# Applying a Single-site Operator to an MPS

In many applications one needs to modify a matrix product 
state (MPS) by multiplying it with an operator that acts 
only on a single site. This is actually a very straightforward
operation and this formula shows you how to do it in ITensor.

Say we have an operator @@G^{s'_3}_{s_3}@@ which
which acts non-trivially on site 3 of our MPS `psi`
as in the following diagram:

<img class="diagram" width="60%" src="docs/VERSION/formulas/operator_app_mps.png"/>

To carry out this operation, contract the operator G with the MPS tensor for site 3,
removing the prime from the @@s'_3@@ index afterward:

<img class="diagram" width="60%" src="docs/VERSION/formulas/operator_contract.png"/>

    auto newA = G*psi(3);
    newA.noPrime();

Finally, put the new tensor back into the MPS to update the third MPS tensor:

    psi.set(3,newA);

Or alternatively `psi.ref(3) = newA;`.
Afterward, we can visualize the modified MPS as:

<img class="diagram" width="60%" src="docs/VERSION/formulas/updated_mps.png"/>

As a technical note, if you are working in a context where gauge or orthogonality
properties of the MPS are important, such as in time evolution using two-site gates
(see the [[two-site gate formula|formulas/gate]]), then you may want to call `psi.position(3)`
before modifying the tensor at site 3, which will ensure that the MPS remains in a 
well-defined orthogonal gauge. Modifying a tensor which is left- or right-orthogonal
(i.e. not the "center" tensor of the gauge) will destroy the gauge condition and 
require extra operations to restore it. (Calling the `.position` method will automatically
fix this but will have to do extra work to do so.)