I think non-locality comes from the mathematical approach used to solve your problem.
Suppose the heat equation similar to Schrödinger's :
With an initial condition, and boundary condition for f to be zero. Moreover we admit an intermediate condition that the function shall vanish on a central interval, the initial condition being non 0 on one side of the intermediate condition.
Analytically there is only 1 solution : an analytical function being 0 on an interval is 0 everywhere, since all derivatives vanish on it and that an entire function is the same as its Taylor expansion. This comes as a corollary of locality and analyticity.
So there is no solution if the initial condition is not 0, analytically, since the question is bad posed and self contradicting.
If you now apply a finite element method in space and finite difference in time you shall write your function at discretized points as : $$f(x,t)=\sum_i a_i(t)\phi_i(x)$$, with special triangle wave basis functions, which by the way are not analytic, but that underspin bothdomains and vanish in the intermediate one (this looks quite artificial and thus the following is to be taken as artificial too)
The weak formulation of FEM implies a matrix equation with two matrices A and B (left to the reader) :
Now a finite difference scheme for the time dependence implies :
- there exist a non vanishing solution
- the inverse of matrix B can be a full one, so
2a) the solution at every point at the next time step depends on the values at all points at the previous time step, hence there is non locality, even if it is not a quantum equation.
2b) the function on the other side of the intermediate condition changes even if it was zero on that domain and even if the solution does not pass through the intermediate interval, which could be called a true tunneling, explained as a quantum space jump.
So holism, or non-locality is not a quantum physical only phenomenon here, but rather a property on how the problem is solved mathematically or numerically.
If this corresponds to a physical transfer is an ongoing problem since EPR in quantum.