Understanding the Identity Operator in 2-Qubit Quantum Systems

[Dragonfall]

Suppose $$\mathbf{U}_f(\left| x\right>\left| y\right> )=\left| x\right>\left| y\oplus f(x)\right>$$ denotes the unitary transformation corresponding to some 1-bit function f.

I'm guessing here that x is the input register, and y the output register.

Now suppose f(0)=0 and f(1)=0.

How is it that $$\mathbf{U}_f=\mathbf{1}$$ the 2-Qbit unit operator?

 

[Dragonfall]

In general, is $$\left| x\right>\left| y\right>$$ a tensor product?

 

[mrandersdk]

don't know about your first question but in general $$\left| x\right>\left| y\right>$$
means tensor product, we are just too lazy to write it.

 

[HallsofIvy]

It might be better to post this under "quantum mechanics".