# What is the matrix for measuring a superposition of general number of qubits in standard basis?

Let's say I have the state of the system of 2 qubits: $$\frac{1}{\sqrt{3}}|00\rangle+\frac{2}{\sqrt{3}}|10\rangle$$, and I want to measure it in the standard basis. How would I write it mathematically? What is the $$n$$ number of qubits version of it?

For measuring 2 qubits, there are 4 possible outcomes, corresponding to projectors $$P_{00}=|00\rangle\langle 00|,\qquad P_{01}=|01\rangle\langle 01|,\qquad P_{10}=|10\rangle\langle 10|,\qquad P_{11}=|11\rangle\langle 11|.$$ So, if you have a state $$|\psi\rangle$$, you get the outcome $$x$$ with probability $$p_x=\langle\psi|P_x|\psi\rangle$$.
This immediately generalises to $$n$$ qubits where you have $$2^n$$ possible outcomes $$x\in\{0,1\}^n$$ using $$P_x=|x\rangle\langle x|$$ and $$p_x=\langle\psi|P_x|\psi\rangle$$.