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\title{Generalized probabilities in statistical theories}

\author{{\sc Federico Holik,}$^{1,\,2}$ \ {\sc C. Massri,} \ {\sc A. Plastino}$^{1,\,3}$ \ {\sc and} {\sc Manuel
S\'{a}enz}}

\maketitle

\begin{center}

\begin{small}
1- Universidad Nacional de La Plata, Instituto
de F\'{\i}sica (IFLP-CCT-CONICET), C.C. 727, 1900 La Plata, Argentina \\
2- Departamento de Matem\'{a}tica - Ciclo B\'{a}sico Com\'{u}n\\
Universidad de Buenos Aires - Pabell\'{o}n III, Ciudad Universitaria
\\ Buenos Aires, Argentina \\
3- Center Leo Apostel for Interdisciplinary Studies and, Department
of Mathematics, Brussels Free University Krijgskundestraat 33, 1160 Brussels, Belgium \\
\end{small}
\end{center}

\vspace{1cm}

\begin{abstract}
\noindent In this review article we present different formal
frameworks for the description of generalized probabilities in
statistical theories. We discuss the particular cases of
probabilities appearing in classical and quantum mechanics, possible
generalizations of the approaches of A. N. Kolmogorov and R. T. Cox
to non-commutative models, and the approach to generalized
probabilities based on convex sets.
\end{abstract}
\bigskip
\noindent

\begin{small}
\centerline{\em Key words:Quantum Probability; Lattice theory;
Information theory; Classical probability; Cox's approach}
\end{small}

\bibliography{pom}

\section{Introduction}

\noindent Which is the nature of the probabilities appearing in
Quantum Mechanics (QM)? The Born's rule tell us how to compute the
probability of a given process using the formalism of QM. But is it
legitimate to speak of something like Quantum Probabilities (QP)?
Some authors react against this possibility, arguing that this is
not possible, mainly because of the non-Kolmogorovian character of
the probabilities appearing in QM. Their central point is that
probabilities in QM cannot be described using classical probability
theory because they do not obey Kolmogorov's axioms.

\nd On the other hand, many authors support the idea that a
quantum probability theory is possible by generalizing
Kolmogorov's axioms to non boolean algebras
\cite{Redei-Summers2006,Davies-Lewies,Srinivas,mackey57,Gudder-StatisticalMethods,mackey-book}.
In this sense, QP exists as a non-Boolean (or non-commutative)
generalization of classical probability theory.

\nd However,  Kolmogorov's approach is not the only one. An
alternative approach to classical probability theory is presented by
R. T. Cox \cite{CoxLibro,CoxPaper}. It is based on a study of the
measure functions compatible with the algebraic properties of the
logic of an intelligent agent trying to make probable inferences out
of available data. A variant of this approach has been used to
describe probabilities in QM
\cite{Caticha-99,GoyalKnuthSkilling,Knuth-2004a,Knuth-2004b,Knuth-2005a,Knuth-2005b,Symmetry,KnuthSkilling-2012,Holik-Plastino-Saenz-2012},
suggesting that they are essentially classical.

\nd We have already mentioned that there exists a general formal
framework which includes CP and QP as particular cases, being a
generalization of  Kolmogorov's approach to non-distributive
algebras. Is there an analogue of this for the  Cox approach? An
affirmative answer to this question was given in
\cite{Holik-Plastino-Saenz-2012}. It is shown there  that the
typical features of QP arise whenever the lattice of propositions
of the Cox's approach is replaced by a non-distributive one. As is
well known, the quantum-logical approach to QM characterizes this
theory using a lattice-theoretical approach in which the lattice
is an orthomodular one
\cite{BvN,dallachiaragiuntinilibro,mikloredeilibro,belcas81,vadar68,vadar70,svozillibro,HandbookofQL,jauch,piron,kalm83,kalm86,greechie81,giunt91,pp91,dvupulmlibro,aertsdaub1,aertsdaub2,aertsjmp83,aertsrepmathphys84a,aertsjmp84b}.
In \cite{Holik-Plastino-Saenz-2012} it is shown that when the
Cox's method is applied to the orthomodular lattice of projections
of a Hilbert space, QP are derived.

\nd Remark that generalized probabilities can also be studied in
what has been called the Convex Operational Models (COM) approach
\cite{Beltrametti.Varadarajan-2000,wilce,Barnum-Wilce-2009,Barnum-Wilce-2010,Barnum-PRL,Barnum-Toner,barnum,Entropy-generalized-II,Barret-2007,Perinotti-2011,GenealizedTeleportationBarnum,
Holik-Plastino-Massri}. In this approach, the properties of the
systems studied and their associated probabilities are encoded in a
geometrical way in the form of a convex set and its space of
observable magnitudes. The quantum formalism and many aspects of
quantum information theory (such as entanglement, discord,
information protocols) can be suitably described using this approach
\cite{barnum,Entropy-generalized-II,Barret-2007,Barnum-Toner,Perinotti-2011,Holik-Plastino-Massri,GenealizedTeleportationBarnum,Barnum-PRL,Barnum-Wilce-2009,Barnum-Wilce-2010,Perinotti-2011}.
Non-linear generalizations of QM where studied using the convex
approach in \cite{Mielnik-68,Mielnik-69,Mielnik-74}.

The present is a review article and no new technical results are
presented. It can be considered as an extension of
\cite{ReviewNOVA}, in which, due to reasons of space, we could not
discuss in detail some interesting topics. This notwithstanding, the
presentation of the topics is original in the sense that the
perspective presented has not been previously published in the
literature, and has conceptual implications for the discussion about
what is the meaning of CP and QP. We start by reviewing lattice
theory and the quantum logical approach to QM in Sections
\ref{s:LatticeTheory} and \ref{s:TheFormalismOfQM}. Next, in Section
\ref{s:CoxReview}, we concentrate on the different approaches to CP,
namely, Kolmogorov's and Cox's, and in Section
\ref{s:COMpreliminaries} we review the COM approach. After these
notions are introduced, we discuss QP in Section
\ref{s:QuantumProbabilities}. We present the generalization of the
Cox's method to general non-distributive lattices in Section
\ref{s:QMDerivation}. Finally, some  conclusions are drawn in
Section \ref{s:Conclusions}.

\section{Lattice Theory}\label{s:LatticeTheory}

\nd Lattices can be defined by using equations, i.e., they can be
characterized as algebraic structures satisfying certain axiomatic
identities. A set $\mathcal{L}$ endowed with two operations
$\wedge$ and $\vee$ will be called a \emph{lattice}, if for al
$x,y,z\in\mathcal{L}$ the following equations are satisfied

\begin{subequations}\label{e:LatticeEquations}
\begin{equation}
x\vee x = x\,\,\, x\wedge x = x\,\,\, \mbox{(idempotence)}
\end{equation}

\begin{equation}
x\vee y = y\vee x\,\,\,x\wedge y = y\wedge x \,\,\,
\mbox{(commutativity)}
\end{equation}

\begin{equation}
x\vee (y\vee z) = (x\vee y)\vee z \,\,\, x\wedge (y\wedge z) =
(x\wedge y)\wedge z \,\,\, \mbox{(associativity)}
\end{equation}

\begin{equation}
x\vee (x\wedge y) = x\wedge (x\vee y) = x \,\,\, \mbox{(absortion)}
\end{equation}

\end{subequations}
\noindent If the extra relationships

\begin{subequations}\label{e:LatticeEquationsDistributive}
\begin{equation}
x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)
\,\,\,\mbox{(distributivity 1)}
\end{equation}

\begin{equation}
x\vee (y\wedge z) = (x\vee y) \wedge (x\vee z)\,\,\,
\mbox{(distributivity 2)}
\end{equation}
\end{subequations}
 are satisfied, the lattice is called \emph{distributive}.

\vskip 3mm

\noindent Lattice theory can also be studied using \textit{partially
ordered sets} (\textit{poset}). A poset is a set $X$ endowed with a
partial ordering relation ``$<$" satisfying

\begin{itemize}

\item For all $x,y\in X$, if $x<y$ and $y<x$, then $x=y$

\item For all $x,y,z\in X$, if $x<y$ and $y<z$, then $x<z$

\end{itemize}

\nd We use the notation ``$x\leq y$" for the case ``$x<y$" or
``$x=y$". A lattice $\mathcal{L}$ will be a poset for which any two
elements $x$ and $y$ have a unique supremum and a unique infimum
with respect to the order structure. The least upper bound of two
given elements ``$x\vee y$" is called the ``join" and their greatest
lower bound ``$x\wedge y$", called their ``meet". A lattice for
which all its subsets have both a supremum and an infimum is called
a \emph{complete lattice}. If furthermore there exists a greatest
element $\mathbf{1}$ and a least element $\mathbf{0}$, the lattice
is called \emph{bounded}. They are usually called the \emph{maximum}
and the \emph{minimum} respectively. Any lattice can be extended
into a bounded lattice by adding a greatest and a least element.
Every non-empty finite lattice is bounded. Complete lattices are
always bounded. An orthocomplementation in a bounded poset $P$ is a
unary operation ``$\neg(\ldots)$" satisfying:

\begin{subequations}\label{e:ComplementationAxioms}
\begin{equation}\label{e:Complement1}
\neg(\neg(a))=a
\end{equation}
\begin{equation}\label{e:Complement2}
a\leq b \longrightarrow \neg b\leq \neg a
\end{equation}
$a\vee \neg a$ and $a\wedge \neg a$ exist and
\begin{equation}\label{e:Complement3}
a\vee \neg a=\mathbf{1}
\end{equation}
\begin{equation}\label{e:Complement4}
a\wedge \neg a=\mathbf{0}
\end{equation}
hold.
\end{subequations}

\nd A bounded poset with orthocomplementation will be called an
\emph{orthoposet}. An \emph{ortholattice}, will be an orthoposet
which is also a lattice. For $a,\,b \in \mathcal{L}$ (an
ortholattice or orthoposet), we say that $a$ is orthogonal to $b$ ($
a \bot b$) iff $a\le \neg b$.  Following \cite{RedeiHandbook}, we
define an \textit{orthomodular lattice} as an ortholattice
satisfying the orthomodular law:

\begin{equation}\label{e:ModularIdentity}
a \leq b \,\,\mbox{and}\,\, \neg a\leq c \Longrightarrow a \vee (b \wedge c) = (a \vee b) \wedge (a\vee c)
\end{equation}
\nd A \textit{modular lattice}, is an ortholattice satisfying the stronger condition (modular law)

\begin{equation}\label{e:ModularIdentity2}
a \leq b \Longrightarrow a \vee (b \wedge c) = (a \vee b) \wedge
(a\vee c),
\end{equation}
\nd and finally, a \textit{boolean lattice} will be an ortholattice satisfying
the still stronger condition (distributive law)

\begin{equation}\label{e:DistributiveLaw}
a \vee (b \wedge c) = (a \vee b) \wedge (a\vee c)
\end{equation}

\nd If $\mathcal{L}$ has a null element $ 0$, then an element $x$ of
$\mathcal{L}$ is an {\it atom} if $0 < x$ and there exists no
element $y$ of $\mathcal{L}$ such that $0 < y < x$. $\mathcal{L}$
will be said to be

\begin{itemize}
\item {\it Atomic}, if for every nonzero element $x$ of $\mathcal{L}$,
there exists an atom $a$ of $\mathcal{L}$ such that $ a \leq x$.
\item Atomistic, if every element of $\mathcal{L}$ is a supremum of
atoms.
\end{itemize}

\vskip 3mm

\noindent In order to illustrate the above notions, let us consider
the following examples:

\begin{description}
  \item[Hilbert lattice:] As an example of an orthomodular lattice which will be of
importance for us, let us consider the set of closed subspaces of a
Hilbert space $\mathcal{H}$. They can be endowed with a lattice
structure as follows \cite{mikloredeilibro}. The operation``$\vee$"
is taken as the direct sum ``$\oplus$'' of subspaces, ``$\wedge$''
as the intersection ``$\cap$'', and ``$\neg$'' as the orthogonal
complement ``$\bot$'', $\textbf{0}=\vec{0}$,
$\textbf{1}=\mathcal{H}$, and denote by $\mathcal{P}(\mathcal{H})$
to the set of closed subspaces. Then, $<\mathcal{P}(\mathcal{H}),\
\cap,\ \oplus,\ \neg,\ 0,\ 1>$ will be a complete bounded
orthomodular lattice (which we will denote simply by
$\mathcal{P}(\mathcal{H})$). As closed subspaces are in one to one
correspondence with projection operators, we will take
$\mathcal{P}(\mathcal{H})$ to be the lattice of closed subspaces or
the lattice of projections interchangeably. One of the most
important features of $\mathcal{P}(\mathcal{H})$ is that the
distributive law \eqref{e:DistributiveLaw} doesn't holds.
$\mathcal{P}(\mathcal{H})$ is modular iff $\mathcal{H}$ is finite
dimensional. If $\mathcal{H}$ is infinite dimensional, then
$\mathcal{P}(\mathcal{H})$ is always orthomodular.

\item[Subsets of a given set] As an example of a boolean lattice,
consider the subsets of a given set endowed with the operations
union ``$\cup$" as ``$\vee$", ``$\cap$'' as ``$\wedge$'', and set
theoretical complement as ``$\neg$''. The propositional calculus of
classical logic also forms a boolean lattice.

\item[Chinesse lantern:] The (chinesse lantern)\cite{svozillibro}
is used in quantum logic to show an example of a system which is not
a quantum one neither a classical one. Its lattice of propositions
corresponds to the following Hasse diagram

  \begin{center}
  \begin{tikzpicture}
  \node (max) at (0,4) {$\mathbf{1}$};

  \node (a) at (-4,2) {$l'$};

  \node (b) at (-2,2) {$r'$};

  \node (c) at (0,2) {$n'$};

  \node (d) at (2,2) {$f'$};

  \node (e) at (4,2) {$b'$};

  \node (f) at (-4,0) {$l$};

  \node (g) at (-2,0) {$r$};

  \node (h) at (0,0) {$n$};

  \node (i) at (2,0) {$f$};

  \node (j) at (4,0) {$b$};

  \node (min) at (0,-2) {$\mathbf{0}$};

  \draw (min) -- (f) -- (b) -- (max);

  \draw (min) -- (g) -- (a) -- (max);

  \draw (min) -- (h) -- (a) -- (max);

  \draw (min) -- (i) -- (e) -- (max);

  \draw (min) -- (j) -- (d) -- (max);

  \draw (h) -- (e) -- (max);

  \draw (h) -- (d) -- (max);

  \draw (h) -- (b) -- (max);

  \draw (h) -- (d) -- (max);

  \draw (f) -- (c) -- (max);

  \draw (i) -- (c) -- (max);

  \draw (j) -- (c) -- (max);

  \draw (g) -- (c) -- (max);

\end{tikzpicture}

\end{center}
\noindent looking at the above diagram, it is easy to see that this
lattice contains two Boolean sublattices.

\item[The lattice of Q-bit:] Given the incredible advances of
quantum information theory in the last decades, the reader may
wonder how does the lattice of a q-bit looks like. Suppose then that
we are given a spin $\frac{1}{2}$ system. As is well known, the set
of all possible pure states is isomorphic to a sphere, namely, the
\emph{Bloch sphere} \cite{BEN}. The different sets of objective
properties, which are of the form ``the particle has spin $+$ (or
$-$) in direction $\hat{z}$", are represented by rays (or points on
the surface of the sphere). They can be given by $\textbf{0}$, the
null element of the space, $\textbf{1}$ (the maximal element of the
space) or a ray. Each ray will have associated a projection operator
$|+\rangle_{\hat{z}}\langle+|$. Thus, the Hasse diagram of a q-bit
will have the form:

\begin{center}
\begin{tikzpicture}

  \node (max) at (0,1) {$\mathbf{1}$};

  \node (f) at (3,0) {$\cdots$};

  \node (a) at (2,0) {$\neg p'$};

  \node (b) at (1,0) {$\neg p$};

  \node (e) at (0,0) {$\cdots$};

  \node (c) at (-1,0) {$p'$};

  \node (d) at (-2,0) {$p$};

  \node (g) at (-3,0) {$\cdots$};

  \node (min) at (0,-1) {$\mathbf{0}$};

  \draw (min) -- (a) -- (max);

  \draw (min) -- (b) -- (max);

  \draw (min) -- (c) -- (max);

  \draw (min) -- (d) -- (max);

\end{tikzpicture}
\end{center}
\end{description}

\section{The Formalism of QM}\label{s:TheFormalismOfQM}

\nd In this Section we discuss some specific features of the
quantum formalism \cite{vN,vadar68,vadar70,mikloredeilibro} which
are relevant for the problem of QP.

\subsection{Elementary measurements and projection operators}\label{s:ElementaryTests}

\nd The rigorous formulation of the formalism of QM was presented in
a series of papers by von Neumann, Jordan, Hilbert and Nordheim
\cite{vN-Hilbert-Nordheim}. It can be said that its definitive form
was accomplished in the book of von Neumann \cite{vN}. In QM,
observable physical magnitudes are represented by compact
self-adjoint operators in a Hilbert space. Due to the spectral
decomposition theorem \cite{ReedSimon,vN}, a key role is played by
\textit{projection valued measures} (PVM): the set of PVM can be put
in a bijective correspondence with the set $\mathcal{A}$ of self
adjoint operators of $\mathcal{H}$. Intuitively speaking, a PVM is a
map which assigns a projection operator to each interval of the real
line. In this sense, projection operators are the building blocks of
any other observable in QM. In order to give a precise definition,
we need first to specify what we mean by ``intervals of the real
line''. The \textit{borel sets} ($B(\mathbb{R})$) are defined as the
family of subsets of $\mathbb{R}$ such that

\begin{itemize}
\item it is closed under set theoretical complements,

\item it is closed under denumerable unions, and

\item it includes all open intervals \cite{ReedSimon}.
\end{itemize}

\noindent Then, a PVM is a map $M$ defined as follows

\begin{subequations}

\begin{equation}
M: B(\mathbb{R})\rightarrow \mathcal{P}(\mathcal{H}),
\end{equation}
\noindent satisfying

\begin{equation}
M(0)=0
\end{equation}
\begin{equation}
M(\mathbb{R})=\mathbf{1}
\end{equation}
\begin{equation}
M(\cup_{j}(B_{j}))=\sum_{j}M(B_{j}),\,\,
\end{equation}
\noindent for any disjoint denumerable family ${B_{j}}$. Also,

\begin{equation}
M(B^{c})=\mathbf{1}-M(B)=(M(B))^{\bot}
\end{equation}

\end{subequations}

\vskip 3mm

\nd Fixing an element $A\in\mathcal{A}$, the intended interpretation
of the associated PVM $M_{A}((a,b))=P_{(a,b)}$ is ``the value of $A$
lies between the interval $(a,b)$". In this sense, projections
represent elementary tests or propositions in QM. As we reviewed in
the previous Section, projection operators can be endowed with a
lattice structure, and thus, also elementary tests. This lattice was
called ``Quantum Logic" by Birkhoff and von Neumann \cite{BvN}. We
will refer to it as the \emph{von Neumann-lattice
($\mathcal{L}_{v\mathcal{N}}(\mathcal{H})$}) \cite{mikloredeilibro}.

\nd As shown in \cite{BvN}, an analogous treatment can be done for
classical systems. In that theory, an observable represented by a
real function $f$ in phase space, assigns to each Borel subset $E$
the subset $f^{-1}(E)$ of the phase space $\Gamma$. As an example,
to the proposition ``the value of $f$ lies between the interval
$(a,b)$" it corresponds a subset $f^{-1}(a,b)$ of $\Gamma$. Thus,
in CM, propositions or elementary tests are associated to subsets
of the phase space, and logical connectives ``$\vee$", ``$\wedge$"
and ``$\neg$", are represented by \emph{union},
\emph{intersection} and (set theoretical) \emph{complement} of
subsets of $\Gamma$. The resulting lattice is a Boolean one.

\vskip 3mm

\nd During the thirties, von Neumann and collaborators, continued
studying formal developments related to the quantum formalism. One
of the results of this investigation was the development of the
theory of \emph{rings of operators} (better known as \emph{von
Neumann algebras}
\cite{mikloredeilibro,RingsOfOperatorsI,RingsOfOperatorsII,RingsOfOperatorsIII,RingsOfOperatorsIV}),
as an attempt of generalizing certain algebraic properties of
Jordan algebras \cite{vN-Hilbert-Nordheim}. The subsequent study
of von Neumann algebras showed that they are closely related to
lattice theory. Murray and von Neumann provided a classification
of factors (von Neumann algebras whose center is formed by the
multiples of the identity) using orthomodular lattices in
\cite{RingsOfOperatorsI,RingsOfOperatorsII,RingsOfOperatorsIII,RingsOfOperatorsIV}.
On the other hand, lattice theory is deeply connected to
projective geometry \cite{ProjectiveGeometries}, and one of the
major discoveries of von Neumann was that of \emph{continuous
geometries}, which do not possess ``points" (or ``atoms") and are
related to type II factors. Far from being a mere mathematical
curiosity, type II factors found applications in statistical
mechanics and type III factors play a key role in the axiomatic
approach to Quantum Field Theory (QFT)
\cite{Redei-Summers2006,mikloredeilibro}.

\nd The quantum logical approach of Birkhoff and von Neumann was
continued by other researchers
\cite{jauch,piron,mackey-book,vadar68,vadar70,Gudder-StatisticalMethods,aertsdaub1,aertsdaub2,FR81}
(and see \cite{HandbookofQL,dallachiaragiuntinilibro,kalm83} for
complete expositions). One of the key results of this approach is
the \emph{representation theorem} of C. Piron \cite{piron}. He
showed that any propositional system can be coordinatized in a
generalized Hilbert space. A later result by Sol\`{e}r shows that
adding extra assumptions, it can only be a Hilbert space over the
fields of the real numbers, complex numbers or quaternions
\cite{Soler-1995}.

\subsection{Quantal Effects}\label{s:QuantalEffects}

\nd Projective measures are not the only way in which observable
quantities can be described in QM. There exists a more general
notion, namely, that of \emph{quantal effect}. And it turns out
that this notion can be generalized to arbitrary statistical
theories. We will start by reviewing the general case before we
consider the quantum one. If the state space of a given
probabilistic theory is given by the set $\Sigma$, let us denote
by $X$ to the set of possible measurement outcomes of an
observable quantity. Then, if the system is in a state $s$, a
probability $p(x,s)$ is assigned to any possible outcome $x\in X$.
This probability should be well defined in order that our theory
be considered as a probabilistic one. In this way, we must have a
function

\begin{eqnarray}
&p:X\times\Sigma\mapsto [0,1]&\nonumber\\
&(x,s)\rightarrow p(x,s)&
\end{eqnarray}

\nd To each outcome $x\in X$ and state $s\in\Sigma$, this function
assigns a probability $p(x,s)$ of $x$ to occur if the system is in
the state $s$. In this way, a triplet $(\Sigma,p(\cdotp,\cdotp),X)$
is assigned for each system of any probabilistic theory
\cite{mackey-book}. Thinking of $s$ as variable, we obtain the
mapping $s\mapsto p(\cdot,s)$ from $\Sigma\rightarrow [0,1]^{X}$.
This implies that all the states of $\Sigma$ can be identified with
maps which generates a canonical vector space. Their closed convex
hull forms a new set $\Omega$ representing all possible
probabilistic mixtures (convex combinations) of states in $\Sigma$.
Given an arbitrary $\alpha\in \Omega$ and any outcome $x\in X$, we
can define an affine evaluation-functional $f_{x}:\Omega\rightarrow
[0,1]$ in a canonical way by $f_{x}(\alpha)=\alpha(x)$. More
generally, we can consider any affine functional
$f:\Omega\rightarrow [0,1]$ as representing a measurement outcome,
and thus use $f(\alpha)$ to represent the probability for that
outcome in state $\alpha$.

\noindent Due to the fact that QM is also a probabilistic theory, it
follows that it can be included in the general framework described
above. In QM, the set of all affine functionals defined as above are
called \textit{effects}. The generalization of the notion of PVM
(which is based on projections) to an observable based on effects
will be called a \emph{positive operator valued measure} (POVM))
\cite{Busch-Lahti-2009,Thesis-Heinonen-2005,Ma-Effects,EffectAlgebras-Foulis-2001,Cattaneo-Gudder-1999,bush,foulis}
and in QM will be represented by a mapping

\begin{subequations}

\begin{equation}
E:B(\mathcal{R})\rightarrow\mathcal{B}(\mathcal{H}).
\end{equation}
\noindent such that

\begin{equation}
E(\mathcal{R})=\mathbf{1}
\end{equation}

\begin{equation}
E(B)\geq 0, \,\,\mbox{for any}\,\, B\in B(\mathcal{R})
\end{equation}

\begin{equation}
E(\cup_{j}(B_{j}))=\sum_{j}E(B_{j}),\,\, \mbox{for any disjoint
familly}\,\, {B_{j}}
\end{equation}

\end{subequations}

\nd A POVM is thus a measure whose values are non-negative
self-adjoint operators on a Hilbert space, and the above definition
reduces to the PVM case when these operators are also projections.
It is the most general formulation of the description of a
measurement in the framework of quantum physics. Positive operators
$E(B)$ satisfyng $0\leq E\leq\mathbf{1}$ are called effects and
generate an \emph{effect algebra}
\cite{EffectAlgebras-Foulis-2001,Cattaneo-Gudder-1999})and we will
denote it by $\mathrm{E}(\mathcal{H})$. In a similar way as in the
general setting, in QM a POVM defines a family of affine functionals
on the quantum state space $\mathcal{C}$ of all positive hermitian
trace-class operators of trace one (which corresponds to $\Omega$ in
the general probabilistic setting):

\begin{subequations}

\begin{equation}
E(B):\mathcal{C}\rightarrow [0,1]
\end{equation}

\begin{equation}
\rho\mapsto \mbox{tr}(E\rho)
\end{equation}

\end{subequations}

\section{Classical Probabilities}\label{s:CoxReview}

\nd This Section is devoted to \emph{classical probability theory}.
But what do we mean with this notion? There exists a vast literature
and tendencies disputing the meaning of CP. We will not give a
detailed survey of the discussion here, but we discuss two of the
most important approaches to CP. These are the one given by A. N.
Kolmogorov \cite{KolmogorovProbability} and the one given by R. T.
Cox \cite{CoxLibro,CoxPaper}.

\subsection{Kolmogorov}\label{s:Kolmogorov}

\noindent Kolmogorov presented his axiomatization of classical
probability theory \cite{KolmogorovProbability} in the 30's as
follows. Given an outcome set $\Omega$, let us consider a
$\sigma$-algebra $\Sigma$ of subsets of $\Omega$. A probability
measure will be a function $\mu$ such that

\begin{subequations}\label{e:kolmogorovian}
\begin{equation}
\mu:\Sigma\rightarrow[0,1]
\end{equation}
\noindent satisfying
\begin{equation}
\mu(\emptyset)=0
\end{equation}
\begin{equation}
\mu(A^{c})=1-\mu(A),
\end{equation}

\noindent where $(\ldots)^{c}$ means set-theoretical-complement and
for any pairwise disjoint denumerable family $\{A_{i}\}_{i\in I}$

\begin{equation}
\mu(\bigcup_{i\in I}A_{i})=\sum_{i}\mu(A_{i})
\end{equation}

\end{subequations}

\noindent Conditions \eqref{e:kolmogorovian} are known as
\emph{axioms of Kolmogorov} \cite{KolmogorovProbability}. The triad
$(\Omega,\Sigma,\mu)$ is called a \emph{probability space}.
Probability spaces obeying Eqs. \eqref{e:kolmogorovian} are usually
referred as Kolmogorovian, classical, commutative or boolean
probabilities \cite{Gudder-StatisticalMethods}.

\nd It is possible to show that if $(\Omega,\Sigma,\mu)$ is a
Kolmogorovian probability space, the \emph{inclusion-exclusion
principle} holds

\begin{equation}\label{e:SumRule}
\mu(A\cup B)=\mu(A)+\mu(B)-\mu(A\cap B)
\end{equation}

\noindent or (as expressed in logical terms)

\begin{equation}\label{e:SumRuleLogical}
\mu(A \vee B)=\mu(A)+\mu(B)-\mu(A \wedge B)
\end{equation}

\noindent As remarked in \cite{RedeiHandbook}, Eq. \eqref{e:SumRule}
was considered as crucial by von Neumann for the interpretation of
$\mu(A)$ and $\mu(B)$ as relative frequencies. If $N_{(A\cup B)}$,
$N_{(A)}$, $N_{(B)}$, $N_{(A\cap B)}$ are the number of times of
each event to occur in a series of $N$ repetitions, then
\eqref{e:SumRule} trivially holds.

\nd This principle does no longer hold in QM, a fact linked to the
non-boolean QM-character. Thus, the relative-frequencies'
interpretation of quantum probabilities becomes problematic. The
QM example shows that non-distributive propositional structures
play an important role in probability theories {\it different
from} that of Kolmogorov.

\subsection{Cox's approach}

\nd Since the beginning of probability theory, there has been a
school of thought known as Bayesianism, which treated probabilities
in a different manner as the one discussed in the previous section.
To them, probabilities were not to be regarded as a property of a
system, but as a property of our knowledge about it. This position
is present as early as in the XIX century in one of the milestones
in the development of this theory \cite{Laplace}. In his work,
Laplace proposed a way to assign probabilities in situations of
ignorance that would eventually be known as ``Laplace principle''.
Later works would try to formalize and give coherence to Bayesian
probability \cite{Jeffreys} \cite{deFinetti}. In this section we
will center our attention on one of this attempts
\cite{CoxLibro,CoxPaper}, due to R. T. Cox.

\nd While attaining equivalent results to Kolmogorov, Cox's approach
is conceptually very different. In the Kolmogorovian approach
probabilities are naturally interpreted (but not necessarily) as
relative frequencies in a sample space. On the other hand, in the
approach developed by Cox, probabilities are considered as a measure
of the degree of belief of an intelligent agent (which may as well
be a machine), on the truth of a proposition $x$ if it is known that
proposition $y$ is true. In this way, Cox intended to find a set of
rules to inferential reasoning that would be coherent with classical
logic and would reduce to it in the case of premises with definite
truth values.


\nd To do this, he departed from two general axioms and presupposed
the calculus of classical propositions which, as is well known,
forms a boolean lattice \cite{Boole}. By doing so, he derived
classical probability theory as an inferential calculus on boolean
lattices. We will sketch here the arguments presented in his book
\cite{CoxLibro}. For a more detailed exposition on the deductions,
the reader is referred to
\cite{CoxPaper,CoxLibro,Jaynes,ReviewCox,Knuth-2004a,Knuth-2004b,Knuth-2005b}.

\nd The two axioms used by Cox \cite{CoxLibro} are

\begin{itemize}

\item C1 - The probability of an inference on given evidence determines the probability of its contradictory on the same evidence.

\item C2 - The probability on a given evidence that both of two inferences are true is determined by their separate probabilities, one on the given evidence, the other on this evidence with the additional assumption that the first inference is true.
\end{itemize}
\vskip 3mm

\nd A real valued function $\varphi$ representing the degree to
which a proposition $h$ (usually called \emph{hypothesis}) implies
another proposition $a$ is postulated. Thus, $\varphi(a|h)$ will
represent the degree of belief of an intelligent agent of how likely
it is that $a$ is true given that he knows that the hypothesis $h$
is true.

\nd Then, requiring the function $\varphi$ to be coherent with the
properties of the calculus of classical propositions, he derives the
rules for manipulating probabilities. Using axiom C2, the
associativity of the conjunction ($a \wedge (b \wedge c) = (a \wedge
b) \wedge c)$), and defining the function
$F[\varphi(a|h),\varphi(a|h)] \equiv \varphi(a \wedge b|h):
\mathbb{R}^2 \rightarrow \mathbb{R}$; he arrives at a functional
equation for $F(x,y)$:

\begin{equation}\label{eq:prod}
 F[x,F(y,z)] = F[F(x,y),z]
\end{equation}

\nd Which, after a rescaling and a proper definition of the
probability $P(a|h)$ in terms of $\varphi(a|h)$, leads to the well
known \emph{product rule} of probability theory:

\begin{equation}
 P(a \wedge b |h) = C P(a | h \wedge b) P(b|h)
\end{equation}

\nd The definition of $P(a|h)$ in terms of $\varphi(a|h)$ is
omitted, as one ultimately ends up using only the function $P(a|h)$
and never $\varphi(a|h)$. In an analogous manner, using axiom C1,
the law of double negation ($\neg \neg a=a$), Morgan's law for
disjunction ($\neg (a \vee b) = \neg a \wedge \neg b$) and defining
the function $f[P(a|h)] \equiv P(\neg a|h): \mathbb{R} \rightarrow
\mathbb{R}$, he arrives at the following functional equation for
$P(a|h)$

\begin{equation}\label{eq:neg}
 P(a|h)^r + P(\neg a|h)^r = 1
\end{equation}

\nd With $r$ an arbitrary constant. Although in principle different
values of $r$ would give rise to different rules for the computation
of the probability of the negation of a proposition, as taking
different values of $r$ account for a rescaling of $P(a|h)$, one
could as well call $P'(a|h) \equiv P(a|h)^r$ \emph{probability} and
work with this function instead of $P(a|h)$. For simplicity, Cox
decides to take $r=1$ and continue using $P(a|h)$.

\nd Using equations \eqref{eq:prod}, \eqref{eq:neg}, the law of
double negation and Morgan's law for conjunction ($\neg (a \wedge b)
= \neg a \vee \neg b$), he arrives to the \emph{sum rule} of
probability theory:

\begin{equation}
 \label{eq:sum}
 P(a \vee b|h) = P(a|h) + P(b|h) - P(a \wedge b|h)
\end{equation}

\nd As it turns out, $P(a|h)$ --if suitably normalized-- satisfies
all the properties of a Kolmogorovian probability (Eqs.
\eqref{e:kolmogorovian}).

\nd Because of the importance of Cox's theorem to the foundations of
probability, it has been target of a thorough scrutiny by many
authors. Some have pointed out inconsistencies behind the implicit
assumptions made during its derivations, most notably the
assumptions behind the validity of equation \ref{eq:prod}. Since
then, there have been different proposals to save Cox's approach by
proving it through less restrictive axioms. In \cite{CriticaCox}, a
discussion of the status of Cox proposal is presented as well as a
counterexample to it. For a review on the subject, it is recommended
to consult \cite{ReviewCox}.

\nd Once the general properties of the function $P(a|h)$ are
established, the next problem is to find a way to determine prior
probabilities (probabilities conditional only to the hypothesis
$h$). Although formally one could assign prior probabilities in any
way coherent with the normalization used \cite{Priors}, in practical
situations one is compelled to assign them in a way that they
reflect the information contained in the hypothesis $h$. A possible
way to do this is by using the MaxEnt
principle\cite{Jaynes}\cite{Jaynes2}, which we will review shortly
in the next Section. Other ways of assigning prior probabilities
include the Laplace principle \cite{Jeffreys} and coherence with
symmetry transformations \cite{Jaynes3}. Nevertheless, the existance
of a general algorithm for assigning prior probabilities is still an
open question.

\subsection{MaxEnt Principle}\label{sec:MaxEnt}

\nd  This principle asserts that the assignment of the prior
probabilities from a hypothesis $h$ should be done by maximizing the
uncertainty associated with its distribution while respecting the
constrains imposed over them by $h$. Although this may sound
paradoxical, by maximizing the uncertainty of the prior
probabilities one avoids to assume more information than the one
strictly contained in $h$.

\nd Taking Shannon's information measure $S[P] = - \sum_i P(a_i|h)
log[P(a_i|h)]$ as the measure of the uncertainty associated with the
distribution $P$, MaxEnt principle can be restated as: the prior
probabilities corresponding to the hypothesis $h$ are given by the
distribution that maximizes $S[P]$ subject to the constraints
imposed by $h$ on $P$. The simplest example is given by the
hypothesis $h$ that imposes no constraints on $P$, in which case $P$
results to be the uniform distribution and the MaxEnt principle
reduces to Laplace's. Different kinds of constraints result on
different prior probability distributions (PPD) \cite{Jaynes2}. In
\cite{TablaMaxEnt} a table of some of the distributions obtained
this way is presented. It must be pointed out that, although given a
set of constraints its corresponding PPD can readily be computed,
there is no general way of translating a hypothesis $h$ into
equivalent constraints.

\nd By means of the MaxEnt principle, classical and quantum
equilibrium statistical mechanics can be formulated on the basis of
information theory \cite{Jaynes}. Assuming that your prior knowledge
about the system is given by the values of $n$ expectation values of
physical quantities $R_j$, i.e., $\langle R_1 \rangle,\ldots,\langle
R_n \rangle$, then the most unbiased probability distribution
$\rho(x)$ is uniquely fixed by extremizing Shannon's logarithmic
entropy $S$ subject to the $n$ constraint

\begin{eqnarray}\label{e:conditionsmean} \langle
R_{i}\rangle=r_{i};\,\, {\rm for \,\,all\,\,} i.
\end{eqnarray}

\nd In order to solve this problem, $n$ Lagrange multipliers
$\lambda_i$ must be introduced.

In the process of employing the MaxEnt procedure one discovers that
the information quantifier $S$ can be identified with the
equilibrium entropy of thermodynamics if our prior knowledge
$\langle R_1 \rangle,\ldots, \langle R_n \rangle$ refers to
extensive quantities \cite{Jaynes}. $S(maximal)$, once determined,
yields complete thermodynamical information with respect to the
system of interest \cite{Jaynes}. The MaxEnt probability
distribution function (PDF), associated to Boltzmann-Gibbs-Shannon's
logarithmic entropy $S$, is given by \cite{Jaynes}

\begin{equation} \label{z1}
\rho_{max}=\exp{[(-\lambda_{0}\mathbf{1}-\lambda_{1}R_{1}-\cdots-\lambda_{n}R_{n})]},
\end{equation}

\noindent where the $\lambda$'s are Lagrange multipliers
guaranteeing that

\begin{equation}  \label{z2}
r_{i}=-\frac{\partial}{\partial\lambda_{i}}\ln Z,
\end{equation}

\noindent while the partition function reads

\begin{equation}  \label{z3}
Z(\lambda_{1}\cdots\lambda_{n})=\mbox{tr}[\exp^{-\lambda_{1}R_{1}-\cdots-\lambda_{n}R_{n}}],
\end{equation}
\noindent and the  normalization condition

\begin{equation}  \label{z4}
\lambda_{0}=\ln Z.
\end{equation}

\nd In a quantum setting, the $R$'s are operators on a Hilbert space
$\mathcal{H}$ while $\rho$ is a density matrix (operator).

\section{Convex Operational Models}\label{s:COMpreliminaries}

\nd As  seen in Section \ref{s:QuantalEffects}, an arbitrary
statistical theory can be endowed with a convex set of states
$\Omega$ and a set of generalized observables (effects) in a
natural way. Generalized observables will be reasonably
represented by affine functionals and we will call $A(\Omega)$ to
the space of all affine functionals. This key observation leads to
a general approach to statistical theories based on the study of
the geometrical properties of convex sets. This is the starting
point of the COM approach. In this Section we concentrate on
elementary notions of COM's, and we refer the reader to
\cite{Barnum-PRL} for en excellent presentation of the subject
(and we follow them in this Section).

\nd We saw that a probability $a(\omega)\in[0,1]$ is well defined
for any state $\omega\in\Omega$ and an observable $a$. In the COM
approach, it is usually assumed that there exists a unitary
observable $u$ such that $u(\omega)=1$ for all $\omega\in\Omega$.
Thus, in analogy with the quantum case, the set of all effects will
be encountered in the interval $[0,u]$ (the order in the general
case is the canonical one in the sapce of affine functionals). A
(discrete) measurement will be represented by a set of effects
$\{a_{i}\}$ such that $\sum_{i}a_{i}= u$. $\Omega$ can be naturally
embedded in the dual space $A(\Omega)^{\ast}$ using the map

\begin{eqnarray}
\omega\mapsto\hat{\omega}\nonumber\\
\quad \hat{\omega}(a):=a(\omega)
\end{eqnarray}

\noindent Let $V(\Omega)$ be the linear span of $\Omega$ in
$A(\Omega)^{\ast}$. Then, it is reasonable to consider $\Omega$
finite dimensional if and only if $V(\Omega)$ is finite dimensional.
For the sake of simplicity, we will restrict ourselves to this case
(and to compact spaces). As is well known, this implies that
$\Omega$ can be expressed as the convex hull of its extreme points.
the extreme points will represent \emph{pure states} (in the QM
case, pure quantum states are indeed the extreme points of
$\mathcal{C}$). It can be shown that for finite dimension $d$, a
system will be classical if and only if it is a simplex\footnote{A
simplex is the convex hull of $d+1$ linearly independent pure
states.}. It is a well known fact that in a simplex a point may be
expressed as a unique convex combination of its extreme points. This
characteristic feature of classical theories no longer holds in
quantum models. Indeed, in the case of QM, there are infinite ways
in which one can express a mixed as a convex combination of pure
states (for a graphical representation, think about the maximally
mixed state in the Bloch sphere).

\noindent Interestingly enough, there is also a connection between
the faces of the convex set of states of a given model and its
lattice of properties (in the quantum-logical sense), providing an
unexpected connection between geometry, lattice theory and
statistical theories. Faces of a convex set are defined as subsets
which are stable under mixing and purification. This is means that F
is a face if each time that

\begin{equation}
x=\lambda x_1+(1-\lambda)x_2, \,\,\,\,\,0\leq\lambda\leq1,
\end{equation}
\nd then $x\in F$ if and only if $x_1\in F$ and $x_2\in F$
\cite{BEN}. The set of faces of a convex set form a lattice in a
canonical way and it can be shown that the lattice of faces of a
classical model is a boolean one. On the other hand, in QM, the
lattice of faces of the convex set of states $\mathcal{C}$ (defined
as the set of positive trace class hermitian operators of trace
one), is isomorphic to the von Neumann lattice of closed subspaces
$\mathcal{P}(\mathcal{H})$ \cite{BEN,belcas81}.

\vskip 1mm

\nd Let us turn now to compound systems. Given a compound system,
its components will have state spaces $\Omega_{A}$ and $\Omega_{B}$.
Let us denote the joint state space by $\Omega_{AB}$ . It is
reasonable to identify $\Omega_{AB}$ with the linear span of
$(V(\Omega_{A})\otimes V(\Omega_{B}))$ \cite{Barnum-PRL}. Then, a
maximal tensor product state space
$\Omega_{A}\otimes_{max}\Omega_{B}$ can be defined as the one which
contains all bilinear functionals $\varphi:A(\Omega_{A})\times
A(\Omega_{B})\longrightarrow\mathbb{R}$ such that $\varphi(a,b)\geq
0$ for all effects $a$ and $b$ and $\varphi(u_{A},u_{B})=1$. The
maximal tensor product state space has the property of being the
biggest set of states in $(A(\Omega_{A})\otimes
A(\Omega_{B}))^{\ast}$ which assigns probabilities to all product-
measurements. The minimal tensor product state space
$\Omega_{A}\otimes_{min}\Omega_{B}$ is simply defined by the convex
hull of all product states. A product state will then be a state of
the form $\omega_{A}\otimes\omega_{B}$ such that

\begin{equation}
\omega_{A}\otimes\omega_{B}(a,b)=\omega_{A}(a)\omega_{B}(b),
\end{equation}
\noindent for all pairs $(a,b)\in A(\Omega_{A})\times
A(\Omega_{B})$. Given a particular compound system of a general
statistical theory, its set of states $\Omega_{AB}$ ---we will call
it $\Omega_{A}\otimes\Omega_{B}$ from now on--- will satisfy

\begin{equation}
\Omega_{A}\otimes_{min}\Omega_{B}\subseteq\Omega_{A}\otimes\Omega_{B}\subseteq\Omega_{A}\otimes_{max}\Omega_{B}
\end{equation}

\noindent As expected, for classical compound systems (because of
the absence of entangled states), we will have
$\Omega_{A}\otimes_{min}\Omega_{B}=\Omega_{A}\otimes_{max}\Omega_{B}$.
For the quantum case we have the strict inclusions
$\Omega_{A}\otimes_{min}\Omega_{B}\subseteq\Omega_{A}\otimes\Omega_{B}\subseteq\Omega_{A}\otimes_{max}\Omega_{B}$.
The general definition of a separable state in an arbitrary COM is
made in analogy with that of \cite{Werner}, i.e., as one which can
be written as a convex combination of product states
\cite{Barnum-Toner,Perinotti-2011}:

\begin{definition}\label{d:generalseparable}
A state $\omega\in\Omega_{A}\otimes\Omega_{B}$ will be called
\emph{separable} if there exist $p_{i}\in\mathbb{R}_{\geq 0}$,
$\omega^{i}_{A}\in\Omega_{A}$ and $\omega^{i}_{B}\in\Omega_{B}$ such
that

\begin{equation}
\omega=\sum_{i}p_{i}\omega^{i}_{A}\otimes\omega^{i}_{B},\quad \sum_i
p_i =1
\end{equation}

\end{definition}

\nd If $\omega\in\Omega_{A}\otimes\Omega_{B}$ is not separable,
then, it will be reasonably called \emph{entangled}
\cite{BEN,Schro1,Schro2}. As expected, entangled states exist only
if $\Omega_{A}\otimes\Omega_{B}$ is strictly greater than
$\Omega_{A}\otimes_{min}\Omega_{B}$.

\vskip 3mm

\nd The COM approach already shows that, given an arbitrary
statistical theory, there is a generalized notion of probabilities
of measurement outcomes. These probabilities are encoded on the
states in $\Omega$. And we have seen that there are many
differences between classical state spaces and non-classical ones:
this is expressed in the geometrical properties of their convex
state spaces and in the correlations appearing when compound
systems are considered. Indeed, QM and classical probability
theories are just particular COMs among a vast family of
possibilities. In the next Section, we will study with more detail
the axiomatization of the probabilities appearing in non-classical
models.

\section{Quantum Probabilities}\label{s:QuantumProbabilities}

In this Section we discuss QP. We start first by reviewing the usual
approach, in which Kolmogorov's axioms are extended to non-Boolean
lattices (or algebras) \cite{Redei-Summers2006}. Next, we review a
relatively recent approach to the probabilities appearing in QM
which uses distributive lattices
\cite{Symmetry,GoyalKnuthSkilling,KnuthSkilling-2012}.

\subsection{The Orthodox Approach}\label{s:QProbabilities}

\nd As we have seen in Section \ref{s:ElementaryTests} , elementary
tests in QM are represented by closed subspaces of Hilbert space.
These subspaces form an orthomodular atomic lattice
$\mathcal{P}(\mathcal{H})$. In order to assign probabilities to
these elementary tests or processes, many texts proceed by
postulating axioms which are similar to those of Kolmogorov
\cite{mikloredeilibro,vN,belcas81}. The Boolean $\Sigma$-algebra
appearing in Kolmogorov's axioms (Eqn. \eqref{e:kolmogorovian}) is
replaced by $\mathcal{P}(\mathcal{H})$:

\begin{subequations}\label{e:nonkolmogorov}
\begin{equation}
s:\mathcal{P}(\mathcal{H})\rightarrow [0;1]
\end{equation}
\noindent such that:
\begin{equation}\label{e:Qprobability1}
s(\textbf{0})=0 \,\, (\textbf{0}\,\, \mbox{is the null subspace}).
\end{equation}
\begin{equation}\label{e:Qprobability2}
s(P^{\bot})=1-s(P),\end{equation} \noindent and, for a denumerable
and pairwise orthogonal family of projections
\begin{equation}\label{e:Qprobability3}
\,\, {P_{j}}, \,\,s(\sum_{j}P_{j})=\sum_{j}s(P_{j}).
\end{equation}
\end{subequations}

\noindent In this way, a real number between $0$ and $1$ is assigned
to any elementary test. Despite of the similarity with Kolmogorov's
axioms, the probabilities defined above are very different, due to
the non-Boolean character of the lattice involved. This feature led
to many authors to assert that there are no quantum probabilities
(just because they are not Kolmogorovian). Gleason's theorem
\cite{Gleason,Gleason-Dvurechenski-2009} asserts that if the
dimension of $\mathcal{H}\geq 3$, any measure $s$ satisfying
\eqref{e:nonkolmogorov} can be put in correspondence with a trace
class operator (of trace one) $\rho_{s}$:

\begin{equation}\label{e:bornrule2}
s(P):=\mbox{tr}(\rho_{s} P)
\end{equation}

\nd On the other hand, using equation \eqref{e:bornrule2} any trace
class operator of trace one defines a measure as in
\eqref{e:nonkolmogorov}. In this way, equations
\eqref{e:nonkolmogorov} define the usual probabilities of QM.

\nd We have mentioned above that \eqref{e:kolmogorovian} and
\eqref{e:nonkolmogorov} are not equivalent probability theories. For
example, Eq. \eqref{e:SumRule} is no longer valid in QM. Indeed, for
suitably chosen quantum events $A$ and $B$ we have

\begin{equation}
s(A)+s(B)\leq s(A\vee B)
\end{equation}

\noindent The probability theory defined by \eqref{e:nonkolmogorov}
may be considered as a generalization of classical probability
theory in the following sense: while in an arbitrary statistical
theory a state will be a normalized measure over a suitable
$C^{\ast}$-algebra, the classical case is recovered when the algebra
is \emph{commutative}
\cite{Gudder-StatisticalMethods,Redei-Summers2006}. Difficulties
which appear when one tries to define a quantum conditional
probability. For a complete discussion about these matters and a
comparison between classical and quantum probabilities, see
\cite{Gudder-StatisticalMethods} and \cite{Redei-Summers2006}.

\subsection{Cox's Method Applied To Physics}\label{s:KnuthGoyal}

\noindent A novel derivation of Feynman's rules for quantum
mechanics was presented in refs. \cite{Symmetry},
\cite{GoyalKnuthSkilling}, and \cite{KnuthSkilling-2012}. An
experimental \emph{logic of processes} for quantum systems is
presented, and this is done in such a way that the resulting lattice
is a distributive one. This is a major difference with the approach
described in Section \ref{s:QProbabilities}, because the lattice of
projections in a Hilbert space is non-distributive.

\noindent The logic of processes is constructed as follows. Given a
sequence of measurements $M_1$,$\ldots$,$M_n$ on a quantum system,
yielding results $m_1$, $m_2$, $\ldots$, $m_n$, a particular process
is represented as a \emph{measuring sequence}
$A=[m_1,m_2,\ldots,m_n]$.

\noindent Next, conditional (logical) propositions
$[m_2,\ldots,m_n|m_1]$ are introduced. Using them, a probability is
naturally associated to a sequence $A$ with the formula

\begin{equation}
P(A)=Pr(m_n,\ldots,m_2|m_1)
\end{equation}

\noindent representing the probability of obtaining outcomes $m_2$,
$\ldots$, $m_n$ conditional upon obtaining $m_1$.

\noindent Let us see how this works with a concrete example in which
the $m_i$'s has two possible values, $0$ and $1$. Then $A_1=[0,1,1]$
and $A_2=[0,0,1,1]$ represent measuring sequences of three and four
measurements respectively. P(A)=Pr(1,1|0), represents the
probability of obtaining outcomes $m_2=1$ and $m_3=1$ conditional
upon obtaining $m_1=0$.

\noindent Measurements can be coarse grained as follows. Suppose
that we want to coarse grain $M_2$. Then, we can unite the two
outcomes $0$ and $1$ in a joint outcome $(0,1)$. Then, a new
measurement $\widetilde{M}_2$ is created. Thus, a possible sequence
obtained by the replacement of $M_2$ by $\widetilde{M}_2$ could be
$[1,(1,2),1]$. Analogous constructions can be done for other
measurements. In this way an operation can be defined for sequences:

\begin{equation}
[m_1,\ldots,(m_i,m'_i),\ldots,m_n]:=[m_1,\ldots,m_i,\ldots,m_n]\vee[m_1,\ldots,m'_i,\ldots,m_n]
\end{equation}

\noindent And another operation can be defined reflecting the fact
that sequences can be compounded as follows

\begin{equation}
[m_1,\ldots,m_j,\ldots,m_n]:=[m_1,\ldots,m_j]\cdot[m_j,\ldots,m_n]
\end{equation}

\noindent With these operations at hand, it is easy to show that if
$A$, $B$ and $C$ are measuring sequences, then

\begin{subequations}\label{e:ExperimentalLogicKnuth}

\begin{equation}
A\vee B=B\vee A
\end{equation}

\begin{equation}
(A\vee B)\vee C=A\vee(B\vee C)
\end{equation}

\begin{equation}
(A\cdot B)\cdot C=A\cdot(B\cdot C)
\end{equation}

\begin{equation}
(A\vee B)\cdot C=(A\cdot C)\vee(B\cdot C)
\end{equation}

\begin{equation}
C\cdot(A\vee B)=(C\cdot A)\vee(C\cdot B),
\end{equation}

\end{subequations}

\noindent Equations \eqref{e:ExperimentalLogicKnuth} show explicitly
that ``$\vee$" is commutative and associative, ``$\cdot$" is
associative, and that there is right- and left-distributivity of
``$\cdot$" over ``$\vee$".

\noindent Equations \eqref{e:ExperimentalLogicKnuth} define the
algebraic ``symmetries" of the experimental logic of processes. As
in the approach of Cox to classical probability, these symmetries
are used to derive Feynman's rules \cite{Symmetry}. But at this
step, a crucial assumption is made: each measuring sequence will be
represented by a pair of real numbers \cite{Symmetry}. This
assumption is justified in \cite{Symmetry} by appealing to Bohr's
complementarity principle.

\noindent If measuring sequences $A$, $B$, etc. induce pairs of real
numbers $\mathbf{a}$, $\mathbf{b}$, etc., then, due to equations
\eqref{e:ExperimentalLogicKnuth}, the associated real numbers should
satisfy

\begin{subequations}\label{e:ExperimentalLogicComplex}

\begin{equation}
\mathbf{a}\vee \mathbf{b}=\mathbf{b}\vee \mathbf{a}
\end{equation}

\begin{equation}
(\mathbf{a}\vee \mathbf{b})\vee
\mathbf{c}=\mathbf{a}\vee(\mathbf{b}\vee \mathbf{c})
\end{equation}

\begin{equation}
(\mathbf{a}\cdot
\mathbf{b})\cdot\mathbf{c}=\mathbf{a}\cdot(\mathbf{b}\cdot
\mathbf{c})
\end{equation}

\begin{equation}
(\mathbf{a}\vee \mathbf{b})\cdot \mathbf{c}=(\mathbf{a}\cdot
\mathbf{c})\vee(\mathbf{b}\cdot \mathbf{c})
\end{equation}

\begin{equation}
\mathbf{c}\cdot(\mathbf{a}\vee \mathbf{b})=(\mathbf{c}\cdot
\mathbf{a})\vee(\mathbf{c}\cdot \mathbf{b})
\end{equation}

\end{subequations}

\noindent The reader can easily verify that equations
\eqref{e:ExperimentalLogicComplex} are satisfied by the field of
complex numbers (provided that the operations are interpreted as sum
and product of complex numbers). How can we assure that complex
numbers are the only field which satisfies equations
\eqref{e:ExperimentalLogicComplex}? In order to single out complex
numbers among other possible fields, additional assumptions must be
added, namely, pair symmetry, additivity and symmetric bias (see
\cite{Symmetry,GoyalKnuthSkilling,KnuthSkilling-2012} for details).
Once these conditions are assumed, the path is clear to derive
Feynman's rules by applying a deduction similar to that of Cox, to
the experimental logic defined by equations
\eqref{e:ExperimentalLogicKnuth}.

\section{Generalization of Cox's method}\label{s:QMDerivation}

\noindent As we have seen in previous Sections, there are two
versions of CP, namely, the approach of R. T. Cox
\cite{CoxLibro,CoxPaper} and the one of A. N. Kolmogorov
\cite{KolmogorovProbability}. The Kolmogorovian approach can be
generalized in order to include non-Boolean models. In what follows,
we will see that Cox's method can also be generalized to
non-distributive lattices, and thus the non-commutative character of
QP can be captured in this framework
\cite{Holik-Plastino-Saenz-2012,HolikRingsOfOperators}.

\subsection{Generalized probability calculus using Cox's method}

\nd As we have seen in Section \ref{s:CoxReview}, Cox studies the
functions defined over a distributive lattices and derives classical
probabilities. In \cite{Holik-Plastino-Saenz-2012} it is shown that
if the lattice is assumed to be non-distributive, the properties of
QP described in Section \ref{s:QProbabilities} can be derived by
applying a variant of the Cox's method as follows (see
\cite{Holik-Plastino-Saenz-2012}). Suppose that propositions of our
system are represented by the lattice of elementary tests of QM,
i.e., the lattice $\mathcal{L}_{v\mathcal{N}}(\mathcal{H})$. This
lattice isomorphic to the lattice of projections
$\mathcal{P}(\mathcal{H})$ of the Hilbert space $\mathcal{H}$. The
goal is to show that the ``degree of implication" measure
$s(\cdots)$ demanded by Cox's method satisfies Eqs.
\eqref{e:nonkolmogorov}. This means that we are looking for a
function to the real numbers $s$, such that it is non-negative and
$s(P)\leq s(Q)$ whenever $P\leq Q$.

\noindent The operation ``$\vee$" in $\mathcal{P}(\mathcal{H})$ is
associative. Then, if $P$ and $Q$ are orthogonal projections, the
relationship between $s(P)$, $s(Q),$ and $s(P\vee Q)$ must be of the
form

\begin{equation}\label{e:s(or)}
s(P\vee Q)=F(s(P),s(Q)),
\end{equation}

\noindent with $F$ a function to be determined. If a third
proposition``$\vee$" and following a similar procedure to that of
Cox, the following functional equation is found

\begin{equation}\label{e:FunctEq1}
F(F(s(P),s(Q)),s(R))=F(s(P),F(s(Q),s(R))).
\end{equation}

\noindent The above equation can be solved up to rescaling
\cite{Knuth-2004a,Knuth-2004b,Knuth-2005b,aczel-book}, and we find

\begin{equation}
s(P\vee Q)=s(P)+s(Q).
\end{equation}

\nd whenever $P\perp Q$. It can be shown that for any finite family
of orthogonal projections $P_j$, $1\leq j\leq n$
\cite{Holik-Plastino-Saenz-2012}:

\begin{equation}
s(\bigvee_{j=1}^{\infty}P_j)=\sum_{j=1}^{\infty}s(P_j),
\end{equation}

\nd and we  recover condition \eqref{e:Qprobability3} of the axioms
of quantum probability. By exploiting the properties of the
orthogonal complement acting on subspaces, it can also be shown
\cite{Holik-Plastino-Saenz-2012} that

\begin{equation}
s(P^{\perp})=1-s(P),
\end{equation}

\nd which is nothing but condition \eqref{e:Qprobability2}. On the
other hand, as $\mathbf{0}=\mathbf{0}\vee\mathbf{0}$ and
$\mathbf{0}\bot\mathbf{0}$, then
$s(\mathbf{0})=s(\mathbf{0})+s(\mathbf{0})$, and thus,
$s(\mathbf{0})=0$, which is condition \eqref{e:Qprobability1}. In
this way, the Cox's method applied to the non-distributive lattice
$\mathcal{L}_{v\mathcal{N}}(\mathcal{H})$ (or
$\mathcal{P}(\mathcal{H})$) yields the same probability theory as
the one provided by equations \eqref{e:nonkolmogorov} for the
quantum case.

\noindent What happens if the Cox's method is applied to an
arbitrary atomic orthomodular lattice $\mathcal{L}$? Now we must
define a function $s:\mathcal{L}\longrightarrow \mathbb{R}$, such
that it is always non-negative $s(a)\geq 0\,\,\forall
a\in\mathcal{L}$ and is also order preserving $a\leq b
\longrightarrow s(a)\leq s(b)$. In \cite{Holik-Plastino-Saenz-2012}
it is shown that under these rather general assumptions, in
\emph{any} orthomodular lattice and for any orthogonal denumerable
family $\{a_i\}_{i\in\mathbb{N}}$, $s$ must satisfy (up to
rescaling)

\begin{subequations}\label{e:GeneralizedProbability}
\begin{equation}
s(\bigvee\{a_i\}_{i\in\mathbb{N}})=\sum_{i=1}^{\infty}s(a_i)
\end{equation}
\begin{equation}
s(\neg a)= 1-s(a)
\end{equation}
\begin{equation}
s(\mathbf{0})=0.
\end{equation}
\end{subequations}

\vskip3mm

\nd In this way, a generalized probability theory is derived (as in
\eqref{e:nonkolmogorov}). Equations \eqref{e:GeneralizedProbability}
define non-classical (non-Kolmogorovian) probability measures, due
to the fact that in any non-distributive orthomodular lattice, there
always exist elements $a$ and $b$ such that

\begin{equation}
(a\wedge b)\vee (a\wedge\neg b)<a,
\end{equation}

\noindent But in any classical probability theory, $s(a\wedge\neg
b)+s(a\wedge b)= s(a)$ is always satisfied.

\section{Conclusions}
\label{s:Conclusions}

\nd We have presented a new approach to probabilities appearing in
QM. While there exist two alternative formalisms to CP (the
Kolmogorovian and the one due to R. T. Cox), we have also shown
that these two approaches can be extended to the non-commutative
case. In this way, we find that CP are just a particular case of a
more general mathematical framework in which, the lattice is
distributive. QP is also a particular case of a vast family of
theories for which the propositional lattice is non-distributive.
Thus, we have a precise formal expression of the notion of QP. Of
course, these formal frameworks does not exhaust the philosophical
debate around the existence or not of a well defined notion of QP;
notwithstanding, the extension of the Cox's method to the
non-distributive case and the possibility of including a
description of the probabilities in QM in it, constitutes a
precise step towards understanding the notion of QP offering a new
point of view of this notion.

\vskip1truecm

\noindent {\bf Acknowledgements} \noindent This work was partially
supported by the  grants PIP N$^o$ 6461/05 amd 1177 (CONICET). Also
by the projects FIS2008-00781/FIS (MICINN) - FEDER (EU) (Spain, EU).

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