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Quantum Kolmogorov complexity based on classical descriptions
 IEEE Trans. Inform. Theory
, 2001
"... Abstract—We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on ..."
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Abstract—We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approximated from above under certain conditions. With high probability a quantum object is incompressible. Upper and lower bounds of the quantum complexity of multiple copies of individual pure quantum states are derived and may shed some light on the nocloning properties of quantum states. In the quantum situation complexity is not subadditive. We discuss some relations with “nocloning ” and “approximate cloning ” properties. Keywords — Algorithmic information theory, quantum; classical descriptions of quantum states; information theory, quantum; Kolmogorov complexity, quantum; quantum cloning. I.
Quantum algorithmic entropy
 in Proc. 16th IEEE Conf. Computational Complexity
, 2001
"... ABSTRACT. We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (“universal probability”) as a starting point, and define complexity (an operator) as its negative logarithm. A number of properties of Kolmogorov complexity extend naturally to ..."
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ABSTRACT. We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (“universal probability”) as a starting point, and define complexity (an operator) as its negative logarithm. A number of properties of Kolmogorov complexity extend naturally to the new domain. Approximately, a quantum state is simple if it is within a small distance from a lowdimensional subspace of low Kolmogorov complexity. The von Neumann entropy of a computable density matrix is within an additive constant from the average complexity. Some of the theory of randomness translates to the new domain. We explore the relations of the new quantity to the quantum Kolmogorov complexity defined by Vitányi (we show that the latter is sometimes as large as 2n − 2 log n) and the qubit complexity defined by Berthiaume, Dam and Laplante. The “cloning ” properties of our complexity measure are similar to those of qubit complexity. 1.
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.
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.
Article Quantum Dynamical Entropies and Gács Algorithmic Entropy
, 2012
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Quantum Optimization Problems
, 2002
"... Abstract. Krentel [J. Comput. System. Sci., 36, pp.490–509] presented a framework for an NP optimization problem that searches an optimal value among exponentiallymany outcomes of polynomialtime computations. This paper expands his framework to a quantum optimization problem using polynomialtime q ..."
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Abstract. Krentel [J. Comput. System. Sci., 36, pp.490–509] presented a framework for an NP optimization problem that searches an optimal value among exponentiallymany outcomes of polynomialtime computations. This paper expands his framework to a quantum optimization problem using polynomialtime quantum computations and introduces the notion of an “universal ” quantum optimization problem similar to a classical “complete ” optimization problem. We exhibit a canonical quantum optimization problem that is universal for the class of polynomialtime quantum optimization problems. We show in a certain relativized world that all quantum optimization problems cannot be approximated closely by quantum polynomialtime computations. We also study the complexity of quantum optimization problems in connection to wellknown complexity classes. Keywords: 1
To my FAMILY who have always loved me KOLMOGOROV COMPLEXITY FOR PROBABILISTIC COMPUTATIONS: TOWARDS RESOURCEBOUNDED KOLMOGOROV COMPLEXITY FOR QUANTUM COMPUTING
, 2002
"... (discard this page) iv The notion of Kolmogorov complexity is very useful in areas ranging from data compression to cryptography to foundations of physics. Some applications use resource bounded Kolmogorov complexity, a version that takes into consideration the running time of the corresponding pro ..."
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(discard this page) iv The notion of Kolmogorov complexity is very useful in areas ranging from data compression to cryptography to foundations of physics. Some applications use resource bounded Kolmogorov complexity, a version that takes into consideration the running time of the corresponding program. The notion of Kolmogorov complexity was originally proposed for traditional (deterministic) algorithms. Lately, it has been shown that in many important problems, probabilistic (in particular quantum) algorithms perform much better than the deterministic ones. It is therefore desirable to generalize the notion of Kolmogorov complexity to probabilistic algorithms. In this thesis, we proposed such a generalization. The resulting definition (partially) explains heuristic quantum versions of Kolmogorov complexity that have been proposed by P. Vitányi, S. Laplante, and others.
FAMILY who have always loved meKOLMOGOROV COMPLEXITY FOR PROBABILISTIC COMPUTATIONS: TOWARDS RESOURCEBOUNDED KOLMOGOROV COMPLEXITY FOR QUANTUM COMPUTING
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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.