# For two-qubit systems, do we have $\langle 01|01\rangle = \langle 0|0\rangle\langle 1|1\rangle$?

I am new to quantum computing and I want to know the following: If I have a 2 qubit system in state e.g. $$\left|01\right>$$ and I want to calculate the probability of measuring e.g. $$\left<01\right|$$ I can write it as following: $$|\left<01|01\right>|^2$$. Now is the notation $$\left<01|01\right>$$ equal to this notation $$\left<0|0\right>\left<1|1\right>$$?

## 2 Answers

Yes.

In general, given two Hilbert space $$H_1$$ and $$H_2$$ with inner product $$\langle \cdot |\cdot \rangle_1$$ and $$\langle \cdot | \cdot \rangle_2$$ then for $$u_1, v_1 \in H_1$$ and $$u_2,v_2 \in H_2$$ we have that

$$\langle u_1 \otimes u_2 | v_1 \otimes v_2 \rangle = \langle u_1|v_1\rangle_1 \cdot \langle u_2|v_2\rangle_2$$

A more general rule is called the mixed-product property: $$(A \otimes B) (C \otimes D) = AC \otimes BD,$$ which holds for any matrices $$A,B,C,D$$ of such sizes that we can form the matrix products $$AC$$ and $$BD$$.