# What does the unitary $[|0\rangle\langle 0|\otimes I+|1\rangle\langle1|\otimes(|1\rangle\langle 0|+|0\rangle\langle1|)]\otimes I$ represent?

Consider the following unitary defined for a system $$A$$ interacting with a bipartite system $$BB^\prime$$

$$U_{AB} = \Big[|0\rangle \langle 0|_{A} \otimes \mathbf{I}_{B} + |1\rangle \langle 1|_{A} \otimes \big(|1\rangle \langle 0|_{B} + |0\rangle \langle 1|_{B} \big) \Big] \otimes \mathbf{I}_{B^\prime},$$ with $$\mathbf{I}_i$$ being the identity.

My question:

1. What is the physical meaning of operation $$U_{AB}$$?

2. Can one represent $$U_{AB}$$ in terms of quantum logic gates ( a circuit diagram)?

• The expression inside the square brackets is just a CNOT from A to B. It is usually considered to be an elementary gate, so it's representation is just itself. Jul 31 '20 at 12:49
• I guess the "key" observation is that $| 1 \rangle \langle 0 |_{B} + | 0 \rangle \langle 1 |_{B} = \sigma^x_{B}$ Aug 1 '20 at 1:58

This gate will take a 3 qubit state and flip the second qubit of this state when the first is $$|1\rangle$$ It will take $$|110\rangle \to|100\rangle, |111\rangle \to |101\rangle$$ and vice-versa.
Edit: important to note is that $$|1\rangle\langle 0|+|0\rangle\langle 1|$$ is the Pauli X gate