# Generalized Probabilities in Statistical Theories

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## Abstract

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## 1. Introduction

“The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.”

“I should say, that in spite of the implication of the title of this talk the concept of probability is not altered in quantum mechanics. When I say the probability of a certain outcome of an experiment is p, I mean the conventional thing, that is, if the experiment is repeated many times one expects that the fraction of those which give the outcome in question is roughly p. I will not be at all concerned with analyzing or defining this concept in more detail, for no departure of the concept used in classical statistics is required. What is changed, and changed radically, is the method of calculating probabilities.”

- States of classical probabilistic systems can be suitably described by Kolmogorovian measures. This is due to the fact that each classical state defines a measure in the Boolean sigma-algebra of measurable subsets of phase space.
- Contrarily to classical states, quantum states cannot be reduced to a single Kolmogorovian measure. A density operator representing a quantum state defines a measure over an orthomodular lattice of projection operators, which contains (infinitely many) incompatible maximal Boolean subalgebras. These represent different and complementary—in the Bohrian sense—experimental setups. The best we can do is to consider a quantum state as a family of Kolmogorovian measures, pasted in a harmonic way [20]; however, there is no joint (classical) probability distribution encompassing all possible contexts.

## 2. Classical Probabilities

#### 2.1. Kolmogorov

#### 2.2. Random Variables and Classical States

#### 2.3. Cox’s Approach

- C1—The probability of an inference on given evidence determines the probability of its contradiction from the same evidence.
- C2—The probability on a given evidence that both of two inferences are true is determined by their separate probabilities, one from the given evidence and the other from this evidence with the additional assumption that the first inference is true.

#### 2.4. MaxEnt Principle

## 3. The Formalism of QM

#### 3.1. Elementary Measurements and Projection Operators

#### 3.2. Quantum States and Quantum Probabilities

#### 3.3. Some Examples

**Finite Probability model: a dice.**Consider the throw of a dice. The possible outcomes are given by $\mathsf{\Omega}=\{1,2,3,4,5,6\}$. A probabilistic state of the dice is determined by assigning real numbers ${p}_{i}$, $i=1,...,6$, to each element of $\mathsf{\Omega}$. If the dice is not loaded, then ${p}_{i}=\frac{1}{6}$ for all i; however, a realistic dice will not satisfy this. An event will be represented by a subset of $\mathsf{\Omega}$. As examples, consider the event “the outcome is even” or “the outcome is greater than 2”. These are represented by $\{2,4,6\}$ and $\{3,4,5,6\}$, respectively. All possible subsets of $\mathsf{\Omega}$ form a Boolean lattice (see Apendix Appendix A), with regard to the set-theoretical operations: “∪” (interpreted as “∨”), “∩” (interpreted as “∧”), and the set theoretical complement (interpreted as “¬”). The example of a $\sigma -$algebra associated to a measurable space $(\mathsf{\Omega},\Sigma ,\mu )$ works in a similar way.

**Hilbert lattice:**As discussed above, the events associated to quantum systems can be put in one to one correspondence with an orthomodular lattice: the one formed by the set of closed subspaces of a Hilbert space $\mathcal{H}$. They can be endowed with a lattice structure as follows [21]. The operation“∨” is taken as the closure of the direct sum “⊕” of subspaces, “∧” as the intersection “∩”, and “¬” as the orthogonal complement “⊥”, $\mathbf{0}=\overrightarrow{0}$, $\mathbf{1}=\mathcal{H}$, and we denote by $\mathcal{P}\left(\mathcal{H}\right)$ the set of closed subspaces. The order “≤” is defined by subspace inclusion: we say that $\mathbb{S}\le \mathbb{T}$, whenever $\mathbb{S}\subseteq \mathbb{T}$. The subspaces $\mathbf{0}$ and $\mathbf{1}$ play the role of the bottom and top elements of the lattice, since, for any subspace $\mathbb{S}$, we have $\mathbf{0}\le \mathbb{S}\le \mathbf{1}$.

Kolmogorov Probability | Quantum Probability | |

Lattice: | $\Sigma $ | $\mathcal{P}\left(\mathcal{H}\right)$ |

(Boolean-algebra) | (orthomodular, non-Boolean) | |

States: | Measures over $\Sigma $ | Measures over $\mathcal{P}\left(\mathcal{H}\right)$ |

Events: | Subsets of $\mathsf{\Omega}$ | Closed subspaces of $\mathcal{H}$ |

**Firefly Model:**The firefly model [40] is used in quantum logic to show an example of a system that is not a full quantum model but has certain features that serve to illustrate what happens with quantum systems. It consists of a firefly that is freed inside a box. We are asked to perform an experiment to detect the location of the firefly but with constrains. We are only allowed to look at two different faces of the box (and we can only choose one on each run of the experiment). The first one is to measure face ${C}_{1}$ with three possible outcomes: the “firefly is detected on the left” (l), on the right (r), and “no-signal” (n) (which means that the light of the firefly was off).

**The lattice of Q-bit:**Given the incredible advances of quantum information theory in recent decades, the reader may wonder what the lattice of a q-bit looks like. It is the simplest quantum model conceivable. Suppose then that we are given a spin $\frac{1}{2}$ system. As is well known, the set of all possible states of a qubit is isomorphic to a sphere, namely, the Bloch sphere [91]. Each pure state of a qubit corresponds to a one dimensional subspace of a two dimensional complex Hilbert space, and can be represented as a point in the surface of the Bloch sphere.

**Kochen–Specker theorem (in a four dimensional model):**A nice example of how the different contexts of a quantum system are intertwined was presented in [97] (of course, for the original version of the Kochen–Specker theorem, the reader is referred to [96]). Given a four-dimensional quantum system, each measurement context has four possible outcomes. Each one of them is mathematically represented by a one dimensional subspace of a four-dimensional Hilbert space.

#### 3.4. Quantal Effects

## 4. Generalization to Orthomodular Lattices

**Definition**

**1.**

**Definition**

**2**

## 5. Convex Operational Models

**Definition**

**3.**

## 6. Cox’s Method Applied To Physics

## 7. Generalization of Cox’s Method

#### Generalized Probability Calculus Using Cox’s Method

## 8. Conclusions

- If the lattice of events that the agent is facing is Boolean (as in Cox’s approach), then, the measures of degree of belief will obey laws equivalent to those of Kolmogorov.
- On the contrary, if the state of affairs that the agent must face presents contextuality (as in standard quantum mechanics), the measures involved must be non-Kolmogorovian [27].

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Lattice Theory

- For all $x,y\in X$, if $x<y$ and $y<x$, then $x=y$.
- For all $x,y,z\in X$, if $x<y$ and $y<z$, then $x<z$.

**complemented distributive lattice**. We use the terms Boolean lattice and Boolean algebra interchangeably.

- Atomic, if, for every nonzero element x of $\mathcal{L}$, there exists an atom a of $\mathcal{L}$ such that $a\le x$.
- Atomistic, if every element of $\mathcal{L}$ is a supremum of atoms.

Boolean | Modular | Orthomodular |

$x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$ | $a\le b\u27f9a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)$ | $a\le b\u27f9a\vee (\neg a\wedge b)=b$ |

$x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$ |

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Holik, F.; Massri, C.; Plastino, A.; Sáenz, M. Generalized Probabilities in Statistical Theories. *Quantum Rep.* **2021**, *3*, 389-416.
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Holik F, Massri C, Plastino A, Sáenz M. Generalized Probabilities in Statistical Theories. *Quantum Reports*. 2021; 3(3):389-416.
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Holik, Federico, César Massri, Angelo Plastino, and Manuel Sáenz. 2021. "Generalized Probabilities in Statistical Theories" *Quantum Reports* 3, no. 3: 389-416.
https://doi.org/10.3390/quantum3030025