---
title: '2-out-of-n CHSH Game: Generalized Nonlocality'
url: https://www.emergentmind.com/topics/2-out-of-n-chsh-game
type: topic
---

# 2-out-of-n CHSH Game: Generalized Nonlocality

The expression “2-out-of-\(n\) CHSH game” does not denote a single universally standardized object in the arXiv literature. In the most explicit usage represented here, it refers to the family denoted \(\mathrm{CHSH}(n)\), in which the referee first chooses a pair of distinct indices \(i<j\) out of \(\{1,\dots,n\}\) and then asks questions derived from that chosen pair; in that precise sense, \(\mathrm{CHSH}(n)\) is a “choose 2 out of \(n\)” CHSH-type XOR game [1506.00607]. Other papers use closely related but inequivalent constructions, including graph-based pair-selection CHSH extensions [2405.20286], the \(n\)-player binary-output instance of \(\mathrm{CHSH}_n\text{-}d\) [1510.09210], parallel or sequential repetitions of ordinary CHSH [1609.03687], [1209.0449], and the iterated CHSH game arising from entanglement swapping on a repeater chain [2405.13819]. The topic is therefore best understood as a family of non-equivalent CHSH generalizations whose common feature is the embedding of ordinary CHSH structure into a larger \(n\)-indexed setting.

## 1. Terminological scope and competing formalizations

The literature distinguishes several objects that are easy to conflate. One is the explicit “choose-2-from-\(n\)” XOR game \(\mathrm{CHSH}(n)\), where two indices are selected from \(n\) and the resulting pair defines a CHSH-type test [1506.00607]. A second is the graph-based extension \(\mathcal G^H\), where \(n\) players sit on the vertices of a graph \(H=(V,E)\), an edge is sampled uniformly, and the original two-player CHSH game is played only on that selected pair [2405.20286]. A third is the multiplayer linear game \(\mathrm{CHSH}_n\text{-}2\), whose binary-output winning condition is an \(n\)-player XOR constraint rather than a pair-selection rule [1510.09210]. A fourth is the iterated CHSH game, in which a line of repeater nodes performs successive entanglement swaps and the final CHSH test is applied only to the two surviving endpoints [2405.13819].

| Formal object | Core definition | Relation to “2-out-of-\(n\)” |
|---|---|---|
| \(\mathrm{CHSH}(n)\) | Choose \(i<j\), then play a CHSH-type XOR test on that pair | Exact choose-2-of-\(n\) interpretation |
| \(\mathcal G^H\) | Choose an edge of a graph and play CHSH on that pair | Pair-selection on prescribed pairs |
| \(\mathrm{CHSH}_n\text{-}2\) | \(n\)-player XOR game with parity constraint | “2 outcomes for \(n\) players,” not choose-2 |
| Iterated CHSH | \(n\)-round entanglement-swapping chain, final CHSH on endpoints | Two endpoints after an \(n\)-stage process |

This terminological divergence matters. Some papers explicitly state that they do **not** study a “2-out-of-\(n\)” object even when their scenario has two distinguished endpoints or two outputs. The most precise encyclopedia usage therefore treats \(\mathrm{CHSH}(n)\) as the canonical choose-2-of-\(n\) CHSH game, and then places alternative usages around it as related but distinct constructions [1506.00607], [2405.13819].

## 2. The explicit choose-2-of-\(n\) game: \(\mathrm{CHSH}(n)\)

For \(\mathrm{CHSH}(n)\), Alice’s question set is
\[
S=\{1,\dots,n\},
\]
and Bob’s question set is the set of ordered pairs
\[
T=\{ij: i,j\in\{1,\dots,n\},\ i\neq j\}.
\]
The referee first chooses a pair \(i,j\) uniformly among all \(\binom n2\) pairs with \(1\le i<j\le n\), and then chooses uniformly one of the four question pairs
\[
(i,ij),\quad (i,ji),\quad (j,ij),\quad (j,ji).
\]
Each of these four pairs therefore occurs with probability
\[
\frac{1}{4\binom n2}.
\]
The winning predicate is CHSH-like: for the chosen pair \(i<j\), the players must give matching answers on
\[
(i,ij),\quad (i,ji),\quad (j,ij),
\]
and opposite answers on
\[
(j,ji).
\]
Equivalently,
\[
V(i,ij)=V(i,ji)=V(j,ij)=1,\qquad V(j,ji)=-1.
\]
This is the formal sense in which the game first chooses “2 out of \(n\)” and then imposes a CHSH-type XOR condition on the chosen pair [1506.00607].

The corresponding game matrix is
\[
G=\frac{1}{4\binom n2}\sum_{1\le i<j\le n} \bigl(|i\,ij|+|j\,ij|+|i\,ji|-|j\,ji|\bigr).
\]
For a quantum strategy with state \(|\psi\rangle\) and \(\pm1\) observables \(A_i\) for Alice and \(B_{jk}\) for Bob, the bias is
\[
\beta(\mathrm{CHSH}(n),\mathcal S) = \frac{1}{4\binom n2} \sum_{1\le i<j\le n} \langle\psi| (A_i\otimes B_{ij}+A_i\otimes B_{ji}+A_j\otimes B_{ij}-A_j\otimes B_{ji}) |\psi\rangle.
\]
The optimal quantum bias is
\[
\beta(\mathrm{CHSH}(n))=\frac{1}{\sqrt2},
\]
so the optimal winning probability is
\[
\omega(\mathrm{CHSH}(n))=\frac12+\frac{1}{2\sqrt2}.
\]
The first member of the family, \(\mathrm{CHSH}(2)\), is the usual CHSH game after relabeling Bob’s two observables [1506.00607].

## 3. Algebraic structure of optimal and nearly optimal strategies

The exact optimality conditions for \(\mathrm{CHSH}(n)\) take an unusually rigid form. For every optimal strategy and every \(1\le i<j\le n\),
\[
\frac{A_i+A_j}{\sqrt2}\otimes I\,|\psi\rangle = I\otimes B_{ij}|\psi\rangle,\qquad
\frac{A_i-A_j}{\sqrt2}\otimes I\,|\psi\rangle = I\otimes B_{ji}|\psi\rangle.
\]
Equivalently,
\[
A_i\otimes I\,|\psi\rangle = I\otimes \frac{B_{ij}+B_{ji}}{\sqrt2}\,|\psi\rangle,\qquad
A_j\otimes I\,|\psi\rangle = I\otimes \frac{B_{ij}-B_{ji}}{\sqrt2}\,|\psi\rangle.
\]
From these relations one derives, on the support of the state, the anti-commutation relations
\[
\{A_i,A_j\}=A_iA_j+A_jA_i=0 \qquad (i\ne j).
\]
The optimal-strategy analysis is therefore controlled by the Clifford-algebra structure generated by \(n\) pairwise anti-commuting reflections [1506.00607].

This algebraic structure yields a full classification of exact optimal strategies. The irreducible block size is
\[
2^{\lfloor n/2\rfloor}.
\]
On the support of \(|\psi\rangle\), Alice’s observables are block diagonal, with each block equal to a canonical family of pairwise anti-commuting observables \(\sigma_{k,1},\dots,\sigma_{k,2k+1}\) built from tensor products of Pauli matrices. If \(n\) is even, each block is
\[
A_i^{(j)}=\sigma_{\lfloor n/2\rfloor,i}.
\]
If \(n=2k+1\) is odd, then for \(i=1,\dots,2k\),
\[
A_i^{(j)}=\sigma_{k,i},
\]
while for \(i=2k+1\), each block is either
\[
A_{2k+1}^{(j)}=\sigma_{k,2k+1}
\quad\text{or}\quad
A_{2k+1}^{(j)}=-\sigma_{k,2k+1}.
\]
Bob’s optimal observables are fixed blockwise by
\[
B_{jk}^{(\ell)}=\frac{(A_j^{(\ell)}+A_k^{(\ell)})^T}{\sqrt2},\qquad
B_{kj}^{(\ell)}=\frac{(A_j^{(\ell)}-A_k^{(\ell)})^T}{\sqrt2}.
\]
The Schmidt support rank is a multiple of \(2^{\lfloor n/2\rfloor}\), and unused orthogonal sectors may carry arbitrary \(\pm1\) observables without affecting optimality [1506.00607].

Near-optimal strategies satisfy robust analogues of the same state-dependent identities. If a strategy is \(\varepsilon\)-optimal, then
\[
\sum_{1\le i<j\le n} \left\| \frac{A_i+A_j}{\sqrt2}\otimes I|\psi\rangle - I\otimes B_{ij}|\psi\rangle \right\|^2
+
\sum_{1\le i<j\le n} \left\| \frac{A_i-A_j}{\sqrt2}\otimes I|\psi\rangle - I\otimes B_{ji}|\psi\rangle \right\|^2
\le 2n(n-1)\varepsilon.
\]
A key caveat is that ordinary support-based rigidity fails: tiny Schmidt blocks can be modified arbitrarily while preserving near-optimality. The robust substitute is an approximate intertwining operator
\[
T= \frac{1}{\sqrt{2^n}} \sum_{(j_1,\dots,j_n)\in\{0,1\}^n} A_1^{j_1}\cdots A_n^{j_n}\otimes I\,|\psi\rangle \langle \widetilde\psi| \widetilde A_1^{j_1}\cdots \widetilde A_n^{j_n}\otimes I,
\]
where \(\widetilde A_i\) and \(|\widetilde\psi\rangle\) form a canonical ideal strategy. This \(T\) satisfies
\[
\|(A_i\otimes I)T - T(\widetilde A_i\otimes I)\|_F < 12 n^2\sqrt{\varepsilon}\,\|T\|_F,
\]
and
\[
\|(I\otimes B_{jk})T - T(I\otimes \widetilde B_{jk})\|_F < 17 n^2\sqrt{\varepsilon}\,\|T\|_F.
\]
A plausible implication is that \(\mathrm{CHSH}(n)\) supplies a rigidity theory whose natural invariant is an approximately embedded Clifford representation rather than full-support operator closeness [1506.00607].

## 4. Pair-selection networks and multiplayer CHSH variants

A different formalization of “2-out-of-\(n\)” arises in graph-based multiplayer extensions. For a simple connected graph \(H=(V,E)\) on \(n\) vertices, the game \(\mathcal G^H\) is defined by sampling an edge uniformly and playing the original two-player game only on that pair. Formally,
\[
v^{H}(x_{1}, \dots, x_{n}, a_{1}, \dots, a_{n}) \coloneqq \frac{1}{|E|} \sum_{(i,j) \in E} v(x_{i}, x_{j}, a_{i}, a_{j}).
\]
For CHSH, this is the natural “pair-selection on a prescribed set of pairs” extension. The main conclusion is exceptionally restrictive:
\[
\textbf{CHSH nonlocality appears only on } P_2 \text{ and } P_4.
\]
Equivalently, in this graph model, quantum players can outperform the classical CHSH value only in the original two-player case and on the four-player line \(P_4\) [2405.20286].

The obstruction is Bell monogamy. On the three-party path \(P_3\),
\[
\mathcal B_{AB}+\mathcal B_{BC}\le 4,
\]
so the average CHSH winning probability over the two edges cannot exceed \(3/4\). On the four-party path \(P_4\),
\[
\mathcal B_{AB}+\mathcal B_{BC}+\mathcal B_{CD}\le 2\sqrt{10},
\]
which gives
\[
\omega^*(\mathrm{CHSH}^{P_4}) = \frac12+\frac{\sqrt{10}}{12}\approx 0.76 > \frac34.
\]
For six-party graphs, the paper reports numerical monogamy relations such as
\[
\mathcal B_{AB}+\mathcal B_{BC}+\mathcal B_{CD}+\mathcal B_{DE}+\mathcal B_{EF}\le 10,
\]
again excluding any quantum advantage. Thus, under edge-based pair selection, almost all network topologies destroy CHSH nonlocality [2405.20286].

A different multiplayer generalization replaces pair selection by an \(n\)-player linear constraint. The game denoted \(\mathrm{CHSH}_{n}\text{-}d\) has winning condition
\[
a_1+\cdots+a_n = \sum_{i<j} x_i\cdot x_j
\]
over \(\mathbb F_d\). In the binary-output case \(d=2\),
\[
a_1\oplus \cdots \oplus a_n = \bigoplus_{i<j}(x_i x_j),
\]
which is an \(n\)-player XOR game. Its quantum value is bounded by
\[
\omega_q(\mathrm{CHSH}_{n}\text{-}d) \le \frac{1}{d} + \frac{d-1}{d\sqrt d},
\]
and therefore
\[
\omega_q(\mathrm{CHSH}_{n}\text{-}2)\le \frac12+\frac{1}{2\sqrt2},
\]
independent of \(n\). This construction matches the phrase “2 outcomes for \(n\) players,” but it is not a choose-2-of-\(n\) game [1510.09210].

## 5. Parallel, sequential, and iterated CHSH constructions

Parallel repetition of ordinary CHSH leads to a different kind of \(n\)-indexed generalization. In the parallel self-testing framework, Alice and Bob receive \(n\)-bit questions and return \(n\)-bit answers, with one nominal CHSH subtest per coordinate. If each subtest \(i\) satisfies
\[
\left\langle V_i\big(V'_{n+i}+W'_{n+i}\big) + W_i\big(V'_{n+i}-W'_{n+i}\big) \right\rangle = 2\sqrt{2},
\]
then there exists a local isometry extracting
\[
|\Phi^+\rangle^{\otimes n}.
\]
In the robust version, if every subtest is at least \(2\sqrt2-\epsilon\), the extracted state is \(O(n^{3/2}\sqrt\epsilon)\)-close to \(n\) EPR pairs. The proof isolates each coordinate by conditioning on the other \(n-1\) question bits, constructs large sets of “good” conditional subtests, and then uses pairwise commutation to glue the coordinatewise CHSH rigidities into a tensor-product structure [1609.03687].

Sequential repetition addresses memoryful, transcript-dependent devices rather than simultaneous copies. In the sequential rigidity theorem for \(n\) CHSH games, there exists a constant \(\kappa\) such that any \(\epsilon\)-structured strategy is \(\kappa n^\kappa \epsilon^{1/\kappa}\)-simulated by an ideal strategy. Up to local isometries,
\[
|\varphi\rangle^{\otimes n}
\]
is embedded in the devices, and round \(j\) uses the ideal CHSH observables on the \(j\)-th qubit. This is not a choose-2-of-\(n\) game either, but it shows that repeated near-optimal CHSH performance forces a fixed tensor-product register of EPR pairs [1209.0449].

The iterated CHSH game is different again. It is defined on a repeater chain
\[
A - B_1 - B_2 - \cdots - B_n - C,
\]
where nearest-neighbor pairs share bipartite resources, each Bob performs a bipartite measurement, the Bob outcomes are broadcast, Alice and Charlie may apply local corrections, and a final CHSH test is performed on the effective endpoint state. If \(\vec b=(b_1,\dots,b_n)\) is the Bob outcome string, with branch probabilities \(p_{\vec b}\) and endpoint CHSH values \(\beta_{\vec b}\), the game value is
\[
\beta = \sum_{\vec b \in (\mathbb Z_k)^{n}} p_{\vec b}\beta_{\vec b}.
\]
The paper emphasizes that this is **not** a combinatorial “pick any 2 out of \(n\)” Bell game. Its main theorem is that oblate stabilizer theory reaches
\[
\beta=4
\]
for arbitrary \(n\), while quantum theory reaches \(2\sqrt2\) for all \(n\) using Bell states, Bell measurements, and local Pauli corrections [2405.13819].

## 6. Other generalized CHSH families and the boundaries of the concept

The finite-field game \(\mathrm{CHSH}_q\) is another generalized CHSH family often confused with “2-out-of-\(n\)” terminology. Here Alice and Bob receive uniformly random inputs
\[
x,y\in\mathbb F_q,
\]
produce outputs
\[
a,b\in\mathbb F_q,
\]
and win iff
\[
a+b=xy
\quad\text{in }\mathbb F_q.
\]
For prime \(q=p\), the paper gives an explicit deterministic classical strategy with winning probability
\[
\omega(\mathrm{CHSH}_p)>\frac{1}{22}p^{-2/3}.
\]
This is a multi-input, multi-output finite-field generalization of CHSH, not a choose-2-of-\(n\) Bell game [1510.07431].

A second nearby family is the \(d\)-outcome CHSH-type SATWAP inequality. It is a two-input, \(d\)-output scenario with generalized observables
\[
A_x^{(k)} = \sum_{a=0}^{d-1} \omega^{ak} M_{a|x}, \qquad
B_y^{(l)} = \sum_{b=0}^{d-1} \omega^{bl} N_{b|y},
\qquad
\omega=\exp\!\left(\frac{2 \pi i}{d}\right),
\]
and generalized correlators
\[
\langle A_x^{(k)} B_y^{(l)} \rangle
=
\sum_{a=0}^{d-1}\sum_{b=0}^{d-1}\omega^{ak+bl}p(ab|xy).
\]
The SATWAP Bell expression has classical bound
\[
\left(\beta_d^{\mathrm{SATWAP}}\right)_l
=
\frac{1}{2} \left(3 \cot\left(\frac{\pi}{4d}\right) - \cot\left(\frac{3\pi}{4d}\right)\right) - 2,
\]
and quantum bound
\[
\left(\beta_d^{\mathrm{SATWAP}}\right)_q = 2(d-1).
\]
Again, this is “2 inputs and \(d\) outcomes,” not “choose 2 out of \(n\)” [2403.05502].

An LCS-based alternative generalization is the family \(\mathcal G_n\), built from the inconsistent equations
\[
x_0x_1=1,\qquad x_0x_1=\omega_n,
\]
with answer alphabet \(Z_n=\{1,\omega_n,\ldots,\omega_n^{n-1}\}\). Its verifier predicate is
\[
V(0,0,a,b)=1 \iff a=b,\qquad
V(0,1,a,b)=1 \iff ab=1,
\]
\[
V(1,0,a,b)=1 \iff a=b,\qquad
V(1,1,a,b)=1 \iff ab=\omega_n.
\]
This family satisfies \(\mathcal G_2=\mathrm{CHSH}\), has classical value \(3/4\), and admits the explicit quantum strategy value
\[
\nu(\mathcal G_n,\mathcal S_n) = \frac{1}{2} + \frac{1}{2n\sin(\pi/2n)}.
\]
It is a \(\mathbb Z_n\)-valued CHSH generalization with two questions per player and \(n\)-ary outputs, not the standard choose-2-of-\(n\) game [1911.01593].

Taken together, these variants show that “2-out-of-\(n\) CHSH game” has a narrow exact meaning only in the \(\mathrm{CHSH}(n)\) family, and a broader heuristic meaning in several adjacent constructions. The exact choose-2-of-\(n\) object is the XOR game \(\mathrm{CHSH}(n)\); graph-based pair selection supplies a network version; multiplayer, finite-field, repeater-chain, and \(d\)-outcome models instead generalize other aspects of CHSH—number of parties, algebraic alphabet, causal structure, or outcome cardinality—rather than the combinatorics of choosing two indices from \(n\).

Source: https://www.emergentmind.com/topics/2-out-of-n-chsh-game