Now, here is the problem. (Capital letters indicate operators, lower letters are states, * indicates Hermitian conjugate) Say we know that state | p > = cos(a) |0> + sin(a) |1> (0<a<PI, a is in R) Two operators : M1= |0><0| , M2=|1><1|, apperatantly they satisfy the completeness equation. M1*M1+M2*M2 = I According to some textbooks, if we perform ONE opeartor on |p>, we should have TWO outcomes |0> or |1>, with certain probabilities. Here is the details: If I use one operator on the state, say M1, then: the state AFTER the operation is: M1 |p> / sqrt(<p | M*M |>p) = cos(a) |0> / cos(a) = |0> ........(1) the probability of this happening is PM1= <p | M*M |>p = cos(a)^2 .........(2) NOW, according to (1), it is impossible to get state (outcome) |1> after the operation. SO, where do the after-state (outcome) |1> go? I can get the outcome of |1> only with the other operator M2. M2 |p> / sqrt(<p | M*M |>p) = sin(a) |1> / sin(a) = |1> .......(3) with the probability PM2=sin(a)^2..............(4) To me, it seems that : The language of "use operator on state |p>" is not correct. A measurement can NOT contain ONLY a subset of the operators (in this case, contains only M1, not M2), A measurement have to be composed of a SET of operators which satisfies completeness equation. Is this the case in real-world experiments? We can not say "perform M1 on p", but have to say "perform the measurement which has M1 and M2 on p" or "measure the spin in the |0> to |1> axis"? BTW, put it in another way. If we keep M1 unchanged, but change M2 to | 1/sqrt(2)( |0> - |1>) > < 1/sqrt(2)( <0| - <1|)|, (they do not satisfy the compleness equation). SO, a measurement containing ONLY M1 and M2 can not be made in real life. Is this right?