I'm reading Quantum Computation and Quantum Information by Nielsen and Chuang (http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf). Equation 2.92 on page 85 is a bit confusing to me

if the state of the quantum system is $|\psi\rangle$ immediately before the measurement, then the probability that result $m$ occurs is given by $$p(m) = \langle\psi| M^†_m M_m |\psi\rangle$$

On page 62, they define $\langle\phi | A | \psi\rangle$ as the inner product between $|\phi\rangle$ an $A|\psi\rangle$

So, in this case, A = $M^†_m M_m$, which means the operator $M_m$ is pre-multiplied by it's Hermitian adjoint (the conjugate of the transpose) Then it is matrix-multiplied by the vector $|\psi\rangle$. Then the result of that is inner-producted with $|\psi\rangle$

is that correct?

  • 1
    $\begingroup$ Note that $\langle \psi | M_m^\dagger M_m | \psi \rangle = \langle M_m \psi|M_m \psi\rangle$ i.e. you're taking the inner product of $M_m |\psi\rangle$ with itself. $\endgroup$ – Beta Decay Feb 21 '20 at 19:08
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    $\begingroup$ For future reference: use \langle \rangle instead of < > $\endgroup$ – Superfast Jellyfish Feb 21 '20 at 19:11

In this case they are taking the inner product of $M_m | \psi \rangle$ with itself. The adjoint of that is $\langle \psi | M_m^\dagger$, so the inner product is $\langle \psi | M_m^\dagger M_m | \psi \rangle$.

  • $\begingroup$ ah, and it seems equation 2.18 on page 67 defines the inner product of |a> and |b> as the conjugate-transpose-of-~| a> times |b>. I think the problem is that I did so much linear algebra in R instead of C. That makes it more clear, thanks! $\endgroup$ – Mohammad Athar Feb 21 '20 at 19:07

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