Quantum Bayesian Networks

September 19, 2008

Bell’s Inequalities for Bayesian Statisticians

Filed under: Uncategorized — rrtucci @ 7:29 pm

My Ph.D. is in physics, so the famous Bell’s Inequalities of quantum mechanics have been hammered into my mind by the educational system since an early age. But maybe I can pique the interest of Bayesian statisticians with little or no exposure to “the other probability theory”, “the other red meat”, i.e., quantum mechanics, by introducing Bell’s inequalities in a language familiar to them, that of Bayesian networks. Yes, indeedy, you heard right. Although seldom done, Bell’s inequalities can be explained simply and intuitively using the language of Bayesian networks, as follows.

Henceforth, I will underline letters that stand for random variables.

A simple Bayesian “model” might have a “parameter” \theta such that \theta \sim P(\theta) and i.i.d. “data” x_1, x_2, \ldots x_n such that x_j\sim P(x_j|\theta). This can be represented by the Bayesian net shown in Fig.1. The data is observed (evidence) and the parameter is inferred from this.

Fig.1

Fig.1

Now consider two point particles that start off at the same point, and then fly apart without interacting with anything else. Let’s assume that both particles are spin-1/2 fermions (like electrons, protons and neutrons), and that they start off in a state of zero angular momentum. In a “Local Realistic” theory, this situation can be represented by the classical Bayesian net shown in Fig.2.

Fig.2

Fig.2

In Fig.2, node \underline{\lambda} represents the “hidden variables”. For j \in\{ 1, 2\}, node \underline{x^{\alpha_j}_j} represents the outcome of a spin measurement \alpha_j performed on particle j. \alpha_j represents the measurement axis. Node \underline{x^{\alpha_j}_j} may assume two possible states, + or −, depending on whether the measurement finds the spin to be pointing up or down along the \alpha_j axis . For example, \underline{x^{A}_1}=+ if a measurement of the spin of particle 1 along the A axis yields “up”.

It is convenient to define the following probability functions

Eq.(1)\qquad P^{\alpha_j}_j(x_j|\lambda) = P(\underline{x^{\alpha_j}_j}= x_j | \underline{\lambda} = \lambda),

Eq.(2)\qquad P^{\alpha_j}_j(x_j) = P(\underline{x^{\alpha_j}_j}= x_j ),

Eq.(3)\qquad P^{\alpha_1\alpha_2}_{12}(x_1,x_2|\lambda) = P(\underline{x^{\alpha_1}_1}= x_1, \underline{x^{\alpha_2}_2}= x_2 | \underline{\lambda} = \lambda),

Eq.(4)\qquad P^{\alpha_1\alpha_2}_{12}(x_1,x_2) = P(\underline{x^{\alpha_1}_1}= x_1, \underline{x^{\alpha_2}_2}= x_2 ),

where j\in \{1, 2 \}.

Fig.2 implies the following equation:

Eq.(5)\qquad P^{\alpha_1\alpha_2}_{12}(x_1,x_2) = \sum_\lambda P^{\alpha_1}_1(x_1|\lambda)P^{\alpha_2}_2(x_2|\lambda)P(\lambda).

The assumption that the particles start off in a state of zero angular momentum
means that

Eq.(6) \qquad P^\alpha_1 (x|\lambda) = P^\alpha_2 (\overline{x}|\lambda),

where x \in \{+,  - \}, and \overline{x} is the opposite of x, so \overline{+} =- and \overline{-} =+ .

It can be shown (see Ref.2 for a proof) that Eqs.(5) and (6) imply that

Eq.(7)\qquad P^{AC}_{12}(x,z) \leq P^{AB}_{12}(x,y) + P^{BC}_{12}(y,z),

and the 5 other inequalities one gets by permuting the symbols A,B and C.

Assume axes A,B and C are coplanar and that angle(A,B) = angle(B,C) = \theta. Also let x = +, y = − and z = + in Eq.(7). Quantum mechanics gives an expression for P^{\alpha_1\alpha_2}_{12}(x_1,x_2) as a function of \theta. Combining Eq.(7) and the expression given by quantum mechanics for P^{\alpha_1\alpha_2}_{12}(x_1,x_2) yields:

Eq.(8)\qquad 0\leq 1 + \cos(2\theta) + 2 \cos(\theta),

which is false for \theta = 135^o, for example.

Thus, quantum mechanics and Local Realism are incompatible. Quantum mechanics tells us that if you measure the spin of particle 1 along the A axis and the spin of 2 along C, where angle(A, C) = 270 degs., and if you do this many times, you will get a probability P^{AC}_{12}(+,+) that is greater than that predicted by Local Realism. Somehow the particles know more about each other than one would have expected from Local Realism alone.

This blog post is an abridged version of Ref.2. Look in there for more details.

References:

  1. here. Wikipedia article on Bell’s inequalities. Explains them in the conventional way, in terms of expected values, without alluding to Bayesian networks
  2. here. An excerpt (pages 43-49) from QFogLibOfEssays.pdf, which is part of the Quantum Fog documentation

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