Quantum Kolmogorov complexity and quantum correlations in deterministic-control quantum Turing machines

TL;DR

The paper develops a quantum algorithmic information framework based on deterministic-control quantum Turing machines (dcq-TMs), extending the model to mixed states and defining dcq-computable quantum states and their (conditional) Kolmogorov complexity $K$. It proves that the complexity of a quantum state equals the complexity of its density-matrix representation up to an additive constant, and shows that this complexity is machine-independent. It then derives a quantum analogue of the chain rule, identifies limitations for a universal chain-rule for arbitrary bipartite states, and introduces a correlation-aware complexity measure for quantum correlations that satisfies a symmetry property up to logarithmic terms. The results illuminate the role of classical representations as resources, establish a quantum no-cloning flavor bound in algorithmic terms, and lay groundwork for a theory of algorithmic information in multipartite quantum systems with potential implications for quantum information processing and cryptography.

Abstract

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.

Figures (2)

• Figure 1: Starting configuration of the dcq-Turing machine with a classical and quantum input.$x = x_{1}\cdot x_{2} \cdots x_{k}$ is the classical input, while $\rho_{\langle1,n\rangle}$ is the quantum input "written" on $n$ cells of the quantum tape, where $\rho_{i}$ is the local state of the $i-$th cell, represented by $\mathbold{\rho}_i=\textnormal{Tr}_{(1,\ldots, i-1, i+1, \ldots, n)}[\mathbold{\rho}_{\langle1,n\rangle}]$. The string $x\Box n$ is surrounded by blank symbols extending to infinity in both directions. The quantum input is surrounded by cells in the local zero state ${\vert{\text{0}}\rangle}\!{\langle{\text{0}}\vert}$ extending to infinity in both directions.
• Figure 2: Final state of the dcq-Turing machine with a classical and quantum outputs.$y = y_{1}\cdot y_{2} \cdots y_{r}$ is the classical output, while $\sigma_{\langle1,s\rangle}$ is the quantum output, where $\sigma_{i}$ is the local state of the $i-$th cell, represented by $\mathbold{\sigma}_i=\textnormal{Tr}_{(1,\ldots, i-1, i+1, \ldots, n)}[\mathbold{\sigma}_{\langle1,s\rangle}]$. The size (in number of qubits) of $\sigma_{(1,s)}$ is specified by the string $s$ in the classical tape. The working cells, which are used during the computation but are not part of the output in the classical and quantum tape are denoted by $w$ and $\mu$, respectively.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark
• Definition 3.1
• Definition 4.1
• Definition 4.2
• Theorem 4.1
• proof
• Definition 5.1