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Quantum State Complexity Measure

Yuri Campbell, José Roberto Castilho Piqueira

Published Oct 27, 2011• 0 citations
OpenAlex

Abstract

The complexity measures role has become much clearer in recent years as they help to better understand complex systems dynamical behavior. Even though the large number of measures proposed to tackle this issue for classical systems, for quantum systems only Kolmogorov’s algorithm complexity extensions have been proposed. Hence, the present approach makes use of a new and mathematically well-established complexity measure for classical systems and extends it to assess quantum states complexity as well. Then the proposed extension is applied to a mixed state constructed with a W-state together with controlled white noise, showing a convex behavior of quantum state complexity. Thus, this reinforces the differences from previous known quantum complexities.

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The complexity measures role has become much clearer in recent years as they help to better understand complex systems dynamical behavior. Even though the large number of measures proposed to tackle this issue for classical systems, for quantum systems only Kolmogorov’s algorithm complexity extensions have been proposed. Hence, the present approach makes use of a new and mathematically well-established complexity measure for classical systems and extends it to assess quantum states complexity as well. Then the proposed extension is applied to a mixed state constructed with a W-state together with controlled white noise, showing a convex behavior of quantum state complexity. Thus, this reinforces the differences from previous known quantum complexities.
In the past decades, all of the efforts at quantifying systems complexity with a general tool has usually relied on using Shannon's classical information framework to address the disorder of the system through the Boltzmann–Gibbs–Shannon entropy, or one of its extensions. However, in recent years, there were some attempts to tackle the quantification of algorithmic complexities in quantum systems based on the Kolmogorov algorithmic complexity, obtaining some discrepant results against the classical approach. Therefore, an approach to the complexity measure is proposed here, using the quantum information formalism, taking advantage of the generality of the classical-based complexities, and being capable of expressing these systems' complexity on other framework than its algorithmic counterparts. To do so, the Shiner–Davison–Landsberg (SDL) complexity framework is considered jointly with linear entropy for the density operators representing the analyzed systems formalism along with the tangle for the entanglement measure. The proposed measure is then applied in a family of maximally entangled mixed state.
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.
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.
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.
In quantum mechanics, a state is an element of a Hilbert space whose dimension exponentially grows with the increase of the number of particles (or qubits, in quantum computing). The vague question "is this huge Hilbert space really there?" has been rigorously formalized inside the computational complexity theory; the research suggests a positive answer to the question. Along this line, I give a definition of the (classical) information content of a quantum state, taking inspiration from the Kolmogorov complexity. I show that, for some well-known quantum circuits (having a number of gates polynomial in the number of qubits), the information content of the output state, evaluated according to my definition, is polynomial in the number of qubits. On the other hand, applying known results, it is possible to devise quantum circuits that generate much more complex states, having an exponentially-growing information content. A huge amount of classical information can be really present inside a quantum state, however, I show that this property is not necessarily exploited by quantum computers, not even by quantum algorithms showing an exponential speed-up with respect to classical computation.
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.