Francisco Javier Fraile Peláez

arXiv2026

Emergence of the Partial Trace from Classical Probability Theory

The partial trace is commonly introduced in quantum mechanics as an algebraic operation used to define reduced states of composite systems. However, the probabilistic origin of this operation goes systematically unnoticed in the literature. Here, we show that the partial trace emerges naturally from the requirement of consistency between the Born rule for measurement probabilities and the classical marginalization of probability mass functions. Starting from the classical marginalization rule relating joint and marginal probability distributions, we impose that the reduced density operator of a subsystem must reproduce the local measurement statistics derived from the global state. We show that this requirement directly leads to the standard expression of the partial trace. From this perspective, the reduced density operator appears not as an ad hoc algebraic construction, but as a natural consequence of the probabilistic structure of quantum mechanics.

quant-ph↗

arXiv2025

Analog Programmable-Photonic Information

The limitations of digital electronics in handling real-time matrix operations for emerging computational tasks - such as artificial intelligence, drug design, and medical imaging - have prompted renewed interest in analog computing. Programmable Integrated Photonics (PIP) has emerged as a promising technology for scalable, low-power, and high-bandwidth analog computation. While prior work has explored PIP implementations of quantum and neuromorphic computing, both approaches face significant limitations due to misalignments between their mathematical models and the native capabilities of photonic hardware. Building on the recently proposed Analog Programmable-Photonic Computation (APC) - a computation theory explicitly matched to the technological features of PIP - we introduce its critical missing component: an information theory. We present Analog Programmable-Photonic Information (API), a mathematical framework that addresses fundamental concepts beyond APC by examining the amount of information that can be generated, computed and recovered in a PIP platform. API also demonstrates the robustness of APC against errors arising from system noise and hardware imperfections, enabling scalable computation without the extensive error-correction overhead required in quantum computing. Together, APC and API provide a unified foundation for on-chip photonic computing, offering a complementary alternative to digital, quantum and neuromorphic paradigms, and positioning PIP as a cornerstone technology for next-generation information processing.

physics.optics↗