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==Foundations of classical and quantum probability.==
Quantum theory emerged as a new mechanics, but it was soon realized that it was also a new [[Probability theory|probability theory]]. The difference between classical and [[Quantum probability|quantum probability]] is usually taken to be the fact that in the former probabilities of disjoint events add, while in the latter amplitudes of disjoint events add, the amplitude of an event being a complex number whose square is the probability of the event. A more precise way of saying this is that the classical theorem of composite probabilities

$$ \tag{a1 }
\sum _ { j } {\mathsf P} ( a\mid  b _ {j} ) {\mathsf P} ( b _ {j} )  =  {\mathsf P} ( a)
$$

is replaced in the quantum case by the theorem of composite amplitudes

$$ \tag{a2 }
\sum _ { j } \psi ( a\mid  b _ {j} ) \psi ( b _ {j} )  =  \psi ( a) ,
$$

where the two identities have to be interpreted as follows: Given the amplitudes  $  \psi ( a\mid  b _ {j} ) $, 
$  \psi ( b _ {j} ) $
and the corresponding classical probabilities  $  {\mathsf P} ( a\mid  b _ {j} )= | \psi ( a \mid  b _ {j} ) |  ^ {2} $, 
$  {\mathsf P} ( b _ {j} )= | \psi ( b _ {j} ) |  ^ {2} $, 
the classical prediction of the value  $  {\mathsf P} ( a) $
is the left-hand side of (a1), while the quantum prediction is the square modulus of the left-hand side of (a2) (the  $  b _ {j} $
are mutually disjoint indecomposable events of which at least one happens with certainty). The difference between the two predictions is a sum of terms called interference terms.

The validity of the theorem of composite probabilities can be shown to be equivalent to the validity of Bayes' definition of classical probability (cf. [[Bayesian approach|Bayesian approach]]). This definition can be considered as one of the axioms defining the classical probabilistic model. Since this axiom cannot be true in the quantum probabilistic model, the following problems naturally arise:

i) From which physically plausible axioms can the quantum probabilistic model be deduced?

ii) Can one produce statistical invariants which would uniquely distinguish among the different probabilistic models on the basis of experimentally measurable data (just as geometric invariants distinguish the different models of space)?

An answer to i) can be given by constructing a system of axioms for probability based not on the notion of event, but on that of measurement. It was shown by J. Schwinger [[#References|[a13]]] that the natural operations one can perform on the elementary measurements define on these a structure of a  $  * $-
algebra of very special type (Schwinger algebras). A well-known classification theorem describes the structure of Schwinger algebras, hence of models for these axioms. These include the usual quantum models, but new interesting models arise.

Unlike for geometrical invariants, there is no single universal method to compute statistical invariants. However, in certain particular cases of direct physical meaning, these invariants have been computed and compared with experimentally measured probabilities, showing that the quantum models are non-Kolmogorovian. Using this technique, the difference between the use of real or complex Hilbert spaces in the quantum model has also been reduced to an experimentally verifyable fact.

Several physicists relate the non-existence of a single classical probabilistic model which would describe a given set of statistical data with the breaking of certain physical properties such as locality or causality.

==Algebraic probability theory.==
Certain probabilistic results (like the [[Law of large numbers|law of large numbers]], the [[Central limit theorem|central limit theorem]], De Finetti's theorem, etc.) have their origin in very general algebraic and combinatorial properties of the notion of statistical [[Independence|independence]]. Certain basic properties of stochastic processes, in particular of Markov and Gaussian processes, are purely algebraic. Finally, certain fundamental techniques and results in probability theory (Doob–Meyer decomposition, the notions of mutual quadratic variation, [[Stochastic differential equation|stochastic differential equation]] and [[Langevin equation|Langevin equation]], etc.) are best understood when their algebraic content is separated from their analytic content. The study of the algebraic properties is the object of algebraic probability theory.

An algebraic probability space is a pair  $  \{ {\mathcal A} , \phi \} $, 
where  $  {\mathcal A} $
is a  $  * $-
algebra and  $  \phi $
is a state on  $  {\mathcal A} $. 
An algebraic probability space  $  \{ {\mathcal A} , \phi \} $
is called classical if  $  {\mathcal A} $
is a commutative algebra, and quantum if  $  {\mathcal A} $
is non-commutative. The main example of a non-classical algebraic probability space is a pair  $  \{ {\mathcal B} ( H) , \phi \} $, 
where  $  \phi $
is a normal state on the algebra  $  {\mathcal B} ( H) $
of bounded operators on a Hilbert space  $  H $. 
Any such state has the form

$$ \tag{a3 }
\phi ( b)  =   \mathop{\rm Tr} ( wb),\  b \in {\mathcal B} ( H),
$$

where  $   \mathop{\rm Tr} $
is the trace on  $  {\mathcal B} ( H) $
and  $  w $
is a positive operator of trace  $  1 $, 
called a density operator (cf. [[Trace on a C*-algebra|Trace on a  $  C  ^ {*} $-
algebra]]). Other examples can be obtained by replacing  $  {\mathcal B} ( H) $
by the polynomial algebra in the quantum fields of a representation.

The notions of a random variable and a stochastic process are introduced in analogy with the classical case (cf. [[Quantum probability|Quantum probability]]).

The notion of independence of a family of random variables, which in classical probability is expressed in terms of factorization of the joint correlation functions, is much more involved in quantum probability, since various forms of non-commutativity of random variables may occur. The notions of boson independence and fermion independence, which are of crucial importance in [[Quantum field theory|quantum field theory]] and quantum statistical mechanics (cf. [[Statistical mechanics, mathematical problems in|Statistical mechanics, mathematical problems in]]), and the notion of free independence (recently introduced by Voiculescu) are examples of the variety of possibilities.

==Laws of large numbers and central limit theorems.==
Let  $  {\mathcal A} $, 
$  {\mathcal B} $
be  $  * $-
algebras. For each  $  n \in \mathbf N $, 
let there be given a random variable  $  j _ {n} :  {\mathcal B} \rightarrow {\mathcal A} $(
cf. [[Quantum probability|Quantum probability]]). The sum of the first  $  N $
random variables is

$$ 
S _ {N} ( b)  =  \sum _ { k= } 1 ^ { N }  j _ {k} ( b) ,\  b \in {\mathcal B} ,\ \ 
N \in \mathbf N .
$$

In the law of large numbers and the quantum central limit theorem one is interested in the limit, as  $  N \rightarrow \infty $, 
of expressions of the form

$$ 
{\mathsf E} \left ( P \left [ 
\frac{S _ {N} ( b _ {1} ) }{N  ^  \alpha  }
 \dots

\frac{S _ {N} ( b _ {h} ) }{N  ^  \alpha  }
 \right ] \right ) ,
$$

where  $  \alpha > 0 $
is a scalar,  $  b _ {1} \dots b _ {h} \in {\mathcal B} $
and  $  P $
is a polynomial in the  $  h $
non-commuting indeterminates  $  X _ {1} \dots X _ {h} $. 
In the classical case one considers functions  $  P $
of a more general form (continuous, bounded, etc.), but if the  $  b _ {j} $
do not commute, such expressions have no natural meaning unless  $  P $
is a polynomial. The law of large numbers corresponds to the case  $  \alpha = 1 $, 
and the central limit theorem to the case  $  \alpha = 1/2 $. 
This formulation of the quantum central limit theorem, due to W. von Waldenfels, has the advantage of being independent of commutation relations among the random variables. Under assumptions of independence (boson, fermion or free) or weak dependence one can prove laws of large numbers and quantum central limit theorems.

In particular, the quasi-free states of quantum field theory (both boson and fermion) can be shown to arise from quantum central limit theorems, just as the usual Gaussian measures arise from classical central limit theorems. For this reason these quasi-free states are also called quantum Gaussian states. The Heisenberg commutation relations are also a quantum central limit effect, in the sense that the cyclic representation associated with a Gaussian state is a representation of the CCR (CAR in the fermion case, cf. [[Commutation and anti-commutation relationships, representation of|Commutation and anti-commutation relationships, representation of]]) with possibly degenerate bilinear form.

Finally, the quantum analogue of the invariance principle (functional central limit theorem) leads to the quantum Brownian motions introduced in the 1960s in laser theory [[#References|[a9]]].

==Conditioning.==
In von Neumann's unifying scheme for classical and quantum probability an important ingredient was missing: conditioning. In order to study non-trivial statistical dependences, in particular, to construct Markov chains, this gap had to be filled. A first attempt in this direction led to the notion of the Umegaki conditional expectation from a  $  * $-
algebra  $  {\mathcal A} $
onto a  $  * $-
subalgebra  $  {\mathcal A} _ {0} $
as a completely-positive linear mapping  $  E:  {\mathcal A} \rightarrow {\mathcal A} _ {0} $
such that

$$ 
a \geq  0  \Rightarrow  E( a) \geq  0,\  a \in {\mathcal A} ;
$$

$$ 
E( a _ {0} a)  =  a _ {0} E( a),\  a _ {0} \in {\mathcal A} _ {0} ,\  a \in {\mathcal A};
$$

$$ 
E( 1)  =  1;
$$

$$ 
E( a)  ^ {*} E( a)  \leq   E( a  ^ {*} a) ,\  a \in {\mathcal A}.
$$

A Umegaki conditional expectation is said to be compatible with a state  $  \phi $
if  $  \phi \circ E= \phi $. 
In the classical case any subalgebra of a [[Von Neumann algebra|von Neumann algebra]]  $  {\mathcal A} $
is the range of a Umegaki conditional expectation  $  E $, 
while if  $  \phi $
is a state on  $  {\mathcal A} $, 
then  $  E $
can be chosen compatible with  $  \phi $. 
Neither of these properties is true in the quantum case (and a subalgebra that is the range of a Umegaki expectation is called expected). A more general notion of conditioning can be introduced, and the corresponding notion of generalized conditional expectation proved to be a valuable tool in the solution of certain open problems in operator theory (the split property, the Stone–Weierstrass conjecture for factors, canonical morphism of local algebras, etc.). Deeply related to these problems is the Cecchini–Petz theory of state extensions, as well as Cecchini's analysis of various notions of Markovianity in von Neumann algebras.

==Markov chains.==
Let  $  {\mathcal B} _ {0} $, 
$  {\mathcal B} _ {1} $
be  $  C  ^ {*} $-
algebras. A transition expectation from  $  {\mathcal B} _ {0} \otimes {\mathcal B} _ {1} $
to  $  {\mathcal B} _ {0} $
is a completely-positive mapping  $  E :  {\mathcal B} _ {0} \otimes {\mathcal B} _ {1} \rightarrow {\mathcal B} _ {0} $
satisfying

$$ 
E( 1\otimes 1)  =  1.
$$

If  $  {\mathcal B} _ {0} = {\mathcal B} _ {1} = {\mathcal B} $, 
one says that  $  E $
is a transition expectation on  $  {\mathcal B} $.

Given a transition expectation  $  E $
from  $  {\mathcal B} \otimes {\mathcal B} _ {0} $
to  $  {\mathcal B} _ {0} $
and a state  $  \phi _ {0} $
on  $  {\mathcal B} _ {0} $, 
the (homogeneous) generalized Markov chain (cf. [[Markov chain, generalized|Markov chain, generalized]]) associated to the pair  $  \{ \phi , E \} $
is the state  $  \phi $
on  $  \otimes _ {\mathbf N }  {\mathcal B} $
characterized by the property that for each integer  $  n $
and all  $  a _ {0} \dots a _ {n} \in {\mathcal B} $
one has

$$ 
\phi _ {0} (
a _ {0} \otimes \dots \otimes a _ {n} \otimes 1 \otimes \dots )  =
$$

$$ 
\phi _ {0} ( a _ {0} \otimes E( a _ {1} \otimes \dots \otimes E( a _ {n} \otimes 1) )).
$$

M. Fannes, B. Nachtergaele and R.F. Werner [[#References|[a10]]] did realize that the valence bond states, introduced by Anderson in an attempt to explain the phenomenon of high temperature superconductivity and studied in detail by many authors, are a particular class of quantum Markov chains. This allowed them, using the general theory of quantum Markov chains, to generalize the construction of these states to arbitrary dimensions, to drastically simplify the proofs and to obtain new results.

==Quantum Markov processes with continuous parameter.==
Let  $  {\mathcal A} $
be a  $  C  ^ {*} $-
algebra. A past filtration on  $  {\mathcal A} $
is a family  $  \{ {\mathcal A} _ { {} t ] }  \} $
of  $  C  ^ {*} $-
algebras contained in  $  {\mathcal A} $
such that  $  {\mathcal A} _ { {} s ] }  \subset  {\mathcal A} _ { {} t ] }  $
if  $  s \leq  t $. 
Similarly one defines a future filtration  $  \{ {\mathcal A} _ {[ t {} }  \} $. 
For each closed interval  $  [ s, t] $, 
the local algebra  $  {\mathcal A} _ {[ s,t] }  $
is defined to be  $  {\mathcal A} _ {[ s{} }  \cap {\mathcal A} _ { {} t] }  $; 
the algebra  $  {\mathcal A} _ {[ t,t] }  $
is called the present algebra at time  $  t $, 
and is denoted by  $  {\mathcal A} _ {\{ t \} }  $.

A time shift for a filtration is a one-parameter endomorphism group  $  \mu _ {t} $
of  $  {\mathcal A} $
such that  $  \mu _ {t} ( {\mathcal A} _ { {} t] }  ) \subset  {\mathcal A} _ { {}} t+ s] $(
and similarly for the past and present algebras) and each  $  \mu _ {t} $
has a left inverse, denoted by  $  \mu _ {t}  ^ {*} $. 
A completely-positive mapping  $  E _ { {}} t] :  {\mathcal A} _ { {}} t] \rightarrow {\mathcal A} _ { {}} s] $
is called Markovian if

$$ 
E _ { {}} s] ( {\mathcal A} _ {[} s,t] )  \subset   {\mathcal A} _ {\{ s \} }  .
$$

It is called projective if

$$ 
E _ { {}} s] \cdot E _ { {}} t]  =  E _ { {}} s] ,\  s< t.
$$

The mappings  $  E _ { {}} t] $
are called covariant with respect to the time shift if

$$ 
\mu _  \tau  \circ E _ { {}} t]  =  E _ { {} t+ \tau ] }  \circ \mu _  \tau  .
$$

These assumptions imply that all the algebras at a fixed time,  $  {\mathcal A} _ {t} $, 
are isomorphic to a fixed algebra  $  {\mathcal B} $, 
independent of  $  t \in T $; 
$  {\mathcal B} $
is often called the initial algebra. Under the assumptions above, the completely-positive mappings

$$ 
P _ {t}  ^ {0}  =  E _ { {}} 0] \circ \mu _ {t} \mid  _ { {\mathcal A} _ {0}  }
$$

form a completely-positive identity-preserving one-parameter semi-group on  $  {\mathcal A} _ {0} $, 
called a Markovian semi-group, or, when  $  {\mathcal B} $
is non-commutative, a quantum dynamical evolution. The triple  $  \{ {\mathcal B} , ( \mu _ {t} ) , j _ {0} \} $, 
where  $  j _ {0} $
is the imbedding of  $  {\mathcal B} $
into  $  {\mathcal A} _ {0} $, 
is called a dilation of the pair  $  \{ {\mathcal B} , ( P _ {t}  ^ {0} ) \} $. 
If the mapping  $  E _ { {}} 0] $
is a Umegaki conditional expectation, one speaks of a dilation in the sense of Kümmerer.

A two-parameter family of completely-positive mappings from  $  {\mathcal A} _ {0} $
into itself satisfying, for every  $  r< s< t $, 
the conditions:

$$ 
m _ {r,s} \circ m _ {s,t}  =  m _ {r,t,}  $$

$$ 
m _ {s,t,} ( {\mathcal A} _ {[ s,u] }  )  \subset   {\mathcal A} _ {[ s,u] }  ,
$$

$$ 
m _ {r,s} \circ E _ { {}} t]  =  E _ { {}} t] \circ m _ {r,s} ,
$$

$$ 
m _ {s,t} ( 1)  =  1,
$$

$$ 
\mu _ {r} \circ m _ {s,t,\ } =  m _ {s+ r,t+ r }  \circ \mu _ {r}  $$

is called a homogeneous multiplicative functional, in analogy with classical probability. In particular, the one-parameter family

$$ 
P  ^ {t}  =  E _ { {}} 0] \circ m _ {0,t}  $$

is a semi-group, called a Feynman–Kac perturbation of the semigroup  $  P _ {t}  ^ {0} $. 
This perturbation technique is called the Feynman–Kac formula, and is at the basis of all known constructions of continuous-time quantum Markov processes. The homogeneous multiplicative functionals satisfying a stochastic differential equation are called stochastic flows, or Evans–Hudson flows.

==Quantum stochastic calculus.==
The random variables of the standard real-valued [[Wiener process|Wiener process]]  $  W _ {t} $
act by multiplication on the space  $  L _ {2} $
of the process, and give rise to a self-adjoint multiplication operator  $  Q _ {t} $(
$  Q _ {t} f ( \omega ) = W _ {t} ( \omega ) \cdot f ( \omega ) $, 
$  f \in L _ {2} $). 
Rotating this process  $  Q _ {t} $, 
regarded as an operator-valued process, by a unitary operator  $  U $, 
gives a new process  $  U  ^ {*} Q _ {t} U = P _ {t} $
which is unitarily isomorphic to the original Wiener process. If  $  U $
is properly chosen (as the infinite-dimensional analogue of the Gauss–Fourier transform considered by Wiener and Segal), the pair of processes  $  Q _ {t} , P _ {t} $
satisfies the Heisenberg commutation relations:  $  [ Q _ {s} , P _ {t} ] = i s \wedge t $. 
A pair  $  Q _ {s} , P _ {t} $
of operator-valued processes with the above properties is called a standard quantum Brownian motion (cf. also [[Brownian motion|Brownian motion]]). It was introduced in the 1960s in the study of dissipative quantum systems and in quantum optics, in connection with laser theory [[#References|[a9]]]. Classical stochastic calculus has been generalized to this process by R.L. Hudson and K.R. Parthasarathy [[#References|[a12]]], which gave rise to a number of important applications in quantum physics. The same authors introduced the number (or gauge) process as a quantum generalization of the usual Poisson process.

The quantum Brownian motions arise in physics as natural approximations of the free quantum electromagnetic field or of thermal reservoirs. More precisely: in the weak coupling limit of a system interacting with a Gaussian quantum field, the usual Heisenberg equations are approximated by a quantum stochastic differential equation driven by a quantum Brownian motion. Some of the basic features of this approximation already appear at the level of second-order perturbation theory (precisely these effects were discovered in the 1960s by laser theorists).

Nothing similar can be said for the quantum Poisson process which approximates the description of a system interacting with a very dilute gas (in the low density limit). However, in this approximation, any effect related to the quantum Poisson process receives contributions from the whole perturbation series, and it would have been impossible to isolate these contributions without the intuitive guide provided by quantum stochastic calculus.

In the probabilistic analogy, the weak coupling limit approximation arises from a sum of uniformly infinitesimal quantum fields — a situation strongly reminiscent of classical central limit theorems, while the low density limit approximation corresponds to rare individual events (low density), each of which, however, has a finite intensity, a situation reminiscent of the classical Poisson limit theorems.

Presently, various quantum stochastic calculi have been constructed, corresponding to different quantum fields. These have created a bridge between classical and quantum probability, of which more and more classical probabilists are taking advantage.

There are quantum generalizations of such classical probabilistic results as the Lévy martingale representation theorem, the Kunita–Watanabe and Doob–Meyer theorems, theorems on the structure of multiplicative functionals, stop times, etc.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> , ''Quantum probability and applications to the quantum theory of irreversible processes (Proc. Arco Felice (1978))''</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Accardi (ed.)  W. von Waldenfels (ed.) , ''Quantum probability and applications to the quantum theory of irreversible processes (Proc. Villa Mondragone (1982))'' , ''Lect. notes in math.'' , '''1055''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Accardi (ed.)  W. von Waldenfels (ed.) , ''Quantum probability and applications II (Proc. Heidelberg (1984))'' , ''Lect. notes in math.'' , '''1136''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Accardi (ed.)  W. von Waldenfels (ed.) , ''Quantum probability and applications III (Proc. Oberwolfach (1986))'' , ''Lect. notes in math.'' , '''1303''' , Springer  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Accardi (ed.)  W. von Waldenfels (ed.) , ''Quantum probability and applications IV (Proc. Rome (1987))'' , ''Lect. notes in math.'' , '''1396''' , Springer  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Accardi (ed.)  et al. (ed.) , ''Quantum probability and applications V (Proc. Heidelberg (1988))'' , ''Lect. notes in math.'' , Springer  (To appear)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  "Quantum probability and applications VI (Proc. Trento (1989))" , ''Quantum Probability and Related Fields'' , World Sci.  (1991)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  "Quantum probability and applications VII (Proc. New Delhi (1990))" , ''Quantum Probability and Related Fields'' , World Sci.  (1991)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  H. Haken,   "Laser theory" , Springer  (1984)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Fannes,   B. Nachtergaele,   R.F. Werner,   "Finitely correlated states on quantum spin chains"  ''Preprint''</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  R.P. Feynman,   "Lectures on physics" , '''III''' , Addison-Wesley  (1966)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  R.L. Hudson,   K.R. Parthasarathy,   "Quantum Ito's formula and stochastic evolutions"  ''Comm. Math. Phys.'' , '''93'''  (1984)  pp. 301–323</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  J. Schwinger,   "Quantum kinematics and dynamics" , Acad. Press  (1970)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J. von Neumann,   "Mathematical foundations of quantum dynamics" , Princeton Univ. Press  (1955)</TD></TR></table>