Physics – Quantum Physics
Scientific paper
2000-11-12
J. Phys. A: Math. Gen., 34:1--22, 2001
Physics
Quantum Physics
20 pages. Minor Tex problem corrected
Scientific paper
10.1088/0305-4470/34/35/312
We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (``universal probability'') as a starting point, and define complexity (an operator) as its negative logarithm. A number of properties of Kolmogorov complexity extend naturally to the new domain. Approximately, a quantum state is simple if it is within a small distance from a low-dimensional subspace of low Kolmogorov complexity. The von Neumann entropy of a computable density matrix is within an additive constant from the average complexity. Some of the theory of randomness translates to the new domain. We explore the relations of the new quantity to the quantum Kolmogorov complexity defined by Vitanyi (we show that the latter is sometimes as large as 2n - 2log n and the qubit complexity defined by Berthiaume, Dam and Laplante. The ``cloning'' properties of our complexity measure are similar to those of qubit complexity.
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