# Quantum Bayesian Networks

## June 12, 2012

### The Importance of Being Consistent

Filed under: Uncategorized — rrtucci @ 4:40 pm

Full Title: The Importance of Being Earnest, A Trivial Comedy For Serious People

Ralph Waldo Emerson said “A foolish consistency is the hobgoblin of little minds” in his essay “Self-Reliance”. I think I know what he meant. Being stuck in the same classical rut all the time is no fun.

Recently I came across the following paper

• Decoherent histories of quantum searching, by Wim van Dam, Hieu D. Nguyen, arXiv:1206.1946

I didn’t look at the paper too closely, but it made me learn a bit about a subject I had never studied before: the decoherent histories (or consistent histories) approach to (or interpretation of) quantum mechanics. This approach is due to Griffiths, Gell-Mann, Hartle, Omnes, and others. Here is a review:

• J.J.Halliwell, A Review of the Decoherent Histories Approach to Quantum Mechanics, arXiv:gr-qc/9407040
• The Net Advance of Physics: DECOHERENT HISTORIES (Interpretation of Quantum Mechanics) here

It occurred to me that quantum Bayesian nets (QB nets) can be used to study consistent histories. Here is how. The following are not very deep observations. What did you expect?! this is a trivial blog for serious people.

Let $Z_{1,N} = \{1,2,\cdots, N\}$. Consider a QB net with $N$ nodes $\underline{x}_1,\underline{x}_2, \cdots, \underline{x}_N$, where for each $j\in Z_{1,N}$, the random variable $\underline{x}_j$ takes on values $x_j\in S_{\underline{x}_j}$.

Let $J\subset Z_{1,N}$ and let $J^c= Z_{1,N}-J$ be its complement. Let $x_J = (x)_{j\in J}$.

Suppose $\rho_{meta}$ is the meta density matrix of the QB net. (See my mixology paper for a definition of meta density matrices.)

Suppose $x_J$ and $y_J$ are both elements of $S_{\underline{x}_J}$. Note that

$[\langle x_J| + \langle y_J|\;]\rho_{meta}[{\rm h.c.}] =$

$= \langle x_J|\rho_{meta}| x_J \rangle + \langle y_J|\rho_{meta}| y_J\rangle + 2 {\rm Re} \langle x_J|\rho_{meta}| y_J\rangle$

Define $x_J$ and $y_J$ to be consistent histories if

${\rm tr}_{\underline{x}_{J^c}}\;\;{\rm Re}\langle x_J|\rho_{meta}| y_J\rangle= 0$

We could define it differently. For example, we could ask instead that

$Av\;\;{\rm Re}\langle x_J|\rho_{meta} | y_J\rangle = 0$

where $Av(\cdot) = \langle f_{J^c}|\cdot|f_{J^c}\rangle$ for a particular, fiducial state $f_{J^c}\in S_{\underline{x}_{J^c}}$.