# CY gate
**CY gate** (controlled-Y) is a two-qubit gate that applies a [[y-gate|Pauli Y]] to the target qubit if and only if the control qubit is $\lvert 1\rangle$, and does nothing otherwise.

$$\text{CY} = \begin{pmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & -i\\ 0 & 0 & i & 0\end{pmatrix}$$

The rows and columns are ordered $\lvert 00\rangle, \lvert 01\rangle, \lvert 10\rangle, \lvert 11\rangle$, with the first qubit as control and the second as target:

$$\text{CY}\lvert 00\rangle = \lvert 00\rangle \qquad \text{CY}\lvert 01\rangle = \lvert 01\rangle \qquad \text{CY}\lvert 10\rangle = i\lvert 11\rangle \qquad \text{CY}\lvert 11\rangle = -i\lvert 10\rangle$$

## Creating entanglement

A Hadamard gate on the control followed by CY produces an entangled state from $\lvert 00\rangle$. The resulting state differs from the [[bell-states|Bell states]] by a local phase on the target: $\frac{1}{\sqrt{2}}(\lvert 00\rangle + i\lvert 11\rangle)$. Like the [[cx-gate|CX gate]], CY is an entangling gate — the output cannot be written as a product of two single-qubit states.

## Properties

The CY gate is its own inverse up to global phase: $\text{CY}^2 = I$. It is equivalent to the [[cx-gate|CX gate]] up to single-qubit rotations on the target: $\text{CY} = (I \otimes S) \cdot \text{CX} \cdot (I \otimes S^\dagger)$, where $S$ is the [[s-gate|S gate]]. Together with all single-qubit gates it forms a universal gate set for quantum computation.