# Quantum Bayesian Networks

## January 12, 2013

### Conditional Ageing Inequality or Me Raising CAIN

Filed under: Uncategorized — rrtucci @ 7:52 pm

As mentioned in my previous post, the one entitled “The devil and Reverend Bayes“, this week I published two papers on arXiv. Below, you’ll find an excerpt taken from the introduction section of one of those papers. This excerpt sums up nicely what I’ve been doing in the last 3 months. Thanks to Henning Dekant for suggesting that I look at the work of Sagawa and Ueda.

This paper originated as an attempt to understand a series of papers (Refs.[5] to [11]) by Sagawa, Ueda and coworkers (S-U) in which they claim that the standard Second Law of thermodynamics does not apply to non-equilibrium processes with feedback (i.e., Maxwell demon type processes). They give a generalization of the Second Law that they claim does apply to such processes. Although I agree in spirit with much of what S-U are trying to do, and I profited immensely from reading their papers, I disagree with some of the details of their theory. I discuss my disagreements with the S-U theory in a separate paper, Ref.[12]. The goal of this paper is to report on my own theory for generalizing the Second Law so that it applies to processes with feedback. My theory agrees in spirit with the S-U theory, but differs from it in some important details.

Let me explain the rationale behind my theory.

Suppose we want to consider a system in thermal contact but not necessarily in equilibrium with a bath at temperature $T$. Let $\underline{X}$ denote all non-thermal variables (fast changing, not in thermal equilibrium) and let $\underline{\Theta}$ denote all thermal variables (slow changing, in thermal equilibrium) describing both the system and bath. Let $\tau$ denote time. For any operator $\Omega_\tau$, define $\Omega_\tau|_{\tau=\tau_1}^{\tau_2}= \Omega_{\tau_2}-\Omega_{\tau_1}$. My slight generalization of the Second Law is

$S_\tau(\underline{\Theta}_\tau|\underline{X}_\tau)|_{\tau=0}^{\tau}\geq 0,$       (1)

where $S(\underline{a}|\underline{b})$ is the conditional entropy (i.e., conditional spread) of $\underline{a}$ given $\underline{b}$. I call Eq.(1) the conditional ageing inequality (CAIN). The standard Second Law corresponds to the special case when there are no $\underline{X}_\tau$ variables, in which case Eq.(1) reduces to

$S_\tau(\underline{\Theta}_\tau)|_{\tau=0}^{\tau}\geq 0 .$          (2)

The standard Second Law could be described as unconditional ageing, or simply as ageing.

Now, what is the justification for the CAIN? The justification for the Second Law Eq.(2) is that the superoperator that evolves the overall probability distribution in the classical case (or the overall density matrix in the quantum case), from time 0 to $\tau$, increases entropy because it can be shown to be doubly stochastic in the classical case (or unital\footnote{A superoperator is unital if it maps the identity matrix to itself.} in the quantum case). The justification for the CAIN is the same, except that the evolution superoperator is doubly stochastic (or unital) only if the non-thermal variables are held fixed during the evolution. The CAIN is not true for all evolution superoperators. Our hope is that it applies to systems of interest that commonly occur in nature.

The goal of this paper is to study the consequences of the CAIN. In particular, we apply the CAIN to four cases of the Szilard engine: for a classical or a quantum system with either one or two correlated particles.

Besides proposing this new inequality that we call the CAIN, another novel feature of this paper is that we use quantum Bayesian networks for our analysis of Maxwell demon type processes.