What does the unitary $[|0ranglelangle 0|otimes I+|1ranglelangle1|otimes(|1ranglelangle 0|+|0ranglelangle1|)]otimes I$ represent?

Quantum Computing Asked on August 15, 2020

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

$$U_{AB} = Big[|0rangle langle 0|_{A} otimes mathbf{I}_{B} + |1rangle langle 1|_{A} otimes big(|1rangle langle 0|_{B} + |0rangle langle 1|_{B} big) Big] otimes mathbf{I}_{B^prime},$$ 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)?

One Answer

The gate/operator in the brackets is the non-local CNOT gate, frequently used to create bipartite entanglement. Given it itself is a 2 qubit gate, then the tensor of this with the identity is simply a gate that acts on 3 qubits.

This gate will take a 3 qubit state and flip the second qubit of this state when the first is $|1rangle$ It will take $|110rangle to|100rangle, |111rangle to |101rangle$ and vice-versa.

Edit: important to note is that $|1ranglelangle 0|+|0ranglelangle 1|$ is the Pauli X gate

Answered by GaussStrife on August 15, 2020

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