Purifications and Fidelity | Understanding Quantum Information & Computation | Lesson 12

This is part of the Understanding Quantum Information & Computation series. Watch the full playlist here:    • Understanding Quantum Information & Comput...  

This lesson explores the concept of a purification in quantum information, where an arbitrary quantum state is represented by a pure state of a larger system that leaves the original state when the rest of the larger system is discarded. This is an incredibly useful concept that allows us to reason interesting things about density matrices by essentially viewing them as parts of pure states, which are simpler in some respects. The lesson also introduces fidelity, a measure of similarity between two quantum states, which plays a key role in quantum computing. Along the way we'll also discuss Schmidt decompositions, the unitary equivalence of purifications, and Uhlmann's theorem, which links fidelity to purifications.

0:00 — Introduction
1:29 — Overview
2:23 — Purifications
4:02 — Existence of purifications
7:33 — Schmidt decompositions
12:54 — Unitary equivalence of purifications
16:21 — Example: superdense coding
18:04 — Cryptographic implications
20:47 — HJW theorem
25:48 — Definition of fidelity
31:32 — Properties of fidelity
35:24 — Gentle measurement lemma
39:18 — Uhlmann’s theorem
43:45 — Conclusion

Find the written content for this lesson on IBM Quantum Learning: https://learning.quantum.ibm.com/cour...

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