--- title: Unanimous Bell Inequalities url: https://www.emergentmind.com/topics/unanimous-bell-inequalities type: topic --- # Unanimous Bell Inequalities Unanimous Bell inequalities are Bell inequalities built only from full-equality events, that is, probabilities of the form \(p(a_1=a_2=\cdots=a_n\mid \mathbf{x})\) in multipartite Bell scenarios. In the framework of equality-comparison or “smell” Bell inequalities, they form a particularly simple subclass that retains only the event that all parties report the same outcome, yet still yields tight inequalities, deterministic nonlocal games, quantum violations, and device-independent witnesses of dimension and genuine multipartite nonlocality [2603.17030]. In a broader, nonstandard usage, related Bell-type constraints have also been described as “unanimous” when they are identically satisfied by every jointly indexed data table obeying specified algebraic conditions; that alternative usage concerns universal joint-distribution constraints rather than the equality-event subclass defined explicitly in current multipartite work [1903.02116]. ## 1. Definition within equality-comparison Bell scenarios The ambient setting is an \((n,m,k)\) Bell scenario with \(n\) parties, \(m\) measurement settings per party, and \(k\) outcomes per measurement. A standard behavior is the conditional distribution \[ p(\mathbf{a}\mid \mathbf{x}) = p(a_1,\dots,a_n\mid x_1,\dots,x_n), \] with \(a_i\in\{0,\dots,k-1\}\) and \(x_i\in\{0,\dots,m-1\}\). The equality-comparison program replaces the full outcome record by the equality pattern among parties’ outputs. For each input tuple \(\mathbf{x}\) and partition \(\sigma\) of \(\{1,\dots,n\}\), one defines \(p(=_{\sigma}\mid \mathbf{x})\), the probability that the observed outcomes induce that partition. If \(\Pi_n\) denotes the set of partitions and \(B_n\) the \(n\)-th Bell number, then the smell behavior space has dimension \[ \dim \mathcal{P}_\Sigma = m^n(B_n-1) \] and its local subpolytope is the convex hull of projected local behaviors [2603.17030]. Unanimous Bell inequalities arise by retaining only the single partition in which all parties belong to one equivalence class. The corresponding full-equality probability is \[ p(=_{ABC\ldots}\mid\mathbf{x}) := p(a_1=a_2=\cdots=a_n\mid \mathbf{x}) = \sum_a p(a,a,\dots,a\mid \mathbf{x}). \] A unanimous Bell inequality is any linear inequality of the form \[ I_{\mathrm{un}}=\sum_{\mathbf{x}} \beta_{\mathbf{x}}\, p(=_{ABC\ldots}\mid \mathbf{x}) \le L_{\mathrm{un}}. \] This restricts the observable behavior to at most one parameter per input, so the unanimous behavior space has dimension at most \(m^n\). The associated local object is the unanimous polytope \(\mathcal{L}_{\mathrm{un}}\), obtained by projecting local deterministic assignments onto these full-equality probabilities [2603.17030]. A basic structural simplification occurs in the bipartite case. For \(n=2\), there is only one nontrivial equality partition, so the unanimous and smell descriptions coincide. Consequently, every bipartite smell inequality is also unanimous in this sense [2603.17030]. ## 2. Polytope structure, saturation, and deterministic-game form The unanimous polytope sits inside the smell polytope and the full local polytope as \[ \mathcal{L}_{\mathrm{un}} \subset \mathcal{L}_\Sigma \subset \mathcal{L}. \] This nesting is central to the usefulness of unanimous inequalities: they discard most of the conditional distribution, but the remaining coordinates still retain nontrivial polyhedral structure and nonlocal content [2603.17030]. A decisive result is that every unanimous Bell inequality can be written as a deterministic nonlocal game. Given \[ I_{\mathrm{un}}=\sum_{\mathbf{x}} \beta_{\mathbf{x}}\, p(=_{ABC\ldots}\mid \mathbf{x}) \le L_{\mathrm{un}}, \] there is an equivalent game, up to an affine rescaling, with input distribution \[ \mu(\mathbf{x})=\frac{|\beta_{\mathbf{x}}|}{\sum_{\mathbf{y}} |\beta_{\mathbf{y}}|} \] and winning predicate \[ V(\mathbf{a},\mathbf{x})= \begin{cases} \delta_{a_1,a_2,a_3,\ldots} & \text{if } \beta_{\mathbf{x}}\ge 0,\\ 1-\delta_{a_1,a_2,a_3,\ldots} & \text{if } \beta_{\mathbf{x}}<0. \end{cases} \] Thus unanimous inequalities admit an especially transparent operational reading: for each input, the players are required either to be unanimous or not unanimous, depending only on the sign of the corresponding coefficient [2603.17030]. Outcome-cardinality saturation is also unusually sharp. For smell inequalities in an \((n,m,k)\) scenario, the local smell polytope saturates at \[ k^\ast=\left\lfloor \frac{n(m+1)}{2}\right\rfloor, \] and in bipartite scenarios this reduces to \(k^\ast=m+1\). For unanimous inequalities there is a stronger theorem: in an \((n,\mathbf{m},k)\) scenario with \(\mathbf{m}=(m_A,m_B,m_C,\dots)\), the unanimous local polytope saturates already at \[ k^\ast = m_{\min}+1,\qquad m_{\min}:=\min_l m_l. \] For the family \((N,2,k)\), this gives \(k^\ast=3\) [2603.17030]. The same theorem provides an explicit formula for the number of local deterministic vertices of \(\mathcal{L}_{\mathrm{un}}\). The existence of that closed counting formula is itself significant, because it reflects the combinatorial tractability created by coarse-graining to full-equality events alone [2603.17030]. ## 3. Canonical unanimous families and representative inequalities The simplest explicit multipartite family introduced so far is \[ \mathcal{F}_{N2} = \sum_{x_1,\dots,x_N} (-1)^{x_1\oplus\cdots\oplus x_N}\, p(=_{1,\dots,N}\mid x_1,\dots,x_N) \le 1+(N+1)_{\mathrm{mod}\,2}. \] It lives in the \((N,2,k)\) scenario with binary inputs and at least three outcomes. The local bound is \[ L_{\mathrm{un}}(\mathcal{F}_{N2})= \begin{cases} 1,& N\text{ odd},\\ 2,& N\text{ even}. \end{cases} \] Via the game correspondence, this family is equivalent to a uniform-input game in which the players win iff they are unanimous when the input parity is even and win iff they are not unanimous when the parity is odd [2603.17030]. The local bound for \(\mathcal{F}_{N2}\) is derived combinatorially. Writing \(E\) and \(O\) for the even- and odd-parity input sets and \(A\) for the inputs on which a deterministic strategy outputs unanimously, the quantity \[ D := |E\cap A| - |O\cap A| \] satisfies \[ D\le \begin{cases} 1,& N\text{ odd},\\ 2,& N\text{ even}, \end{cases} \] and these bounds are achievable. This yields the game score \[ \omega_c(N)=\frac{1}{2}+\frac{D}{2^N}, \] which translates back to the stated Bell bound [2603.17030]. A second compact representative is the four-party unanimous facet \[ \mathcal{U}_4 = p(=_{ABCD}\mid 0000) - p(=_{ABCD}\mid 1100) + p(=_{ABCD}\mid 1101) + p(=_{ABCD}\mid 1110) - p(=_{ABCD}\mid 1111) \le 1. \] This inequality is a facet of \(\mathcal{L}_{\mathrm{un}}\) in the \((4,2,k)\) setting and later becomes a genuine multipartite nonlocality witness [2603.17030]. In the bipartite case, the equality-probability formulation directly recovers CHSH-type structure. CHSH can be written purely in terms of equality events as \[ p(=_{AB}\mid 00)+p(=_{AB}\mid 01)+p(=_{AB}\mid 10)-p(=_{AB}\mid 11)\le 2. \] This exhibits the unanimous construction as a higher-output and multipartite generalization of XOR or full-correlator inequalities: for two parties, equality probabilities play the role ordinarily played by binary correlators [2603.17030]. ## 4. Quantum violations and device-independent witnessing tasks Most unanimous Bell inequalities studied so far admit quantum violations. For the odd-\(N\) family \(\mathcal{F}_{N2}\), numerical optimization yields, for \(N=3\), \[ Q_2(\mathcal{F}_{32}) \ge 1.0688,\qquad Q_3(\mathcal{F}_{32}) \ge 1.2568, \] and for \(N=5\), \[ Q_2(\mathcal{F}_{52}) \ge 1.0016,\qquad Q_3(\mathcal{F}_{52}) \ge 1.2448. \] The observed gap \(Q_3>Q_2\) for these odd-\(N\) cases is why \(\mathcal{F}_{N2}\) is proposed as a candidate dimension-witness family, with the conjecture that maximal violations require an \(N\)-qutrit state and suitable measurements [2603.17030]. The same framework yields witnesses of genuine multipartite nonlocality. For \(\mathcal{U}_4\), the bilocal-nonsignaling bound equals the local bound, \[ L_{\text{bilocal-NS}}(\mathcal{U}_4)=1, \] while a four-qubit state achieves \[ Q_2(\mathcal{U}_4)=\frac54>1. \] Because no bilocal-nonsignaling model can exceed 1, any experimental value above 1 certifies genuine 4-partite nonlocality [2603.17030]. Bipartite unanimous inequalities are also relevant to dimension witnessing and to the comparison with CHSH. Since bipartite smell and unanimous inequalities coincide, the inequality \[ \begin{aligned} \mathcal{S}_{33}=\;& p(=_{AB}\mid 00)+p(=_{AB}\mid 01)+p(=_{AB}\mid 02)\\ &+p(=_{AB}\mid 10)-p(=_{AB}\mid 12)-p(=_{AB}\mid 20)\\ &+p(=_{AB}\mid 21)-p(=_{AB}\mid 22)\le 3 \end{aligned} \] is also unanimous. Its local bound is \(L_k(\mathcal{S}_{33})=3\) for all \(k\ge 2\), while numerically \[ Q_2=3.5\ \text{(exact)},\qquad Q_3\ge 3.6325,\qquad Q_4\ge 3.6879. \] This makes \(\mathcal{S}_{33}\) a dimension witness. The same paper identifies two-qubit entangled states that do not violate CHSH but do violate \(\mathcal{S}_{33}\); one explicit example is the rank-2 state \[ \rho_{p,\theta}=p\,|\Psi_1\rangle\langle\Psi_1|+(1-p)\,|\Psi_2\rangle\langle\Psi_2|, \] with \(|\Psi_1\rangle=\cos\theta\,|00\rangle+\sin\theta\,|11\rangle\), \(|\Psi_2\rangle=|01\rangle\), \(p=0.955\), and \(\theta=\pi/14\), for which \[ \mathcal{S}_{33}(\rho_{0.955,\pi/14})\ge 3.0023>3. \] This establishes that equality-based, and hence bipartite unanimous, inequalities can detect nonlocality beyond what CHSH sees in some state families [2603.17030]. ## 5. Foundational antecedents and alternative uses of unanimity Although the term “unanimous Bell inequality” is explicit only in recent equality-event work, several earlier frameworks isolate closely related ideas. Abramsky and Hardy introduced logical Bell inequalities from the principle that a classical or noncontextual hidden-variable model must admit a single global truth assignment to all relevant propositions. If formulas \(\Phi_1,\dots,\Phi_N\) are jointly contradictory, then any classical model obeys \[ \sum_{i=1}^N p(\Phi_i)\le N-1, \] and they prove that, in a precise sense, all Bell inequalities are equivalent to logical Bell inequalities of this kind [1203.1352]. This places unanimity at the level of global logical consistency: not all local propositions can be true simultaneously under one hidden-variable assignment. Santos gave a metric formulation in which Bell inequalities arise from triangle inequalities for the distance \[ d(a_j,a_k)=P(a_j\neq a_k) = P(a_j)+P(a_k)-2P(a_j a_k). \] In this view, Bell and Clauser–Horne–Shimony–Holt inequalities are necessary conditions for the existence of a single joint distribution, or equivalently for a complete noncontextual random-variables representation [1410.4935]. A plausible implication is that unanimous inequalities can be interpreted as especially symmetric instances of such global consistency constraints. The same logic appears in Hardy-type constructions. In the \((n,2,2)\) scenario, Hardy’s conditions induce the Bell–Hardy inequality \[ \sum_{j\in I} P(1\mid 1_j)+P(0\mid 1)-P(1\mid 0)\ge 0, \] and Yu proved that this inequality is tight for arbitrary \(n\), hence facet-defining for the Bell polytope, and dual to an extremal nonsignaling Hardy box [1403.0790]. This supplies an analytically controlled example in which a unanimity-like logical pattern of forbidden and allowed events becomes both a facet inequality and a canonical extremal nonlocal box. A more controversial, nonstandard usage appears in Sica’s re-derivation of Bell inequalities as finite-data identities. For three \(\{\pm1\}\)-valued sequences cross-correlated by a common run index, \[ |E_N(a,b)-E_N(a,c)|\le 1-E_N(b,c), \] and for four jointly indexed sequences a CHSH-type finite-data inequality also holds identically. Sica’s claim is that such inequalities are universally satisfied by any jointly existing triplets or quadruplets of data and that what is physically testable is therefore the form of the correlations and the existence of a joint distribution, especially once quantum non-commutation is treated explicitly [1903.02116]. This broader use of “unanimous” differs conceptually from the equality-event subclass of [2603.17030], but both usages emphasize universality across an admissible class of joint models. ## 6. Geometry, classification, and current status Unanimous inequalities are attractive partly because their coarse-grained behavior spaces are much smaller than the full local polytope, which makes facet enumeration feasible in scenarios where the full problem is intractable. In bipartite scenarios, unanimous and smell polytopes coincide, and a substantial fraction of their facets are also facets of the full local polytope. For all bipartite smell or unanimous facets except positivity, the observed relation is \[ L