# FeaturedA H-theorem in quantum information theory

1. Nov 1, 2016

### Staff: Mentor

An interesting paper has appeared on nature.com:

http://www.nature.com/articles/srep32815

The abstract:

I expect this to spawn plenty of pop science claims about "scientists say we can reverse entropy". But the paper itself looks like a good discussion of how the second law actually works for quantum systems.

2. Nov 1, 2016

### Staff: Mentor

A preprint of the paper is on arxiv as well:

https://arxiv.org/abs/1407.4437

It has some appendices that don't appear in the published version at nature.com.

3. Nov 4, 2016

4. Nov 4, 2016

### atyy

"In contrast to the classical formulation of the second law where any isolated classical system evolves with the non-diminishing entropy, its literal extension onto the quantum case is meaningless since the entropy of any isolated quantum system does not change ... Hence, to bring the thermodynamic meaning to the consideration of quantum evolution one has to allow an interaction with the environment and establish the notion of the quasi-isolated system."

Is that right? Aren't they talking about the fine grained entropy which is also unchanging in classical mechanics for an isolated system?

5. Nov 4, 2016

### Demystifier

Perhaps they are talking about entanglement entropy in the quantum case, but even that has a classical analogue.

6. Nov 4, 2016

### Jimster41

Just trying to get any kind of handle on what this paper is saying - ended up trying to grok Boltzman's equation (never a waste of time ).
(mod I get this may need to be moved - since it is a question about support for the concepts in this paper, not about the paper per se')

Dividing (3) by dt and substituting into (2) gives:

${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {\mathbf {p} }{m}}\cdot \nabla f+\mathbf {F} \cdot {\frac {\partial f}{\partial \mathbf {p} }}=\left({\frac {\partial f}{\partial t}}\right)_{\mathrm {coll} }}$

After moving all but first term to the right side is the following a correct description?

The change in state (of a system of particles) per unit time is equal to...
the change is state per unit time due to collisions
minus
the product of particle momenta over mass times the gravitational field gradient (this term really confuses me. Is that gradient the gravitational gradient?)
minus
the effect of outside force on change of state per change in momenta (could use more clarity on this term also - is this saying the outside force F changes the state in different ways depending on the current momentum of the particles. that would make sense to me but I'm not sure I am understanding it right)

Last edited: Nov 4, 2016
7. Nov 4, 2016

### Jimster41

Does an unobserved (non-interacting, isolated) quantum system increase in entropy? By what mechanism?

My cartoon (perhaps very seriously incorrect) was that QM was a way of calculating by perfect conservation of probabilities and real uncertainty, the future of an system that doesn't yet "exist".

8. Nov 5, 2016

### vanhees71

Could somebody enlighten me, what's new in this paper? Of course, entropy in classical and quantum theory in the non-coarse-grained description is conserved (Liouville, von Neumann). Only through "throwing away information" you get an entropy gain. In the derivation of the Boltzmann equation from classical Hamiltonian mechanics in the usual dilute-gas limit when only considering $2 \rightarrow 2$ collisions, what's "thrown away" are the correlations in the two-body phase-space distribution when it is substituted by the product of the single-particle phase-space distributions ("Stoßzahl ansatz", "molecular-chaos hypothesis"), thus truncating the BBGKY hierarchy at the lowest order. In the derivation of transport equations from QT, you throw away information by employing the gradient expansion and introducing the Markov approximation, i.e., again you throw away correlations, and in this way you get a positive semidefinite one-particle phase-space distribution function out of the single-particle Wigner transform of the one-body Green's function.

It seems to me that in the present paper there is done not much more than employing such techniques too. It's just dubbed "quantum information", but what's different from good old Kadanoff-Baym or, in the argument with reservoirs ("environment") Feynman-Vernon?

9. Nov 7, 2016

### Demystifier

Yes it does, if you define entropy by some appropriate coarse graining.

10. Nov 7, 2016

### Demystifier

Even if there is nothing new at the conceptual level, I think they have some new general quantitative theorems.

11. Nov 7, 2016

### Jimster41

The BBGKY reference was helpful.

If you were a super-computer with an obsessive personality, no care for final prediction, a limitless imagination and - you just wanted to imagine the evolution of an s-body Quantum system out with utter fidelity far as you could - you would not be coarse-graining right? Rather you would be - what, continuously adding the next (s+1) particle?

Until someone came along and demanded your prediction you would not truncate the BBGKY chain, correct?

12. Nov 7, 2016

### Demystifier

Correct. But why s-body, rather than n-body? This is like denoting the wave function with $\theta$ rather than $\psi$.

13. Nov 7, 2016

### Jimster41

I had the same question - but s is what wiki used (but then they also used N). Confusing.
https://en.wikipedia.org/wiki/BBGKY_hierarchy

14. Nov 7, 2016

### Jimster41

If you were that computer and you had started with probability distribution information about q and p for N particles, you pictured that chain in your vast 1e42 horsepower imagination - where would you get information for the n+1 particle?

I guess it would have had to do truncation (and throw away information) at the outset or else it would explode from infinite recursion - adding p's and q's for ever more particles... just to get BBGKY set up?

Last edited: Nov 7, 2016
15. Nov 7, 2016

### Demystifier

Well, if you are a computer, there is a lot of "trivial" stuff you cannot do. For instance, you cannot calculate numerically (exactly) the circumference of a unit circle in a continuum space (because you don't know the infinite number of decimals for $\pi$), and you cannot prove the Godel theorem. But it does not mean that space is not continuous or that you cannot prove the Godel theorem. It only means that nature (including you) may not be a digital computer.

16. Nov 8, 2016

### Jimster41

So is the the part people are getting kooky over on page three where they describe a system evolving in a non-unital way due to the specific interaction occurring in "non-energy" degrees of freedom (spin) which are also non-commuting?

So spin degrees of freedom are considered non-energy degrees of freedom? I didn't know this.

Is the idea that in such an interaction the reservoir acts as kind of an "information reservoir" that can do "energy free" quantum mechanical work constraining spins (is that right?) because information is not energy?

But then how did that information reservoir get created if not through energy. Is there a rule of QM interactions that says you have to add an energy dof to non-energy dof interactions at some point. I mean how many of these interactions could be chained together before you have to pay the energy bill due to QM group rules? Or what makes it impossible to create such an information reservoir without any energy?

Also, is the down side to just calling information energy - that then you have to say spin-statistics itself does free work.

17. Nov 9, 2016

### vanhees71

I don't know what you mean with "non-energy degrees of freedom". Of course, the spin can enter the Hamiltonian, e.g., in the Pauli equation, where it describes the interaction of the magnetic moment associated with the spin of the spin-1/2 particle (in atomic physics usually electrons) with the magnetic field.

Also what do you mean with "information energy"? Are you referring to the Maxwell demon or something related? Indeed, recently a Maxwell demon has been realized on the quantum level, and everything works as expected:

Phys. Rev. Lett. 115, 260602 (2015)
https://arxiv.org/abs/1507.00530

18. Nov 9, 2016

### Jimster41

"Let us consider a fixed energy subspace E of the system Hilbert space spanned by the orthonormal basis states |ψi,Ei, Hˆ S|ψi,Ei = E|ψi,Ei, where index i denotes all the remaining non-energy system’s degrees of freedom and Hˆ S is the system Hamiltonian" p2

Re non-energy degrees of freedom I was trying to understand the above sentence and others in the paper ref by the OP which seem to be distinguishing QM interactions between the particle and reservoir which involve exchange of energy and those that don't.

And I am struggling with the idea that you can have information that does not imply energy. To me a bit being either zero or one, or a change from one to the other seems to require some action (or constraint of action) and therefore, naively at least, suggests work.

If I'm understanding the paper and the hubbub at all it seems to be about energy isolated QM systems evolving in with decreasing entropy.
Really it seems more just about looking at how different cases of such evolution relate to the change in entropy. They seem to focus as much on the cases that imply increasing entropy as they do the one that suggests it is possible to have a case where entropy is decreasing.

I haven't looked at it yet but the paper you linked to does seem pretty relevant - are you saying this paper (in the OP) is actually somewhat old news?

Last edited: Nov 9, 2016
19. Nov 9, 2016

### Jimster41

That is a pretty interesting paper.