Entanglement, Bell States, and Purification in Quantum Photonic Systems | MIT OpenCourseWare

Overview

In this MIT OpenCourseWare lecture, the instructor explains quantum entanglement with photons and two qubit states. The session shows how entangled states cannot be written as a simple product of subsystems and introduces Bell states as maximally entangled two qubit states. It then describes entanglement as a resource and how weakly entangled states can be distilled into high quality Bell states through purification, using measurements on multiple copies. The talk covers the role of nonlocal correlations, decoherence, and the possibility of creating entangled states for atoms by measuring photons from the atoms, with Hong-Ou-Mandel interference used to scramble the photons. The material ties together fundamental concepts in quantum information with practical photonic experiments.

Introduction

Entanglement is presented as a fundamental quantum resource that cannot be factorized into a product of subsystems. The lecture uses photonic two-qubit states to illustrate how polarization modes can yield maximally entangled Bell states, and discusses the nonlocal correlations that arise when two subsystems are separated.

Bell States and Entanglement as a Resource

The four Bell states, such as 00+11 and 01+10, are introduced as canonical maximally entangled two-qubit states. Entanglement is highlighted as a resource that enables protocols like teleportation and enhanced metrology, and the fragility of entanglement under decoherence is discussed. The Schmidt number and entropy of entanglement provide ways to quantify how strong the entanglement is in a given state.

PURIFICATION: Distilling Strong Entanglement

A purification protocol is outlined where two copies of a weakly entangled state are processed using a controlled NOT operation and measurements on a subset of photons. If the measurement outcomes are favorable, the remaining photons are projected into a maximally entangled Bell state. If not, those copies are discarded. The success probability depends on the initial amplitudes A and B of the state, and the method illustrates how entanglement can be treated as a distillable resource rather than a fixed property.

Measuring Entanglement

The reduced density matrix of a subsystem is used to characterize entanglement via von Neumann entropy. Starting from a pure Bell state, tracing out one qubit yields a completely mixed state with entropy equal to one bit, whereas a product state yields entropy zero. This provides a practical metric to compare different entangled states and to relate entanglement to observable properties of subsystems.

Atoms and Photons: Generating Entanglement

The lecture moves to atoms where entanglement is harder to realize than with photons. A probabilistic scheme uses light emitted by two atoms to perform a measurement on photons. A favorable measurement outcome leaves the atoms in an entangled state, enabling potential tasks such as quantum teleportation. The Hong-Ou-Mandel interference plays a central role in scrambling photons so that the source of each photon cannot be distinguished, which is essential for creating entanglement between distant atoms.

Hong-Ou-Mandel Interference

Identical photons incident on a balanced beam splitter exhibit Hong-Ou-Mandel interference, where the two photons bunch into the same output port. The interference signature is a direct indication of photon indistinguishability and underpins the probabilistic entanglement generation schemes that rely on joint photon detection events rather than which-path information.

Outlook

The talk emphasizes that entanglement is a fragile, nonlocal resource that can be distilled via purification, while also noting the no cloning constraint. It concludes with reflections on practical implementations for photonic and atomic systems and the value of MIT OCW as a resource for learning quantum information concepts.