## Quantum Kolmogorov complexity based on classical descriptions (2001)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 19 - 1 self |

### BibTeX

@ARTICLE{Vitányi01quantumkolmogorov,

author = {Paul M. B. Vitányi},

title = {Quantum Kolmogorov complexity based on classical descriptions},

journal = {IEEE Trans. Inform. Theory},

year = {2001},

pages = {2464--2479}

}

### Years of Citing Articles

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### Abstract

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 no-cloning properties of quantum states. In the quantum situation complexity is not sub-additive. We discuss some relations with “no-cloning ” and “approximate cloning ” properties. Keywords — Algorithmic information theory, quantum; classical descriptions of quantum states; information theory, quantum; Kolmogorov complexity, quantum; quantum cloning. I.

### Citations

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Citation Context ...antum states; information theory, quantum; Kolmogorov complexity, quantum; quantum cloning. I. Introduction QUANTUM information theory, the quantum mechanical analogue of classical information theory =-=[6]-=-, is experiencing a renaissance [2] due to the rising interest in the notion of quantum computation and the possibility of realizing a quantum computer [16]. While Kolmogorov complexity [12] is the ac... |

1680 | An introduction to Kolmogorov complexity and its applications
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Citation Context ...cate the success of classical Kolmogorov complexity as a general proof method in applications ranging from combinatorics to the analysis of algorithms, and from pattern recognition to learning theory =-=[13]-=-. We propose a theory of quantum Kolmogorov complexity based on classical descriptions and derive the results given in the abstract. A preliminary partial version appeared as [19]. What are the proble... |

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Citation Context ...re the problems and choices to be made developing a theory of quantum Kolmogorov complexity? Quantum theory assumes that every complex vector of unit length represents a realizable pure quantum state =-=[17]-=-. There arises the question of how to design the equipment that prepares such a pure state. While there are continuously many pure states in a finite-dimensional complex Partially supported by the EU ... |

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Citation Context ...ion theory [6], is experiencing a renaissance [2] due to the rising interest in the notion of quantum computation and the possibility of realizing a quantum computer [16]. While Kolmogorov complexity =-=[12]-=- is the accepted absolute measure of information content in a individual classical finite object, a similar absolute notion is needed for the information content of an individual pure quantum state. O... |

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Citation Context ...fectiveness on such descriptions leads to constructive procedures. The most general such procedures satisfying universally agreed-upon logical principles of effectiveness are quantum Turing machines, =-=[3]-=-. To define quantum Kolmogorov complexity by way of quantum Turing machines leaves essentially two options: 1. We want to describe every quantum superposition exactly; or 2. we want to take into accou... |

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Citation Context ...y) in practice. Thus, in applications an algorithmic random number generator is used. Strictly speaking this invalidates the analysis based on mathematical randomized computation. As John von Neumann =-=[15]-=- put it: “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number—... |

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Citation Context ...tive property: K(|φ, ψ〉) + < K(|φ〉) + K(|ψ〉 | |φ〉). But, in general, quantum Kolmogorov complexity is not sub-additive. Although “cloning” of non-orthogonal states is forbidden in the quantum setting =-=[21]-=-, [7], m copies of the same quantum state have combined complexity that can be considerable lower than m times the complexity of a single copy. In fact, quantum Kolmogorov complexity appears to enable... |

137 | A single quantum cannot be cloned - Wootters, Zurek - 1982 |

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Citation Context ...f all contemplated machines to a single primitive rotation θ with cosθ = 3 4 5 and sinθ = 5 , then there are only countably many Turing machines and the universal machine simulates the others exactly =-=[1]-=-. Every quantum Turing machine computation, using arbitrary real rotations to obtain a target pure quantum state, can be approximated to every precision by machines with fixed rotation θ but in genera... |

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Citation Context ...uantum; Kolmogorov complexity, quantum; quantum cloning. I. Introduction QUANTUM information theory, the quantum mechanical analogue of classical information theory [6], is experiencing a renaissance =-=[2]-=- due to the rising interest in the notion of quantum computation and the possibility of realizing a quantum computer [16]. While Kolmogorov complexity [12] is the accepted absolute measure of informat... |

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Citation Context ...roperty: K(|φ, ψ〉) + < K(|φ〉) + K(|ψ〉 | |φ〉). But, in general, quantum Kolmogorov complexity is not sub-additive. Although “cloning” of non-orthogonal states is forbidden in the quantum setting [21], =-=[7]-=-, m copies of the same quantum state have combined complexity that can be considerable lower than m times the complexity of a single copy. In fact, quantum Kolmogorov complexity appears to enable us t... |

29 |
Minimum Description Length Induction
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Citation Context ...any real numbers from Cauchy sequences of rational numbers, the rationals being effectively enumerable. Kolmogorov complexity: We summarize some basic definitions in Appendix A (see also this journal =-=[20]-=-) in order to establish notations and recall the notion of shortest effective descriptions. More details can be found in the textbook [13]. Shortest effective descriptions are “effective” in the sense... |

18 | Quantum algorithmic entropy - Gács - 2001 |

15 | Quantum Kolmogorov complexity
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- 2001
(Show Context)
Citation Context ...ubit Descriptions: The most straightforward way to define a notion of quantum Kolmogorov complexity is to consider the shortest effective qubit description of a pure quantum state which is studied in =-=[4]-=-. (This qubit complexity can also be formulated in terms of the conditional version of bit complexity as in [19].) An advantage of qubit complexity is that the upper bound on the complexity of a pure ... |

15 | Three Approaches to the Quantitative Definition of Information in an Individual Pure Quantum State”, e-print quant-ph/9907035 at http://arXiv.org
- Vitányi
(Show Context)
Citation Context ...n to learning theory [13]. We propose a theory of quantum Kolmogorov complexity based on classical descriptions and derive the results given in the abstract. A preliminary partial version appeared as =-=[19]-=-. What are the problems and choices to be made developing a theory of quantum Kolmogorov complexity? Quantum theory assumes that every complex vector of unit length represents a realizable pure quantu... |

8 |
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Citation Context ...put state against the target state destroys the latter. It is possible to prepare approximate copies from the target state, but the more copies one prepares the less they approximate the target state =-=[10]-=-, and this deterioration appears on the surface to prevent use in our application below. To sidestep the fragility of the pure quantum target state, we simply require that it is an outcome, in as many... |

7 |
Quantum cloning without signaling, Phys
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(Show Context)
Citation Context ...tain, the more the fidelity of the obtained clones deteriorates with respect to the original copy. This stands to reason since high fidelity cloning would enable both superluminal signal transmission =-=[11]-=- and extracting essentially unbounded information concerning the probability amplitude from the original qubits. The approximate cloning possibility suggests that in our setting the approximation pena... |

5 | The Boltzmann entropy and randomness tests
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(Show Context)
Citation Context ...f Kolmogorov complexity to randomness of infinite sequences (such as binary expansions of real numbers) in terms of coarse-graining and sequential Martin-Löf tests, analogous to the classical case in =-=[9]-=-, [13]. But this is outside the scope of the present paper. II. Quantum Turing Machine Model We assume the notation and definitions in Appendices A, B. Our model of computation is a quantum Turing mac... |

1 | Boltzmann entropy and randomness tests - “The - 1994 |