# Constructing an arbitrary 2-Qbit state

I am reading a book on quantum computing. The author is constructing an arbitrary 2-Qbit state from unitary transformations. I need help understanding on step in his logic.

He starts by noting that the general 2-Qbit state has the form

$$|\Psi\rangle = a_{00}|00\rangle + a_{01}|01\rangle + a_{10}|10\rangle + a_{11}|11\rangle$$

This can also be written as:

$$|\Psi\rangle = |0\rangle \otimes |\psi\rangle + |1\rangle \otimes |\phi\rangle$$ $$|\psi\rangle = a_{00}|0\rangle + a_{01}|1\rangle$$ $$|\phi\rangle = a_{10}|0\rangle + a_{11}|1\rangle$$

So far so good. Next the author says apply $\textbf{u} \otimes \textbf{1}$ to $|\Psi\rangle$, where $\textbf{u}$ is a linear transformation, whose action on the computational basis is of the form:

$$\textbf{u}|0\rangle = a|0\rangle + b|1\rangle, \quad \textbf{u}|1\rangle = -b^*|0\rangle + a^*|1\rangle; \quad |a|^2 + |b|^2 = 1$$

The author doesn't state this, but I assume $a,b \in \mathbb{C}$ and $a^*$ and $b^*$ are the complex conjugates of $a$ and $b$ respectively. Also unstated but assumed by me is that $\textbf{u}$ is a $2x2$ matrix and $\textbf{1}$ is the $2x2$ identity.

Now comes the part I don't understand. The author states:

$$(\textbf{u} \otimes \textbf{1})|\Psi\rangle = (a|0\rangle + b|1\rangle) \otimes |\psi\rangle + (-b^*|0\rangle + a^*|1\rangle) \otimes |\phi\rangle$$

In order for the above to be true, it would seem that

$$(\textbf{u} \otimes \textbf{1})|\Psi\rangle = \textbf{u} |0\rangle \otimes |\psi\rangle + \textbf{u}|1\rangle \otimes |\phi\rangle$$

But I don't understand why.

• It's the distributive law, along with the tensor product identity $(\mathbf{u}\otimes\mathbf{1})(|0\rangle \otimes |\psi\rangle) = \mathbf{u} |0\rangle \otimes \mathbf{1} |\psi\rangle.$ – Peter Shor Oct 9 '17 at 2:11
• Thanks! I was not aware that the definition of tensor products on linear maps was the distributive law you mentioned. – Max Oct 9 '17 at 2:30

In general, $(A\otimes B)(C \otimes D)=(AC) \otimes (BD)$, in quantum computing this is interpreted as applying a transformation to only one state of a composite system.