# Nielsen & Chuang, 10th anniversary edition, page 85, equation 2.92

I don't understand why the stated equation is not equal to the expectation value of $$m^2$$

• Are the matrices $M$ promised to have eigenvalues/eigenvectors? Nov 2, 2022 at 11:28
• Yes, that is specified in the text. Nov 2, 2022 at 13:49
• Are you sure? It's not in my older version of the book (and shouldn't be) Nov 2, 2022 at 14:01
• Hopefully that explains my misunderstanding. Here is the text: Postulate 3: Quantum measurements are described by a collection {Mm} of measurement operators. These are operators acting on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ immediately before the measurement then the probability that result m occurs is given by ... and the equation follows Nov 2, 2022 at 14:06
• I see that in the following text, they do a simple example where M^2=M, but that does not answer my question Nov 2, 2022 at 14:25

You seem to have changed a collection of operators $$\{M_m\}$$ into a single operator $$M=\sum_mmM_m$$ (this is actually an observable).
The whole point of the collection is that you handle each of the cases $$m$$ separately - you do a measurement and your measurement apparatus gives you one value $$m$$ (corresponding to a specific $$M_m$$) and you calculate the consequences of that.
• The sum is what I've had to introduce to recreate your calculation in the notation of N&C. Note that if you think you're doing the calculation with a single operator, that single operator is something that has to square to identity because it must satisfy the completeness relation. Then all the $m_i^2=1$. Nov 2, 2022 at 15:23