Does pure state dephase or decohere? I know that from the fundamental postulates of quantum mechanics, the norm of state must be preserved under time evolution.
But today, I heard some mysterious comment that pure state, the state which is NOT ensemble-averaged, DOES dephase or decohere. I also heard that the Bloch vector length of the pure state does decrease in the Bloch sphere.
This statement is completely contradictory to what I have learned in quantum mechanics. I do not understand this comment and personally think that it is a wrong statement. But since it is also possible that my thinking and what I have learned from the textbook are wrong, I hope someone gives any rebuttal/proof of the statement.
 A: So you might have heard about entanglement and entanglement entropy.
(I will assume you have some knowledge about pure and mixed states, as well as the tensor product.)
First, consider a pure quantum state, for example a many body system (maybe a spin chain) in a closed system. If the initial state is somehow prepared to be pure and you have exact knowledge, meaning the (von Neumann) entropy is 0, you call that a pure state, which will evolve under unitary time evolution and stay pure all the time. It doesn't need to be an Eigenstate to the underlying Hamiltonian.
Interesting things might happen to the subsystems of the closed system. If they are free to interact, so they are not closed with respect to each other (for example the spins in a spin chain, that might be governed by nearest neighbor interaction) their states will almost certainly start to entangle under the influence of the global Hamiltonian.
We would describe the global state by a state in the tensor product of the Hilbert space of the subsystems. Even though this state will remain pure all the time, when the subsystems entangle, their states become mixed. This seems unintuitive, when the initial state is a (pure) product states of (pure) states in each subsystem.
To see this, I encourage you to check that for a spin chain, that is
$\mathcal{H}=\underset{i=1,\dots N}{\otimes}\mathcal{H}_\text{spin}$, where $\mathcal{H}_\text{spin}$ is the usual 2d spin hilbert space one should be familiar with.
We can for example consider 2 spins with the Hamiltonian (ignoring constants)
$H=\lvert\uparrow\downarrow\rangle\langle\downarrow\uparrow\rvert+\lvert\downarrow\uparrow\rangle\langle\uparrow\downarrow\rvert$ and the initial state $\lvert\psi_0\rangle=\lvert\uparrow\downarrow\rangle$. The initial state is quite obviously a pure product state, but when you compute the time evolution (which is easily done by plugging this into the Schrödinger equation), you will see that the state $\lvert\psi(t)\rangle$ cannot be represented as a product state anymore, which is then called entanglement.
To make the formalism complete one needs to introduce density and reduced density matrices. If you don't know them already, a complete introduction to that is required anyway, but to say the least, a reduced density matrix describes a subsystem of a global system by "ignoring" all degrees of freedom of the complement of that subsystem, which is done by computing the partial trace of all other subsystems except the one that we want to describe. Here also comes in the Bloch sphere that you mentioned: The most general (reduced) density matrix of a single spin can be represented by a 3d vector on or inside a sphere, its length will decrease when the (reduced) density matrix becomes mixed, which happens as it entangles with its environment.
The key insight is the following: Even if the global density matrix always remains pure for this situation, the reduced density matrices (that might have also started purely, given a pure product initial state) will become mixed by entanglement. This mixedness can experimentally be measured by what is called entanglement entropy, it is basically the von Neumann entropy of reduced density matrices, but there are other functions like the Renyi entropies that are also a good measure for entanglement. The concept one should have in mind is that information, which for a pure initial prodcut state is stored locally in each subsystem (every reduced density matrix is initially pure, all the local observables/expectation values can be determined locally) will distribute over the whole system via entanglement, so after a while all the reduced density matrices are mixed and local observables must be described by ensembles, just like in usual thermodynamics (even though this is not to be mixed with all the 2nd law of thermodynamics stuff). The information about the initial state is not lost, but can be reconstructed by knowing the whole system at once. It is somehow stored "in the entanglement" between subsystems.
Of course, in nature nothing is perfectly isolated from its environment, therefore all the pure states will somehow interact and entangle with their environment and after a while the entanglement makes their initial information unaccessible.
