Quantum information distance

2021 ◽  
Author(s):  
Songsong Dai
Keyword(s):  
Quantum Information ◽  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Transition Program ◽  
Information Distance ◽  
Universal Turing Machine ◽  
Classical Setting ◽  
Universal Quantum ◽  
Definition Of ◽  
Quantum Turing Machine

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.

scholarly journals Fuzzy Kolmogorov Complexity Based on a Classical Description

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 66
Author(s):  
Songsong Dai
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Classical Description ◽  
Fuzzy Language ◽  
Classical Setting

In this paper, we give a definition for fuzzy Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a single finite string is the length of the shortest program that produces this string. We define the fuzzy Kolmogorov complexity as the minimum classical description length of a finite-valued fuzzy language through a universal finite-valued fuzzy Turing machine that produces the desired fuzzy language. The classical Kolmogorov complexity is extended to the fuzzy domain retaining classical descriptions. We show that our definition is robust, that is to say, the complexity of a finite-valued fuzzy language does not depend on the underlying finite-valued fuzzy Turing machine.

THE SECOND QUANTIZED QUANTUM TURING MACHINE AND KOLMOGOROV COMPLEXITY

2008 ◽  
Vol 22 (12) ◽  
pp. 1203-1210 ◽  
Author(s):  
CAROLINE ROGERS ◽  
VLATKO VEDRAL
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Physical State ◽  
Physical Reality ◽  
Physical Resources ◽  
Quantum Turing Machine

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 the second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities, it is continuous. We give examples where the second quantized and quantum Kolmogorov complexity differ.

Quantum theory, the Church–Turing principle and the universal quantum computer

1985 ◽  
Vol 400 (1818) ◽  
pp. 97-117 ◽  
Keyword(s):  
Quantum Theory ◽  
Complexity Theory ◽  
Turing Machine ◽  
Physical System ◽  
Quantum Computer ◽  
Physical Principle ◽  
Universal Turing Machine ◽  
Universal Quantum ◽  
Computing Machines ◽  
The Church

It is argued that underlying the Church–Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizible physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Classical physics and the universal Turing machine, because the former is continuous and the latter discrete, do not obey the principle, at least in the strong form above. A class of model computing machines that is the quantum generalization of the class of Turing machines is described, and it is shown that quantum theory and the 'universal quantum computer’ are compatible with the principle. Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine. These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it. The intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than Everett’s. Some of the numerous connections between the quantum theory of computation and the rest of physics are explored. Quantum complexity theory allows a physically more reasonable definition of the ‘complexity’ or ‘knowledge’ in a physical system than does classical complexity theory.

Note on a universal quantum Turing machine

2008 ◽  
Vol 372 (31) ◽  
pp. 5120-5122 ◽  
Author(s):  
Satoshi Iriyama ◽  
Takayuki Miyadera ◽  
Masanori Ohya
Keyword(s):  
Turing Machine ◽  
Universal Quantum ◽  
Quantum Turing Machine

Quantum turing machine and brain model represented by Fock space

2016 ◽  
Vol 14 (04) ◽  
pp. 1640008
Author(s):  
Satoshi Iriyama ◽  
Masanori Ohya
Keyword(s):  
Turing Machine ◽  
Quantum Channel ◽  
Fock Space ◽  
Adaptive Dynamics ◽  
Quantum Algorithm ◽  
Psychological Phenomena ◽  
Definition Of ◽  
Complex Phenomena ◽  
Math Phys ◽  
Quantum Turing Machine

The adaptive dynamics is known as a new mathematics to treat with a complex phenomena, for example, chaos, quantum algorithm and psychological phenomena. In this paper, we briefly review the notion of the adaptive dynamics, and explain the definition of the generalized Turing machine (GTM) and recognition process represented by the Fock space. Moreover, we show that there exists the quantum channel which is described by the GKSL master equation to achieve the Chaos Amplifier used in [M. Ohya and I. V. Volovich, J. Opt. B 5(6) (2003) 639., M. Ohya and I. V. Volovich, Rep. Math. Phys. 52(1) (2003) 25.]

scholarly journals Teorias da Aleatoriedade

2004 ◽  
Vol 11 (2) ◽  
pp. 75-98
Author(s):  
Carlos A. P. Campani ◽  
Paulo Blauth Menezes
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Random Sequence ◽  
The Other ◽  
Turing Machines ◽  
Intuitive Concept ◽  
Theory Of Information ◽  
Definition Of

This work is a survey about the definition of “random sequence”. We emphasize the definition of Martin-Löf and the definition based on incompressibility (Kolmogorov complexity). Kolmogorov complexity is a profound and sofisticated theory of information and randomness based on Turing machines. These two definitions solve all the problems of the other approaches, satisfying our intuitive concept of randomness, and both are mathematically correct. Furthermore, we show the Schnorr’s approach, that includes a requisite of effectiveness (computability) in his definition. We show the relations between all definitions in a critical way. Keywords: randomness, Kolmogorov complexity, Turing machine, computability, probability.

scholarly journals Undecidability in quantum thermalization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Naoto Shiraishi ◽  
Keiji Matsumoto
Keyword(s):  
Turing Machine ◽  
Thermal Equilibrium ◽  
General Theorem ◽  
Nearest Neighbour ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Universal Turing Machine ◽  
Many Body Systems ◽  
Neighbour Interaction

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

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