Lesson 09: Density Matrices | Understanding Quantum Information & Computation
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- Опубликовано: 9 июн 2024
- In the general formulation of quantum information, quantum states are represented by a special class of matrices called density matrices. This lesson describes the basics of how density matrices work and explains how they relate to quantum state vectors. It also introduces the Bloch sphere, which provides a useful geometric representation of qubit states, and discusses different types of correlations that can be described using density matrices.
0:00 - Introduction
1:46 - Overview
2:55 - Motivation
4:40 - Definition of density matrices
9:55 - Examples
12:58 - Interpretation
15:37 - Connection to state vectors
20:13 - Probabilistic selections
25:23 - Completely mixed state
28:41 - Probabilistic states
32:03 - Spectral theorem
37:36 - Bloch sphere (introduction)
38:36 - Qubit quantum state vectors
41:30 - Pure states of a qubit
43:52 - Bloch sphere
47:38 - Bloch sphere examples
51:36 - Bloch ball
55:40 - Multiple systems
56:46 - Independence and correlation
1:00:55 - Reduced states for an e-bit
1:04:16 - Reduced states in general
1:08:53 - The partial trace
1:12:23 - Conclusion
Find the written content for this lesson on IBM Quantum Learning: learning.quantum.ibm.com/cour...
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I am a first-year university student in Japan, and I recently became interested in quantum computing after watching videos of IBM's researchers. I hope to specialize in quantum computing and work at IBM in the future. Since my university only offers classes in Japanese, these videos are truly invaluable to me. I plan to study abroad extensively to improve my English skills and fulfill my dream. Thank you very much, Professor John Watrous. I look forward to the next video!
I'm a simple man, I see John's lecture and I click it
Professor John Watrous is well-known in his field, and it's great that he took the time to upload this informative video on RUclips. It's a wonderful resource for the community. Thanks Qiskit
Thanks John, long awaited lecture.
If I succeed my exams it will definitely be thanks to you. You made me understand quickly many concepts I struggeled to figure out before. Thank you for all your work in making this complex subject a little bit more accessible !
Thank you so very much for the extremely valuable lesson! Much success to all!
Awesome, thx! This series, plus the Qiskit summer school last year, brought me on speed on QC in roughly one year. I don't think I would have understood this very lesson here without all the gentle introduction to the concepts over the months. I'm very thankful for your guidance and l'm looking forward to the next lesson!
It's great to see you again, Prof. Watrous, after that somewhat extended hiatus/break. As always, these lectures are top-notch.
Every time I see the Bloch sphere/ball, I'm reminded of SU(2) and spinors. So I can't help but ask: has anyone made an attempt at expressing the math in a spinor formalism? And if so, does it help to express the concepts more easily (or more economically, at any rate)?
I, for one, still have trouble getting my head around spinors, in spite of mining all the source materials I've been able to find (e.g., the Dover book on the topic gets into the deep end pretty quickly, for me anyway...).
There's definitely a known connection along the lines you're suggesting, but I will have to leave the question of whether it helps to others. I know very little (essentially nothing, really) about spinors, so I'm afraid I'm not the right person to answer your question.
Great lesson, thank you so much.
I'm all for it 🙌🏼.
Wonderful!
Are the slides for lesson 09 available anywhere? Thanks.
Schrödinger on Heisenberg
“After matrices are thus constructed in a very general way, so as to satisfy the general rules, I will show the following in Section 4. The special system of algebraic equations, which, in a special case, connects the matrices of the position and impulse co-ordinates with the matrix of the Hamiltonian function, and which the authors call “equations of motion”, will be completely solved by assigning the auxiliary role to a definite orthogonal system, namely, to the system of proper functions of that partial differential equation which forms the basis of my wave mechanics.”
On the Relation of the Quantum Mechanics of Heisenberg,
Born, and Jordan, and that of Schrödinger.
Anallen der Physik (4), vol. 79, 1926.
English, 1928 Blackie and Sons LTD..
Alice and Bob detangled in spacetime ... but reunited both completely mixed up in the center of a Ball 🤑
I was following everything until the equations displayed, approximately 25 seconds
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