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Unification of stochastic matrices and quantum operations for N-level systems

Bilal Canturk

TL;DR

The paper addresses how to unify classical stochastic dynamics with quantum dynamics in finite dimensions by embedding stochastic matrices into quantum operations through a quantum operational representation. It shows that CP-divisibility naturally extends the Chapman-Kolmogorov equation, making it a necessary but not sufficient condition for quantum Markovianity, as demonstrated in a dichotomic Markov-process example. The work establishes that quantum states and operations generalize one-point probability vectors and stochastic matrices, enabling a common framework for analyzing both classical and quantum evolution and enabling quantum tools to model classical stochastic processes. This embedding provides a concrete bridge between classical and quantum theories of memoryless dynamics and suggests avenues for exploring CP-divisible non-Markovian quantum dynamics in discrete time.

Abstract

The time evolution of the one-point probability vector of stochastic processes and quantum processes for $N$-level systems have been unified. Hence, quantum states and quantum operations can be regarded as generalizations of the one-point probability vectors and stochastic matrices, respectively. More essentially, based on the unification, it has been proven that completely positive divisibility (CP-divisibility) for quantum operations is the natural extension of the Chapman-Kolmogorov equation. It is thus shown that CP-divisibility is a necessary but insufficient condition for a quantum process to be specified as Markovian. The main results have been illustrated through a dichotomic Markov process.

Unification of stochastic matrices and quantum operations for N-level systems

TL;DR

The paper addresses how to unify classical stochastic dynamics with quantum dynamics in finite dimensions by embedding stochastic matrices into quantum operations through a quantum operational representation. It shows that CP-divisibility naturally extends the Chapman-Kolmogorov equation, making it a necessary but not sufficient condition for quantum Markovianity, as demonstrated in a dichotomic Markov-process example. The work establishes that quantum states and operations generalize one-point probability vectors and stochastic matrices, enabling a common framework for analyzing both classical and quantum evolution and enabling quantum tools to model classical stochastic processes. This embedding provides a concrete bridge between classical and quantum theories of memoryless dynamics and suggests avenues for exploring CP-divisible non-Markovian quantum dynamics in discrete time.

Abstract

The time evolution of the one-point probability vector of stochastic processes and quantum processes for -level systems have been unified. Hence, quantum states and quantum operations can be regarded as generalizations of the one-point probability vectors and stochastic matrices, respectively. More essentially, based on the unification, it has been proven that completely positive divisibility (CP-divisibility) for quantum operations is the natural extension of the Chapman-Kolmogorov equation. It is thus shown that CP-divisibility is a necessary but insufficient condition for a quantum process to be specified as Markovian. The main results have been illustrated through a dichotomic Markov process.

Paper Structure

This paper contains 9 sections, 6 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical P-divisible processes
  3. Quantum operations
  4. Quantum operational representation of stochastic matrices
  5. Extension of classical P-divisibility
  6. Application to dichotomic Markov Process
  7. Conclusion
  8. The proof of Theorem \ref{['theorem2']}
  9. P-divisible Non-Markovian Process

Key Result

Theorem 1

(Existence Theorem) There exists a quantum operational representation $\Phi_{c}(t,t_{1})$ of a general stochastic matrix $\Lambda(t,t_{1})$ acting on $\mathbb{R}^{N}$ whose Kraus representation is given by the Kraus operators with $s,j,k\in\{0,1,\ldots,N-1\}$ and $\lambda_{jk}=\Lambda_{jk}(t,t_{1})$.

Figures (1)

  • Figure 1: $(a)$ (red): The time evolution of the initial probability vector $\mathbf{p}(t_{1})$ under the stochastic matrix $\Lambda(t,t_{1})$ to the final probability vector $\mathbf{p}(t)$ is equivalent to the operation in $(b)$ (blue), which states that $\mathbf{p}(t_{1})$ is first mapped by $\mathit{F}$ to the quantum state $\rho(t_{1})$ and then, $\rho(t_{1})$ evolves under the quantum operation $\Phi_{c}(t,t_{1})$ to the diagonal quantum state $\rho(t)$, and finally, the probability vector corresponding to $\rho(t)$ under the action of $\mathit{F}^{-1}$ is the final probability vector $\mathbf{p}(t)$.

Theorems & Definitions (16)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • proof
  • Remark 2
  • Lemma 1
  • proof
  • Definition 3
  • Theorem 2
  • ...and 6 more