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arXiv:0707.4559

Information-Disturbance theorem and Uncertainty Relation

Takayuki Miyadera, Hideki Imai

Published Jul 31, 2007• 4 citations
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Abstract

It has been shown that Information-Disturbance theorem can play an important role in security proof of quantum cryptography. The theorem is by itself interesting since it can be regarded as an information theoretic version of uncertainty principle. It, however, has been able to treat restricted situations. In this paper, the restriction on the source is abandoned, and a general information-disturbance theorem is obtained. The theorem relates information gain by Eve with information gain by Bob.

DOI
10.48550/arxiv.0707.4559

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It has been shown that Information-Disturbance theorem can play an important role in security proof of quantum cryptography. The theorem is by itself interesting since it can be regarded as an information theoretic version of uncertainty principle. It, however, has been able to treat restricted situations. In this paper, the restriction on the source is abandoned, and a general information-disturbance theorem is obtained. The theorem relates information gain by Eve with information gain by Bob.
In this paper, a representation of the information-disturbance theorem based on the quantum Kolmogorov complexity that was defined by P. Vit´anyi has been examined. In the quantum information theory, the information-disturbance relationship, which treats the trade-off relationship between information gain and its caused disturbance, is a fundamental result that is related to Heisenberg’s uncertainty principle. The problem was formulated in a cryptographic setting and the quantitative relationships between complexities have been derived.
When an observer wants to identify a quantum state, which is known to be one of a given set of nonorthogonal states, the act of observation causes a disturbance to that state. We investigate the trade-off between the information gain and that disturbance. This issue has important applications in quantum cryptography. The optimal detection method, for a given tolerated disturbance, is explicitly found in the case of two equiprobable nonorthogonal pure states. \textcopyright{} 1996 The American Physical Society.
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the ``Heisenberg uncertainty principle.'' A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for isolated systems are unitary. In this paper we shall explore several aspects of the information--disturbance principle in an attempt to make it firmly quantitative and flesh out its significance for quantum theory as a whole.
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the “Heisenberg uncertainty principle.” A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for isolated systems are unitary. In this paper we shall explore several aspects of the information-disturbance principle in an attempt to make it firmly quantitative and flesh out its significance for quantum theory as a whole.
The engine that powers quantum cryptography is the principle that there are no physical means for gathering information about the identity of a quantum system's state (when it is known to be prepared in one of a set of nonorthogonal states) without disturbing the system in a statistically detectable way. This situation is often mistakenly described as a consequence of the ``Heisenberg uncertainty principle.'' A more accurate account is that it is a unique feature of quantum phenomena that rests ultimately on the Hilbert space structure of the theory along with the fact that time evolutions for isolated systems are unitary. In this paper I explore several aspects of the ``information / disturbance principle'' in an attempt to make it firmly quantitative for both pure and mixed states. The final section briefly explores the extent to which such a principle can be taken as a foundation for unitary dynamics rather than as a consequence.