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

Tadaki, Kohtaro

doi:10.1002/malq.200410061

Cited by 10 publications

(18 citation statements)

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“…The first attempt to define a quantum Kolmogorov complexity was by Svozil [29] in 1996. The attempts to define a quantum Kolmogorov complexity [3,7,14,27,29,30,31,32] assume that the descriptions have determinate lengths. Second quantised Kolmogorov complexity differs from the others in that it uses indeterminate length quantum strings.…”

Section: Previous Workmentioning

confidence: 99%

“…Gacs [7] and Tadaki [30,31] both defined quantum Kolmogorov complexity like measures based on probability theory. Gacs attempted to define a universal probability measure, which unfortunately does not correspond to any length measure [32].…”

Section: E a Priori Probabilitymentioning

confidence: 99%

“…Gacs attempted to define a universal probability measure, which unfortunately does not correspond to any length measure [32]. Tadaki [31] extended the semi-POVM to infinite dimensions and derived a quantum generalisation of Chaitin's halting probability [5] as the probability of any measurement outcome on the maximal infinite dimensional semi-POVM.…”

Section: E a Priori Probabilitymentioning

confidence: 99%

“…There have been many attempts to define a quantum analogy of classical Kolmogorov complexity [3,7,14,27,29,30,31,32]. In this paper, a modification of the scheme proposed in Berthiaume, van Dam and Laplante [3] is taken [22,24].…”

Section: Second Quantised Kolmogorov Complexitymentioning

confidence: 99%

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Second Quantized Kolmogorov Complexity

Rogers

Vedral

Nagarajan

2008

Int. J. Quantum Inform.

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The Kolmogorov complexity of a string is the length of its shortest description. We define a second quantised Kolmogorov complexity where the length of a description is defined to be the average length of its superposition. We discuss this complexity's basic properties. We define the corresponding prefix complexity and show that the inequalities obeyed by this prefix complexity are also obeyed by von Neumann entropy.

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Section: Previous Workmentioning

confidence: 99%

Section: E a Priori Probabilitymentioning

confidence: 99%

Section: E a Priori Probabilitymentioning

confidence: 99%

Section: Second Quantised Kolmogorov Complexitymentioning

confidence: 99%

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Second Quantized Kolmogorov Complexity

Rogers

Vedral

Nagarajan

2008

Int. J. Quantum Inform.

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“…In particular, we show that universal operator has certain nontrivial form if it exists. * Most part of this research was carried out without knowing about Tadaki's work [15]. Quite recently, in March 2014, Prof. Tadaki draw our attention to his work and we noticed that there are substantial overlaps between Tadaki's work and ours.…”

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confidence: 99%

On Gács' quantum algorithmic entropy

Takisaka

2014

Electron. Proc. Theor. Comput. Sci.

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We define an infinite dimensional modification of lower-semicomputability of density operators by Gács with an attempt to fix some problem in the paper. Our attempt is partly achieved by showing the existence of universal operator under some additional assumption. It is left as a future task to eliminate this assumption. We also see some properties and examples which stimulate further research. In particular, we show that universal operator has certain nontrivial form if it exists. PreliminariesKolmogorov complexity is the notion of actual information content of finite string in computational point of view. This notion has been proposed by Kolmogorov -Solomonoff -Chaitin in 1960s and used in various areas as a basic tool to represent descriptive complexity. On the other hand, Since Shor's algorithm [1] has been discovered, the research on quantum information has made a great progress and produced various proposals on application to quantum information technology.Quantum Kolmogorov complexity is one of these branches appeared in early 2000s. Several different definitions are proposed so far [2][3][4], and some applications to quantum information are recently emerging [5][6]. However, it seems that there is very little progress in this area despite a decade has passed since these suggestions have been made, and a number of elementary facts are still not investigated.In particular, relationships between them are not clarified. In classical domain, there are several definitions of descriptive complexity and some of them are known as equivalent notions (Levin's coding theorem). This theorem, in some sense, guarantees that these notions are reliable.It naturally leads us to the following question: can we find any good relationship between these quantum complexities? In particular, if it turns out that some of them are equivalent, it would be helpful to make these notions more reliable and more applicable to other research subject such like quantum information theory.We particularly have interest on those by Berthiaume et al.[2] and Gács [3] since they are the quantum extension of plain Kolmogorov complexity and universal semimeasure, respectively. Levin's coding theorem claims that prefix Kolmogorov complexity and universal semimeasure are equivalent, so they are expected to be nearly equivalent.For Berthiaume's definition, there are several results about fundamental facts such like its invariance and relation between classical complexity [7][8][9]. As compared to this, there are not so much subsequent research of Gács' approach, so we mainly treat his definition. * Most part of this research was carried out without knowing about Tadaki's work [15]. Quite recently, in March 2014, Prof. Tadaki draw our attention to his work and we noticed that there are substantial overlaps between Tadaki's work and ours. As far as we know, his work seems to be the first one providing an extension of Gacs' work to the infinite-dimensional setting. In the present paper, we tried to reflect Tadaki's results as possible as we can. Interes...

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Partial Randomness and Dimension of Recursively Enumerable Reals

Tadaki

2009

Mathematical Foundations of Computer Science 2009

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Abstract. A real α is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin Ω number, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real T ∈ (0, 1], where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. As a result, we present ten equivalent characterizations of the partial randomness of an r.e. real. The resultant characterizations of partial randomness are powerful and have many important applications. One of them is to present equivalent characterizations of the dimension of an individual r.e. real. The equivalence between the notion of Hausdorff dimension and compression rate by program-size complexity (or partial randomness) has been established at present by a series of works of many researchers over the last two decades. We present ten equivalent characterizations of the dimension of an individual r.e. real.

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

Cited by 10 publications

References 12 publications

Second Quantized Kolmogorov Complexity

Second Quantized Kolmogorov Complexity

On Gács' quantum algorithmic entropy

Partial Randomness and Dimension of Recursively Enumerable Reals

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