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arXiv:0707.2924

On the Quantum Complexity of Classical Words

Markus P. Müller

Published Jul 19, 2007• 1 citation
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Abstract

We show that classical and quantum Kolmogorov complexity of binary words 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 word. 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 some analytical estimates and a classical program for the simulation of a universal quantum computer.

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We show that classical and quantum Kolmogorov complexity of binary words 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 word. 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 some analytical estimates and a classical program for the simulation of a universal quantum computer.
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.
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 P. Vitanyi (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.
The purpose of this thesis is to give a formal definition of quantum Kolmogorov complexity (QC), and rigorous mathematical proofs of its basic properties. The definition used here is similar to that by Berthiaume, van Dam, and Laplante. It defines the QC of some indeterminate-length qubit string ρas the minimum base length of any quantum input which makes a quantum Turing machine (QTM) halt and output ρ, up to some error tolerance. First, we prove that there is a QTM which is universal in the sense of input base length. Furthermore, we show several general properties of QTMs, including a result on mutually orthogonal ``halting spaces'', and a way to transform every almost-halting input into a deterministically-halting input by adding at most a constant number of qubits. Afterwards, we apply these results to QC. In particular, we show that QC is invariant, incompressible, agrees with classical Kolmogorov complexity for classical strings, and is closely related to von Neumann entropy for ergodic quantum information sources.
In analogy of classical Kolmogorov complexity we develop a theory of the algorithmic information in bits contained in any one of continuously many pure quantum states: quantum Kolmogorov complexity. Classical Kolmogorov complexity coincides with the new quantum Kolmogorov complexity restricted to the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approximated from above. With high probability a quantum object is incompressible. There are two alternative approaches possible: to define the complexity as the length of the shortest qubit program that effectively describes the object, and to use classical descriptions with computable real parameters.
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 Vitányi (we show that the latter is sometimes as large as 2 n − 2 log 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.
Extends algorithmic information theory to quantum mechanics, taking a universal semi-computable density matrix ("universal probability") as a starting point, and defines 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 P.M.B. Vita/spl acute/nyi (1999) (we show that the latter is sometimes as large as 2n-2 log n) and the qubit complexity defined by A. Berthiaume et al. (2000). The "cloning" properties of our complexity measure are similar to those of qubit complexity.