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arXiv:quant-ph/0506266 · Modern Physics Letters B

THE SECOND QUANTIZED QUANTUM TURING MACHINE AND KOLMOGOROV COMPLEXITY

Caroline Rogers, Vlatko Vedral

Published May 20, 2008• 7 citations
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Abstract

The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach — our measure of the second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities, it is continuous. We give examples where the second quantized and quantum Kolmogorov complexity differ.

DOI
10.1142/s021798490801464x

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The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach — our measure of the second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities, it is continuous. We give examples where the second quantized and quantum Kolmogorov complexity differ.
Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore, we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.
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.
We give a definition for the Kolmogorov complexity of a pure quantum state. In classical information theory, the algorithmic complexity of a string is a measure of the information needed by a universal machine to reproduce the string itself. We define the complexity of a quantum state by means of the classical description complexity of an (abstract) experimental procedure that allows us to prepare the state with a given fidelity. We argue that our definition satisfies the intuitive idea of complexity as a measure of "how difficult" it is to prepare a state. We apply this definition to give an upper bound on the algorithmic complexity of a number of known states. Furthermore, we establish a connection between the entanglement of a quantum state and its algorithmic complexity.
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.
This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and define dcq-computable states as those that can be approximated by a dcq-TM. Moreover, we introduce (conditional) Kolmogorov complexity of quantum states and use it to study three particular aspects of the algorithmic information contained in a quantum state: a comparison of the information in a quantum state with that of its classical representation as an array of real numbers, an exploration of the limits of quantum state copying in the context of algorithmic complexity, and study of the complexity of correlations in quantum systems, resulting in a correlation-aware definition for algorithmic mutual information that satisfies symmetry of information property.