Quantum algorithmic entropy

Gács, Péter

doi:10.1088/0305-4470/34/35/312

Cited by 52 publications

(70 citation statements)

References 3 publications

(4 reference statements)

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“…Gacs [6] and Tadaki [7] defined quantum Kolmogorov complexity without reference to a computation device by generalizing classical universal semi-measures to quantum universal semi-POVM's. Tadaki [8] went on to derive Ω Q , a quantum generalization of Chaitin's halting probability [9].…”

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confidence: 99%

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Rogers

Vedral

2008

Mod. Phys. Lett. B

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The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach -our measure of second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities it is continuous. We give examples where second quantized and quantum Kolmogorov complexity differ.PACS numbers: 03.67.LxQuantum physics, as far as we know, is the most accurate and universal description of all physical phenomena in the universe. If we therefore wish to speak about the complexity of some processes or some physical states in nature, we need to use quantum physics as our best available theory. An important question is (in very simple terms), given a physical system in some state, how difficult is it for us to reproduce it. If we wish to have a universal measure of this difficulty (which applies to all systems and states) a way to proceed is to follow the prescription of Kolmogorov. First, we define a universal computer (which is capable of simulating all other computers) and then we look for the shortest input (another physical state) to this computer that reproduces as the output the desired physical state. This way, all the complexity is defined with respect to the same universal computer, and we can thus speak about universal complexity. The universal computer (such as a universal Turing machine) needs to be fully quantum mechanical in order to capture all the possibilities available in nature. In this letter, we define a fully quantized Turing machine which we use to define a fully quantum Kolmogorov complexity and apply it to a number of different problems. Our approach is different from others in that we consider indeterminate length input programs whose expected length is our measure of complexity. Others, whose work is discussed in detail below, have perhaps avoided our approach, as allowing programs in superposition can lead to a superposition of halting times. Since traditionally computation is viewed as giving a deterministic output after a fixed amount of time, having a superposition of halting times seems to contradict the very concept of computation. We, on the other hand, view the superposition of different length inputs which can lead to a superposition of halting times as a necessity dictated by a fully quantum mechanical description of nature. Moreover allowing superpositions of different programs can lead to a program which has on average a smaller Kolmogorov complexity than when programs are restricted to having a variable but determinate length (we give examples of this in this letter). Since we want to know what is physically the shortest input which produces a given output, we need allow the computation with superpositions of different length programs. This letter is organized as follows. First we discuss the concept of a Turing machine and review previous work. Then we define the...

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confidence: 99%

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Rogers

Vedral

2008

Mod. Phys. Lett. B

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“…It cannot be merely considered as a description of properties defined at any moment but rather as a predictive tool relative to some experimental situation. Nevertheless, some authors (Zurek 1989;Caves 1990;Vitányi 2001;Gács 2001); (Mora and Briegel 2004) have proposed to apply the notion of algorithmic complexity to quantum states according to two main approaches: Mora and Briegel (2004) have proposed a concept of preparation complexity which characterizes "how difficult" it is to prepare a quantum state. These authors have proposed to define the algorithmic complexity of a quantum state as the classical algorithmic complexity of its preparation process, which can always be described as a finite sequence of quantum gates-which express simple quantum unitary operations, like for example, the Hadamard gate or the CNOT gate (see (Nielsen 1998) for a review).…”

Section: What Is "Information?"mentioning

confidence: 99%

“…4 Indeed, the notion of a quantum computer is not essential since its work can always be simulated by a classical computer provided the question of the computation time is not taken into account(Vitányi 2001, p. 5;Gács 2001).…”

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Informational Branching Universe

Uzan

2009

Found Sci

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This paper suggests an epistemic interpretation of Belnap's branching spacetimes theory based on Everett's relative state formulation of the measurement operation in quantum mechanics. The informational branching models of the universe are evolving structures defined from a partial ordering relation on the set of memory states of the impersonal observer. The totally ordered set of their information contents defines a linear "time" scale to which the decoherent alternative histories of the informational universe can be referredwhich is quite necessary for assigning them a probability distribution. The "historical" state of a physical system is represented in an appropriate extended Hilbert space and an algebra of multi-branch operators is developed. An age operator computes the informational depth of historical states and its standard deviation can be used to provide a universal information/energy uncertainty relation. An information operator computes the encoding complexity of historical states, the rate of change of its average value accounting for the process of correlation destruction inherent to the branching dynamics. In the informational branching models of the universe, the asymmetry of phenomena in nature appears as a mere consequence of the subject's activity of measuring, which defines the flow of time-information.

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“…There are precedent works which make an attempt to extend the universal probability to operators in quantum systems [6,15].…”

Section: Related Workmentioning

confidence: 99%

“…The purpose of [6] is mainly to define the information content of an individual pure quantum state, i. e., to define the quantum Kolmogorov complexity of the quantum state, while such an attempt is not the purpose of both [15] and the present paper. [6] generalized the universal probability to a matrix-valued function µ, called the quantum universal semi-density matrix. The function µ maps any positive integer N to an N × N positive semi-definite Hermitian matrix µ(N ) with its trace less than or equal to 1.…”

Section: Related Workmentioning

confidence: 99%

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

Tadaki

2006

Mathematical Logic Qtrly

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This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H(s) of a given finite binary string s. In the standard way, H(s) is defined as the length of the shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H(s) is defined as − log 2 m(s) without reference to the concept of program-size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure (POVM) in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed.In what follows, we introduce an operator versionĤ(s) of H(s) in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties.

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Quantum algorithmic entropy

Cited by 52 publications

References 3 publications

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

The Second Quantized Quantum Turing Machine and Kolmogorov Complexity

Informational Branching Universe

An extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system

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