Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.

The interference manifested in the two-slit experiments with small particles is one of the best known and most typical quantum phenomena. It is somewhat surprising therefore that quantum mechanics rules out third-order interference. This was discovered by Sorkin [

In the present paper, Sorkin’s interference terms

In this general framework, the identity

Besides the identity

The next two sections summarize those parts of [

A quantum logic is the mathematical model of a system of quantum events or propositions. Logical approaches use the name “proposition”, while the name “event” is used in probability theory and will also be preferred in the present paper. The concrete quantum logic of standard quantum mechanics is the system of closed linear subspaces of a Hilbert space or, more generally, the projection lattice in a von Neumann algebra.

Usually, an abstract quantum logic is assumed to be an orthomodular partially ordered set and, very often, it is also assumed that it is lattice. For the purpose of the present paper, however, a more general and simpler mathematical structure without order relation is sufficient. Only an orthocomplementation, an orthogonality relation, and a sum operation defined for orthogonal events are needed. The orthocomplementation represents the logical negation, orthogonality means mutual exclusivity, and the sum represents the logical and operation in the case of mutual exclusivity. The precise axioms were presented in [

The quantum logic

Then

The supremum exists in this case due to the orthomodularity.

A state is a map

If these axioms are satisfied,

Note that the following identity which will be used later holds for convex combinations of states

A typical example of the above structure is the projection lattice

For the proof of (

Equation (

A quantum logic with a sufficiently rich state space as postulated by (UC1) can be embedded in the unit interval of an order-unit space. In the present section, it will be shown that the existence and the uniqueness of the conditional probabilities postulated by (UC2) give rise to some important additional structure on this order-unit space, which was originally presented in [

A partially ordered real vector space

The order-unit space

For any set

Suppose that A is an order-unit space with order unit

For each

For

Suppose

Note that the linear extension

Each UCP space E is a subset of the interval

Define

Now the positive projection

If

Therefore

If e and f are events in a UCP space E with

Suppose

Now assume that

Moreover,

The projections

An element

With two disjoint events

For instance, consider the two-slit experiment and let

With three orthogonal events

Similar to the two-slit experiment, now consider an experiment where the screen has three instead of two slits. Let

Considering experiments with three and more slits, this third-order interference term and a whole sequence of further higher order interference terms were introduced by Sorkin [

Using the identity

This interference term shall now be studied in a von Neumann algebra where the conditional probability has the shape

Therefore, in a von Neumann algebra and in standard quantum mechanics, the identity

Suppose that

Suppose

In a von Neumann algebra,

Alfsen and Shultz introduced the following symmetry condition for the conditional probabilities in [

The first summand on the left-hand side

Figure

Condition

The symmetry condition for the conditional probabilities

Condition

Reconsidering the von Neumann algebras where

In addition to the

A second reason is that

Some further important characteristics of these linear maps will now be collected. Suppose

The norm of the operator

If two events e and f in an UCP space

By Lemma

An important link between the linear maps

The interference term

Suppose that

If

If

(i)

(ii) With

Suppose that E is a UCP space. Then the following three conditions are equivalent.

The implication (i)

It now becomes clear that the symmetry condition

In this section, the orthogonal additivity of

Consider functions

Note that the projection lattices of von Neumann algebras have the Jordan decomposition property and the

Suppose that

Consider

For each

Under the assumptions of the last lemma, a product can now be defined on the order-unit space

Suppose that

The inclusion

Suppose

Finally

Thus

Suppose that E is a UCP space with the

Suppose

So far it has been seen that the Jordan decomposition property and the absence of third-order interference entail a product

The product

The elements of a UCP space

First, one would like to identify the elements

Second, the expectation value of the square

Third, one would like to have the usual polynomial functional calculus allocating an element

The following theorem shows that these requirements make

Suppose that

Suppose

A second commutative product is now introduced on

Since

Suppose

When the starting point is the projection lattice

A rich theory of Jordan algebras is available and most of them can be represented as a Jordan subalgebra of the self-adjoint linear operators on a Hilbert space. The major exception is the Jordan algebra consisting of the

A reconstruction of quantum mechanics up to this point has thus been achieved from a few basic principles. The first one is the absence of third-order interference and the second one is the postulate that the elements of the constructed algebra exhibit a behaviour which one would expected from observables. The third one, the

The combination of a simple quantum logical structure with the postulate that unique conditional probabilities exist provides a powerful general theory which includes quantum mechanics as a special case. It is useful for the reconstruction of quantum mechanics from a few basic principles as well as for the identification of typical properties of quantum mechanics that distinguish it from other more general theories. Three such properties have been studied: a novel bound for quantum interference, a symmetry condition for the conditional probabilities, and the absence of third-order interference

In the framework of the quantum logics with unique conditional probabilities, the absence of third-order interference

As the identity

The possibility that no such examples exist is not anticipated but cannot be ruled out as long as no one has been found. It would mean that every quantum logic with unique conditional probabilities can be embedded in the projection lattice in a Jordan algebra and that third-order interference never occurs. Moreover, the further postulates concerning the behaviour of the observables (power associativity, positive squares) would then as well become redundant in the reconstruction of quantum mechanics. If this could be proved, the reconstruction process could be cleared up considerably.

In any case, it seems that third-order interference can play a central role in the reconstruction of quantum mechanics from a few basic principles, in an axiomatic access to quantum mechanics with a small number of interpretable axioms, as well as in the characterisation of the projection lattices in von Neumann algebras or their Jordan analogue, the JBW algebras, among the quantum logics. Mathematically, the symmetry property

The author would like to thank the Perimeter Institute and Philip Goyal for hosting the workshop “Reconstructing Quantum Theory” in August 2009 and Howard Barnum for the inspiring discussions during this workshop and for directing my attention to Sorkin’s higher order interference.