[Paper Review] Sorkin's third-order interference term in quantum logics with unique conditional probabilities

Gerd Niestegge|arXiv (Cornell University)|Dec 1, 2009

Quantum Mechanics and Applications4 citations

TL;DR

This paper generalizes Sorkin's third-order interference term I3 to quantum logics with unique conditional probabilities, showing that I3 = 0 is deeply connected to the existence of a non-associative product in the order-unit space. The key contribution is that when observables behave quantum-mechanically (positive squares, functional calculus), the space becomes a Jordan algebra, enabling a reconstruction of quantum mechanics from the absence of third-order interference and additional principles.

ABSTRACT

Considering not only the well-known two-slit experiment, but also experiments with three slits, Sorkin introduced the third-order interference term I3 and discovered that the absence of third-order interference (I3=0) is typical of quantum mechanics where only second-order interference occurs. In the present paper, the interference term I3 is ported to the quantum logics with unique conditional probabilities. In this framework, the identity I3=0 does not hold in general and its consequences are analysed. A first result reveals a close link between this identity and the existence of a product in the order-unit space generated by the quantum logic. In the general case, this product is neither commutative nor associative. By a second result, the order-unit space becomes a Jordan algebra, if each element behaves like one would expect from an observable (i.e., its square is positive and there is a polynomial functional calculus). Almost all such Jordan algebras can be represented as operator algebras on a Hilbert space, and a reconstruction of quantum mechanics up to this point is thus achieved from the absence of third-order interference and a few other principles. Besides the identity I3=0, two further interesting properties of quantum mechanics distinguishing it from more general theories are studied. These are a novel bound for quantum interference and a symmetry condition for the conditional probabilities.

Motivation & Objective

• To extend Sorkin's third-order interference term I3 to the framework of quantum logics with unique conditional probabilities.
• To investigate the physical and algebraic consequences of the identity I3 = 0 in this generalized setting.
• To determine under what conditions the order-unit space generated by the quantum logic becomes a Jordan algebra.
• To identify additional principles—beyond I3 = 0—that distinguish quantum mechanics from more general probabilistic theories.
• To explore the role of conditional probabilities and their symmetry properties in characterizing quantum behavior.

Proposed method

• Adapting Sorkin's interference term I3 to quantum logics equipped with unique conditional probabilities, using a probabilistic operational framework.
• Analyzing the algebraic structure of the order-unit space generated by the quantum logic, particularly the existence and properties of a product operation.
• Establishing a link between the vanishing of I3 and the existence of a product in the order-unit space, which is generally non-commutative and non-associative.
• Imposing physical constraints on elements (e.g., positivity of squares, existence of polynomial functional calculus) to force the product to satisfy Jordan algebra axioms.
• Using the structure of Jordan algebras to show that almost all such algebras can be represented as operator algebras on a Hilbert space.
• Deriving a novel bound on quantum interference and a symmetry condition for conditional probabilities as distinguishing features of quantum mechanics.

Experimental results

Research questions

• RQ1 What is the generalization of Sorkin’s third-order interference term I3 in quantum logics with unique conditional probabilities?
• RQ2 How does the condition I3 = 0 relate to the algebraic structure of the order-unit space generated by the quantum logic?
• RQ3 Under what conditions does the order-unit space become a Jordan algebra, and what physical assumptions are required for this?
• RQ4 What additional principles—beyond I3 = 0—distinguish quantum mechanics from more general probabilistic theories in this framework?
• RQ5 What are the implications of the symmetry condition and the novel interference bound for the structure of quantum probability?

Key findings

• The identity I3 = 0 is equivalent to the existence of a product in the order-unit space generated by the quantum logic, which is generally non-commutative and non-associative.
• When elements behave like quantum observables (positive squares and polynomial functional calculus), the product structure becomes a Jordan algebra.
• All such Jordan algebras can be represented as operator algebras on a Hilbert space, enabling a reconstruction of quantum mechanics up to this point.
• The absence of third-order interference, combined with the functional calculus and positivity conditions, leads to a structure isomorphic to standard quantum mechanics.
• Two additional distinguishing features of quantum mechanics are identified: a novel upper bound on interference strength and a symmetry condition for conditional probabilities.
• These additional features—beyond I3 = 0—help to uniquely characterize quantum theory within a broader class of probabilistic theories.