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