Relation between a measurement and the operators

[ouacc]

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?

 

[Doc Al]

I'd say that if the operator representing the quantity that you are measuring is M, and it's eigenvectors are |0> (eigenvalue = M1) and |1> (eigenvalue = M2) (and that's the complete set of eigenvectors for that operator), then for any initial state | p > = cos(a) |0> + sin(a) |1>, the projection operators acting on that state can give you the probability amplitude of a measurement of M giving any particular value (M1 or M2).

 

[denian]

I can confirm that your understanding is correct. The measurement operators M1 and M2 must satisfy the completeness equation in order for them to represent a valid measurement. This means that they must be able to account for all possible outcomes of the measurement, in this case, the states |0> and |1>. If we only use one of the operators, we can only obtain one of the outcomes and the other outcome is not accounted for. This is why it is important to have a set of operators that satisfy the completeness equation in order to perform a valid measurement.

In real-world experiments, we cannot simply say "perform M1 on p" as this would not be a complete measurement. Instead, we must specify the set of operators being used, such as "measure the spin in the |0> to |1> axis" or "perform the measurement which has M1 and M2 on p". This ensures that all possible outcomes are accounted for and the measurement is valid.

Additionally, your example of changing M2 to |1/sqrt(2)(|0> - |1>)><1/sqrt(2)(<0| - <1|)| shows that not all sets of operators will satisfy the completeness equation, and therefore cannot be used for a valid measurement. This further emphasizes the importance of using a complete set of operators in measurements.

In conclusion, your understanding of the relation between a measurement and the operators is accurate. A valid measurement must be composed of a set of operators that satisfy the completeness equation in order to account for all possible outcomes. This is important to keep in mind when designing and conducting experiments.