Using the Schmidt Decomposition to Determine Quantum Entanglement

TL;DR

The paper addresses detecting and representing quantum entanglement in bipartite systems using the Schmidt decomposition. It develops two concrete procedures—singular value decomposition of a coefficient matrix and partial tracing of the density matrix—to extract Schmidt coefficients and basis vectors, enabling a clean Schmidt form $|Z\rangle=\sum_k \sigma_k|u_k\rangle\otimes|v_k\rangle$ and a clear separability criterion via the Schmidt number. Through explicit examples, including a separable two-qubit state and a three-qubit entangled state, it demonstrates how to identify entanglement and reconstruct the simplest state representation; it then applies the framework to quantum teleportation, showing how entanglement underpins the protocol. Finally, it discusses extensions to entanglement witnesses and operator Schmidt decompositions, broadening the toolkit to multi-particle and mixed-state scenarios. The work provides practical, technique-driven insights for diagnosing entanglement and for enabling reliable quantum communication primitives like teleportation, with a path toward generalization beyond bipartite systems.

Abstract

Quantum information theory is a rapidly growing area of math and physics that combines two independent theories, quantum mechanics and information theory. Quantum entanglement is a concept that was first proposed in the EPR paradox. In quantum mechanics, particles can be in superposition, meaning they are in multiple different states at once. It is not until the particle is measured that it is forced into a single state. However, it is possible that particles can be tied to other particles, meaning that the measurement of one particle will determine the measurement of the other particle. Entanglement is at the very core of quantum information theory. It is one of the core pieces that allows for the massive increase in computing power. For this paper, we decided to focus on demonstrating the mathematical method (the Schmidt decomposition) for determining if a system is entangled, and a demonstration of quantum entanglement's use (quantum teleportation) as well as a quick look at how to extend the uses of the Schmidt decomposition.