Phase groups and the origin of non-locality for qubits

• Quantum-like theories can display selected quantum features, helping distinguish which properties are interlinked and which are independent.Source: §1 · p. 2
• A shared mathematical framework would allow theories with different formalisms to be compared and their differing physical features localized structurally.Source: §1 · p. 2
• The framework is extended with an abstract GHZ state and correlations to analyze non-locality across theories.Source: §1 · p. 2
• The paper asks which structure in Hilbert-space quantum mechanics generates non-locality, understood through the failure of local hidden-variable models.Source: §1 · p. 2
• Comparing stabiliser quantum mechanics with Spekkens’s toy theory isolates the phase group: Z4 for quantum mechanics versus Z2 × Z2 for the toy theory, with the former supporting a hidden-variable no-go argument.Source: §1 · p. 2

• States are morphisms from the unspecified unit system to a system, while sequential composition requires matching intermediate systems and tensoring represents compound systems and parallel processes.Source: §2 · p. 3
• A dagger symmetric monoidal category combines tensor composition with a dagger operation, and identifies unitary arrows through inverse-like dagger equations.Source: §2 · p. 3
• Dagger compositional theories interpret objects as systems, morphisms as processes, states and effects as special morphisms, composition as sequential execution, and tensor products as parallel composition.Source: §2 · p. 4
• The framework can represent Hilbert-space quantum theory with linear maps and relational possibilistic theory with relations, subsets, and Boolean scalars.Source: §2 · p. 4
• The W-construction removes global phases and converts inner products into Born-rule transition probabilities, while leaving relational theory equivalent to its possibilistic form.Source: §2 · pp. 4–5

3.1 Observables in †C-theories

• 3.1 Observables in †C-theories: Observables are commutative isometric dagger Frobenius comonoids, corresponding to orthonormal bases in FHilb and supporting diagrammatic spider calculus.Source: §3.1 · pp. 5–6
• 3.1 Observables in †C-theories: Observables lift to tensor-product systems, induce compact structures, and define eigenstates that can likewise lift across subsystems.Source: §3.1 · pp. 6–7

3.2 GHZ states in †C-theories

• 3.2 GHZ states in †C-theories: GHZ structures are in bijective correspondence with observables, providing an abstract counterpart to GHZ states.Source: §3.2 · p. 8

3.3 Phase groups in †C-theories

• 3.3 Phase groups in †C-theories: Unbiased states and corresponding unitary actions form isomorphic Abelian groups, called the phase group.Source: §3.3 · p. 9

3.4 GHZ correlations in †C-theories

• 3.4 GHZ correlations in †C-theories: Correlation triples formed from phase-group states remain correlation triples under permutation, connecting GHZ correlations to phase-group structure.Source: §3.4 · pp. 9–10
• 3.4 GHZ correlations in †C-theories: A GHZ correlation triple operationally fixes the third outcome from the first two, and algebraically consists of triples (x, x′, (x ⊙ x′)∗).Source: §3.4 · pp. 9–10

4 The key examples: Stab and Spek

• The framework compares stabiliser qubit quantum mechanics and Spekkens’s toy theory despite their different mathematical formalisms, revealing striking structural similarity.Stabiliser theory uses qubit states and Clifford operations, while the toy theory uses finite phase spaces, subsets, and relations.Source: §4 · p. 10, §4.1 · p. 11

4.1 Stabiliser qubit quantum mechanics

• Stabiliser theory includes six single-qubit stabiliser states, Clifford dynamics generated by single-qubit Cliffords and CNOT, and projective Pauli measurements.It also contains entangled states, including the three-qubit GHZ state, and supports local-hidden-variable no-go proofs using three qubits.Source: §4.1 · pp. 10–11

4.2 Spekkens’s toy theory

• Spekkens’s toy theory represents a qubit-like system with four ontic states and epistemic states constrained by the knowledge balance principle.Its six epistemic single-system states correspond to subsets of two ontic states, and the theory is constructed as a local hidden variable theory.Source: §4.2 · p. 11

4.3 The †C-theories Stab and Spek

• The categorical GHZ construction yields the standard GHZ state in Stab, linking the shared framework to the theory’s non-locality analysis.The same framework applies correlation-triple results to both theories, enabling comparison of their physical differences.Source: §4.3 · pp. 12–13
• The †C-theories Stab and Spek are generated constructively, and each has exactly three mutually unbiased observables on its qubit-like object.Stab is generated within FHilb by Clifford operations and an observable, while Spek is generated within FRel by permutations and relational structure.Source: §4.3 · pp. 12–13
• The relational construction of Spek is possibilistic rather than probabilistic, recording whether outcomes are possible; positive-real scalar modifications can restore probabilities.This is sufficient for the later non-locality arguments, which use measurement probabilities of 0 and 1.Source: §4.3 · p. 13

4.4 Pinpointing the difference between Spek and Stab

• Although both theories share the same abstract generators and composition patterns, their interactions with the observable differ through the phase group.The phase group is Z4 for Stab and Z2 × Z2 for Spek, pinpointing the mathematical difference associated with their distinct physical behaviour.Source: §4.4 · p. 14

5 Mutually unbiased qubit theories

• MUQTs formalize theories with three mutually unbiased two-outcome observables, six qubit states, and states that are eigenstates or unbiased for each observable.Stab and Spek both satisfy the MUQT definition.Source: §5 · pp. 14–15

5.1 Classification

• In a MUQT, the phase group has four elements, so only Z4 and Z2 × Z2 can determine the GHZ-correlation structure.The observable monoid and corresponding GHZ correlations are determined by the phase group.Source: §5.1 · pp. 15, 17
• The two possible GHZ-correlation forms are exactly those realized by stabiliser theory and Spekkens’s toy theory.Stab realizes Z4, while Spek realizes Z2 × Z2.Source: §5.1 · pp. 15, 17, 18

5.2 Link to non-locality

• GHZ correlations connect the abstract classification to non-locality because the stabiliser no-go proof uses a GHZ state and Pauli measurements.The same structural ingredients appear categorically in MUQTs.Source: §5.2 · p. 18
• No MUQT with a Z4 qubit basis structure admits a local realist model, whereas a local realist model can be constructed for the GHZ state in any Z2 × Z2 MUQT.The Z2 × Z2 conclusion does not establish that every such theory is globally local, because other states may lack a local interpretation.Source: §5.2 · pp. 18–19
• Thus, within this framework, Z4 basis structure embodies non-locality, while Z2 × Z2 basis structure cannot itself endow a theory with non-locality.The claim concerns the contribution of the basis structure, not necessarily all states or structures in a Z2 × Z2 theory.Source: §5.2 · p. 19

Comparison of phase groups

• The paper contrasts this direct Z4 argument with the more indirect conclusion that Z4 GHZ states are non-local while Z2 × Z2 GHZ states can be local.The indirect route identifies the phase groups through theories already known to be non-local or local, respectively.Source: §6 · pp. 19–20

Local realist representations

• A local realist representation assigns subsystem eigenstates through hidden variables and reproduces joint eigenstate probabilities for every observable choice.The theory admits such a representation when every state does.Source: §6 · p. 19

Abstract no-go argument

• The no-go argument applies to qubit GHZ states in Stab and FHilb without reference to Hilbert space, deriving the contradiction directly from the Z4 phase-group structure.The argument is presented as a generalization of the usual GHZ reasoning.Source: §6 · p. 20

Forbidden triples

• A forbidden triple consists of states whose GHZ correlation would require distinct eigenstates of the same observable, forcing its amplitude to vanish.Proposition 6.3 derives this vanishing condition categorically.Source: §6 · p. 20

Abstract no-go argument

• Any GHZ state whose phase group contains Z4, with the specified eigenstate partition across two observables, does not admit a local realist representation.The identity and involutive elements form one observable’s eigenstates, while the other two elements form the second observable’s eigenstates.Source: §6 · p. 20

Correlation constraints

• Z4 correlations forbid triples with an odd number of |−⟩ states and triples containing two {|♯⟩, |=⟩} states plus one {|+⟩, |−⟩} state when the relevant count is even.These forbidden patterns supply the correlation constraints used in the contradiction.Source: §6 · pp. 20–21

Conclusions

• Z4 distinguishes stabiliser QM from Spekkens’s Z2 × Z2 theory and suffices to establish non-locality, while the toy theory is local.The framework accommodates both theories and relates their phase-group difference to their physical locality properties.Source: §7 · p. 21

Mutually unbiased qubit theories

• Mutually unbiased qubit theories have GHZ correlations determined by phase groups Z4 and Z2 × Z2.These are the phase groups identified for the special class of mutually unbiased qubit theories.Source: §7 · p. 21

Generalization

• The phase group continues to determine GHZ correlations when mutually unbiased theories are generalized beyond qubits.The proposed generalization allows more observables and eigenstates while retaining mutual unbiasedness.Source: §7 · p. 21

Higher-dimensional theories

• For qutrits, the two nine-element phase groups are Z9 and Z3 × Z3, while stabiliser and toy theories coincide with the local group Z3 × Z3.The paper leaves open whether a theory with phase group Z9 exists and what its properties would be.Source: §7 · pp. 21–22

Scope

• Mixed states and operations can be added to the pure fragments through Selinger’s CPM-construction.The conclusions otherwise concern the pure fragments of both theories.Source: §7 · p. 21

Further work

• Future work includes classifying locality across higher-dimensional mutually unbiased theories and determining whether non-mutually-unbiased observables are needed for super-quantum correlations.The paper also asks whether abstract locality proofs can be established for Z2 × Z2 MUQTs.Source: §7 · p. 22