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}}.

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: