A theorem about states on quantum logics. States on Jordan algebras

Gerd Niestegge

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.

M. S. Matveichuk

M. S. Matveichuk

Ian Richard C Buckley

Wiesner's quantum money protocol was conceived around 1970, over a decade before Wootters and Zurek established the No-Cloning Theorem. Its security was argued operationally; No-Cloning later supplied the proof-theoretic foundation. This paper asks whether the Exceptional Jordan Algebra $\mathfrak{J}_3(\mathbb{O})$ (the Albert algebra) admits a non-associative No-Cloning theorem that is strictly stronger than the standard one, and what it implies for quantum money. The standard No-Cloning proof uses only linearity and the bilinear tensor structure; it does not use associativity, so a direct translation to $\mathfrak{J}_3(\mathbb{O})$ recovers only the standard result in new notation. The genuinely non-associative obstruction is of a different character: it concerns not the impossibility of a cloning map but the impossibility of a cloning target. We prove two theorems. Theorem 3.1 (NA No-Cloning --- Target Obstruction): the two-copy system of $\mathfrak{J}_3(\mathbb{O})$ does not exist as a Jordan algebra. Any Jordan algebra containing two commuting copies of $\mathfrak{J}_3(\mathbb{O})$ would be special (embeddable in an associative algebra), contradicting the exceptionality of $\mathfrak{J}_3(\mathbb{O})$ (Albert). There is no valid target into which an octonionic state can be cloned while remaining in the same algebraic category. Theorem 3.3 (NA No-Cloning --- Map Collapse): any Jordan algebra homomorphism $\Phi\colon \mathfrak{J}_3(\mathbb{O}) \to A_{sa}$ into the self-adjoint part of an associative algebra $A$ satisfies $\Phi([x, y, z]) = 0$ for all $x, y, z \in \mathfrak{J}_3(\mathbb{O})$. Since non-Fano associators span the 16-dimensional Peirce-$\tfrac{1}{2}$ subspace of $\mathfrak{J}_3(\mathbb{O})$, $\Phi$ collapses 16 of the 27 dimensions of $\mathfrak{J}_3(\mathbb{O})$. A dimension-collapsing map cannot clone. Together these theorems say: for octonionic quantum states, No-Cloning holds for a reason the standard theorem does not reach. The obstruction is structural, not just linear-algebraic. The Fano-Token protocol encodes a quantum banknote as a rank-1 projector in $\mathfrak{J}_3(\mathbb{O})$, the Furey projector $P_\ell$ stabilised by the $G_2$ automorphism group. As a Corollary 4.1 of Theorem 3.3 , any associative measurement operator $M$ applied to $P_\ell$ produces a defect equal to the associator $[P_\ell, M, P_\ell]$, with norm $2$ for non-Fano-compatible $M$. This converts Wiesner's passive defense into an active alarm: unauthorized measurement is detectable instantaneously, without token retrieval. The physical claim that this defect manifests as a measurable energy transition in the 731-QPU hardware is stated as Conjecture and kept separate from the algebraic results.

Krystyna Bugajska

Gerd Niestegge

In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.