89 research outputs found
Quantum Kolmogorov Complexity
In this paper we give a definition for quantum Kolmogorov complexity. In the
classical setting, the Kolmogorov complexity of a string is the length of the
shortest program that can produce this string as its output. It is a measure of
the amount of innate randomness (or information) contained in the string.
We define the quantum Kolmogorov complexity of a qubit string as the length
of the shortest quantum input to a universal quantum Turing machine that
produces the initial qubit string with high fidelity. The definition of Vitanyi
(Proceedings of the 15th IEEE Annual Conference on Computational Complexity,
2000) measures the amount of classical information, whereas we consider the
amount of quantum information in a qubit string. We argue that our definition
is natural and is an accurate representation of the amount of quantum
information contained in a quantum state.Comment: 14 pages, LaTeX2e, no figures, \usepackage{amssymb,a4wide}. To appear
in the Proceedings of the 15th IEEE Annual Conference on Computational
Complexit
Quantum Kolmogorov Complexity and Quantum Key Distribution
We discuss the Bennett-Brassard 1984 (BB84) quantum key distribution protocol
in the light of quantum algorithmic information. While Shannon's information
theory needs a probability to define a notion of information, algorithmic
information theory does not need it and can assign a notion of information to
an individual object. The program length necessary to describe an object,
Kolmogorov complexity, plays the most fundamental role in the theory. In the
context of algorithmic information theory, we formulate a security criterion
for the quantum key distribution by using the quantum Kolmogorov complexity
that was recently defined by Vit\'anyi. We show that a simple BB84 protocol
indeed distribute a binary sequence between Alice and Bob that looks almost
random for Eve with a probability exponentially close to 1.Comment: typos correcte
Quantum Kolmogorov Complexity Based on Classical Descriptions
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.Comment: 17 pages, LaTeX, final and extended version of quant-ph/9907035, with
corrections to the published journal version (the two displayed equations in
the right-hand column on page 2466 had the left-hand sides of the displayed
formulas erroneously interchanged
On the Quantum Kolmogorov Complexity of Classical Strings
We show that classical and quantum Kolmogorov complexity of binary strings
agree up to an additive constant. Both complexities are defined as the minimal
length of any (classical resp. quantum) computer program that outputs the
corresponding string.
It follows that quantum complexity is an extension of classical complexity to
the domain of quantum states. This is true even if we allow a small
probabilistic error in the quantum computer's output. We outline a mathematical
proof of this statement, based on an inequality for outputs of quantum
operations and a classical program for the simulation of a universal quantum
computer.Comment: 10 pages, no figures. Published versio
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem
In classical information theory, entropy rate and Kolmogorov complexity per
symbol are related by a theorem of Brudno. In this paper, we prove a quantum
version of this theorem, connecting the von Neumann entropy rate and two
notions of quantum Kolmogorov complexity, both based on the shortest qubit
descriptions of qubit strings that, run by a universal quantum Turing machine,
reproduce them as outputs.Comment: 26 pages, no figures. Reference to publication added: published in
the Communications in Mathematical Physics
(http://www.springerlink.com/content/1432-0916/
Quantum Kolmogorov Complexity and Information-Disturbance Theorem
In this paper, a representation of the information-disturbance theorem based
on the quantum Kolmogorov complexity that was defined by P. Vitanyi 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 quantitative relationships between complexities have
been derived.Comment: Special issue: Kolmogorov Complexit
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