# Pure and mixed thermal states

• camipol89
In summary, the conversation discusses the concept of thermal states and whether they are always mixed states or if there are also pure thermal states. It is mentioned that in the limit of zero temperature, a pure state can be reached as long as the ground state is not degenerate. However, in reality, it is not possible to reach absolute zero temperature. It is also mentioned that pure states can still display thermal properties and can be approximated by looking at small subsystems. The conversation also touches on the definition of a pure state and the identity operator not being a proper statistical operator due to the infinite dimension of the state space.

#### camipol89

Hey guys,
I was reading about thermal states and now I have a doubt: is a thermal state always a mixed state with density matrix ρ=exp(-βH)/Tr(exp(-βH)), or is there also a pure thermal state?
Thank you

Think about what happens for $T=1/\beta \rightarrow 0$!

ρ→1 and we get a pure state, is that correct?
So there are pure thermal state, they are just thermal state at an (ideal) zero temperature?
If I am right, do such pure thermal states exist in nature?
Thanks a lot for your help!

As long as the ground state is not degenerate, the zero temperature limit does give a pure state. However, one can never really reach zero temperature by cooling in the physical world. On the other hand, there are other ways to prepare a nearly isolated pure state.

In fact, there is a sense in which pure states may display thermal properties. Imagine a closed system begun in some pure initial state which evolves unitarily. If the initial state is, on average, highly excited (e.g. has a finite energy density above the ground state), then one would expect on general grounds that the system should "thermalize" in some sense. Yet the state of the whole system must remain pure. However, as long as we look at small pieces of the whole system, we may imagine that the rest of the system acts as an effective bath, and the state of the small subsystem may look thermal.

A pure state is by definition described by a statistical operator that is a projection operator
$$\hat{\rho}=|\psi \rangle \langle \psi|$$
with a normalized state vector $|\psi \rangle$, i.e., with
$$\langle \psi|\psi \rangle.$$
Indeed if the ground state is not degenerate, then
$$\lim_{T \rightarrow 0} \hat{\rho}_{\text{can}} = |\Omega \rangle \langle \Omega|,$$
where $|\Omega \rangle$ is the energy-eigenvector for the lowest energy-eigenvalue, and this is uniquely defined (up to a phase, which cancels in the statistical operator).

To stress it again: The identity operator generally cannot be a proper statistical operator, because for usual physical systems the state space is a true infinitely-dimensional Hilbert space, and a statistical operator must have trace 1. The identity operator in a proper Hilbert space has no finite trace and thus cannot represent a mixed state.

## 1. What is the difference between pure and mixed thermal states?

Pure thermal states refer to a state of a system that is in thermal equilibrium with a heat bath at a specific temperature, while mixed thermal states are a combination of different pure thermal states with different temperatures.

## 2. How are pure and mixed thermal states represented mathematically?

Pure thermal states are represented by density matrices that are diagonal with respect to the energy eigenbasis, while mixed thermal states are represented by density matrices that are a linear combination of density matrices for pure thermal states at different temperatures.

## 3. What is the physical significance of pure and mixed thermal states?

Pure thermal states represent the state of a system that is in perfect thermal equilibrium with its environment, while mixed thermal states represent a more realistic scenario where a system is in contact with multiple heat baths at different temperatures.

## 4. How do pure and mixed thermal states relate to entropy?

Pure thermal states have zero entropy, as they represent a perfectly ordered system. Mixed thermal states, on the other hand, have a non-zero entropy value, which quantifies the degree of disorder in the system due to its contact with multiple heat baths.

## 5. Can pure and mixed thermal states be experimentally observed?

Yes, both pure and mixed thermal states can be experimentally observed through various techniques such as measuring the energy distribution of particles in a system or observing the fluctuations in temperature of a system over time.