I’ve written an awful lot on this topic in the past couple of years. My latest publication condenses, consolidates, and improves on all of this and notes a distinct clash between the classical definition of entropy and the behavior of quantum systems. In particular, I think I have finally succeeded at showing that Bell-type inequalities are simply another way of stating the second law of thermodynamics. As such, I think one could derive a Bell-type inequality (of the Cerf-Adami type) for merging black holes based on all the work with black hole thermodynamics. Of course, I have no idea what that would tell you, but hey, I’m a theorist…

# Bayesian statistics and Bell’s inequalities

I was surfing the web and stumbled across a fascinating example of the application of Bayesian statistics that I thought had some pedagogical power to it. The original post, which is self-admittedly excruciating, is here. In any case, here’s the data for the example:

1.) 1% of women over the age of 40 who participate in routine screening have breast cancer

2.) 80% of women with breast cancer will get positive mammographies

3.) 9.6% of women *without* breat cancer also get positive mammographies

Question: A woman in this age group has a positive mammography. What is the likelihood she has cancer?

Apparently most doctors get the answer to this question wrong. Perhaps surprisingly (depending upon how your brain naturally processes these statistics) most doctors answer that the likelihood that the woman has breast cancer is somewhere between 70% and 80%. Before I show you the correct answer I will note that only 15% of doctors actually get this right as it is worded. Rather, if it is worded as follows, 46% of doctors get it correct:

1.) 100 out of 10,000 women over the age of 40 who participate in routine screening have breast cancer

2.) 80 of 100 women with breast cancer will get a positive mammography

3.) 950 of 9900 women *without* breast cancer will also get positive mammographies

Question: A woman in this age group has a positive mammography. What is the likelihood she has cancer?

The correct answer is 7.8%. How do you get that result? Well, the total number of women who get *positive* mammographies is 950 + 80 = 1030 (notice that it doesn’t really matter how many get a mammography at all, just how many who had one got a positive result). But only 80 of them actually turned out to have cancer. Therefore:

Pr(cancer if MMO +) = 80/1030 = 0.07767 or 7.8%

What’s the key to Bayesian statistics? The key is prior knowledge. Bayesian probabilities can easily be modified if the given information changes. In a sense it is because there is some correlation or link between certain quantities. The way in which a typical probability is interpreted is as a measure of how frequent an event is. So if you roll a pair of dice 10 times in a row and come up with a roll of four more times than any other roll you might be tempted to think that four is the most likely roll on any pair of dice which is patently false (theoretically a roll of seven is the most likely). This is the frequentist interpretation of probabilities. The Bayesian interpretation assigns probabilities to propositions that are uncertain since it is in some sense a measure of the degree of certainty. Certainly there are plenty of instances when the two give the same result but often cases where they do not. In the analysis above it is important not to care about frequencies, rather just exact data for the given situation. In the dicing example the number of rolls wouldn’t make a difference. Rather a Bayesian analysis might look at the full situation and make and argument from that (in that sense I argue that the idea of counting microstates and macrostates in order to determine probabilities is Bayesian since it has nothing to do with how frequently an event *occurs* but rather how many possible combinations are available, that is to say the amount of knowledge one has).

What does this have to do with Bell’s inequalities? Well, in Wigner’s derivation of Bell’s inequalities he *clearly* uses the frequentist approach to probabilities. Are Bell’s inequalities inherently frequentist then? Not necessarily since it is quite clear that one could consider even the Wigner form and assume that information about two of the systems both independently depend on (or can be informationally updated from) a third. Plenty of authors have considered this point of view but the details are beyond this current post.

**Note to my students:** Think you’ve found an elusive macroscopic violation of (A, not B) + (B, not C) ≥ (A, not C)? Post it here!

# Preparing for March Meeting of APS

Well, I’ve been lax in posting as I’ve been busy getting ready for the March Meeting of the APS. But I have an idea that’s been percolating in my head in relation to the March Meeting and it forms the basis of an idea that I have that is the at the core of a hypothesis I’m presenting there.

The short end of it is that I have a theory based around the idea that Bell’s inequalities are merely another statement of the Second Law of Thermodynamics, the latter actually *not* a fundamental law, but rather merely a strong argument about probabilities (see, for example, Dan Schroeder’s *Thermal Physics*). In short (and there will be a paper up on the arXiv about this soon), here’s argument:

1. The entropy of mixing represents the entropy created when mixing two systems. It is always zero or positive since it is essentially another way of counting the configurations of the system. It is zero when the two systems are the same. As such it is a measure of ‘separation’ of the probability distributions of the two systems.

2. The relative (Shannon) entropy and its other forms the conditional and mutual entopies (see Nielsen and Chuang’s *Quantum Computation and Quantum Information* are also measures of the separation of probability distributions of two systems.

3. The data pipelining inequality implies that, if X -> Y -> Z is a Markov chain, then based on fundamental properties of Shannon entropies (again see Nielsen and Chuang) we can write H(*X*:*Y*) ≥ H(*X*:*Z*) as well as H(*Y*:*Z*) ≥ H(*X*:*Z*)

4. It follows trivially from 3. that

H(*X*:*Y*) + H(*Y*:*Z*) ≥ H(*X*:*Z*)

It is possible to form inequalities of the type derived by Cerf and Adami from this inequality. In addition, if we speak entirely in terms of relative information, by dividing through by the *total* relative information available, represented by a *sum* of each of these, we can write an inequality similar to Wigner’s form of Bell’s inequalities (see Wigner’s paper or Sakurai’s *Modern Quantum Mechanics*):

Pr(*X*:*Y*) + Pr(*Y*:*Z*) ≥ Pr(*X*:*Z*)

Is it possible Bell’s inequalities, which are entirely classical obviously (which is why they are violated), are just another way of writing the second law? The Shannon entropies are purely classical and the inequality above represents, essentially, the positivity condition for entropy inherent in the second law.

The thing is that, for quantum systems, we would want to speak in terms of the *von Neumann* entropy which is not wholly classical. In that sense it is not clear that we could even write such an inequality in these terms. But this begs the question, then: shouldn’t the definition of entropy be consistent across regimes?

That’s the question I want feedback on.