The theory of probability used in QM is intrinsically different from the one commonly used for the following reason: The space of events is non-commutative (more properly non-Boolean) and this fact deeply affects the conditional probability theory. The probability that A happens if B happened is computed differently in classical probability theory and in quantum theory, when A and B are quantistically incompatible events. In both cases probability is a measure on a lattice, but, in the classical case, the lattice is a Boolean one (a $\sigma$-algebra), in the quantum case it is not.
To be clearer, classical probability is a map $\mu: \Sigma(X) \to [0,1]$ such that $\Sigma(X)$ is a class of subsets of the set $X$ including $\emptyset$, closed with respect to the complement and the countable union, and such that $\mu(X)=1$ and:
$$\mu(\cup_{n\in \mathbb N}E_n) = \sum_n \mu(E_n)\quad \mbox{if $E_k \in \Sigma(X)$ with $E_p\cap E_q= \emptyset$ for $p\neq q$.}$$
The elements of $\Sigma(X)$ are the events whose probability is $\mu$. In this view, for instance, if $E,F \in \Sigma(X)$, $E\cap F$ is logically interpreted as the event "$E$ and $B$".
Similarly $E\cup F$ corresponds to "$E$ or $F$" and $X\setminus F$ has the meaning of "not $F$" and so on.
The probability of $P$ when $Q$ is given verifies $$\mu(P|Q) = \frac{\mu(P \cap Q)}{\mu(Q)}\:.\tag{1}$$
If you instead consider a quantum system, there are "events", i.e. elementary "yes/no" propositions experimentally testable, that cannot by joined by logical operators.
An example is $P=$"the $x$ component of this electron is $1/2$" and $Q=$"the $y$ component of this electron is $1/2$". There is no experimental device able to assign a truth value to $P$ and $Q$ simultaneously, so that elementary propositions as "$P$ and $Q$" make no sense. Pairs of propositions like $P$ and $Q$ above are physically incompatible.
In quantum theories (the most elementary version due to von Neumann), the events of a physical system are represented by the orthogonal projectors of a separable Hilbert space $H$. The set ${\cal P}(H)$ of those operators replaces the classical $\Sigma(X)$.
In general, the meaning of $P\in {\cal P}(H)$ is something like
"the value of the observable $Z$ belongs to the subset $I \subset \mathbb R$" for some observable $Z$ and some set $I$. There is a procedure to integrate such class of projectors to construct a self-adjoint operator, and this is the physical meaning of the spectral theorem.
If $P, Q \in {\cal P}(H)$, there are two possibilities: $P$ and $Q$ commute or they do not.
Von Neumann's fundamental axiom states that commutability is the mathematically corresponding of physical compatibility.
When $P$ and $Q$ commutes $PQ$ and $P+Q-PQ$ still are orthogonal projectors, that is elements of ${\cal P}(H)$.
In this situation, $PQ$ corresponds to "$P$ and $Q$", whereas $P+Q-PA$ corresponds to "$P$ or $Q$" and so on, and classical formalism holds true this way.
As a matter of fact, a maximal set of pairwise commuting projectors has formal properties identical to those of classical logic: is a Boolean $\sigma$-algebra.
In this picture, a quantum state is a map assigning the probability $\mu(P)$ that $P$ is experimentally verified for every $P\in {\cal P}(H)$.
It has to satisfy: $\mu(I)=1$ and
$$\mu(\sum_{n\in \mathbb N}P_n) = \sum_n \mu(P_n)\quad \mbox{if $P_k \in {\cal P}(H)$ with $P_p P_q= P_qP_q =0$ for $p\neq q$.}$$
Celebrated Gleason's Theorem, establishes that, if $dim(H)>2$, the measures $\mu$ are all of the form $\mu(P)= tr(\rho_\mu P)$ for some mixed state $\rho_\mu$ (a positive trace-class operator with unit trace), biunivocally determined by $\mu$.
In the convex set of states, the extremal elements are the standard pure states. They are determined, up to a phase, by unit vectors $\psi \in H$, so that, with some trivial computation (completing $\psi_\mu$ to an orthonormal basis of $H$ and using that basis to compute the trace),
$$\mu(P) = \langle \psi_\mu | P \psi_\mu \rangle = ||P \psi_\mu||^2\:.$$
(Nowadays, there is a generalized version of this picture where the set ${\cal P}(H)$ is replaced by the class of bounded positive operators in $H$ (the so-called "effects") and Gleason's theorem is replaced by Busch's theorem with a very similar statement.)
Quantum probability is therefore given by the map, for a given generally mixed state $\rho$,
$${\cal P}(H) \ni P \mapsto \mu(P) =tr(\rho_\mu P) $$
It is clear that, as soon as one deals with physically incompatible
propositions, (1) cannot hold just because there is nothing like $P \cap Q$ in the set of physically sensible quantum propositions.
All that is due to the fact that the space of events ${\cal P}(H)$ is now noncommutative.
ADDENDUM. Actually, it is possible to extend the notion of logical operators $AND$ and $OR$ even for ${\cal P}(H)$ and that was the program of von Neumann and Birkhoff (the quantum logic). In fact just the lattice structure of ${\cal P}(H)$ permits it, or better is it. The point is that the physical interpretation of this extension of $AND$ and $OR$ is not clear. The resulting lattice is however non-Boolean. In other words, for instance, these extended $AND$ and $OR$ are not distributive as the standard $AND$ and $OR$ are (this reveals their quantum nature). However, the found structure is well known: A $\sigma$-complete, orthomodular, separable, atomic, irreducible and verifying the covering property, lattice. At the end of 1900 it was definitely proved, by Solèr, a conjecture due to von Neumann stating that there are only three possibilities for practically realizing such lattices: The lattice of orthogonal projectors on a separable complex Hilbert space, the lattice of orthogonal projectors on a separable real Hilbert space, the lattice of orthogonal projectors on a separable quaternionic Hilbert space. Requiring the existence time reversal symmetry introduces a complex structure on real Hilbert spaces giving rise to a complex Hilbert space. Conversely, still nowadays, it is not obvious if it is possible to rule out quaternionic Hilbert spaces to describe quantum physics.
This post imported from StackExchange Physics at 2014-06-03 16:29 (UCT), posted by SE-user V. Moretti
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