# Decoherence paper, incomprehensible claims

I am reading the paper "The quantum-to-classical transition and decoherence" by Maximilian Schlosshauer (https://arxiv.org/abs/1404.2635). I doubt the most basic claims in this paper.

The story is as follows: an isolated microscopic system $$S$$ (Hilbert space $$H_S$$) is put into a superposition $$\phi=\phi_1+\phi_2$$ of states $$\phi_1,\phi_2\in H_S$$ and then brought into contact with an environment $$E$$ (Hilbert space $$H_E$$) which initially is in state $$e\in H_E$$. Before interaction the composite system $$(S,E)$$ is in state $$\Phi=\phi\otimes e = \phi_1\otimes e+\phi_2\otimes e\in H_S\otimes H_E.$$ Now an interaction between $$S$$ and $$E$$ takes place and it is claimed that this puts the combined system $$(S,E)$$ into a state of the form $$\Psi = \phi_1\otimes e_1+\phi_2\otimes e_2\in H_S\otimes H_E$$ (cited paper, equation (1), p2). This is surprising, since we do not know anything about the interaction (no assumptions made).

I believe that such a principle cannot be maintained: we can write $$\phi$$ in other ways as a superposition $$\phi=\psi_1+\psi_2$$ and it must then follow that $$\Psi$$ is also of the form $$\Psi = \psi_1\otimes f_1 + \psi_2\otimes f_2\in H_S\otimes H_E,$$ for some vectors $$f_1,f_2\in H_E$$.

In other words the vectors $$e_1,e_2\in H_E$$ have the following interesting property: $$\psi_1+\psi_2=\phi_1+\phi_2\ \Rightarrow\ \exists f_1,f_2\in H_E:\thinspace \phi_1\otimes e_1+\phi_2\otimes e_2 = \psi_1\otimes f_1 + \psi_2\otimes f_2.$$ We will now exhibit an example where such vectors $$e_1,e_2$$ do not exist. Let $$H_S=H_E=L^2([0,1])$$. Then the tensor product $$H_S\otimes H_E$$ is isomorphic to the Hilbert space $$L^2([0,1]^2)$$ with a canonical isomorphism mapping $$f(x)\otimes g(y)$$ to the function $$f(x)g(y)\in L^2([0,1]^2)$$.

Now let $$\phi_1(x)=1$$, $$\phi_2(x)=x$$, $$\psi_1(x)=1-\sqrt x$$ and $$\psi_2(x)=x+\sqrt x$$. Then $$\phi_1+\phi_2=\psi_1+\psi_2$$ but there are no functions $$e_1(y),e_2(y)\in H$$ such that the function $$\phi_1(x)e_1(y)+\phi_2(x)e_2(y)$$ can be written in the form $$\phi_1(x)e_1(y)+\phi_2(x)e_2(y) = \psi_1(x)f_1(y)+\psi_2(x)f_2(y),$$ for some functions $$f_1,f_2\in H$$ since this amounts to the equality $$e_1(y)+xe_2(y) = (1+\sqrt x)f_1(y)+(x-\sqrt x)f_2(y),$$ for almost all $$x,y\in[0,1]$$. Simply evaluate the equality at enough different values of $$x$$ to obtain an overdetermined system of equations in $$e_i(y),f_j(y)$$ with no solution. Clearly this same argument works if we replace $$H=L^2([0,1])$$ with a finite dimensional Euclidean space of dimension at least 5.

Why is this not a problem?

• From the paper, $S$ assumed to be two-dimensional. Thus, any state will be of the form $\phi_1\otimes e_1 + \phi_2\otimes e_2$, if $\phi_1$ and $\phi_2$ for a basis for $S$, which means the initial state must evolve into a state of that form. So whatever calculations you've done can't be a problem. Commented Dec 30, 2023 at 18:23
• @march OK, thanks, I overlooked this, since it is not explicitly stated in the paper. If you make it an official answer it will be accepted.
– gcc
Commented Dec 30, 2023 at 20:14

In the paper, the states that OP labels $$\phi_1$$ and $$\phi_2$$ correspond to the states $$\lvert s_1\rangle$$ and $$\lvert s_2 \rangle$$, which correspond to passage through one of two slits in a double-slit experiment. For that reason, the Hilbert space $$S$$ is two-dimensional, and so any state in the tensor product space of $$S$$ and $$E$$ can be written in the form $$\phi_1\otimes e_1 + \phi_2\otimes e_2\,,$$ where $$s_j$$ are some arbitray states in $$E$$. In particular, the state after some time-evolution must be write-able in this way, so there can be no contradiction.
The issue with the OP's "counterexample" is that they are implicitly assuming that $$S$$ is of dimension greater than 2, because they are using the four states $$\phi_1(x)=1$$, $$\phi_2(x)=x$$, $$\psi_1(x)=1-\sqrt x$$, and $$\psi_2(x)=x+\sqrt x$$, which spans a space of dimension 3.