# Help interpret bra-ket notation, quantum measurement equation (neielsen chuang, postulate 3)

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?

• 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. – Beta Decay Feb 21 '20 at 19:08
• For future reference: use \langle \rangle instead of < > – 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 $$\langle \psi | M_m^\dagger M_m | \psi \rangle$$.
• 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! – Mohammad Athar Feb 21 '20 at 19:07