Source
Backens & Duman (2014), LIT-054, read in full in NOTE-066.
What was actually shown
The fragment. Toy bits only, states of maximal knowledge (the analogue of pure states), post-selected measurements and reversible transformations (§1, §4). Each toy bit has four ontic states labelled by (X, Z) ∈ Z₂²; the valid epistemic states know a commuting set of quadratures and are ignorant otherwise (§2.1, from LIT-007). There are 24 reversible maps of one toy bit, all of S₄ (§4.3).
The calculus. Green and red phased spiders and a colour-change node, interpreted as relations; equalities are exact because the only scalars are the empty and the full relation (§4.4). Universality follows from Coecke–Edwards’ generators (§4.3). Soundness is checked rule by rule by computing the relations, by induction, and via Coecke–Edwards–Spekkens for the spider rule (§4.4).
Completeness (Thm 17). Every state diagram rewrites to a reduced graph-state normal form (Thms 10–11). Lemma 15 is proved in full (App. A.2). Theorem 16, that a simplified pair represents the same state iff the diagrams are identical, is given by analogy to Backens’ stabilizer proof. With map–state duality the calculus is complete for the fragment.
The comparison. A binary 2n×n matrix S is a valid toy state iff SᵀJS = 0, and valid reversible transformations are exactly the Q with QᵀJQ = J (Lemma 3, p. 15). These are the qubit stabilizer check-matrix conditions. The phase-free rules (spiders, bialgebra, copy, colour change) are identical to stabilizer ZX. The phase-bearing rules have analogues: the 11-commutation rule and ZX π-commutation are both instances of Eq. 40 (§4.5). The toy phase group is the Klein four-group {00, 01, 10, 11} under XOR; the stabilizer one is Z₄ = {0, π/2, π, −π/2}.
What could have come out otherwise. Some toy-theory equation could have needed a rule with no stabilizer counterpart, or completeness could have failed for the relational semantics. Neither happened.
What this does not say
- That the phase group is proved to be the only difference. “The only difference … is the phase group” is a citation to Coecke–Edwards–Spekkens (§4.5; NOTE-066, C5). This paper exhibits the narrower correspondence above.
- That the toy theory’s locality or failure to violate Bell inequalities is explained by its being ψ-epistemic. The paper asserts a relation in §1 with no argument (C7), and no Bell, GHZ or contextuality scenario is drawn. NOTE-066 answers the seeded question of which operational difference the phase group captures: none, in this paper.
- That the graph-state theorems were proved for the toy theory independently. They are transferred from Van den Nest–Dehaene–De Moor through the identical check-matrix formalism.
- Mixed states, or qudits. Less-than-maximal knowledge is left to a CPM construction not carried out; qudit completeness is open (§6).
Connections
- NOTE-066 links the phase-group difference to LIT-007 §IV.B, where the qubit stabilizer subtheory is contextual and its epistricted counterpart is not. Neither paper states the link.
- The same bit theory, with modes as systems, is the classical model of balanced interference in THEORY-403.
- The toy theory is ψ-epistemic in the overlap sense with product composition, so by PBR it cannot contain PBR’s measurements (NOTE-066’s inference; THEORY-605).