The church of the larger (or higher, or greater) Hilbert space is just a trick that some people like (myself included) for rewriting some operations.

The most general operations that you can write down for a system are described by completely positive maps, while we like describing things with unitaries, which you can always do by moving from the original Hilbert space to a larger one (i.e. adding more qubits). Similarly, for measurements, you can turn general measurements into projective measurements by increasing the size of the Hilbert space. Also, mixed states can be described as pure states of a larger system.

## Example

Consider the map that takes a qubit and with probability $1-p$ does nothing, and with probability, $p$ applies the bit-flip operation $X$:
$$
|\psi\rangle\langle\psi|\mapsto (1-p)|\psi\rangle\langle\psi|+pX|\psi\rangle\langle\psi|X
$$
This is not unitary, but you can describe it as a unitary on two qubits (i.e. by moving from a Hilbert space dimension 2 to Hilbert space dimension 4). This works by introducing an extra qubit in the state $\sqrt{1-p}|0\rangle+\sqrt{p}|1\rangle$ and performing a controlled-not controlled by the new qubit and targeting the original one.
$$
|\psi\rangle(\sqrt{1-p}|0\rangle+\sqrt{p}|1\rangle)\mapsto|\Psi\rangle=\sqrt{1-p}|\psi\rangle|0\rangle+\sqrt{p}\left(X|\psi\rangle\right)|1\rangle.
$$
To get back the action of the system on just the original qubit, you trace out the new qubit:
$$
\rho={\rm Tr}_2\left(|\Psi\rangle\langle\Psi|\right)= (1-p)|\psi\rangle\langle\psi|+pX|\psi\rangle\langle\psi|X.
$$
In other words, you just ignore the existence of the new qubit after you’ve implemented the unitary! Note that as well as demonstrating the church of the larger Hilbert space for operations, this also demonstrates it for states - the mixed state $\rho$ can be made into the pure state $|\Psi\rangle$ by increasing the size of the Hilbert space.

6Can you post a reference for some context? Where have you read this term. Thanks. – Andrew O – 2018-04-10T00:58:12.193