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Please give a complete proof of the principle of Deferred Measurement Principle, including careful notation (preferably explaining the meaning of each symbol) and rigorous mathematical reasoning (preferably giving the rationale or references for each step of the derivation).

Ren-Xin Zhao
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  • I don't think the explanation makes sense of what each symbol means. Also troubled by the derivation process of the response. – Ren-Xin Zhao Aug 22 '23 at 12:58
  • Perhaps consider amending your question to state how the previous explanation troubles your understanding... – Mark Spinelli Aug 22 '23 at 18:02
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    @R-XZhao I can't quite understand what your comment is saying, but if you have specific comments or questions about that post you can ask a question specifying what exactly you're trying to understand and how/why that is not covered in the linked post – glS Aug 23 '23 at 18:58

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For me the intuition is the 'ancillary measurement principle'. Measuring with an ancilla is the same as inline measurement:

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You can then easily see it's possible to move the CNOT from before the measurement to after the measurement, as long as you know CNOTs commute when they only touch at the control:

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Now you might say this is not a proof, because at the end the control is on the output qubit instead of the output bit. And that's right. But you didn't ask for a proof you asked for the intuition.

My intuition is it's much clearer that the qubit has the same value as the bit after the measurement. Because at that point the qubit has been collapsed into the computational basis. So the deferred measurement principle becomes blatantly obvious, and it's just a matter of grinding the algebra to see it's actually equivalent.

Craig Gidney
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  • how about this one : https://quantumcomputing.stackexchange.com/questions/13944/is-there-any-difference-between-a-quantum-and-classically-controlled-gate-if-i – Ren-Xin Zhao Aug 23 '23 at 05:56
  • another one is : https://quantumcomputing.stackexchange.com/questions/22240/show-that-the-effect-of-a-controlled-unitary-on-qubits-followed-by-a-measurement – Ren-Xin Zhao Aug 23 '23 at 05:56
  • But I'm not sure that the above two replies correctly and directly reinstate the problem, and I need more of a mathematical proof. – Ren-Xin Zhao Aug 23 '23 at 05:57
  • there're two similar answer that i don't know whether they are right: First one is. https://quantumcomputing.stackexchange.com/questions/13944/is-there-any-difference-between-a-quantum-and-classically-controlled-gate-if-i The second one is. https://quantumcomputing.stackexchange.com/questions/22240/show-that-the-effect-of-a-controlled-unitary-on-qubits-followed-by-a-measurement – Ren-Xin Zhao Aug 24 '23 at 15:12