Formalization of Ternary Quantum Systems in Lean4

We present a symplectic linear-algebraic proof of the Quantum Singleton Bound for stabiliser quantum error-correcting codes together with a Lean4 formalisation of the linear-algebraic argument. The proof is formulated in the language of finite-dimensional symplectic vector spaces modelling Pauli operators and relies on distance-based erasure correctability and the cleaning lemma. Using a dimension-counting argument within the symplectic stabiliser framework, we derive the bound $k + 2(d-1) \le n$ for any $[[n, k, d]]$ stabiliser code. This approach isolates the algebraic structure underlying the bound and avoids the heavier analytic machinery that appears in entropy-based proofs, while remaining well-suited to formal verification.We present a symplectic linear-algebraic proof of the Quantum Singleton Bound for stabiliser quantum error-correcting codes together with a Lean4 formalisation of the linear-algebraic argument. The proof is formulated in the language of finite-dimensional symplectic vector spaces modelling Pauli operators and relies on distance-based erasure correctability and the cleaning lemma. Using a dimension-counting argument within the symplectic stabiliser framework, we derive the bound $k + 2(d-1) \le n$ for any $[[n, k, d]]$ stabiliser code. This approach isolates the algebraic structure underlying the bound and avoids the heavier analytic machinery that appears in entropy-based proofs, while remaining well-suited to formal verification.

quant-pharXiv:2602.20186Feb 24, 20260 citations

Quantum-Chemical Models

Jacek Karwowski

Springer eBooksJan 1, 199723 citations

How to combine three quantum states

Māris Ozols

We devise a ternary operation for combining three quantum states: it consists of permuting the input systems in a continuous fashion and then discarding all but one of them. This generalizes a binary operation recently studied by Audenaert et al. [arXiv:1503.04213] in the context of entropy power inequalities. Our ternary operation continuously interpolates between all such nested binary operations. Our construction is based on a unitary version of Cayley's theorem: using representation theory we show that any finite group can be naturally embedded into a continuous subgroup of the unitary group. Formally, this amounts to characterizing when a linear combination of certain permutations is unitary.We devise a ternary operation for combining three quantum states: it consists of permuting the input systems in a continuous fashion and then discarding all but one of them. This generalizes a binary operation recently studied by Audenaert et al. [arXiv:1503.04213] in the context of entropy power inequalities. Our ternary operation continuously interpolates between all such nested binary operations. Our construction is based on a unitary version of Cayley's theorem: using representation theory we show that any finite group can be naturally embedded into a continuous subgroup of the unitary group. Formally, this amounts to characterizing when a linear combination of certain permutations is unitary.

arXiv:1508.00860Aug 4, 20153 citations

Symbolic Algebra in Quantum Chemistry

So Hirata

Theoretical Chemistry AccountsJan 5, 200687 citations

Qutrits in the Wild: A Ternary Formalism for Non-Binary Reasoning Across Natural Systems

Trent Slade

Binary logic and qubit-based formalisms have dominated scientific modeling due to their success in deterministic physics and digital computation. However, many natural systems—ecological networks, genetic coding, social dynamics, language, and evolutionary optimization—are fundamentally non-binary, exhibiting cyclic dominance, neutrality, ambiguity, and triadic closure. This work proposes a unified ternary formalism for wild complexity, grounded in the mathematics of the qutrit and the Lie group SU(3), as a native descriptive language for these systems. By extending state spaces from H2 to H3 and dynamics from Pauli to Gell-Mann operators, the framework captures cyclic flow, constructive symmetry, and trialitic coupling that binary models structurally exclude. The analysis demonstrates how antisymmetric f_abc generators encode ecological Rock–Paper–Scissors dynamics and trophic cycles, while symmetric d_abc generators model coalition stability, genetic degeneracy, and social buffering via neutrality. Ternary error correction is shown to underlie the genetic code’s wobble structure and aligns with optimal bounds set by the Ternary Golay Code. In language and artificial intelligence, qutrit-based semantic models and three-valued logic naturally represent ambiguity, polysemy, and epistemic uncertainty, providing a principled route to mitigating hallucination. In evolutionary computation, qutrit genetic algorithms demonstrate superior exploration via rotation and disaster operators unavailable to binary search. Finally, these results are situated within deeper triality principles from Spin(8), Cartan geometry, and teleodynamics, arguing that triadic structure is not incidental but a fundamental organizational law of living, social, and cognitive systems.Binary logic and qubit-based formalisms have dominated scientific modeling due to their success in deterministic physics and digital computation. However, many natural systems—ecological networks, genetic coding, social dynamics, language, and evolutionary optimization—are fundamentally non-binary, exhibiting cyclic dominance, neutrality, ambiguity, and triadic closure. This work proposes a unified ternary formalism for wild complexity, grounded in the mathematics of the qutrit and the Lie group SU(3), as a native descriptive language for these systems. By extending state spaces from H2 to H3 and dynamics from Pauli to Gell-Mann operators, the framework captures cyclic flow, constructive symmetry, and trialitic coupling that binary models structurally exclude. The analysis demonstrates how antisymmetric f_abc generators encode ecological Rock–Paper–Scissors dynamics and trophic cycles, while symmetric d_abc generators model coalition stability, genetic degeneracy, and social buffering via neutrality. Ternary error correction is shown to underlie the genetic code’s wobble structure and aligns with optimal bounds set by the Ternary Golay Code. In language and artificial intelligence, qutrit-based semantic models and three-valued logic naturally represent ambiguity, polysemy, and epistemic uncertainty, providing a principled route to mitigating hallucination. In evolutionary computation, qutrit genetic algorithms demonstrate superior exploration via rotation and disaster operators unavailable to binary search. Finally, these results are situated within deeper triality principles from Spin(8), Cartan geometry, and teleodynamics, arguing that triadic structure is not incidental but a fundamental organizational law of living, social, and cognitive systems.

Dec 5, 20250 citations

Models for Quantum Chemistry

Hinne Hettema

European studies in philosophy of scienceJan 1, 20170 citations

k + 2(d-1) \le n

for any

[[n, k, d]]

stabiliser code. This approach isolates the algebraic structure underlying the bound and avoids the heavier analytic machinery that appears in entropy-based proofs, while remaining well-suited to formal verification.