Introduction

Measurements provide an interface between quantum and classical information. When a measurement is performed on a system in a quantum state, classical information is extracted, revealing something about that quantum state — and generally changing or destroying it in the process. In the simplified formulation of quantum information (as presented in the Basics of quantum information course), we typically limit our attention to projective measurements, including the simplest type of measurement: standard basis measurements. The concept of a measurement, however, can be generalized beyond projective measurements.

In this lesson we'll consider measurements in greater generality. We'll discuss a few different ways that general measurements can be described in mathematical terms, and we'll connect them to concepts discussed previously in the course.

We'll also take a look at a couple of notions connected with measurements, namely quantum state discrimination and quantum state tomography. Quantum state discrimination refers to a situation that arises commonly in quantum computing and cryptography, where a system is prepared in one of a known collection of states, and the goal is to determine, by means of a measurement, which state was prepared. For quantum state tomography, on the other hand, many independent copies of a single, unknown quantum state are made available, and the goal is to reconstruct a density matrix description of that state by performing measurements on the copies.

Lesson video

In the following video, John Watrous steps you through the content in this lesson on general measurements. Alternatively, you can open the YouTube video for this lesson in a separate window.