What is the temperature of a pure quantum state? I was wondering about temperatures and pure quantum states. I'm currently working on thermalization of isolated quantum systems, which can be described by pure quantum states (kets). How do we define a temperature for these? 
Normally you work within the density matrix formalism so you can define an ensemble, which is a mixed state, for a certain temperature. In my scenario we prepare a system in a lab in a pure state, that is isolated, and let it thermalize to its equilibrium state. This must be (in very good approximation) still a pure state if the system is strongly isolated, because of the unitarity of evolution operator.
We can then probably assign a certain temperature to this equilibrium system, as we could bring this system in contact with a heath bath with a certain temperature and monitor if energy flows out of the system or into the system (aka it has a higher or lower temperature). But from a theoretical viewpoint I have no idea how to assign a temperature to a pure quantum state, so I'm quite dazzled by this. 
I know the ground state is supposed to have $T = 0 K$, but that's all I could think of. The Internet is not giving me much more information.
Edit: Ofcourse I am talking about a certain many-body system with a corresponding Hamiltonian, which should be non-integrable and chaotic. This way the system will thermalize (in most cases) following the Eigenstate Thermalization Hypothesis. You could take the 1D spin chain non-integrable Ising model as example.
 A: The same question could be asked in classical physics: What is the temperature of a specific microstate of an ideal gas? Conceptually, the answer to the quantum version is essentially the same as the answer to the classical version.
If the question is how to actually calculate the temperature of a given pure state (at least in principle), then one way is to take a partial trace over half of the system. If the system is large enough so that the energy of interaction between the two halves is negligible, then the result should have the form $\exp(-\beta H)$ where $H$ is the self-Hamiltonian of the remaining half. This assumes that the whole-system pure state has attained "equilibrium", in the sense that it has evolved long enough to be in a "typical" state for the given macro-conditions.
This paper may be of interest:


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*"Canonical Typicality of Energy Eigenstates of an Isolated Quantum System," https://arxiv.org/abs/1511.06680
From the abstract:

Currently there are two main approaches to describe how quantum statistical physics emerges from an isolated quantum many-body system in a pure state...

And here's an older review paper:


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*"Pure State Quantum Statistical Mechanics," http://arxiv.org/abs/1003.5058
