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## Summary:

- How to do Schmidt decomposition in QC?

Hi, there. I have some problems when learning Schmidt decomposition in Nielsen's QC.

The statement of Schmidt decomposition is simple and clear, however, the book doesn't give a clear procedure to do the Schmidt decomposition. I don't know whether the proof under the theorem is the the one I should use or not. After all, it seems practicable to me.

Well, then I got confused again when solving the exercise that ask me to find the Schmidt decompositions of the states

##\frac {\left| 00 \right>+\left | 11 \right >} {\sqrt 2}## ; ##\frac {\left| 00 \right>+\left | 01 \right >+\left | 10 \right >+\left | 11 \right >} 2##; and ##\frac {\left| 00 \right>+\left |01 \right >+\left | 10 \right >} {\sqrt 3}##, because I think the decompositions are themselves.

For example, the decomposition of ##\frac {\left| 00 \right>+\left |01 \right >+\left | 10 \right >} {\sqrt 3}## should be ##\frac {\left| 00 \right>} {\sqrt 3} + \frac {\left |01 \right >} {\sqrt 3} + \frac {\left | 10 \right >} {\sqrt 3}## which is itself. The expression satisfy all the conditions required in the theorem. But I don't think I'm right.

Could you help me point out what part do I make mistakes?

ps: Nielsen's book makes me feel terrible. I have a great time learning QM with Griffth's book. But I struggle in Nielsen's book. It is more abstract and there are not many examples that I can try. I really feel depressed and wonder that QC may be not suitable for me. But I really love computer and physics since high school.

pss: What do the "

The statement of Schmidt decomposition is simple and clear, however, the book doesn't give a clear procedure to do the Schmidt decomposition. I don't know whether the proof under the theorem is the the one I should use or not. After all, it seems practicable to me.

Well, then I got confused again when solving the exercise that ask me to find the Schmidt decompositions of the states

##\frac {\left| 00 \right>+\left | 11 \right >} {\sqrt 2}## ; ##\frac {\left| 00 \right>+\left | 01 \right >+\left | 10 \right >+\left | 11 \right >} 2##; and ##\frac {\left| 00 \right>+\left |01 \right >+\left | 10 \right >} {\sqrt 3}##, because I think the decompositions are themselves.

For example, the decomposition of ##\frac {\left| 00 \right>+\left |01 \right >+\left | 10 \right >} {\sqrt 3}## should be ##\frac {\left| 00 \right>} {\sqrt 3} + \frac {\left |01 \right >} {\sqrt 3} + \frac {\left | 10 \right >} {\sqrt 3}## which is itself. The expression satisfy all the conditions required in the theorem. But I don't think I'm right.

Could you help me point out what part do I make mistakes?

ps: Nielsen's book makes me feel terrible. I have a great time learning QM with Griffth's book. But I struggle in Nielsen's book. It is more abstract and there are not many examples that I can try. I really feel depressed and wonder that QC may be not suitable for me. But I really love computer and physics since high school.

pss: What do the "

**Quantum Physics Workshop**" section do? Should I really post this thread in that section?