walk in terms of entropy rate. We develop analytical methods to calculate and approximate it. This allows us to draw conclusions about the differences between classical stochastic and quantum processes in terms of the classical information theory.

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Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

Cited by 32 publications

References 48 publications

Chaos and complexity of quantum motion

Chaos and complexity of quantum motion

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

Entropy rate of message sources driven by quantum walks

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