Does QED evolve unitarily after the Schwinger limit? If QED becomes nonlinear after the Schwinger limit, shouldn't QED no longer be unitary (above the limit) since linearity is a requirement of a unitary operator (and vice versa)?
Does this mean that superpositions cannot be used to fully describe physics above the limit?
 A: There are two different notions of non-linearity at play. Non-linear QED is still linear in the sense that superpositions of states are solutions of the Schrodinger equation.
Let's consider the Schrodinger equation describing non-relativistic quantum mechanics for a single particle (setting $\hbar=1$)
\begin{equation}
i \frac{\partial}{\partial t} | \Psi \rangle = \hat{H} | \Psi \rangle
\end{equation}
The Hamiltonian operator $\hat{H}$ is a function of the position operator $\hat{x}$ and momentum operator $\hat{p}$.
Now

*

*The Schrodinger equation must be a linear function of the state $|\Psi \rangle$, otherwise we are no longer talking about quantum mechanics.

*The Hamiltonian operator can be a quadratic function of the position and momentum operators, like the harmonic oscillator Hamiltonian: $\hat{H}= \frac{1}{2} \left( \hat{p}^2 + \omega^2 \hat{x}^2\right)$. This is a linear system, in the sense that the equations of motion are linear in $x$ and $p$. However, the Hamiltonian can also include higher powers. For example, one can have an anharmonic oscillator, $\hat{H}= \frac{1}{2} \left( \hat{p}^2 + \omega^2 \hat{x}^2 + \lambda \hat{x}^4\right)$. In this case, one sometimes says the system is non-linear (since the equations of motion are not linear in $x$), even though the Schrodinger equation is still linear in $| \Psi \rangle$.

Non-linear QED obeys the first type of linearity, and includes non-linear interactions in the sense of the second bullet point. There is no contradiction with basic principles of quantum mechanics.
A: Linearity is required in the Hilbert space of quantum states. The QED fields are not elements of the Hibert space  of quantum states and their equations can be non-linear without any problem. Indeed non-linearity is required for interactions. Linear equations only describe non-interacting systems. Linear superposition of the E&M fields fails when they are very strong, and this is due photons interacting with other photons via coupling to virtual electron-positron pairs that are described by Fenyman diagrams with fermion loops.
A: Physically everything must be OK, but mathematically, when you create pairs and do not change the electric field (which normally must decrease due to joining with the born charges of the opposite signes), you may have a perpetuum mobile. It just means your boundary conditions are not physical.
