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Information theory, evolutionary computation, and Dembski’s “complex specified information”
, 2003
"... Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws ..."
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Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws are incapable of generating CSI. In particular, CSI cannot be generated by evolutionary computation. Dembski asserts that CSI is present in intelligent causes and in the flagellum of Escherichia coli, and concludes that neither have natural explanations. In this paper we examine Dembski’s claims, point out significant errors in his reasoning, and conclude that there is no reason to accept his assertions.
Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity
 IEEE Trans. Inf. Th
"... 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 ..."
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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.
Upper bound by Kolmogorov complexity for the probability in computable quantum measurement
 In: Proceedings 5th Conference on Real Numbers and Computers
"... Abstract. We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given classical finite binary string s. In the standar ..."
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Abstract. We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given classical finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal selfdelimiting Turing machine to output s. In the other way, we first introduce the socalled universal probability m, and then define K(s) as − log 2 m(s) without using the concept of programsize. We generalize the universal probability to a matrixvalued function, and identify this function with a POVM (positive operatorvalued measure). On the basis of this identification, we study a computable POVM measurement with countable measurement outcomes performed upon a finite dimensional quantum system. We show that, up to a multiplicative constant, 2 −K(s) is the upper bound for the probability of each measurement outcome s in such a POVM measurement. In what follows, the upper bound 2 −K(s) is shown to be optimal in a certain sense.
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2006
"... In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qub ..."
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In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them as outputs.
The Quantum Computing Challenge
"... . The laws of physics imposes limits on increases in computing power. Two of these limits are interconnect wires in multicomputers and thermodynamic limits to energy dissipation in conventional irreversible technology. Quantum computing is a new computational technology that promises to eliminate pr ..."
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. The laws of physics imposes limits on increases in computing power. Two of these limits are interconnect wires in multicomputers and thermodynamic limits to energy dissipation in conventional irreversible technology. Quantum computing is a new computational technology that promises to eliminate problems of latency and wiring associated with parallel computers and the rapidly approaching ultimate limits to computing power imposed by the fundamental thermodynamics. Moreover, a quantum computer will be able to exponentially improve known classical algorithms for factoring, and quadratically improve every classical algorithm for searching an unstructured list, as well as give various speedups in communication complexity, by exploiting unique quantum mechanical features. Finally, a quantum computer may be able to simulate quantum mechanical systems, something which seems out of the question for classical computers, thus reaching the ultimate goal of replacing actual quantum...
Computational Complexity Measures of Multipartite Quantum Entanglement (Extented Abstract)
, 2003
"... Abstract. We shed new light on entanglement measures in multipartite quantum systems by taking a computationalcomplexity approach toward quantifying quantum entanglement with two familiar notions— approximability and distinguishability. Built upon the formal treatment of partial separability, we me ..."
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Abstract. We shed new light on entanglement measures in multipartite quantum systems by taking a computationalcomplexity approach toward quantifying quantum entanglement with two familiar notions— approximability and distinguishability. Built upon the formal treatment of partial separability, we measure the complexity of an entangled quantum state by determining (i) how hard to approximate it from a fixed classical state and (ii) how hard to distinguish it from all partially separable states. We further consider the Kolmogorovianstyle descriptive complexity of approximation and distinction of partial entanglement. 1 Computational Aspects of Quantum Entanglement Entanglement is one of the most puzzling notions in the theory of quantum information and computation. A typical example of an entangled quantum state is the Bell state (or the EPR pair) (00 〉 + 11〉) / √ 2, which played a major role in, e.g., superdense coding [4] and quantum teleportation schemes [1]. Entanglement can be viewed as a physical resource and therefore can be quantified.
On the Quantum Complexity of Classical Words
, 2007
"... We show that classical and quantum Kolmogorov complexity of binary words agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the corresponding word. It follows that quantum complexity is an extension o ..."
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We show that classical and quantum Kolmogorov complexity of binary words agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the corresponding word. It follows that quantum complexity is an extension of classical complexity to the domain of quantum states. This is true even if we allow a small probabilistic error in the quantum computer’s output. We outline a mathematical proof of this statement, based on some analytical estimates and a classical program for the simulation of a universal quantum computer.
GACS QUANTUM COMPLEXITY AND QUANTUM ENTROPY
"... The development of quantum information theory motivated the extension to the quantum realm of notions from algorithmic complexity theory. Because of the structure of quantum mechanics, several inequivalent generalizations are possible. The same phenomenon characterizes the extension of the dynamical ..."
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The development of quantum information theory motivated the extension to the quantum realm of notions from algorithmic complexity theory. Because of the structure of quantum mechanics, several inequivalent generalizations are possible. The same phenomenon characterizes the extension of the dynamical entropy of Kolmogorov and Sinai. In the following, we shall examine the relations between the quantum complexity introduced by Gacs and the quantum dynamical entropy proposed by Alicki and Fannes. 1.