私は、量子コンピュータの基礎理論を下記に示した英語の本で勉強しています。単なる教養としてではなく、高校数学の範囲で、厳密に式を使ってクリアに説明しているところが非常に気に入っています。易しい英語で貫かれています。
●ノーベル物理学賞2022の量子理論を高校数学で学ぶ
https://sparse-dense.blogspot.com/2022/10/2022.html
読み進めているうちに、ヤマ場となるEntanglement（量子もつれ）とBell's Inequality（Bellの不等式(の不成立)）のところで、どうしても確認したい点が出てきました。著者のChris Bernhardt（米国Fairfield大学の数学の名誉教授）に質問したところ、直ちに回答してくれました。この世界、そういう文化になっているので、思いついたら質問するのがいいですね。
もしかすると、この本で勉強する人にとって、以下の応答は役立つかも知れないと考え掲載します。Prof. Chrisからは、メール応答文のブログ掲載に同意していただいています。
（あなたのブログはとても役立ちそうだ、と言っていただいたのも嬉しい。）
----------------
【山本より】
Hello Chris,
I am a Japanese reader of "Quantum Computing for Everyone".
This book is wonderful because it tries to explain difficult things clearly in high school mathematics.
I have a few questions regarding the Entangled Quantum Clocks on Page 68.
I think this is one of the most important parts of the book.
Thank you for your reply in advance.
Q1
I understand as follows, is it correct?
"There are two humans here (me and you), each with 100 clocks.
In other words, it is not the assumption that 100 people each have a watch."
Q2
Is the First scenario or the second scenario a fact that can be derived from the entangled state shown in the center of this page?
Or rather, if such a scenario were to hold, would it be considered as explained on page 69 below?
----------------
【Chrisより】
Hello Fujio,
You are correct that entanglement is one of the most important parts.
Q1. You are correct. There are 100 pairs of clocks. Each pair of clocks is entangled. For each pair, you have one and I have the other.
Q2. The entangled state shown corresponds to one pair of entangled clocks. As soon as one of us looks at a clock, the state immediately jumps to either
or 
. Both states are unentangled. We either both get YES or both get NO.
Our clocks are correlated, they always agree, but they don’t “decide” what the answer will be until one of us looks at (measures) our clock.
Hope this helps,
Chris
----------------
【山本より】
Dear Chris,
Thank you very much for your reply.
Your answer cleared my mind.
I totally understand!
This takes me to Chapter 5, "Bell's Inequality".
I have already featured this book of yours in my blog post.
It's in Japanese so you may not be able to read it, but here it is:
https://sparse-dense.blogspot.com/2022/10/2022.html
I hope my question and your answer will be useful to other readers.
May I publish this response as-is on my blog?
Add the following information:
A Japanese version of this book was already published by Yuichiro Minato et. al.
In the middle of p.68 of the English version, "and I have the hundred partners." was mistranslated as follows:
"suppose there are 100 people to be paired with me"
I quickly realized that it was a weird translation, but your answer made it clear.
----------------
【Chrisより】
Of course! The translation sounds a little strange. Your blog looks as though it could be very helpful.
----------------
【山本より】
Dear Chris,
With your answers to my previous questions, I was able to read and understand Chapter 5 Bell's Inequality. Just in case, I would like to confirm. I understood as below, is it correct?
In the classical model, the results of Alice's and Bob's agree at least 5/9 (=0.55) in the case of P.82. The existence of such a lower bound is called Bell's inequality.
On the other hand, in the Quantum theory model, both results agree exactly by half (=0.50). The difference between the classical model and the quantum model, that is, 0.55 not equal 0.50, is called Bell's inequality violation.
Best regards,
Fujio Yamamoto
----------------
【Chrisより】
Dear Fujio,
The classical model predicts the proportion of times the measurements agree to be at least 5/9. Quantum theory predicts 1/2. These two numbers are not equal — that’s Bell’s inequality. It means that we can perform experiments to determine which theory is incorrect.
All experiments performed so far, have agreed with quantum theory.
Chris
----------------
【山本より】
Dear Chris,
Thank you very much again for your reply.
The word "inequality" reminded me of ">" or "<", so I focused on the "lower limit" or "upper limit". Anyway, now I have a clear understanding.
I think your book can be divided into chapters 1 to 5 (Bell' Inequality) and chapters 6 onwards. The first half is basic theory and the second half is applied. I understand the first half almost perfectly, so I can work on the second half smoothly.
Incidentally, I was able to find a street in Belfast on Google Maps named "Bell's Theorem Crescent". Great!
----------------
【山本より】
Dear Chris
I really like your book on Quantum Computing and am studying.
I wrote a blog post to remind myself of what I've learned about Entanglement and the Ekert protocol.
#My understanding of the Quantum Key Distribution Protocol Ekerthttps://sparse-dense.blogspot.com/2022/11/my-understanding-of-quantum-key.html
Thanks for your wonderful book.
Regards,
Fujio
----------------
【Chrisより】
Dear Fuji,
Great post — and in English!
Thanks for sending me this,
Chris
----------------
