#### NateTG

Science Advisor

Homework Helper

- 2,449

- 5

This post is motivated by some of the discussion on the "Is QM inherently non-local" thread. The goal is to demonstrate that there is a local hidden interpretation possible for any signal local theory. This interpretation will be Bell local, but not Bell realistic.

So, let's say we have a set up [itex]S[/itex] on which there is a possible list of

individual measurements, indexed by some completely ordered set I. [tex]M=\{m_i \forall i \in I\}[/tex] and, that for each measurement [itex]m_i[/itex], there is a range of possible results [itex]R_i[/itex].

And, there is a set of possible experiments [tex]E \subset P(M)[/tex] which consist of subsets of the list of possible measurments with the additional property that for any [itex]e \in E[/itex], [itex]P(e) \subset E[/itex].

Then, for each experiment [itex]e \in E[/itex], there is a result space [itex]R_e[/itex] which is the cartesian product of the ranges of the individual measurements in [itex]e=\{m_{i_1},m_{i_2},m_{i_3}...\}[/itex]

And, a theory [itex]T[/itex] is a set of probability measures for the result spaces of each of the experiments.

Now, since the experiments are all elements of [itex]P(M)[/itex] there is a natural partial order [itex]\prec[/itex] based on set inclusion, so if we have two experiments [itex]e_1[/itex] and [itex]e_2[/itex], and the set of measurements in [itex]e_1[/itex] is a subset of the measurements in [itex]e_2[/itex] then [itex]e_1 \prec e_2[/itex] and, I will refer to this as [itex]e_2[/itex] is a stricter experiment than [itex]e_1[/itex].

Then, a theory is 'signal local' if for any [itex]e_1 \prec e_2[/itex], the probability probability measure that the theory gives for [itex]R_{e_1}[/itex] is the same as the natural probability subspace generated by restricting [itex]R_{e_2}[/itex] i.e. that for any sub set [itex]s \subset R_{e_1}[/itex] the subset [itex]s \times \prod_{m \in e_2, m \notin e_1} R_m[/itex] has the same measure in [itex]R_{e_2}[/itex]

And, a theory has a local hidden interpretation if there is some probability measur

e on [itex]R_M[/itex] (the cartesian product of the ranges of all individiually possible measurements) that gives the same probability measure as the theory if restricted to [itex]R_e[/itex] for any experiment in [itex]E[/itex].

Now let's say we have a signal local theory [itex]L[/itex], then for every experiment [itex]e \in E[/itex] there is a probability measure on [itex]R_e[/itex]. Now, for every measurable subset [itex]s \subset R_e[/itex] we assign the same value to the subset [itex]s \times \prod_{m \notin e} R_m[/itex] in our prospective measure on [itex]R_M[/tex]. In addition, specify that the measure of the empty set is zero, and the measure of the entire set is 1.

But, because of the properties that were ascribed to signal locality above it's easy to show that this prospective measure is indeed well-defined, and a probability measure.

Now, because this is physics, it's important to discuss whether the mathematical and physical properties match up:

The notion that was described as signal locality is, technically, stronger than signal locality since it's concievable that multiple measurements could occur local to each other, but that concern can be addressed by narowing down the notion of measurement.

The notion that was described as a local hidden interpretation is quite easy - let [itex]\lambda[/itex] be an element of [itex]R_M[/itex] then, since [itex]\lambda[/itex] describes the results of all possible measurements, clearly [\itex]\lambda[itex] can be used as the local hidden state for any particle if it is assigned with the appropriate probability distribution.

So, let's say we have a set up [itex]S[/itex] on which there is a possible list of

individual measurements, indexed by some completely ordered set I. [tex]M=\{m_i \forall i \in I\}[/tex] and, that for each measurement [itex]m_i[/itex], there is a range of possible results [itex]R_i[/itex].

And, there is a set of possible experiments [tex]E \subset P(M)[/tex] which consist of subsets of the list of possible measurments with the additional property that for any [itex]e \in E[/itex], [itex]P(e) \subset E[/itex].

Then, for each experiment [itex]e \in E[/itex], there is a result space [itex]R_e[/itex] which is the cartesian product of the ranges of the individual measurements in [itex]e=\{m_{i_1},m_{i_2},m_{i_3}...\}[/itex]

And, a theory [itex]T[/itex] is a set of probability measures for the result spaces of each of the experiments.

Now, since the experiments are all elements of [itex]P(M)[/itex] there is a natural partial order [itex]\prec[/itex] based on set inclusion, so if we have two experiments [itex]e_1[/itex] and [itex]e_2[/itex], and the set of measurements in [itex]e_1[/itex] is a subset of the measurements in [itex]e_2[/itex] then [itex]e_1 \prec e_2[/itex] and, I will refer to this as [itex]e_2[/itex] is a stricter experiment than [itex]e_1[/itex].

Then, a theory is 'signal local' if for any [itex]e_1 \prec e_2[/itex], the probability probability measure that the theory gives for [itex]R_{e_1}[/itex] is the same as the natural probability subspace generated by restricting [itex]R_{e_2}[/itex] i.e. that for any sub set [itex]s \subset R_{e_1}[/itex] the subset [itex]s \times \prod_{m \in e_2, m \notin e_1} R_m[/itex] has the same measure in [itex]R_{e_2}[/itex]

And, a theory has a local hidden interpretation if there is some probability measur

e on [itex]R_M[/itex] (the cartesian product of the ranges of all individiually possible measurements) that gives the same probability measure as the theory if restricted to [itex]R_e[/itex] for any experiment in [itex]E[/itex].

Now let's say we have a signal local theory [itex]L[/itex], then for every experiment [itex]e \in E[/itex] there is a probability measure on [itex]R_e[/itex]. Now, for every measurable subset [itex]s \subset R_e[/itex] we assign the same value to the subset [itex]s \times \prod_{m \notin e} R_m[/itex] in our prospective measure on [itex]R_M[/tex]. In addition, specify that the measure of the empty set is zero, and the measure of the entire set is 1.

But, because of the properties that were ascribed to signal locality above it's easy to show that this prospective measure is indeed well-defined, and a probability measure.

Now, because this is physics, it's important to discuss whether the mathematical and physical properties match up:

The notion that was described as signal locality is, technically, stronger than signal locality since it's concievable that multiple measurements could occur local to each other, but that concern can be addressed by narowing down the notion of measurement.

The notion that was described as a local hidden interpretation is quite easy - let [itex]\lambda[/itex] be an element of [itex]R_M[/itex] then, since [itex]\lambda[/itex] describes the results of all possible measurements, clearly [\itex]\lambda[itex] can be used as the local hidden state for any particle if it is assigned with the appropriate probability distribution.

Last edited: