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

• 2-out-of-n CHSH game is a generalization of the classic CHSH game where a pair is chosen from n indices to perform a CHSH-type XOR test.
• The optimal quantum strategy achieves a bias of 1/√2 with winning probability 1/2 + 1/(2√2), characterized by Clifford-algebra structures and pairwise anti-commutation relations.
• Variants include graph-based, multiplayer, parallel, sequential, and iterated settings, each highlighting distinct combinatorial and algebraic aspects in nonlocality tests.

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 (Ostrev, 2015). Other papers use closely related but inequivalent constructions, including graph-based pair-selection CHSH extensions (Cui et al., 2024), the $n$-player binary-output instance of $\mathrm{CHSH}_n\text{-}d$ (Murta et al., 2015), parallel or sequential repetitions of ordinary CHSH (Coladangelo, 2016, Reichardt et al., 2012), and the iterated CHSH game arising from entanglement swapping on a repeater chain (Dmello et al., 2024). 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)$0, where two indices are selected from $\mathrm{CHSH}(n)$1 and the resulting pair defines a CHSH-type test (Ostrev, 2015). A second is the graph-based extension $\mathrm{CHSH}(n)$2, where $\mathrm{CHSH}(n)$3 players sit on the vertices of a graph $\mathrm{CHSH}(n)$4, an edge is sampled uniformly, and the original two-player CHSH game is played only on that selected pair (Cui et al., 2024). A third is the multiplayer linear game $\mathrm{CHSH}(n)$5, whose binary-output winning condition is an $\mathrm{CHSH}(n)$6-player XOR constraint rather than a pair-selection rule (Murta et al., 2015). 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 (Dmello et al., 2024).

Formal object	Core definition	Relation to “2-out-of-$\mathrm{CHSH}(n)$7”
$\mathrm{CHSH}(n)$8	Choose $\mathrm{CHSH}(n)$9, then play a CHSH-type XOR test on that pair	Exact choose-2-of-$i<j$0 interpretation
$i<j$1	Choose an edge of a graph and play CHSH on that pair	Pair-selection on prescribed pairs
$i<j$2	$i<j$3-player XOR game with parity constraint	“2 outcomes for $i<j$4 players,” not choose-2
Iterated CHSH	$i<j$5-round entanglement-swapping chain, final CHSH on endpoints	Two endpoints after an $i<j$6-stage process

This terminological divergence matters. Some papers explicitly state that they do not study a “2-out-of-$i<j$7” object even when their scenario has two distinguished endpoints or two outputs. The most precise encyclopedia usage therefore treats $i<j$8 as the canonical choose-2-of-$i<j$9 CHSH game, and then places alternative usages around it as related but distinct constructions (Ostrev, 2015, Dmello et al., 2024).

2. The explicit choose-2-of-$\{1,\dots,n\}$0 game: $\{1,\dots,n\}$1

For $\{1,\dots,n\}$2, Alice’s question set is

$\{1,\dots,n\}$3

and Bob’s question set is the set of ordered pairs

$\{1,\dots,n\}$4

The referee first chooses a pair $\{1,\dots,n\}$5 uniformly among all $\{1,\dots,n\}$6 pairs with $\{1,\dots,n\}$7, and then chooses uniformly one of the four question pairs

$\{1,\dots,n\}$8

Each of these four pairs therefore occurs with probability

$\{1,\dots,n\}$9

The winning predicate is CHSH-like: for the chosen pair $\mathrm{CHSH}(n)$0, the players must give matching answers on

$\mathrm{CHSH}(n)$1

and opposite answers on

$\mathrm{CHSH}(n)$2

Equivalently,

$\mathrm{CHSH}(n)$3

This is the formal sense in which the game first chooses “2 out of $\mathrm{CHSH}(n)$4” and then imposes a CHSH-type XOR condition on the chosen pair (Ostrev, 2015).

The corresponding game matrix is

$\mathrm{CHSH}(n)$5

For a quantum strategy with state $\mathrm{CHSH}(n)$6 and $\mathrm{CHSH}(n)$7 observables $\mathrm{CHSH}(n)$8 for Alice and $\mathrm{CHSH}(n)$9 for Bob, the bias is

$n$0

The optimal quantum bias is

$n$1

so the optimal winning probability is

$n$2

The first member of the family, $n$3, is the usual CHSH game after relabeling Bob’s two observables (Ostrev, 2015).

3. Algebraic structure of optimal and nearly optimal strategies

The exact optimality conditions for $n$4 take an unusually rigid form. For every optimal strategy and every $n$5,

$n$6

Equivalently,

$n$7

From these relations one derives, on the support of the state, the anti-commutation relations

$n$8

The optimal-strategy analysis is therefore controlled by the Clifford-algebra structure generated by $n$9 pairwise anti-commuting reflections (Ostrev, 2015).

This algebraic structure yields a full classification of exact optimal strategies. The irreducible block size is

$n$0

On the support of $n$1, Alice’s observables are block diagonal, with each block equal to a canonical family of pairwise anti-commuting observables $n$2 built from tensor products of Pauli matrices. If $n$3 is even, each block is

$n$4

If $n$5 is odd, then for $n$6,

$n$7

while for $n$8, each block is either

$n$9

Bob’s optimal observables are fixed blockwise by

$\mathrm{CHSH}_n\text{-}d$0

The Schmidt support rank is a multiple of $\mathrm{CHSH}_n\text{-}d$1, and unused orthogonal sectors may carry arbitrary $\mathrm{CHSH}_n\text{-}d$2 observables without affecting optimality (Ostrev, 2015).

Near-optimal strategies satisfy robust analogues of the same state-dependent identities. If a strategy is $\mathrm{CHSH}_n\text{-}d$3-optimal, then

$\mathrm{CHSH}_n\text{-}d$4

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

$\mathrm{CHSH}_n\text{-}d$5

where $\mathrm{CHSH}_n\text{-}d$6 and $\mathrm{CHSH}_n\text{-}d$7 form a canonical ideal strategy. This $\mathrm{CHSH}_n\text{-}d$8 satisfies

$\mathrm{CHSH}_n\text{-}d$9

and

$n$0

A plausible implication is that $n$1 supplies a rigidity theory whose natural invariant is an approximately embedded Clifford representation rather than full-support operator closeness (Ostrev, 2015).

4. Pair-selection networks and multiplayer CHSH variants

A different formalization of “2-out-of-$n$2” arises in graph-based multiplayer extensions. For a simple connected graph $n$3 on $n$4 vertices, the game $n$5 is defined by sampling an edge uniformly and playing the original two-player game only on that pair. Formally,

$n$6

For CHSH, this is the natural “pair-selection on a prescribed set of pairs” extension. The main conclusion is exceptionally restrictive: $n$7 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 $n$8 (Cui et al., 2024).

The obstruction is Bell monogamy. On the three-party path $n$9,

$n$0

so the average CHSH winning probability over the two edges cannot exceed $n$1. On the four-party path $n$2,

$n$3

which gives

$n$4

For six-party graphs, the paper reports numerical monogamy relations such as

$n$5

again excluding any quantum advantage. Thus, under edge-based pair selection, almost all network topologies destroy CHSH nonlocality (Cui et al., 2024).

A different multiplayer generalization replaces pair selection by an $n$6-player linear constraint. The game denoted $n$7 has winning condition

$n$8

over $n$9. In the binary-output case $\mathrm{CHSH}(n)$00,

$\mathrm{CHSH}(n)$01

which is an $\mathrm{CHSH}(n)$02-player XOR game. Its quantum value is bounded by

$\mathrm{CHSH}(n)$03

and therefore

$\mathrm{CHSH}(n)$04

independent of $\mathrm{CHSH}(n)$05. This construction matches the phrase “2 outcomes for $\mathrm{CHSH}(n)$06 players,” but it is not a choose-2-of-$\mathrm{CHSH}(n)$07 game (Murta et al., 2015).

5. Parallel, sequential, and iterated CHSH constructions

Parallel repetition of ordinary CHSH leads to a different kind of $\mathrm{CHSH}(n)$08-indexed generalization. In the parallel self-testing framework, Alice and Bob receive $\mathrm{CHSH}(n)$09-bit questions and return $\mathrm{CHSH}(n)$10-bit answers, with one nominal CHSH subtest per coordinate. If each subtest $\mathrm{CHSH}(n)$11 satisfies

$\mathrm{CHSH}(n)$12

then there exists a local isometry extracting

$\mathrm{CHSH}(n)$13

In the robust version, if every subtest is at least $\mathrm{CHSH}(n)$14, the extracted state is $\mathrm{CHSH}(n)$15-close to $\mathrm{CHSH}(n)$16 EPR pairs. The proof isolates each coordinate by conditioning on the other $\mathrm{CHSH}(n)$17 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 (Coladangelo, 2016).

Sequential repetition addresses memoryful, transcript-dependent devices rather than simultaneous copies. In the sequential rigidity theorem for $\mathrm{CHSH}(n)$18 CHSH games, there exists a constant $\mathrm{CHSH}(n)$19 such that any $\mathrm{CHSH}(n)$20-structured strategy is $\mathrm{CHSH}(n)$21-simulated by an ideal strategy. Up to local isometries,

$\mathrm{CHSH}(n)$22

is embedded in the devices, and round $\mathrm{CHSH}(n)$23 uses the ideal CHSH observables on the $\mathrm{CHSH}(n)$24-th qubit. This is not a choose-2-of-$\mathrm{CHSH}(n)$25 game either, but it shows that repeated near-optimal CHSH performance forces a fixed tensor-product register of EPR pairs (Reichardt et al., 2012).

The iterated CHSH game is different again. It is defined on a repeater chain

$\mathrm{CHSH}(n)$26

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 $\mathrm{CHSH}(n)$27 is the Bob outcome string, with branch probabilities $\mathrm{CHSH}(n)$28 and endpoint CHSH values $\mathrm{CHSH}(n)$29, the game value is

$\mathrm{CHSH}(n)$30

The paper emphasizes that this is not a combinatorial “pick any 2 out of $\mathrm{CHSH}(n)$31” Bell game. Its main theorem is that oblate stabilizer theory reaches

$\mathrm{CHSH}(n)$32

for arbitrary $\mathrm{CHSH}(n)$33, while quantum theory reaches $\mathrm{CHSH}(n)$34 for all $\mathrm{CHSH}(n)$35 using Bell states, Bell measurements, and local Pauli corrections (Dmello et al., 2024).

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

The finite-field game $\mathrm{CHSH}(n)$36 is another generalized CHSH family often confused with “2-out-of-$\mathrm{CHSH}(n)$37” terminology. Here Alice and Bob receive uniformly random inputs

$\mathrm{CHSH}(n)$38

produce outputs

$\mathrm{CHSH}(n)$39

and win iff

$\mathrm{CHSH}(n)$40

For prime $\mathrm{CHSH}(n)$41, the paper gives an explicit deterministic classical strategy with winning probability

$\mathrm{CHSH}(n)$42

This is a multi-input, multi-output finite-field generalization of CHSH, not a choose-2-of-$\mathrm{CHSH}(n)$43 Bell game (Pivoluska et al., 2015).

A second nearby family is the $\mathrm{CHSH}(n)$44-outcome CHSH-type SATWAP inequality. It is a two-input, $\mathrm{CHSH}(n)$45-output scenario with generalized observables

$\mathrm{CHSH}(n)$46

and generalized correlators

$\mathrm{CHSH}(n)$47

The SATWAP Bell expression has classical bound

$\mathrm{CHSH}(n)$48

and quantum bound

$\mathrm{CHSH}(n)$49

Again, this is “2 inputs and $\mathrm{CHSH}(n)$50 outcomes,” not “choose 2 out of $\mathrm{CHSH}(n)$51” (Baroni et al., 2024).

An LCS-based alternative generalization is the family $\mathrm{CHSH}(n)$52, built from the inconsistent equations

$\mathrm{CHSH}(n)$53

with answer alphabet $\mathrm{CHSH}(n)$54. Its verifier predicate is

$\mathrm{CHSH}(n)$55

$\mathrm{CHSH}(n)$56

This family satisfies $\mathrm{CHSH}(n)$57, has classical value $\mathrm{CHSH}(n)$58, and admits the explicit quantum strategy value

$\mathrm{CHSH}(n)$59

It is a $\mathrm{CHSH}(n)$60-valued CHSH generalization with two questions per player and $\mathrm{CHSH}(n)$61-ary outputs, not the standard choose-2-of-$\mathrm{CHSH}(n)$62 game (Cui et al., 2019).

Taken together, these variants show that “2-out-of-$\mathrm{CHSH}(n)$63 CHSH game” has a narrow exact meaning only in the $\mathrm{CHSH}(n)$64 family, and a broader heuristic meaning in several adjacent constructions. The exact choose-2-of-$\mathrm{CHSH}(n)$65 object is the XOR game $\mathrm{CHSH}(n)$66; graph-based pair selection supplies a network version; multiplayer, finite-field, repeater-chain, and $\mathrm{CHSH}(n)$67-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 $\mathrm{CHSH}(n)$68.