Experimental setup

In this section, a quantum mechanical implementation of the counterfactual reasoning presented Example 10 in Section 5.4 is proposed. The example is briefly described again here (cf. [35]): Jack and Jim are old friends. Under normal circumstances they will help each other. But Jim is very proud, so he will never ask for help from someone with whom he has recently quarreled. Jack, on the other hand, is very unforgiving. So he will never help someone with whom he has just quarreled. Now Jack and Jim have a quarrel. Our question is:

If Jim asked Jack for help, then Jack would help him.

This scenario is implemented as follows. First we begin with the facts and construct a quantum mechanical reasoning scheme based upon them. Let $p$ be the proposition ``Jim is very proud,'' $q$ be ``Jack is very unforgiving,'' and $r$ be ``Jim and Jack have a quarrel.'' $p$, $q$, $r$ are eigenstates of an operator $S$ that asserts the states of affairs discussed here. Let the inference operator be a unitary operator $U$ which transforms an initial state of affairs to an end state. Under unambiguous circumstances $U$ should be able to deliver a univocal answer $s$ (whether Jack helps Jim -- $\mathbb{T}$ or $\mathbb{F}$) that is another eigenstate of $S$. In the experiment, each proposition is associated with one assertion eigenstate and one refutation eigenstate, which are respectively denoted by a plus sign ($+$) or a minus sign ($-$) attached to the proposition symbol. Furthermore, we suppose that $\{p\pm ,q\pm ,r\pm ,s\pm\}$ is a complete eigenbasis of the states of affairs presented here. Consequently, any input state of affairs can be represented as:

$$c_{p+}\left\vert p+ \right\rangle +c_{p-}\left\vert p- \right... ...left\vert r+ \right\rangle + c_{r-}\left\vert r- \right\rangle$$

with

$$\sum\limits_{\psi \in \left\{ {p\pm ,q\pm ,r\pm ,s\pm } \right\}} {\left\vert {c_\psi } \right\vert}^2=1,$$

where $c_\psi \in \mathbb{C}$ is the projection of a state of affairs on $\left\vert \psi \right\rangle$. For brevity, a state of affairs is represented by an eight-dimensional complex valued vector

$$(p_-,p_+,q_-,q_+,r_-,r_+,s_-,s_+)^t,$$

where $\phi_-$ ($\phi_+$) is the component of the eigenstate representing proposition $\phi$ being false (true), $\phi \in \{p,q,r,s\}$. For example, a state of affairs in which Jim is not proud and Jack is not unforgiving and they do not have a quarrel is represented by,

$$({1\over {\sqrt 3}},0,{1\over {\sqrt 3}},0,{1\over {\sqrt 3}},0,0,0)^t$$

The training set is constructed according to a variety of possible but coherent situations^6.6. That is, situations in which the inference rule can be unquestionably applied and may lead to a coherent answer. In classical physics, one can have access to an omniscient vantage point as an external observer. This is the position we take in constructing the training set^6.7. We assume that these situations are ``real'' in an ensemble of ``possible worlds,'' and treat them as ``factual'' situations. They are thus constructed only for the sake of training the quantum system to ``implement'' a naive classicization of counterfactual reasoning. These situations are summarized in the following table:

p	q	r	s	p	q	r	s
$\mathbb{F}$	$\mathbb{F}$	$\mathbb{F}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{F}$	$\mathbb{T}$	$\mathbb{T}$
$\mathbb{T}$	$\mathbb{F}$	$\mathbb{F}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$
$\mathbb{F}$	$\mathbb{T}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$	$\mathbb{T}$	$\mathbb{F}$	$\mathbb{T}$

In a sense, a set of these situations can be seen as the coherent experience of an individual in a real world. For example, the situation is unambiguous if Jim and Jack in fact do not have a quarrel: Jim will ask for help and Jack will help him (the first row in the table).

The questionable situations are the following two:

1. Jim is very proud and Jack is not unforgiving and they have a quarrel.
2. Jim is very proud and Jack is unforgiving and they have a quarrel.

The first questionable assertion is in fact not very problematic, for it leads to the same conclusion anyway: If Jim were not proud or they do not quarrel, Jim would ask Jack for help. Under both circumstances, Jack would help him, since Jack is not unforgiving. Thus $s$ should be true. The reason to omit it as questionable is because the antecedent in the original statement is not true, and then one needs counterfactual reasoning (even in an imaginary ``classicized'' situation). The really problematic situation is the second assertion. It is a counterfactual conditional that can be both true and false even in an ensemble of imaginary ``possible worlds.'' In fact, it is our original problem, which is difficult to account for in classical logic.

The training scheme is similar to that in the previous sections. The Hamiltonian has a total of 64 free parameters to be decided. An error function as defined in Equation 6.3, and a standard conjugate gradient method are used to obtain the parameters.