The Stochastic-Quantum Theorem (2309.03085v1)

Abstract: This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called generalized stochastic systems, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any generalized stochastic system and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasiprobability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any generalized stochastic system on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.

• The paper presents the stochastic-quantum theorem, demonstrating how generalized stochastic processes can be embedded into unistochastic quantum systems.
• It employs Hilbert-space dilation methods to map non-Markovian dynamics onto deterministic quantum frameworks.
• The findings pave the way for simulating complex stochastic systems on quantum computers and reinterpret foundational quantum mechanics postulates.

The Stochastic-Quantum Theorem: Bridging Stochastic and Quantum Systems

The paper under consideration, titled "The Stochastic-Quantum Theorem" by Jacob A. Barandes, presents a formal exploration of novel mathematical structures that unify aspects of stochastic processes with quantum systems through a newly formulated theorem. The core assertion of the paper is encapsulated in the stochastic-quantum theorem, which demonstrates a way to embed generalized stochastic systems within quantum systems that evolve unitarily. This exploration provides a new perspective on the foundational aspects of quantum theory, offering implications for both theoretical understanding and practical applications, especially regarding quantum computing.

To properly understand the scope of this research, it is essential to unpack the concepts of generalized stochastic systems as introduced in this work. These are mathematical structures that extend traditional definitions of dynamical systems by incorporating stochastic processes, such as non-Markovian dynamics, which allow for probabilistic state evolution without the constraints of divisibility or strict Markov properties.

The stochastic-quantum theorem posits that every generalized stochastic system can be represented as a subsystem of a unistochastic system. Unistochastic systems are characterized by transition matrices whose entries are the modulus squares derived from unitary quantum time-evolution operators. This realization signifies that stochastic dynamics, which might seem inherently chaotic or random, can be mirrored within the deterministic framework of quantum mechanics. This bridges a conceptual gap by suggesting that the probabilistic nature of quantum theory could have roots in deeper stochastic processes.

A significant implication of this theorem is the potential reinterpretation of foundational quantum mechanics postulates - the use of complex numbers, Hilbert spaces, and the peculiarities of quantum evolution and measurement (Born rule) could be viewed as emergent phenomena from underlying stochastic processes. This brings a fresh perspective to phenomena such as quantum interference and entanglement, which, according to the paper, may reflect the indivisible dynamics of generalized stochastic systems rather than purely quantum-specific effects.

The theoretical construction provided in the paper lays a rigorous foundation for simulating generalized stochastic processes on quantum computers, thus potentially enhancing their utility in modeling complex dynamical systems beyond conventional computational paradigms. Quantum computers could effectively simulate non-Markovian processes, offering new paths for solving problems that involve intricate stochastic behaviors.

Furthermore, this correspondence could facilitate novel interpretations and insights in quantum foundations, especially regarding quantum measurement and reality. By viewing quantum phenomena through the lens of stochastic processes, this work offers a potential reconciliation for some of the paradoxical elements inherent in quantum theory, such as wave-function collapse and non-locality, which have long been debated in philosophical and scientific circles.

One of the paper's substantive contributions is to translate stochastic processes into the language of quantum mechanics by using Hilbert-space formalism. The paper illustrates how constructing an expanded or "dilated" Hilbert space allows for the embedding of a generalized stochastic process into a deterministic quantum framework, illuminated by the Stinespring dilation theorem. This reveals the generality and versatility of quantum theory as a model for stochastic dynamics, posing theoretical challenges as well as opening new vistas for applications.

In conclusion, this paper offers a noteworthy advancement in the theoretical landscape by linking stochastic systems with quantum mechanics in a mathematically rigorous fashion. It invites further exploration into how this stochastic-quantum correspondence can be practically implemented and what it reveals about the nature of reality modeled by quantum mechanics. Future research might explore exploring how this approach might illuminate novel generalizations of quantum theory or how quantum computers might leverage this foundational understanding to achieve unprecedented computational feats.