# What does "commuting operators can be measured simultaneously" mean?

I want to understand better what it means by any commuting set of operators can be measured simultaneously.

Suppose I have an $$n$$-qubit arbitrary pure state $$\rho = \lvert \psi \rangle \langle \psi \rvert$$. Is it always true that all $$X_1 = X\otimes \cdots \otimes I$$, ..., $$X_n = I \otimes \cdots \otimes X$$ can be measured simultaneously? If so, then is it always true that measuring all $$X_i, Y_i, Z_i$$ $$\forall i \in [n]$$ can be done by using $$3$$ copies of quantum states? If not, then what does it mean by "any commuting set of operators can be measured simultaneously"? What are conditions?

• My you expand your claim or put the reference? Dec 23, 2022 at 14:17

For your second question, if you can simultaneously measure different observables (not all hardware can), then yes you could measure all $$X_i$$ then all $$Y_i$$ then all $$Z_i$$ on 3 different copies without one outcome effecting another. Theoretically, you can do other things like measure X on all even qubits and Z on all odd qubits since these commute. These outcomes are still random variables the depend on the underlying state but not on other measurement outcomes.
Finally, a little example to illustrate this. Say you have a 2-qubit state $$|\psi \rangle = |00 \rangle$$. If you measure $$Z_1, X_2$$, then you'd always get a 0 for the result of $$Z_1$$ and you'd get a 0 or 1 with equal probability for the result of $$X_2$$ regardless of the order in which you measure. Now if you measure $$X_2$$ first, and get outcome +1, then the resulting state becomes $$|0\rangle |+\rangle$$. If you then measure $$Z_2$$ then you'd get 0 half the time and 1 half the time. Compare this to measuring $$Z_2$$ first then $$X_2$$. In that case, you'd always get 0 when you measure $$Z_2$$ first then you'd get a 50-50 outcome for $$X_2$$. Since $$X_2, Z_2$$ don't commute, they disturb each other and the order that the measurement is done changes the statistics of the outcomes. Measuring $$Z_2, X_2$$ simultaneously is then difficult to think about: which one happens first?