1 Introduction↩︎

A central organising thrust of quantum computer science is to understand and exploit uniquely quantum resources towards provable advantages over classical information processing. A key role in this program is played by nonlocality [1], [2], a property of correlations that obstructs simulation by classical probabilistic models. Nonlocality can be precisely certificated in terms of games, which serve as core technical tools in quantum information and complexity theory. As structurally clarified by Abramsky–Brandenburger [3], nonlocality (and its generalisation, contextuality) are essentially logical in character. This work addresses the problem of classifying logical proofs of nonlocality under minimal assumptions.

Whereas standard witnesses of nonlocality take the form of probabilistic inequalities satisfied by classical correlations but violable by quantum correlations [4], the maximal form of strong nonlocality is witnessed by a logical paradox. Nonlocality paradoxes arise from restricted joint measurability: quantum theory allows only compatible sets of measurements, called contexts, to be jointly measurable. The canonical example is due to Greenberger–Horne–Zeilinger (GHZ) [5], [6]. They gave a three-qubit state with two measurements per qubit such that: the set of constraints describing the associated possibilistic empirical data, one for each context (i.e.a choice of one measurement on each qubit), is unsatisfiable.

Paradoxes undergird probabilistic nonlocality in that all nonlocal correlations are dilutions of paradoxes by classical randomness. The nonlocal fraction [7], [8] measures nonlocality by quantifying the degree to which correlations are paradoxical. In many information-theoretic tasks, quantum advantage scales directly with the nonlocal fraction of an available resource state. For example, nonlocal games [9] involve players—who may share a distributed resource but cannot directly communicate—giving coordinated responses to a referee’s queries. When classical correlations limit the players’ success probability, deterministic winning strategies necessarily require sharing a nonlocal paradox.

Such unconditional advantages in communication tasks can be converted into rare unconditional complexity-theoretic separations. For example, Bravyi–Gosset–König [10] gave a constant-depth quantum circuit family, capable of generating and playing many nonlocal games, that cannot be simulated by classical circuits of sublogarithmic depth. Nonlocal games also play an essential role in the proof that \(\textsf{MIP}^* = \textsf{RE}\): entangled multi-interactive provers recognise exactly the semidecidable languages [11].

Nonlocality paradoxes directly boost computation in the model of measurement-based quantum computation. Anders–Browne [12] showed how access to measurements on GHZ states can promote a restricted linear computer to classical universality; Raussendorf [13] extended this to general measurement-based computation. Other information-theoretic tasks requiring nonlocality include device-independent quantum cryptography [14], [15] and randomness generation [16].

These results demonstrate that the mere presence of nonlocality is insufficient for realising advantages; the precise structure of the paradoxes involved plays a vital role. Revealing this structure enables systematic investigation of the advantages such paradoxes confer and the mechanisms by which they do so. Moreover, understanding how abundant quantum paradoxes are, and whether simple systems can generate many independent instances of them, bears directly on how nonlocality can be amplified into provable computational separations. These considerations naturally lead to a classification problem for nonlocality paradoxes, initiated in earlier work by the present authors [17] in the three-qubit setting.

1.1 Prior work↩︎

Abramsky et al.[18] began the systematic study of three-qubit nonlocality paradoxes, giving the first examples beyond that of GHZ. They showed that all three-qubit paradoxes involve, up to a natural physical equivalence, equatorial measurements performed on a balanced state. Further, they gave an infinite family of paradoxes, indexed by \(m \in \mathbb{N}\), with a striking novel conditional structure. Both Alice’s and Bob’s \(2m\) measurements (i.e.those on the first two qubits) form half of a regular \(4m\)-polygon, while Charlie’s two measurements (i.e.those on the third qubit) are \(X,Y\) as in the standard GHZ example. The GHZ example corresponds to \(m=1\); for higher \(m\), the resulting parity constraints on satisfying assignments are conditional on Charlie’s outcome when he measures \(Y\).

This was extended to a classification program by the present authors, who introduced several key new technical tools. We conjectured that all paradoxes necessarily require an interpolant state, which lies on the line connecting the GHZ state to the tensor product of the Bell state and the plus state. We exhaustively classified all paradoxes satisfying the following conditions:

1. an interpolant state is used,

2. Charlie has two measurements,

3. Alice and Bob have the same number of measurements,

4. the logical constraints are as strong as physics allows,

5. the contradiction arises by summing all parity constraints.

Our earlier classification subsumed and greatly extended the family of Abramsky et al., introducing several new infinite families that exhibited numerous novel structural features. Removing our prior assumptions (1) through (5) fundamentally alters the possible logico-combinatorial structure of paradoxes and demands significantly stronger methods.

1.2 Summary of main results↩︎

In this work, we give a complete structural classification of three-qubit nonlocality paradoxes admitting a biconditional parity proof (Definition 3). In such a proof, after relabelling the parties if necessary, Charlie has exactly two measurement settings, and each Charlie conditioning—a choice of Charlie measurement and outcome—turns the Alice–Bob possibilistic constraints into parity relations: an Alice–Bob outcome pair is possible if and only if it satisfies the corresponding conditioned \(\mathbb{Z}_2\)-parity equation. This is a vast family of three-qubit nonlocality paradoxes that includes all prior known families including the standard GHZ example, the family of Abramsky et al.[18], and the earlier families of the present authors [17].

We show below that the paradoxes admitting a biconditional parity proof are precisely those that use interpolant states when Charlie is restricted to two measurement settings (Proposition 4); that is, our present classification drops the above assumptions (3), (4), and (5) from our earlier classification. We achieve this by introducing a graph-theoretic framework for studying nonlocality paradoxes in terms of \(2\)-CNF formulae and implication graphs (Definition 5). In the interpolant case, these implication graphs are bidirected, so paradoxicality is witnessed by paths from literals to their complements (Lemma 6). We then show that every paradox with a biconditional parity proof is reducible to a canonical combinatorial description and construct the bijection of Theorem 18: \[\mathsf{BPP} \quad \cong \quad \mathsf{CanonicalTriples}.\] Here, \(\mathsf{BPP}\) is the set of specifications of quantum state and minimal measurement sets for paradoxes admitting biconditional parity proofs, up to permuting parties and local changes of basis (Definition 3). The set \(\mathsf{CanonicalTriples}\) consists of the canonical classification data, i.e.the lexicographically normalised triples that contain exactly one representative of each physical equivalence class (Definition 20). Each such triple is a small tuple of integers and real parameters with membership decided by easily checkable conditions.

In particular, the data that classifies a biconditional parity proof starts with a Charlie clock (Definition 6), consisting of a clock group \(\mathbb{Z}_N\), a choice of three ticks, and a real parameter. The valid clocks—those that are both realisable by quantum data and paradoxical—are classified explicitly in Theorem 11. Once the clock is fixed, Alice’s measurements, and hence Bob’s, are specified by an Alice–Bob completion (Definition 13): a finite choice of cosets of certain subgroups of the clock group. These cosets are exactly the carriers of witness paths in the implication graphs (Proposition 14). The shadow test determines which further witnesses are already forced by a partial choice of cosets (Proposition 15), and the canonical completions are precisely those giving minimal measurement supports (Definition 17 and Theorem 16). Finally, Alice and Bob’s measurements can be partitioned into layers, which are noninteracting copies of the clock group, each offset by a real-valued shift; the free choice of these shifts completes the classifying data (Definition 18).

We conclude by presenting families of new exotic paradoxes that violate all prior assumptions on their structure. Section 5 gives new interpolant-state paradoxes beyond the restriction of Charlie to two measurements, including families with three Charlie measurements (Theorems 20 and 21) and examples with four Charlie measurements (Example 1). In Section 6, we present a new non-interpolant paradox, which provides a counterexample to the conjecture of [17] that every three-qubit nonlocality paradox must arise from an interpolant state.

Together, these results provide several striking new insights into the nature of quantum nonlocality paradoxes. Earlier, the reason why so few examples of three-qubit nonlocality paradoxes were known might have been attributed to them being genuinely rare. We now see, upon performing a deep and systematic search for them, that, while such paradoxes may be difficult to construct, they are incredibly abundant. Indeed, every condition on them that was presumed to be necessary is revealed below not to be. While paradoxicality does impose strict symmetry and structural constraints, there is enough slack to construct increasingly exotic examples.

Equivalently, our classification can be read as a fine-grained study of how quantum interference gives rise to provably nonclassical behaviour. Nonlocality paradoxes require realising many impossible (i.e.probability zero) events arising from exact amplitude cancellations that are sufficiently well-distributed across combinations of measurements to rule out every hidden variable; our results identify the arithmetic and combinatorial patterns by which such local cancellations assemble into a global logical contradiction. An exciting future direction is to consider how this wealth of quantum paradoxes can be leveraged towards demonstrating unconditional quantum advantages in a variety of information processing tasks.

1.2.0.1 Outline.

Section 2 reviews the measurement-scenario formalism for three-qubit nonlocality paradoxes and the amplitude equations governing impossible events. Section 3 recasts paradoxicality in terms of \(2\)-CNF formulae and implication graphs. Section 4 proves the classification of biconditional parity proofs via valid Charlie clocks, canonical Alice–Bob completions, and layer shifts. Section 5 presents new interpolant-state paradoxes beyond the restriction of Charlie to two measurements, including examples with three and four Charlie measurements. Finally, Section 6 constructs a non-interpolant-state paradox lying outside the biconditional parity proof classification.

2 Background↩︎

2.1 Nonlocality↩︎

We briefly review the notion of nonlocality, a special case of contextuality, within the framework of Abramsky–Brandenburger [3]. Contextuality is a property of probabilistic data relative to a given measurement scenario. In this work, we focus on data arising from quantum states and measurements, i.e.quantum scenarios.

Experimental setups are modelled by the abstract framework of measurement scenarios: a triple \((\mathcal{X}, \mathcal{O}, \mathcal{C})\), consisting of a set of measurement labels \(\mathcal{X}\), a set of possible outcomes \(\mathcal{O}\), and a cover \(\mathcal{C}\) of \(\mathcal{X}\), consisting of measurement contexts, which are maximal sets of measurements that can be jointly performed. Since we are only concerned with qubit quantum scenarios, we fix \(\mathcal{O} = \mathbb{Z}_2\). A quantum scenario yields an empirical model \(\mathcal{E}\), consisting of probability distributions \(P_C : \mathcal{O}^C \to [0,1]\) indexed by \(C \in \mathcal{C}\), defined by the Born rule, which satisfy no-signalling: for any \(C, C' \in \mathcal{C}\), the marginal distributions \(P_C\big|_{C\cap C'}\) and \(P_{C'}\big|_{C\cap C'}\) agree.

An empirical model is noncontextual if it admits a global distribution \(\xi: \mathcal{O}^\mathcal{X} \to [0, 1]\), also known as a local hidden variable model, over global assignments that reproduces the observed marginal distributions, i.e.\(\xi|_C = P_C\) for all \(C \in \mathcal{C}\). Such models necessarily satisfy all Bell inequalities [19], [20], thus contextuality is witnessed by violation of a probabilistic inequality. A global assignment \(g: \mathcal{X} \rightarrow \mathcal{O}\) is consistent with an empirical model if, for each \(C\in \mathcal{C}\), the joint outcome predicted by \(g\) is possible, i.e.\(P_C(g|_C) \neq 0\). An empirical model is strongly contextual if it admits no such consistent global assignment. Standard examples include the GHZ paradox [6] and the PR box [21].

Nonlocality is a special type of contextuality in which the measurements have a multipartite structure. In the quantum case, a Bell measurement scenario \(\mathcal{M}\) on \(n\) qubits is an \(n\)-tuple of finite sets \((M_1, \ldots, M_n)\), where each \(M_i\) is a set of measurements on the \(i\)-th qubit. The contexts of \(\mathcal{M}\) are \(\mathcal{C} = \prod_{i=1}^n M_i\), i.e.choices of exactly one measurement per qubit. The set of all measurements is \(\mathcal{X} = \bigsqcup_{i=1}^n M_i\).

A local measurement is given by \[E_{\theta, \varphi} := \sin\left(\theta\right)[\cos\left(\varphi\right) X + \sin\left(\varphi\right) Y] + \cos\left(\theta\right) Z ,\] where \(\theta \in \left[0, \tfrac{\pi}{2}\right)\) and \(\varphi \in [0, 2\pi)\), and \(X,\;Y,\;Z\) denote the Pauli operators. The \(+1\) eigenstate of \(E_{\theta, \varphi}\) is \[\ket{\theta, \varphi} := \cos\tfrac{\theta}{2}\ket{0} + e^{i\varphi}\sin\tfrac{\theta}{2} \ket{1}\] and \(-1\) eigenstate is \(\ket{\pi - \theta, \varphi + \pi}\). Thus, a context may be specified by a tuple \((\boldsymbol{\theta}, \boldsymbol{\varphi}) := ((\theta_1, \varphi_1),\ldots,(\theta_n, \varphi_n))\), and an event by \((\boldsymbol{\theta}, \boldsymbol{\varphi}) \rightarrow \mathbf{o}\), where \(\mathbf{o} \in \mathcal{O}^n\) labels measurement outcomes (with \(+1, -1\) relabelled as \(0, 1\), respectively). We reformulate the definition of strong nonlocality in this setting.

Let \(\ket{(\boldsymbol{\theta}, \boldsymbol{\varphi}) \rightarrow \mathbf{o}}\) denote the eigenstate associated with the event \((\boldsymbol{\theta}, \boldsymbol{\varphi}) \rightarrow \mathbf{o}\). The event is impossible if and only if \(\braket{(\boldsymbol{\theta}, \boldsymbol{\varphi}) \rightarrow \mathbf{o} | \psi} = 0\). We refer to a pair \((\ket{\psi}, \mathcal{M})\) as a (quantum nonlocality) paradox when \(\mathcal{E}(\ket{\psi}, \mathcal{M})\) is strongly nonlocal.

2.2 Nonlocality paradoxes of three qubits↩︎

Two-qubit quantum states do not exhibit strong nonlocality [22], and hence strong nonlocality requires at least three qubits. Abramsky et al.[18] showed the following:

A balanced state is of the form \(\Bstate = \mathcal{N}(\ket{v_{\boldsymbol{\lambda}}} + e^{i \Phi} \ket{w_{\boldsymbol{\lambda}}})\), where \(\boldsymbol{\lambda} = (\lambda_1, \lambda_2, \lambda_3)\) with \(\lambda_j \in \left[0, \tfrac{\pi}{2}\right)\), \(\Phi \in [0,2\pi )\), and \(\ket{v_{\boldsymbol{\lambda}}} = \bigotimes_{j=1}^3 \ket{v_{\lambda_j}},\;\ket{w_{\boldsymbol{\lambda}}} = \bigotimes_{j=1}^3 \ket{w_{\lambda_j}}\) with \[\ket{v_{\lambda_j}} := \cos{\tfrac{\lambda_j}{2}} \ket{0} + \sin{\tfrac{\lambda_j}{2}} \ket{1}, \quad \ket{w_{\lambda_j}} := \sin{\tfrac{\lambda_j}{2}} \ket{0} + \cos{\tfrac{\lambda_j}{2}} \ket{1},\] and \(\mathcal{N}\) is a normalising constant.

An equatorial measurement has the form \(E_\varphi := \cos\left(\varphi\right) X + \sin\left(\varphi\right) Y\), where \(\varphi \in [0,2\pi)\). Since only equatorial measurements contribute to strong nonlocality, we only consider measurement scenarios \(\mathcal{M}= (M_1, M_2, M_3)\) such that each \(M_i \subset [0, 2\pi)\) is a set of angles. We refer to \(M_1,\;M_2,\;\text{and}\;M_3\) as the Alice, Bob, and Charlie measurements, respectively. The following remark yields further simplifications.

The following distinguished subclass of balanced states was originally introduced in [17].

All previously known examples of three-qubit nonlocality paradoxes [17], [18], [22] arise from interpolant states. In Section 6, we present a first example of a nonlocality paradox involving a non-interpolant state.

2.2.1 Equivalences of paradox↩︎

We regard two three-qubit scenarios \((\ket{\psi},\mathcal{M})\) and \((\ket{\psi'},\mathcal{M}')\) as equivalent if they differ only by relabelling the qubits and by local changes of basis. More precisely, this means that there exists a local unitary \(U=U_1\otimes U_2\otimes U_3\) and a permutation \(\sigma\) of the three qubits such that \(U\ket{\psi}=\ket{\psi'}\), and for each \(i=1,2,3\), the local unitary \(U_i\) carries the measurement set \(M_i\) onto \(M'_{\sigma(i)}\), where equatorial measurements are identified modulo \(\pi\).

This equivalence relation is particularly rigid for equatorial measurement scenarios. Indeed, by [17], if a single-qubit unitary sends a set of at least two equatorial measurements to another equatorial set, then, up to an overall phase, it must be of the form \[X^a P_\theta, \quad a\in\mathbb{Z}_2,\; \theta\in[0,2\pi); \qquad P_\theta=\operatorname{diag}(1,e^{i\theta}).\] Thus, the allowed local changes of coordinates are rotations of the equator, possibly followed by the reflection induced by \(X\): \[E_\varphi\longmapsto E_{\varphi+\theta} \quad \text{or} \quad E_\varphi\longmapsto E_{-\varphi-\theta}.\] Equivalently, the only freedom on each local measurement set is to rotate the equatorial circle, and possibly reverse its orientation.

For interpolant states, this freedom is even more restricted. By [17], if \(\Bsimple\) and \(\Bsimple<\lambda'>\) with \(\lambda,\lambda'\neq 0\) are equivalent, then \(\lambda=\lambda'\). Moreover, after fixing the interpolant form, the only remaining local unitaries are, up to phase, \[P_\theta\otimes P_{-\theta}\otimes I \quad \text{or} \quad XP_\theta\otimes XP_{-\theta}\otimes X.\]

Finally, if a balanced state has at least one zero component in \(\boldsymbol{\lambda}\), then, after permuting the qubits, we may assume \(\lambda_1=0\). In that case, the phase \(\Phi\) can be removed by a phase rotation on the first qubit \(P_{-\Phi}\otimes I\otimes I.\)

We also require all nonlocality paradoxes to satisfy the following minimality condition: a nonlocality paradox \((\ket{\psi}, \mathcal{M})\) is minimal if \((\ket{\psi}, \mathcal{M}')\) is not a paradox for any \(\mathcal{M}'\) with strictly fewer measurements than \(\mathcal{M}\), i.e.\(M_i' \subseteq M_i\) for all \(i\), with at least one containment strict.

2.3 Impossible events in three-qubit quantum scenarios↩︎

For \(\boldsymbol{\varphi} = (\varphi_1, \varphi_2, \varphi_3)\), the event \(\boldsymbol{\varphi} \to \mathbf{0}\) is impossible precisely when the amplitude \(\braket{\boldsymbol{\varphi}|\BstateNoKet{}{}}\) is \(0\). As shown in [18], this condition holds exactly when the following equation is satisfied: \[\label{eqn:imposs95} \sum_{i=1}^3 \beta(\lambda_i, \varphi_i) \equiv \pi - \Phi \mod 2\pi,\tag{2}\] where the function \(\beta: \left[0, \tfrac{\pi}{2}\right)\times \mathbb{R}/2\pi\mathbb{Z}\rightarrow \mathbb{R}/2\pi\mathbb{Z}\) is defined as follows: \[\label{eqn:beta} \beta(\lambda, \varphi) := \varphi - 2\arctan \left( \frac{\cos{\frac{\lambda}{2}} \sin{\varphi}}{\sin{\frac{\lambda}{2}} + \cos{\frac{\lambda}{2}} \cos{\varphi}} \right).\tag{3}\] We note that the expression for \(\beta\) differs slightly from that of [18] and is instead due to [17]. This function provides a systematic framework for analysing impossible events in tripartite quantum scenarios. The following two lemmas follow directly.

For convenience, in the rest of this paper, a congruence is assumed to be modulo \({2\pi}\) unless otherwise stated.

To better analyse 4 , the auxiliary function \[\delta(\lambda, \varphi) := \beta(\lambda, \varphi + \pi) - \beta(\lambda, \varphi) \equiv \pi - 2\arctan(\sin\varphi\tan\lambda),\] was introduced in [17], which captures the change in \(\beta\) under a flip of the measurement outcome. The properties of \(\delta\) determine which events in a given context are impossible and were analysed in [17].

2.4 Biconditional parity proofs↩︎

3 A graph-theoretic formalism for paradoxes↩︎

A graph-theoretic formalism encodes the logical constraints of nonlocality paradoxes in a precise combinatorial structure, making their properties accessible to rigorous analysis using standard tools from graph theory. In this section and those that follow, we draw on classical results from graph theory [23] and 2-SAT [24], [25]. In particular, we rely on the correspondence between \(2\)-CNF formulae and implication graphs [26]. For completeness and clarity, we restate these results in a unified graph-theoretic language tailored to nonlocality paradoxes.

3.1 Logical structure of nonlocality paradoxes↩︎

We review the logical structure of the three-qubit interpolant-state paradoxes developed in [17] and extend it to general balanced-state scenarios. The GHZ state is itself an interpolant state (with \(\boldsymbol{\lambda}=\mathbf{0}\)), and the corresponding paradox is characterised by a single unsatisfiable system of \(\mathbb{Z}_2\)-linear equations. By contrast, the family of paradoxes in [17], [18] involves multiple such systems, conditioned on the outcome of Charlie’s measurements.

We fix the quantum scenario \((\Bstate,\mathcal{M})\). By 2 , an event \((A,B,C)\to(a,b,c)\) is impossible if and only if \[\label{eqn:imposs95abc} \beta(\lambda_1, A) + a\,\delta(\lambda_1,A) + \beta(\lambda_2, B) + b\,\delta(\lambda_2 , B) \equiv \pi - T_{C,c} -\Phi ,\tag{5}\] where \[\label{eqn:charlie-tick-def} T_{C,c} := \beta(\lambda_3, C) + c\, \delta(\lambda_3 , C) \in \mathbb{R}/2\pi\mathbb{Z},\tag{6}\] which we refer to as a Charlie tick value.

We refer to the pair \((C,z)\), consisting of a measurement \(C\in M_3\) and an outcome \(c=z\), as a Charlie conditioning. We define \[E_{C,z} := \bigl\{ ((A,a),(B,b)) \mid (A,B,C)\to(a,b,z) \text{ is impossible} \bigr\},\] the set of outcome-labelled measurement pairs on Alice and Bob that are impossible with \((C,z)\). We will refer to the atomic propositions \((A,a)\), \((B,b)\) as literals, defining \(\neg(A,a)\) as \((A,a\oplus 1)\).

Now write \(M_3 = \{C_0,\dots,C_{n-1}\}\) and fix a bit string \(\mathbf{z} = (z_0,\ldots,z_{n-1}) \in \mathbb{Z}_2^n\). We define the set \[E_{\mathbf{z}} := \bigcup_{l=0}^{n-1} E_{C_l,z_l},\] which collects all impossible Alice–Bob literal pairs with the chosen Charlie outcomes \(\mathbf{z}\). We refer to \(\mathbf{z}\) as a total Charlie assignment. Each element \(((A,a),(B,b))\in E_{\mathbf{z}}\) imposes the constraint that Alice and Bob cannot output \(a\) on \(A\) and \(b\) on \(B\), i.e.it gives rise to the clause \[\label{eq:clause} \neg((A, a)\;\land\;(B, b)) \iff (A, a\oplus 1) \vee (B, b\oplus 1).\tag{7}\] Let \(\Omega_l(z_l)\) denote the conjunction of all clauses arising from \(E_{C_l,z_l}\), and define \[\label{eq:2CNF} \Omega(\mathbf{z}) := \bigwedge_{l=0}^{n-1} \Omega_l(z_l).\tag{8}\] So, each choice of \(\mathbf{z}\) determines a \(2\)-CNF formula.

For interpolant-state scenarios, \(((A,a),(B,b))\in E_{C,z}\) if and only if \(((A, a\oplus 1),(B,b\oplus 1))\in E_{C,z}\). The two clauses 7 arising from the pairs \(((A,a),(B,b))\) and \(((A,a\oplus 1),(B,b\oplus 1))\) together yield the constraint on compatible global assignments \(g:\bigsqcup_{i=1}^3 M_i \rightarrow \mathbb{Z}_2\) \[\label{eqn:z2-constraint} g(A)\oplus g(B)=a\oplus b \oplus 1.\tag{9}\] Thus, in the interpolant case, the formula \(\Omega(\mathbf{z})\) reduces to a system of \(\mathbb{Z}_2\)-linear equations, which we denote by \(\Psi(\mathbf{z})\). The paradox condition is then precisely that this system be inconsistent for every total Charlie assignment. Equivalently, Lemma 4 says that \[(\Bsimple,\mathcal{M}) \text{ is a paradox} \quad\Longleftrightarrow\quad \Psi(\mathbf{z})\text{ is inconsistent for all }\mathbf{z}\in \mathbb{Z}_2^n.\]

3.2 Graph-theoretic structure of nonlocality paradoxes↩︎

The logical structure of three-qubit paradoxes can be encoded in a graph-theoretic framework. As before, we fix \((\Bstate,\mathcal{M})\) with \(M_3=\{C_0,\dots,C_{n-1}\}\). For a given Charlie conditioning \((C_l,z)\), we define \(G_l(z)\) to be a simple bipartite graph with the vertex set \[V({G}_l (z_l)) := \{(A,a)\mid A\in M_1,\;a\in\mathbb{Z}_2\} \sqcup \{(B,b)\mid B\in M_2,\;b\in\mathbb{Z}_2\},\] and edge set \(E({G}_l (z_l)) := E_{C_l, z_l}\).

Since \(\beta\) is injective on \([0,2\pi)\), if an Alice–Bob literal pair \((A, a)\) and \((B, b)\) forms an edge (i.e.gives an impossible event) for one Charlie conditioning \((C_l, z_l)\), it cannot form an edge for any other Charlie conditioning \((C_{l'}, z_{l'})\). Therefore, the graph \(G (\mathbf{z})\) is simple. Furthermore, for any given Charlie conditioning \((C_l, z_l)\), each literal \((A,a)\) or \((B,b)\) has degree at most \(1\); hence, each edge set \(E_{C_l, z_l}\) is a partial matching.

Each edge of \(G (\mathbf{z})\) induces a clause of the form 7 , and the conjunction of all such clauses yields the \(2\)-CNF formula \(\Omega(\mathbf{z})\) defined in 8 . The satisfiability of \(\Omega(\mathbf{z})\) admits a natural graph-theoretic characterisation.

The implication graph \(I_\Omega (\mathbf{z})\) is a directed bipartite graph, with bipartition given by the literals over \(M_1\) and those over \(M_2\). The satisfiability of \(\Omega(\mathbf{z})\) can be characterised in terms of the strongly connected components of this graph.

For interpolant-state scenarios, the forbidden pairs arise in complementary pairs. Consequently, the implication graph has an additional reversibility property.

4 Classification of biconditional parity proofs↩︎

In this section, we give a complete classification of minimal biconditional parity proofs of three-qubit strong nonlocality. Recall that, up to equivalence, these are precisely those paradoxes involving an interpolant state \(\Bsimple = \Bstate<(0,0,\lambda)>[0]\) and for which Charlie is restricted to two measurements: \(M_3 = \{C_0, C_1\}\subset [0,\pi)\). We will show that, despite the seemingly continuous nature of the parameters involved, a discrete structure emerges: we will demonstrate a bijection between these paradoxes and tuples of numbers that satisfy simple, easily checkable conditions.

We reserve the scalar subscripts \(l,z\in\mathbb{Z}_2\) for a Charlie measurement \(C_l\) and a corresponding Charlie outcome \(z\). The four pairs \((l,z)\) index Charlie’s four tick values \(T_{C_l,z}\), i.e.the possible \(\beta\)-values he can contribute, as defined in 6 . For convenience, we will write \(T_{l,z} := T_{C_l,z}\). A total Charlie assignment (as in Section 3.1) is a pair of outcomes for \(C_0, C_1\), i.e.a vector \(\mathbf{z}\) in \[Q:=\mathbb{Z}_2^2.\]

Our exposition consists of three key stages:

1. First, in Section 4.1, we show that the data associated with Charlie’s qubit and measurements (\(\lambda \text{ and } M_3\) or, equivalently, his set of four tick values), is equivalent to a choice of discrete clock structure: a cyclic group \(\mathbb{Z}_N\), three nonzero elements of \(\mathbb{Z}_{2N}\), and a real-valued offset \(\mu \in [0, 2N)\). We show that this induced clock leads to natural normal forms for Charlie’s ticks as well as Alice and Bob’s measurements. Alice and Bob’s measurements are best expressed as a pair of an element of the clock group \(\mathbb{Z}_N\) and a discrete layer label that indexes a real-valued shift. Finally, we precisely characterise the discrete clock structures, with simple and easily checkable criteria, that are valid in the sense of being both quantum-realisable and completable by a choice of Alice and Bob measurement sets to a minimal paradox.

2. Next, in Section 4.2, we determine the choices of Alice and Bob measurement sets that complete a valid Charlie clock to yield a minimal biconditional parity proof. We apply Lemma 6 and find that the existence of witness paths in the implication graphs of the paradox yields a choice of coset of \(\mathbb{Z}_N\) for each total Charlie assignment and a layer index for each coset. Together, these layer-labelled cosets yield a cover of Alice and Bob’s measurements. We eliminate the redundancy in this data by reducing this to a choice of coset only for certain selected total Charlie assignments. We define such a partial choice to be canonical if it uniquely defines a full choice of cosets and characterise canonicity with elementary number-theoretic criteria.

3. Finally, in Section 4.3, we show that, upon fixing a valid Charlie clock and canonical Alice–Bob completion of that clock, the only remaining freedom is a choice of real-valued shift for each of the layer indices in the Alice–Bob completion. We are then ready to assemble all the above results to give the desired classification theorem.

The main theorem is a bijection \[\mathsf{BPP} \quad \cong \quad \mathsf{CanonicalTriples} \quad\subset \quad \bigsqcup_{\Gamma\in\mathsf{Clock}}\;\; \bigsqcup_{P\in\mathsf{Comp}(\Gamma)}\; \mathsf{Shift}(P).\] Here \(\mathsf{BPP}\) denotes minimal biconditional parity proofs, up to the physical equivalence of Section 2.2.1; \(\mathsf{Clock}\) denotes valid Charlie clocks; \(\mathsf{Comp}(\Gamma)\) denotes canonical Alice–Bob completions over \(\Gamma\); and \(\mathsf{Shift}(P)\) records the remaining real layer shifts. The set \(\mathsf{CanonicalTriples}\) is a subset of all possible triples of a Charlie clock, an Alice-Bob completion of that clock, and a free choice of layer shifts that contains precisely one representative triple for each physical equivalence class of minimal biconditional parity proofs.

4.1 Charlie clocks↩︎

The purpose of this subsection is to isolate the part of the classification controlled entirely by the state and Charlie’s two measurements. We begin with abstract definitions of the Charlie clock structure and what it means for such a structure to be paradoxical and (quantum-)realisable; in the next subsection, we show the definition of paradoxical allows us to construct witness paths for a paradox. We then show how a biconditional parity proof yields a clock structure, and express all measurements relative to this clock, with offsets. Finally, we classify the clocks that are both paradoxical and realisable.

Geometrically, a Charlie clock places Charlie’s four possible outcome-dependent \(\beta\)-contributions on a common \(2N\)-clock, with the real offset \(\mu\) fixing the absolute rotation and the indices \(t_{l,z}\) recording the discrete positions of the ticks. The parameters \(s_0\) and \(s_1\) are the spacings between opposite outcomes of \(C_0\) and \(C_1\), while \(t\) is the offset between the first tick of \(C_0\) and the first tick of \(C_1\). For a total Charlie assignment \(\mathbf{z}=(z_0,z_1)\), the number \(H_{\mathbf{z}}=t_{1,z_1}-t_{0,z_0}\) will be shown to be how many ticks the clock advances between consecutive Alice literals in a witness path; such a step is obtained by following two consecutive implication edges connected to the intermediate Bob literal, one from each Charlie conditioning. Thus \(d_{\mathbf{z}}=\gcd(N, H_{\mathbf{z}})\) records the period of iterating this advancement on the measurement clock \(\mathbb{Z}_N\), and \(D_{\mathbf{z}}=d_{\mathbf{z}}\mathbb{Z}_N\) is exactly the return orbit subgroup: its cosets are the possible Alice index sets of \(\mathbf{z}\)-witness paths. Finally, \(u_{\mathbf{z}}=t_{0,z_0}\) is the reflection offset for the first Charlie conditioning, so any Alice measurement \(A\) involved in an impossible event in that conditioning forces Bob to use the reflected measurement \(u_{\mathbf{z}}-B\).

Next, we define a class of Charlie clocks that we will, in the next subsection, show are those that allow us to construct the witness paths needed for a paradox.

We now define the class of abstract Charlie clock structures that are actually realised by a choice of interpolant state and a pair of measurements.

Finally, we now define the set of valid clocks to be those that are both quantum-realisable and completable by a choice of Alice-Bob measurements to a paradox. To avoid this set containing two clocks that represent the same paradox up to physical equivalence, we normalise the clock data.

We denote the set of valid Charlie clocks by \(\mathsf{Clock}\).

4.1.1 Clock normal form↩︎

We first explain how to extract the Charlie clock from a biconditional parity proof. Lemma 6 says that in an interpolant-state scenario the relevant implication graphs are bidirected, so a contradiction is equivalent to the existence of a witness path from a literal to its complement. In the biconditional case, a minimal witness path alternates between the edges of two Charlie measurements. Moving two edges along such a path translates Alice’s measurement angle by a difference of two Charlie ticks. Since the path returns to the same Alice measurement with opposite outcome, that tick difference must be a rational multiple of \(\pi\). Repeating this for all total Charlie assignments produces a common finite clock.

Recall [17] that for scenarios involving interpolant states, the impossibility equation 5 has a much simpler form: an event \((A,B,C_l) \to (a,b,z_l)\) is impossible if and only if \[\label{eqn:imposs95interpolant} A + B \equiv T_{l,z} + (1 \oplus a \oplus b) \pi.\tag{13}\] More generally, a context \((A,B,C)\) contains impossible events if and only if \[\label{eqn:imposs95interpolant95mod95pi} A + B \equiv T_{l,z} \mod\pi.\tag{14}\]

Given a clock, we can naturally write Alice and Bob’s measurements relative to it by foliating \([0,\pi)\) into a disjoint union of translates of \(\frac{\pi}{N}\mathbb{Z}_N\).

As a consequence of the above Proposition, each of the measurements involved in a witness path, required to establish a paradox, lives entirely within one layer. Having chosen a layer index \(i \in \mathcal{I}\), we may simplify the notation and write \(A_j := A_{i,j},\, B_k := B_{i,k}\). Moreover, upon fixing a total Charlie assignment \(\mathbf{z}= (z_0, z_1)\), we write \(t_l := t_{l, z_l}\).

4.1.2 Valid Charlie clocks↩︎

We now completely classify the valid Charlie clocks. This is the only point in the proof where quantum realisability, via the analytic functions \(\beta\) and \(\delta\), enter. After this subsection, all remaining arguments are combinatorial and number-theoretic.

First, we define the pairs of possible \(\beta\) contributions Charlie can make with his two outcomes for a single fixed measurement, assuming the state parameter \(\lambda\) is also fixed.

Next, we determine which tick pairs arise from actual choices of \(\lambda\) and Charlie measurement \(C \in [0, \pi)\) and show how to extract \(\lambda, C\) from the tick pair.

Next we characterise when two tick pairs can be realised by two Charlie measurements on a single state. Two distinct tick pairs \(\mathbf{T}_0 = (-\tau_0, \sigma_0 - \tau_0)\) and \(\mathbf{T}_1 = (-\tau_1, \sigma_1 - \tau_1)\) are quantum realisable if and only if they can be realised by the same \(\lambda\). If one of the tick pairs is \((0, \pi)\), then \(\lambda\) is uniquely determined by the other tick pair via ?? . Otherwise, it is necessary and sufficient that \[\begin{gather} \Lambda(\tau_0, \sigma_0) = \Lambda(\tau_1, \sigma_1) \nonumber\\ \implies\quad \frac{\cos(\sigma_0/2)}{\cos(\tau_0 - \sigma_0/2)} = \frac{\cos(\sigma_1/2)}{\cos(\tau_1 - \sigma_1/2)}.\label{eqn:equal-lambdas} \end{gather}\tag{20}\]

Finally, we adjoin the conditions of a Charlie clock to be paradoxical to it being realisable, and characterise the valid clocks.

We have now completed the state-and-Charlie part of the classification. The next subsections fix a valid clock and determine exactly which Alice–Bob measurements complete it to a minimal biconditional parity proof.

4.2 Alice–Bob completions↩︎

Fix a valid Charlie clock \(\Gamma\). For each total Charlie assignment \(\mathbf{z}\), Definition 6 associates a subgroup \(D_{\mathbf{z}} \leq \mathbb{Z}_N\). This subsection shows how the Alice and Bob measurements constituting a witness path in the implication graph \(I_\Psi(\mathbf{z})\) are precisely described by cosets of \(D_\mathbf{z}\). More specifically, if Alice’s measurements lie in the coset \(y+D_{\mathbf{z}}\), then Bob’s measurements are forced to lie in the reflected coset \(u_{\mathbf{z}}-y+D_{\mathbf{z}}\).

While a paradox determines Alice (and therefore Bob) coset choices for each \(\mathbf{z}\), we do not necessarily need a coset for every \(\mathbf{z}\) to reconstruct the paradox. This is because after choosing cosets only for some selected \(S \subset Q\), their union may already contain cosets for each \(\mathbf{z}\in S \setminus Q\). This is decided by the shadow test, a simple congruence-divisibility criterion.

Much of the technical work in this section is devoted to reducing the problem from specifying all coset witnesses to specifying only a distinguished subset of them. The payoff is the notion of a canonical Alice–Bob completion, which identifies the minimal data needed to reconstruct the Alice and Bob measurement sets uniquely, up to arbitrary real-valued labels assigned to the layers.

4.2.1 Completions and measurement supports↩︎

Before the identification of witness paths with cosets in Section 4.2.2, we first introduce a set-theoretic construction that records the measurements that must appear in Alice’s and Bob’s measurement sets in order to contain a coset witness path. We then define an Alice–Bob completion as the data of a selected collection of total Charlie conditionings, a partition of these conditionings into layers, and a choice of coset for each selected conditioning. In the remainder of this subsection, we show that these cosets indeed give witness paths, and we develop a simple shadow test for determining when a partial selection of layer-indexed cosets already contains all witness paths needed to certify a paradox.

Throughout, every coset representative is taken to be the least integer representative of its coset. Every valid clock admits Alice–Bob completions: take \(S=Q\), one layer, and any representatives \(y_{\mathbf{z}}\in\mathbb{Z}_N/D_{\mathbf{z}}\). We can remove one measurement at a time until the paradox is minimal.

4.2.2 Witness paths as paired cosets↩︎

We now prove that witness carriers are exactly the witness paths that demonstrate \(\Omega(\mathbf{z})\) is inconsistent once they have been expressed relative in terms of the Charlie clock.

The following maps will be shown to capture the effect of taking two nontrivial steps in a witness path from an Alice literal, to a Bob literal, to the next Alice literal, as expressed in terms of the clock. We require the effect on both the measurement label and the literal (measurement with outcome).

Observe that, within a fixed layer, a literal can be equivalently written in a clock form as follows.

The following lemma establishes that, assuming that a minimal directed path involves Alice and Bob measurements that have a coset structure consistent with the measurement return map, there is a well-defined lifting from measurement labels in \(\mathbb{Z}_N\) to literals in \(\mathbb{Z}_{2N}\), hence justifying the well-definedness of the literal return map.

Now we show that the structure of a minimal directed path considered in Lemma 11 in fact applies to every possible minimal directed path, allowing us to deduce an equivalence between Alice/Bob measurement cosets and witness paths.

We have now established that, given a valid Charlie clock, a paradox is the same as a choice of representative for a coset of \(D_\mathbf{z}\leq \mathbb{Z}_N\), and layer index, for each \(\mathbf{z}\in Q\). Next, we reduce this information to the minimum necessary and identify those Alice–Bob completions that do not contain redundant information.

4.2.3 The shadow test↩︎

An Alice–Bob completion may select witness carriers only for some total Charlie assignments. Nevertheless, the measurement support generated by those selected carriers may contain further witness carriers. The purpose of the shadow test is to determine exactly which carriers are already present. Since all sets involved are cosets of cyclic groups, this test ultimately reduces to elementary congruence checks.

Fix a valid Charlie clock \(\Gamma\), and let \(P=(S,\mathcal{I},\iota,\mathbf{y})\) be an Alice–Bob completion. For a target assignment \(\mathbf{w}\in Q\), a layer \(i\in \mathcal{I}\), and a representative \(r\in\mathbb{Z}_N/D_{\mathbf{w}}\), we want to decide whether the full carrier \(\mathsf W_{\mathbf{w}}(i,r)\) is contained in \(\mathsf{MSupp}(P)\). This means checking both \[r+D_{\mathbf{w}} \subseteq \mathcal{J}_i(P)\] on Alice’s side, and \[u_{\mathbf{w}}-r+D_{\mathbf{w}} \subseteq \mathcal{K}_i(P)\] on Bob’s side.

It is useful to compare Alice and Bob in the same coordinates. Reflect Bob’s clock through \(u_{\mathbf{w}}\), i.e.apply \[x\longmapsto u_{\mathbf{w}}-x.\] Then the target Bob coset \(u_{\mathbf{w}}-r+D_{\mathbf{w}}\) becomes \(r+D_{\mathbf{w}}\). A selected Bob source coset \[u_{\mathbf{z}}-y_{\mathbf{z}}+D_{\mathbf{z}}\] is correspondingly reflected to \[y_{\mathbf{z}}+u_{\mathbf{w}}-u_{\mathbf{z}}+D_{\mathbf{z}}.\] Thus both Alice and reflected Bob ask the same question: which parts of the target coset \(r+D_{\mathbf{w}}\) are covered by selected source cosets?

The divisibility condition in Definition 16 says that a selected source coset is large enough to contain a parity fibre of the target coset. The congruence says which fibre it contains.

It remains to justify that no finer pieces are needed. A priori, a target coset might be covered by thirds, quarters, or still smaller intersections. The following two lemmas show that, in the case where \(|M_3| = 2\) and hence \(|Q| = 4\), any such cover is detected already on the quotient \(D_{\mathbf{w}}/2D_{\mathbf{w}}\).

Thus the shadow test computes the witness carriers already present in an Alice–Bob completion using only the elementary checks \[d_{\mathbf{z}}\mid 2d_{\mathbf{w}} \qquad\text{and}\qquad c^p_{\mathbf{w}\mathbf{z}}\equiv r+\varepsilon d_{\mathbf{w}}\mod{d_{\mathbf{z}}}.\]

4.2.4 Canonical Alice–Bob completions↩︎

We now show that every minimal biconditional parity proof, which we have shown to determine a valid Charlie clock, also determines a unique canonical Alice–Bob completion.

Equivalently, \(P\) is canonical precisely when every total Charlie assignment has at least one solution and \[S= \{\mathbf{w}\in Q \mid \operatorname{Sol}_P(\mathbf{w}) \text{ is a singleton}\}.\] This equivalence is perhaps the most useful way to remember the definition: the selected assignments are exactly the uniquely witnessed assignments.

Thus, a canonical completion is precisely the finite coset data that minimally and uniquely specifies the Alice–Bob measurement support of the proof.

4.3 Classification of biconditional parity proofs↩︎

We now assemble the classification. The valid clock gives the state and Charlie measurements. The canonical Alice–Bob completion gives the finite measurement support. The only remaining freedom is the assignment of real numbers to each layer. These have the effect of translating each layer, to a different disjoint copy of the clock and do not affect the essential structure of the paradox. This freedom is recorded by a shift tuple.

4.3.1 Layer shifts and measurement sets↩︎

Thus, after the clock and canonical completion are fixed, the shift tuple is exactly the remaining continuous parameter in the proof.

4.3.2 The bijection↩︎

We can now define the combinatorial data that classifies a biconditional parity proof up to the physical equivalence of Section 2.2.1  eliminating the possible redundancy created by any freedom of how to label the three parties.

This completes the classification. Every minimal biconditional parity proof is encoded by a valid finite Charlie clock, a canonical Alice–Bob completion in terms of coset representatives, and a normalised tuple of real-valued layer shifts.

5 Interpolant-state paradoxes with more than two Charlie measurements↩︎

In this section, we investigate interpolant-state scenarios with at least three Charlie measurements and identify new classes of previously unknown, more exotic forms of nonlocality paradox. These were discovered by using computer-aided searches applied to the novel graph-theoretic formalism developed in Section 3. Examples of minimal three-qubit nonlocality paradoxes with more than two Charlie measurements were not known to exist prior to this.

5.1 A new GHZ-state paradox↩︎

For the GHZ state, the impossibility equation 4 simplifies to \[\label{eq:beta95GHZ} A+B+C \equiv (a\oplus b\oplus c \oplus 1) \pi .\tag{28}\] Thus, if \(A+B+C\) is an integer multiple of \(\pi\), the outcomes given by a compatible global assignment must satisfy the possible parity constraint \[\label{eq:GHZ-outcome} a \oplus b \oplus c \equiv \frac{A+B+C}{\pi} \mod{2},\tag{29}\] whereas if \(A+B+C\) is not a multiple of \(\pi\), the context has no impossible events, hence contributes no parity constraint.

Let us consider the quantum scenario \((\ket{\mathrm{GHZ}}, \mathcal{M})\) with \[M_i = \{ \tfrac{\pi}{4},\, \tfrac{\pi}{2},\, \tfrac{3\pi}{4} \} \qquad \text{for each } i=1,2,3.\] Given a global assignment, we denote the corresponding outcomes by \(a_0,a_1,a_2\) for Alice, and similarly by \(b_k\) and \(c_l\) for Bob and Charlie. By 29 , the associated system of \(\mathbb{Z}_2\)-linear equations for this scenario is as follows: \[\label{eq:GHZ-system2} \widetilde{\Psi}_{\mathrm{GHZ}} = \begin{cases} a_0 \oplus b_0 \oplus c_1 = 1, \quad a_0 \oplus b_1 \oplus c_0 = 1,\\ a_1 \oplus b_0 \oplus c_0 = 1, \quad a_1 \oplus b_2 \oplus c_2 = 0, \\ a_2 \oplus b_1 \oplus c_2 = 0, \quad a_2 \oplus b_2 \oplus c_1 = 0. \end{cases}\tag{30}\] Summing all six equations in 30 yields \(0 = 1\), establishing paradoxicality. Notably, the scenario \((\ket{\mathrm{GHZ}}, \mathcal{M})\) is not maximally impossible in the sense of [17], illustrating the rich structure of nonlocality paradoxes even for the simple GHZ state.

5.2 Three Charlie measurements: symmetric Alice–Bob case↩︎

We establish the existence of new exotic families of nonlocality paradoxes involving interpolant states \(\Bsimple\) with \(\lambda \neq 0\). We refer to a scenario \((\Bsimple, \mathcal{M})\) as an \((N_1, N_2, N_3)\)-scenario if \(|M_i| = N_i\) for \(i=1,2,3\). In this subsection, we will deal with the case \(N_1 = N_2\).

Fix an even integer \(N:=2n \ge 4\). For an integer \(0< m\le n-2\), define the interpolant-state parameter \[\lambda_{N,m} := \frac{\pi}{2} - \frac{m\pi}{N} \in (0,\tfrac{\pi}{2}).\] Note that different pairs \((N,m)\) may yield the same value of \(\lambda_{N,m}\) (for example, \(\lambda_{12,4}=\lambda_{24,8}=\frac{\pi}{6}\)).

Define the set $ J := _n$ and the measurement scenario \[\begin{align} M_1 := J \cup (u + J), \quad M_2 := v - M_1 , \quad \text{and} \quad M_3 := \{ C_0, C_1, C_2 \} ; \end{align}\] where all angles are taken modulo \(\pi\) with \[\begin{align} u := \big(v - \beta(\lambda_{N,m},C_2)\big) \mod{\tfrac{2\pi}{N}},\quad v := \beta\big(\lambda_{N,m},\tfrac{\pi}{2}\big) \mod{\pi} ,\\ C_0 (N, m) := \arcsin\!\left(\frac{\tan\lambda_{N,m+1}}{\tan\lambda_{N,m}}\right), \quad C_1 := \tfrac{\pi}{2}, \quad \text{and} \quad C_2 = \pi - C_0. \end{align}\] This choice of \(u\) ensures that \(|M_1|=N\), and by symmetry \(|M_2|=N\). Since \(m\le n-2\), we have \(\lambda_{N,m+1}\in(0,\frac{\pi}{2})\) and \(\tan\lambda_{N,m} > \tan\lambda_{N,m+1}\), so the argument of \(\arcsin\) lies in \((0,1)\) and \(C_0\) is well defined.

5.3 Three Charlie measurements: asymmetric Alice–Bob case↩︎

Fix an integer \(p\ge 1\), \(N := 4p\), an odd integer \(1\le m < 2p\), and set \[\theta := \frac{m\pi}{N} \in (0, \tfrac{\pi}{2} ) .\] Define the interpolant parameter \(\lambda\) and the angle \(C\) by \[\label{eq:param95intp953A} \begin{align} u &:= \sqrt{\tfrac{\tan{\theta}}{\tan{(\theta/2)}}}, \qquad v:= \sqrt{\tan{\theta} \tan{(\theta/2)}}, \\ \lambda_{p,m} &:= 2\arctan \big( \tfrac{u-1}{u+1} \big), \qquad C:= 2\arctan{v} . \end{align}\tag{31}\] Since \(u>1\) and \(v>0\), it follows that \(\lambda \in \big( 0, \tfrac{\pi}{2})\) and \(C\in (0,\pi)\).

Define the measurement sets (modulo \(\pi\)): \[\begin{align} M_1 &:= \big\{j\tfrac{\pi}{N} \mid j \in \mathbb{Z}_{N} \big\}, \\ M_2 &:= \big\{k\tfrac{\pi}{N} \mid k\in2\mathbb{Z}_{2p}\big\}, \\ M_3 &:= \{C_0:=0,\;C_1:=C,\;C_2:=\pi-C \}, \end{align}\] so that \(|M_1| = 4p = N\) and \(|M_2|= 2p = \frac{N}{2}\).

5.4 Four Charlie measurements↩︎

In contrast to the previous cases, we do not obtain a uniform family here, primarily because the parameter \(\lambda\) and the angles \(C_l\) do not admit a closed-form description in this setting.

We take \(M_1, M_2 \subseteq \frac{\pi}{12}\mathbb{Z}_{12} \subset [0,\pi)\) and write \(A_j :=j\frac{\pi}{12}\) and \(B_k := k\frac{\pi}{12}\) for the Alice and Bob measurement angles, respectively. By numerical search, we find the following parameters \[\label{eq:param95495msnts} \begin{gather} \lambda_* \approx 0.5440881066818782, \\ C_* = 0.4588205874371110, \qquad C_*' = 1.2673000748575629 \end{gather}\tag{32}\] such that with \(\lambda = \lambda_*\), \(C = C_*\), and \(C' = C_*'\), we have \[\label{eq:tick95values95495msnts} \begin{align} T_{C, 0} &\equiv \tfrac{23\pi}{12} , \quad T_{C, 1} \equiv \tfrac{3\pi}{4},\quad T_{\pi-C, 0} \equiv \tfrac{5\pi}{4}, \quad T_{\pi-C, 1} \equiv\tfrac{\pi}{12},\\[0.5em] T_{C', 0} &\equiv \tfrac{7\pi}{4}, \quad T_{C', 1} \equiv \tfrac{5\pi}{12},\quad T_{\pi-C', 0} \equiv \tfrac{19\pi}{12}, \quad T_{\pi-C', 1} \equiv \tfrac{\pi}{4} . \end{align}\tag{33}\]

The parameters in 32 are determined only up to a numerical error of order \(10^{-13}\). We fix \[M_3 := \{ C_0 :=C_*,\;C_1 :=C_*',\;C_2 :=\pi-C_*,\;C_3 :=\pi-C_*' \}.\]

6 A nonlocality paradox beyond interpolant states↩︎

In this section, we present the first example of a nonlocality paradox arising from a state outside the interpolant family; indeed, the existence of a non-interpolant-state paradox was previously conjectured to not hold. Since the measurement scenario includes only two Charlie measurements, the paradox remains biconditional. However, its proof does not admit a reformulation as a parity proof, and therefore lies outside the classification of biconditional parity proofs given in Section [sec:two-charlie].

For this section, it is convenient to rewrite the additive impossibility condition 2 in multiplicative form. Under this reformulation, the quantum condition that an event has probability zero becomes a product equation rather than an additive congruence. To this end, we identify each \(\beta\) value with its corresponding point on the circle. To witness paradoxicality, we construct finite implication cycles that force an Alice literal and its complement to lie in the same strongly connected component.

6.1 The unit circle reformulation↩︎

Recall that a literal is a pair \((\varphi, z)\) with \(\varphi \in [0,\pi)\) an equatorial measurement angle and \(z\in \mathbb{Z}_2\).

It is straightforward to see that the quotient in 34 has modulus \(1\), since its numerator and denominator have the same modulus. Indeed, \[\left|\sin\tfrac{\lambda}{2} \pm e^{-i\varphi} \cos\tfrac{\lambda}{2} \right|^2 = \sin^2\tfrac{\lambda}{2}+\cos^2\tfrac{\lambda}{2} \pm 2\sin\tfrac{\lambda}{2} \cos\tfrac{\lambda}{2} \cos\varphi = \left|\cos\tfrac{\lambda}{2} \pm e^{-i\varphi} \sin\tfrac{\lambda}{2} \right|^2 .\] Thus, each such quotient defines a point on the unit circle.

Moreover, each tick determines a unique literal, since the map \[w \longmapsto \frac{\sin{\frac{\lambda}{2}}+ w \cos{\frac{\lambda}{2}}}{\cos{\frac{\lambda}{2}} + w \sin{\frac{\lambda}{2}} }\] is a fractional-linear automorphism of \(\mathbb{T}\). We can now restate the impossibility condition in this formalism.

Note that \(\beta(\lambda, \varphi)\), where \(\beta\) is as in 3 , is the argument of \(T_{\lambda} (\varphi)\). Hence, 4 is exactly the argument form of 35 .

It will also be useful to relate the tick \(T_\lambda(\varphi)\) to the tick of the complementary literal, namely \(T_\lambda(\varphi+\pi)\).

The complement map \(F_{\lambda}\) on the unit circle corresponds to an additive involution \(\mathsf{F}_{\lambda}: \mathbb{R}/2\pi \mathbb{Z}\to \mathbb{R}/2\pi \mathbb{Z}\) on phases defined by \(e^{i\mathsf{F}_{\lambda} (\xi)} = F_{\lambda} (e^{i\xi})\). Equivalently, \[\mathsf{F}_{\lambda} (\xi) = \xi + \hat{\delta} (\xi),\] where \(\hat{\delta} (\xi)\) is is the \(\delta\)-shift associated to the unique \(\varphi\) such that \(\beta(\lambda, \varphi) = \xi\).

Fix a Charlie conditioning \((C_l,z_l)\). An Alice tick \(T_1\) and a Bob tick \(T_2\) determine an impossible event precisely when \[T_1\, T_2 = -e^{i\Phi} (T_{\lambda_3} (C_l + z_l \pi))^{-1} =: \Gamma_{l,z}.\] Since the example has two Charlie measurements, each total Charlie assignment \(\mathbf{z}\in \mathbb{Z}_2^2\) determines two associated target products: \(\Gamma_{0,z_0}\) and \(\Gamma_{1,z_l} \in \mathbb{T}\).

For \(\Gamma \in \mathbb{T}\), define the reflection \(R_{\Gamma} : \mathbb{T}\to \mathbb{T}\) by \[R_{\Gamma} (\zeta) := \overline{\zeta} \Gamma.\] Note that \(\zeta R_\Gamma (\zeta) = \Gamma\). Thus \(R_\Gamma (\zeta)\) is the unique tick paired with \(\zeta\) to hit the target product \(\Gamma\).

We also assume that the balanced-state parameters for Alice and Bob coincide, as will be the case in the example below. Thus, set \(\alpha := \lambda_1 = \lambda_2\). Fix a total Charlie assignment \(\mathbf{z}\), and write \(\Gamma_0 := \Gamma_{0,z_0}\) and \(\Gamma_1 := \Gamma_{1,z_1}\). We define the Alice return map \(P_{\mathbf{z}}: \mathbb{T}\to \mathbb{T}\) as follows:

\[\label{eq:A-A-map} P_{\mathbf{z}} := F_{\alpha} \circ R_{\Gamma_1} \circ F_{\alpha} \circ R_{\Gamma_0} .\tag{37}\] An application of \(P\) corresponds to passing through a first impossible event, taking the complementary tick, then passing through a second impossible event and taking the complementary tick once more.

6.2 The non-interpolant paradox example↩︎

6.2.1 The balanced state and Charlie’s measurements↩︎

Let us set \[\rho:=\sqrt{2}-1, \qquad \alpha:=\arcsin(\rho).\] We take Alice and Bob to have the same balanced-state parameter: \(\lambda_1=\lambda_2:=\alpha\). Thus, their common complement map is \[F(\zeta):= F_\alpha(\zeta) = \frac{\rho-\zeta}{1-\rho\zeta}.\] For Charlie, we take the balanced-state parameter \(\lambda_3:=\frac{\pi}{6}\), and the two equatorial measurements \[C_0:=0, \qquad C_1:=\frac{\pi}{2}.\] Finally, take the balanced-state phase to be \(\Phi:= \pi\). The state is therefore \[\label{eq:bal-state-counterex} \ket{\mathrm{B} ((\alpha, \alpha, \tfrac{\pi}{6}), \pi)} = \frac{1}{\sqrt{2(1-\rho^2/2)}} \left( \ket{v_\alpha}\ket{v_\alpha}\ket{v_{\pi/6}} - \ket{w_\alpha}\ket{w_\alpha}\ket{w_{\pi/6}} \right) .\tag{39}\] Charlie’s ticks are as follows: \[T_{\frac{\pi}{6}}(0) = 1,\;T_{\frac{\pi}{6}}(\pi) = -1 , \quad T_{\frac{\pi}{6}}(\tfrac{\pi}{2}) = e^{-i\frac{\pi}{3}},\;T_{\frac{\pi}{6}}(\tfrac{\pi}{2}+\pi) = e^{i\frac{\pi}{3}}.\] Since \(\Phi=\pi\), the impossibility condition 35 reduces to the requirement that the product of the three ticks be equal to \(1\). Thus, after fixing a Charlie literal with tick \(\chi\), the corresponding target product for the Alice and Bob ticks is \(\chi^{-1}\). Therefore the target products for \(C_0\) are \(\Gamma_{0,0} = 1, \;\Gamma_{0,1} = -1\), and for \(C_1\) are \(\Gamma_{1,0} = e^{i\frac{\pi}{3}} ,\;\Gamma_{1,1} = e^{-i\frac{\pi}{3}}\).

6.2.2 The four Alice return maps↩︎

We will verify that, for each \(\mathbf{z}\in \mathbb{Z}_2^2\), the map \(P_{\mathbf{z}}\) has finite order and that there exists a tick \(\zeta_{\mathbf{z}}\in\mathbb{T}\) satisfying 38 . We also note that the reflection map \(R_\Gamma\) may be written as \(\zeta \longmapsto \Gamma/\zeta\), since, for \(\zeta\in\mathbb{T}\), one has \(\overline{\zeta}=\zeta^{-1}\). Thus \(R_\Gamma\) may be viewed as a Möbius transformation, represented by the linear fractional matrix \[M_{\Gamma} = \begin{pmatrix} 0 & \Gamma \\ 1 & 0 \end{pmatrix} ,\] where matrices are understood projectively. Similarly, the complement map \(F\) is represented by \[M_{F} = \begin{pmatrix} -1 & \rho \\ -\rho & 1 \end{pmatrix} .\] Consequently, the Alice return map \(P_{\mathbf{z}}\) is represented projectively by \[\label{eq:matrix-return-map} M_{\mathbf{z}} := M_F M_{\Gamma_{1,z_1}} M_F M_{\Gamma_{0,z_0}} .\tag{40}\]

Let \(\Gamma_{0,z_0} = e^{i\theta_{0,z_0}}\) and \(\Gamma_{1,z_1} = e^{i\theta_{1,z_1}}\), using representatives \[\theta_{0,z_0} \in \{0, \pi\}, \qquad \theta_{1,z_1} \in \{\tfrac{\pi}{3}, -\tfrac{\pi}{3} \}.\] Multiplying the matrices in 40 and dividing the trace by \(\sqrt{\det M_{\mathbf{z}}}\) gives the normalised trace of \(M_\mathbf{z}\): \[\label{eq:norm-trace} \tau_{z_0,z_1} := \frac{\operatorname{Tr}(M_\mathbf{z})}{\sqrt{\operatorname{det}(M_\mathbf{z})}} = 2\,\frac{\cos{\left( \frac{\theta_{0,z_0}-\theta_{1,z_1}}{2} \right)} - \rho^2 \cos{\left(\frac{\theta_{0,z_0}+\theta_{1,z_1}}{2}\right)}}{1-\rho^2} .\tag{41}\] Substituting \(\rho^2 = 3 - 2\sqrt{2}\) and \(\frac{1+\rho^2}{1-\rho^2} = \sqrt{2}\) into 41 gives \[\tau_{0,0} = \tau_{0,1} = 2\cos{\tfrac{\pi}{6}}, \qquad \tau_{1,0} = - 2\cos{\tfrac{\pi}{4}}, \qquad \tau_{1,1} = 2\cos{\tfrac{\pi}{4}}.\] Let \(M\) be a determinant-one representative of a projective Möbius transformation. If \(\operatorname{tr}(M)=2\cos\theta\), then the characteristic polynomial of \(M\) is \(\xi^2-2\cos\theta\,\xi+1\), so the eigenvalues are \(e^{i\theta}, \;e^{-i\theta}\). The projective order is the smallest positive integer \(n\) such that \(M^n\) is a scalar multiple of the identity. Equivalently, \(e^{in\theta}=e^{-in\theta}\) or \(n\theta\in \pi\mathbb{Z}\). Hence, if \(\theta/\pi=k/N\) in lowest terms, the projective order is \(N\). Therefore \[\operatorname{ord} (P_{0,0}) = \operatorname{ord} (P_{0,1}) = 6, \qquad \operatorname{ord} (P_{1,0}) = \operatorname{ord} (P_{1,1}) = 4.\]

6.2.3 The starting tick↩︎

It remains to set the starting ticks \(\eta_\mathbf{z}\in \mathbb{T}\) that satisfy 38 for each \(\mathbf{z}\in \mathbb{Z}_2^2\). For \(\mathbf{z}\in \{ (0,0), (0,1)\}\), we take \(\eta_{0,0} = \eta_{0,1} = 1\). Note that \[M_{0,0} = M_{0,1} = 8\begin{pmatrix} 10-7\sqrt{2} & 7-5\sqrt{2} \\ -7+5\sqrt{2} & -10+7\sqrt{2}. \end{pmatrix}\] Hence, \[P_{0,0}^3 (1) = P_{0,1}^3 (1) = \frac{(10-7\sqrt{2}) + (7-5\sqrt{2})}{(-7+5\sqrt{2}) + (-10+7\sqrt{2})} = -1\] Since \(F(1) = \frac{\rho -1}{1- \rho} = -1\), we have \[\label{eq:P00} P_{0,0}^3 (\eta_{0,0}) = F(\eta_{0,0}), \qquad P_{0,1}^3 (\eta_{0,1}) = F(\eta_{0,1}).\tag{42}\]

For \(\mathbf{z}\in \{ (1,0), (1,1) \}\), i.e.the two order-four maps \(P_{1,0}\) and \(P_{1,1}\), we choose the starting ticks by solving 38 explicitly. Let \[h(t):=(3-2\sqrt2)t^2+\sqrt3(10-7\sqrt2)t+(7-5\sqrt2).\] Since its discriminant \(\Delta=430-304\sqrt2\) is positive, \(h\) has real roots. We choose \[t_* := \frac{-\sqrt3(10-7\sqrt2)+\sqrt{430-304\sqrt2}}{2(3-2\sqrt2)},\] so that \(h(t_*)=0\).

Let us set \[\eta(t):=\frac{1+it}{1-it}.\] For \(t\in \mathbb{R}\), \(|1+it|=|1-it|\) and hence \(|\eta(t)|=1\). We set \[\eta_{1,0}:=\eta(t_*), \qquad \eta_{1,1}:=\eta(-t_*) = \overline{\eta_{1,0}}.\] It remains to verify 38 . Write \[M_{1,0}^2= \begin{pmatrix} a & b \\ c & d \end{pmatrix}.\] Since \(F(\zeta)=(\rho-\zeta)/(1-\rho \zeta)\), the equation \(P_{1,0}^2(\zeta)=F(\zeta)\) is equivalent, after clearing denominators, to \[(-\rho a+c)\zeta^2 + (a-\rho b-\rho c+d)\zeta + (b-\rho d)=0.\] Substituting \(\zeta=\eta(t)\) and multiplying by \((1-it)^2\) simplifies the left-hand side to \[4(1+i\sqrt3)h(t).\] Thus, \(h(t_*)=0\) implies \[\label{eq:P10} P_{1,0}^2(\eta_{1,0})=F(\eta_{1,0}).\tag{43}\] Finally, \(P_{1,1}\) is obtained from \(P_{1,0}\) by complex conjugating the target products. Since \(F\) has real coefficients, conjugating 43 gives \[\label{eq:P11} P_{1,1}^2(\eta_{1,1})=F(\eta_{1,1}).\tag{44}\] Collecting 42 –44 , each total Charlie assignment \(\mathbf{z}\) has the data required by the conditions of Lemma 18:

6.2.4 Proof of paradoxicality↩︎

For a tick \(\zeta \in \mathbb{T}\), let \(\mathsf{m}(\zeta) \in [0,\pi)\) denote the equatorial measurement whose two literal ticks are \(\zeta\) and \(F(\zeta)\). This is well-defined because \(T_\alpha\) is a bijection of the unit circle. We now define the measurement scenario for Alice and Bob. As above, Charlie’s measurement set is \[\label{eq:M953} M_3 = \left\{0,\, \frac{\pi}{2} \right\} .\tag{45}\] For Alice, we take all measurements appearing in the four Alice return orbits: \[\label{eq:M951} M_1 := \big\{ \mathsf{m}(P_{\mathbf{z}}^r(\eta_{\mathbf{z}})) \mid \mathbf{z}\in \mathbb{Z}_2^2,\;0\le r < \operatorname{ord} (P_{\mathbf{z}}) \big\} .\tag{46}\] For Bob, we take the corresponding reflected ticks for the first Charlie conditioning: \[\label{eq:M952} M_2 := \big\{ \mathsf{m}(R_{\Gamma_{0,z_0}} \circ P_{\mathbf{z}}^r(\eta_{\mathbf{z}})) \mid \mathbf{z}\in \mathbb{Z}_2^2,\;0\le r < \operatorname{ord} (P_{\mathbf{z}}) \big\} .\tag{47}\] These sets are finite because each return map \(P_{\mathbf{z}}\) has finite order.

We now verify that the balanced state 39 and the above measurement scenario indeed witness a nonlocality paradox.

We have shown that the above scenario is paradoxical. We now prove that the underlying state is not equivalent to an interpolant state.

Acknowledgments↩︎

The authors acknowledge the use of AI throughout the research process for discussion and drafting; they are solely responsible for all results, proofs, and exposition. The authors acknowledge support from the Canada Research Chair program, NSERC Discovery Grant RGPIN-2022-03103, and the NSERC-European Commission project FoQaCiA.

7 Proofs of select results↩︎

7.1 Proofs of results in Section [sec:two-charlie]↩︎

7.1.1 Proof of Lemma 9↩︎

7.1.2 Remainder of proof of Proposition 9↩︎

7.2 Proofs of results in Section 5↩︎

7.2.1 Proof of Theorem 20↩︎

7.2.2 Proof of Lemma 15↩︎

7.2.3 Proof of Theorem 21↩︎

7.3 Proofs of results in Section 6↩︎

7.3.1 Proof of Lemma 17↩︎

7.3.2 Proof of Lemma 18↩︎

7.3.3 Proof of Proposition 23↩︎

8 Lists of witness paths for Example 1↩︎

We provide further details for the four scenarios in Example 1. Note that in each scenario, the Charlie tick indices \(\mathsf{t}= (t_{l,z})_{l \in \mathbb{Z}_4,\, z \in \mathbb{Z}_2}\) are \[\mathsf{t}= \begin{pmatrix} 23 & 9 \\ 21 & 5 \\ 15 & 1 \\ 19 & 3 \end{pmatrix}.\]

For each scenario, we first specify the Alice and Bob measurement sets. We then exhibit, for every total Charlie assignment \(\mathbf{z}\in \mathbb{Z}_2^4\), a witness path in the corresponding implication graph \(I_{\Psi}(\mathbf{z})\). This proves that the conditioned formula is inconsistent in each case, and hence establishes the desired nonlocality paradox. By Lemma 6, it suffices to exhibit a path from some literal to its complement. Throughout this section, we write \(A^a\) for the literal \((A,a)\), and similarly write \(B^b\) for the literal \((B,b)\).

8.1 Example 1(1)↩︎

For the \((4,4,4)\)-scenario, we have the following sets of Alice and Bob measurements. \[\begin{align} M_1^{(4,4)} &:= \{ A_0,\;A_2,\;A_4,\;A_6\}, \\ \text{and} \quad M_2^{(4,4)} &:=\{B_1,\;B_3,\;B_5,\;B_9 \}. \end{align}\]

We give an example of the implication graph for \(\mathbf{z}=(0,0,0,0)\):

For each of the remaining \(\mathbf{z}\) values not covered by Remark 26, we list a witness path in the corresponding implication graph. Here, in the last line, \(z\in\mathbb{Z}_2\); thus the displayed witness path applies to both \(\mathbf{z}=(1,1,1,0)\) and \(\mathbf{z}=(1,1,1,1)\).

• \(\mathbf{z}= (0,0,0,0):\;A_0^0 \Longrightarrow B_3^1 \Longrightarrow A_6^0 \Longrightarrow B_9^0 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (0,0,1,0):\;A_0^0 \Longrightarrow B_1^0 \Longrightarrow A_6^1 \Longrightarrow B_5^0 \Longrightarrow A_4^1 \Longrightarrow B_9^0 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (0,0,1,1):\;A_0^0 \Longrightarrow B_3^0 \Longrightarrow A_6^1 \Longrightarrow B_9^0 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (0,1,0,0):\;A_2^0 \Longrightarrow B_9^1 \Longrightarrow A_6^1 \Longrightarrow B_5^0 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,1,1,0):\;A_2^0 \Longrightarrow B_9^1 \Longrightarrow A_4^0 \Longrightarrow B_1^0 \Longrightarrow A_6^1 \Longrightarrow B_5^0 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,1,1,1):\;A_2^0 \Longrightarrow B_9^1 \Longrightarrow A_6^0 \Longrightarrow B_5^1 \Longrightarrow A_0^1 \Longrightarrow B_3^1 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (1,1,0,0):\;A_0^0 \Longrightarrow B_9^0 \Longrightarrow A_6^0 \Longrightarrow B_1^1 \Longrightarrow A_4^1 \Longrightarrow B_5^1 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (1,1,1,z):\;A_0^0 \Longrightarrow B_9^0 \Longrightarrow A_4^1 \Longrightarrow B_5^1 \Longrightarrow A_0^1\).

8.2 Example 1(2)↩︎

For the \((6,6,4)\)-scenario, we have \[\begin{align} M_1^{(6,6)} &:= \{ A_0,\;A_1,\;A_2,\;A_3,\;A_6,\;A_7\}, \\ \text{and} \quad M_2^{(6,6)} &:=\{B_0,\;B_1,\;B_2,\;B_5,\;B_6,\;B_9 \}. \end{align}\]

Again, we list witness paths from the remaining \(\mathbf{z}\) values not covered in Remark 26.

• \(\mathbf{z}= (0,0,0,0):\; A_2^0 \Longrightarrow B_9^1 \Longrightarrow A_6^1 \Longrightarrow B_5^0 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,0,1,0):\;A_0^0 \Longrightarrow B_1^0 \Longrightarrow A_6^1 \Longrightarrow B_5^0 \Longrightarrow A_2^1 \Longrightarrow B_9^0 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (0,0,1,1):\;A_1^0 \Longrightarrow B_2^0 \Longrightarrow A_7^1 \Longrightarrow B_6^0 \Longrightarrow A_3^1 \Longrightarrow B_0^1 \Longrightarrow A_1^1\).

• \(\mathbf{z}= (0,1,0,0):\;A_2^0 \Longrightarrow B_9^1 \Longrightarrow A_6^1 \Longrightarrow B_5^0 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,1,1,0):\;A_1^0 \Longrightarrow B_0^0 \Longrightarrow A_7^1 \Longrightarrow B_6^0 \Longrightarrow A_1^1\).

• \(\mathbf{z}= (0,1,1,1):\;A_0^0 \Longrightarrow B_1^0 \Longrightarrow A_2^0 \Longrightarrow B_9^1 \Longrightarrow A_6^0 \Longrightarrow B_5^1 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (1,1,0,0):\;A_0^0 \Longrightarrow B_5^0 \Longrightarrow A_2^1 \Longrightarrow B_1^0 \Longrightarrow A_6^1 \Longrightarrow B_9^1 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (1,1,1,z):\;A_3^0 \Longrightarrow B_2^0 \Longrightarrow A_7^0 \Longrightarrow B_6^1 \Longrightarrow A_3^1\).

8.3 Example 1(3)↩︎

For the \((6,2,4)\)-scenario, we have

\[\begin{align} M_1^{(6,2)} &:= \{ A_0,\;A_2,\;A_4,\;A_6,\;A_8,\;A_{10}\}, \\ \text{and} \quad M_2^{(6,2)} &:=\{ B_1,\;B_7 \}. \end{align}\]

Again, we list witness paths from the remaining \(\mathbf{z}\) values not covered in Remark 26.

• \(\mathbf{z}= (0,0,0,0):\;A_2^0 \Longrightarrow B_7^1 \Longrightarrow A_8^1 \Longrightarrow B_1^0 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,0,1,0):\;A_0^0 \Longrightarrow B_7^1 \Longrightarrow A_6^0 \Longrightarrow B_1^1 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (0,0,1,1):\;A_2^0 \Longrightarrow B_7^1 \Longrightarrow A_8^0 \Longrightarrow B_1^1 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,1,0,0):\;A_4^0 \Longrightarrow B_1^0 \Longrightarrow A_{10}^1 \Longrightarrow B_7^0 \Longrightarrow A_4^1\).

• \(\mathbf{z}= (0,1,1,0):\;A_4^0 \Longrightarrow B_1^0 \Longrightarrow A_{10}^1 \Longrightarrow B_7^0 \Longrightarrow A_4^1\).

• \(\mathbf{z}= (0,1,1,1):\;A_4^0 \Longrightarrow B_1^0 \Longrightarrow A_{10}^1 \Longrightarrow B_7^0 \Longrightarrow A_4^1\).

• \(\mathbf{z}= (1,1,0,0):\;A_2^0 \Longrightarrow B_7^0 \Longrightarrow A_8^0 \Longrightarrow B_1^0 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (1,1,1,0):\;A_0^0 \Longrightarrow B_7^1 \Longrightarrow A_6^0 \Longrightarrow B_1^1 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (1,1,1,1):\;A_2^0 \Longrightarrow B_7^0 \Longrightarrow A_8^1 \Longrightarrow B_1^1 \Longrightarrow A_2^1\).

8.4 Example 1(4)↩︎

For the \((7,5,4)\)-scenario, we have

\[\begin{align} M_1^{(7,5)} &:= \{ A_0,\;A_1,\;A_2,\;A_3,\;A_6,\;A_7,\;A_8 \}, \\ \text{and} \quad M_2^{(7,5)} &:=\{B_0,\;B_1,\;B_4,\;B_7,\;B_8 \}. \end{align}\]

Again, we list witness paths from the remaining \(\mathbf{z}\) values not covered in Remark 26.

• \(\mathbf{z}= (0,0,0,0):\;A_3^0 \Longrightarrow B_4^1 \Longrightarrow A_7^0 \Longrightarrow B_8^0 \Longrightarrow A_3^1\).

• \(\mathbf{z}= (0,0,1,0):\;A_1^0 \Longrightarrow B_8^1 \Longrightarrow A_3^0 \Longrightarrow B_4^1 \Longrightarrow A_7^0 \Longrightarrow B_0^1 \Longrightarrow A_1^1\).

• \(\mathbf{z}= (0,0,1,1):\;A_2^0 \Longrightarrow B_7^1 \Longrightarrow A_8^0 \Longrightarrow B_1^1 \Longrightarrow A_2^1\).

• \(\mathbf{z}= (0,1,0,0):\;A_3^0 \Longrightarrow B_4^1 \Longrightarrow A_7^0 \Longrightarrow B_8^0 \Longrightarrow A_3^1\).

• \(\mathbf{z}= (0,1,1,0):\;A_0^0 \Longrightarrow B_1^0 \Longrightarrow A_6^1 \Longrightarrow B_7^0 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (0,1,1,1):\;A_1^0 \Longrightarrow B_4^0 \Longrightarrow A_7^1 \Longrightarrow B_8^0 \Longrightarrow A_3^1 \Longrightarrow B_0^1 \Longrightarrow A_1^1\).

• \(\mathbf{z}= (1,1,0,0):\;A_1^0 \Longrightarrow B_4^0 \Longrightarrow A_3^1 \Longrightarrow B_0^0 \Longrightarrow A_7^1 \Longrightarrow B_8^1 \Longrightarrow A_1^1\).

• \(\mathbf{z}= (1,1,1,0):\;A_0^0 \Longrightarrow B_1^0 \Longrightarrow A_6^1 \Longrightarrow B_7^0 \Longrightarrow A_0^1\).

• \(\mathbf{z}= (1,1,1,1):\;A_2^0 \Longrightarrow B_7^0 \Longrightarrow A_8^1 \Longrightarrow B_1^1 \Longrightarrow A_2^1\).

References↩︎