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arXiv:2608.00930

Parallel Repetition in the Two-Player Quantum Cloning Game

Eli Coe Naig, Stephen A. Fenner

quant-phSubmitted Aug 2, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

We study parallel repetition in the two-player quantum cloning game, a monogamy-of-entanglement game motivated by quantum position verification. Colisson Palais, Escolà-Farràs, and Speelman bounded the value of $n$ copies between $(3/4)^n$ and $\cos^{2n}(π/8)$. For two copies, we prove that neither bound is tight. An explicit challenge-dependent strategy achieves value $(5+\sqrt{17})/16>9/16$, so strong parallel repetition fails for the unrestricted game. A block Gram matrix argument gives the upper bound $(11+\sqrt{65})/32<\cos^4(π/8)$ and strictly improves the previous parallel-repetition upper bound for every $n$. For every $n$, challenge-independent strategies have optimal value $(3/4)^n$.

AI Summary

The paper studies parallel repetition of the unrestricted two-player quantum cloning game, motivated by quantum position verification. A referee announces which of two noncommunicating players must share a Bell pair with it; in parallel repetitions, responses may depend on the entire challenge string. The single-copy value is $3/4$, while previous bounds for $n$ copies were $(3/4)^n\leq\omega^*(\mathrm{QCG}_2^{\times n})\leq\cos^{2n}(\pi/8)$. For two copies, the paper establishes $\frac{5+\sqrt{17}}{16}\leq\omega^*(\mathrm{QCG}_2^{\times2})\leq\frac{11+\sqrt{65}}{32}$. The lower bound strictly exceeds $9/16$, disproving strong parallel repetition for this game. It is attained by explicit challenge-dependent local two-qubit gates and a shared six-qubit state, verified through the spectrum of the acceptance operator. In contrast, challenge-independent strategies have optimal value exactly $(3/4)^n$ for every $n$, by tensorization of the averaged Bell-test operator. The upper bound expresses the game value through a block Gram matrix indexed by challenge strings. It retains the direction of differing challenge bits: block norms are bounded by $2^{-\max(t_A,t_B)}$, where $t_A,t_B$ count switches toward testing the respective players. The resulting scalar matrix $N_n$ gives $\omega^*\leq2^{-n}\|N_n\|$, strictly improving the previous upper bound for every $n$. The exact two-copy value remains open. The counterexample allows an unrestricted initial state and referee marginal; it does not itself provide an attack on the routing position-verification protocol without pre-shared entanglement, whose state constraints differ.

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We study parallel repetition in the two-player quantum cloning game, a monogamy-of-entanglement game motivated by quantum position verification. Colisson Palais, Escolà-Farràs, and Speelman bounded the value of nnn copies between (3/4)n(3/4)^n(3/4)n and cos⁡2n(π/8)\cos^{2n}(π/8)cos2n(π/8). For two copies, we prove that neither bound is tight. An explicit challenge-dependent strategy achieves value (5+17)/16>9/16(5+\sqrt{17})/16>9/16(5+17​)/16>9/16, so strong parallel repetition fails for the unrestricted game. A block Gram matrix argument gives the upper bound (11+65)/32<cos⁡4(π/8)(11+\sqrt{65})/32<\cos^4(π/8)(11+65​)/32<cos4(π/8) and strictly improves the previous parallel-repetition upper bound for every nnn. For every nnn, challenge-independent strategies have optimal value (3/4)n(3/4)^n(3/4)n.
The paper studies parallel repetition of the unrestricted two-player quantum cloning game, motivated by quantum position verification. A referee announces which of two noncommunicating players must share a Bell pair with it; in parallel repetitions, responses may depend on the entire challenge string. The single-copy value is 3/43/43/4, while previous bounds for nnn copies were (3/4)n≤ω∗(QCG2×n)≤cos⁡2n(π/8)(3/4)^n\leq\omega^*(\mathrm{QCG}_2^{\times n})\leq\cos^{2n}(\pi/8)(3/4)n≤ω∗(QCG2×n​)≤cos2n(π/8).
For two copies, the paper establishes 5+1716≤ω∗(QCG2×2)≤11+6532\frac{5+\sqrt{17}}{16}\leq\omega^*(\mathrm{QCG}_2^{\times2})\leq\frac{11+\sqrt{65}}{32}165+17​​≤ω∗(QCG2×2​)≤3211+65​​. The lower bound strictly exceeds 9/169/169/16, disproving strong parallel repetition for this game. It is attained by explicit challenge-dependent local two-qubit gates and a shared six-qubit state, verified through the spectrum of the acceptance operator. In contrast, challenge-independent strategies have optimal value exactly (3/4)n(3/4)^n(3/4)n for every nnn, by tensorization of the averaged Bell-test operator.
The upper bound expresses the game value through a block Gram matrix indexed by challenge strings. It retains the direction of differing challenge bits: block norms are bounded by 2−max⁡(tA,tB)2^{-\max(t_A,t_B)}2−max(tA​,tB​), where tA,tBt_A,t_BtA​,tB​ count switches toward testing the respective players. The resulting scalar matrix NnN_nNn​ gives ω∗≤2−n∥Nn∥\omega^*\leq2^{-n}\|N_n\|ω∗≤2−n∥Nn​∥, strictly improving the previous upper bound for every nnn.
The exact two-copy value remains open. The counterexample allows an unrestricted initial state and referee marginal; it does not itself provide an attack on the routing position-verification protocol without pre-shared entanglement, whose state constraints differ.
We present a strong parallel repetition theorem for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our result is the first parallel repetition theorem for entangled games involving more than two players. Furthermore, our theorem applies to games where the players are allowed to output (possibly entangled) quantum states as answers. More specifically, let GGG be a kkk-player free game, with entangled value val∗(G)=1−ε\mathrm{val}^*(G) = 1 - εval∗(G)=1−ε. We show that the entangled value of the nnn-fold repetition of GGG, val∗(G⊗n)\mathrm{val}^*(G^{\otimes n})val∗(G⊗n), is at most (1−ε)Ω(n/k2)(1 - ε)^{Ω(n/k^2)}(1−ε)Ω(n/k2). In the traditional setting of k=2k=2k=2 players, our parallel repetition theorem is optimal in terms of its dependence on εεε and nnn. For an arbitrary number of players, our result is nearly optimal: for all kkk, we exhibit a kkk-player free game GGG and n>1n > 1n>1 such that val∗(G⊗n)≥val∗(G)n/k\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}val∗(G⊗n)≥val∗(G)n/k. Hence, exponent of the repeated game value cannot be improved beyond Ω(n/k)Ω(n/k)Ω(n/k). Our parallel repetition theorem improves on the prior results of [Jain, et al. 2014] and [Chailloux, Scarpa 2014] in a number of ways: (1) our theorem applies to a larger class of games (arbitrary number of players, quantum outputs); (2) we demonstrate that strong parallel repetition holds for the entangled value of free games: i.e., the base of the repeated game value is 1−ε1 - ε1−ε, rather than 1−ε21 - ε^21−ε2; and (3) there is no dependence of the repeated game value on the input and output alphabets of GGG. In contrast, it is known that the repeated game value of classical free games must depend on the output size. Thus our results demonstrate a seperation between the behavior of entangled games and classical games.
We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a kkk-player free game GGG with entangled value val∗(G)=1−ε\mathrm{val}^*(G) = 1 - εval∗(G)=1−ε, the nnn-fold repetition of GGG has entangled value val∗(G⊗n)\mathrm{val}^*(G^{\otimes n})val∗(G⊗n) at most (1−ε3/2)Ω(n/sk4)(1 - ε^{3/2})^{Ω(n/sk^4)}(1−ε3/2)Ω(n/sk4), where sss is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G⊗n)≤(1−ε2)Ω(n/s)\mathrm{val}(G^{\otimes n}) \leq (1 - ε^2)^{Ω(n/s)}val(G⊗n)≤(1−ε2)Ω(n/s), due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition. Our second theorem handles the broader class of free games GGG where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1−ε2)Ω(n/sk2)(1 - ε^2)^{Ω(n/sk^2)}(1−ε2)Ω(n/sk2). We also show that the dependence of the exponent on kkk is necessary: we exhibit a kkk-player free game GGG and n≥1n \geq 1n≥1 such that val∗(G⊗n)≥val∗(G)n/k\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}val∗(G⊗n)≥val∗(G)n/k. Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.
A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than (1−ϵ)(1-\epsilon)(1−ϵ), what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141–172], who showed that the players cannot win all copies in the repeated game with probability better than (1−ϵ/2)Ω(nϵ2/c)(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}(1−ϵ/2)Ω(nϵ2/c) (here c is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is (1−ϵ/2)Ω(ϵn)(1-\epsilon/2)^{\Omega(\epsilon n)}(1−ϵ/2)Ω(ϵn) as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant 0<δ<ϵ0<\delta <\epsilon0<δ<ϵ, the probability that the players win a (1−ϵ+δ)(1-\epsilon+\delta)(1−ϵ+δ) fraction of the games in the parallel repetition is at most exp⁡(−Ωϵ(δ3n/c))\exp\left(-\Omega_{\epsilon}(\delta^3 n/c)\right)exp(−Ωϵ​(δ3n/c)) (here the constant may depend on ϵ\epsilonϵ); our result has applications to testing Bell inequalities, since it implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap. Our first bound is independent of the answer length and has a better dependence on ϵ\epsilonϵ. By the recent work of Raz [Proceedings of the 494949th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 369–373], this bound is tight. Our bound gives a generic way to improve the soundness of a probabilistically checkable proof (PCP), in a way that is independent of the answer length of the PCP. Using it, for every k, one can convert any q query PCP with answer length c, size scscsc, and soundness (1−ϵ)(1-\epsilon)(1−ϵ) into a two-query PCP with answer length ckckck, size O(ck(2s)k)O(ck (2s)^k)O(ck(2s)k), and soundness (1−ϵ/2q)Ω(ϵk/q)(1-\epsilon/2q)^{\Omega(\epsilon k/q)}(1−ϵ/2q)Ω(ϵk/q). Another consequence of our bound is that the unique games conjecture of Khot [Proceedings of the 343434th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] can now be shown to be equivalent to the following a priori weaker conjecture: There is an unbounded increasing function f:R+→R+f:\mathbb{R}^+\rightarrow \mathbb{R}^+f:R+→R+ such that for every ϵ>0\epsilon> 0ϵ>0, there exists an alphabet size M(ϵ)M(\epsilon)M(ϵ) for which it is NP-hard to distinguish a unique game with alphabet size M in which a (1−ϵ2)(1-\epsilon^2)(1−ϵ2) fraction of the constraints can be satisfied from one in which a (1−ϵf(1/ϵ))(1-\epsilon f(1/\epsilon))(1−ϵf(1/ϵ)) fraction of the constraints can be satisfied.
We prove that parallel repetition of the (3-player) GHZ game reduces the value of the game polynomially fast to 0. That is, the value of the GHZ game repeated in parallel ttt times is at most t−Ω(1)t^{-Ω(1)}t−Ω(1). Previously, only a bound of ≈1α(t)\approx \frac{1}{α(t)}≈α(t)1​, where ααα is the inverse Ackermann function, was known. The GHZ game was recently identified by Dinur, Harsha, Venkat and Yuen as a multi-player game where all existing techniques for proving strong bounds on the value of the parallel repetition of the game fail. Indeed, to prove our result we use a completely new proof technique. Dinur, Harsha, Venkat and Yuen speculated that progress on bounding the value of the parallel repetition of the GHZ game may lead to further progress on the general question of parallel repetition of multi-player games. They suggested that the strong correlations present in the GHZ question distribution represent the "hardest instance" of the multi-player parallel repetition problem. Another motivation for studying the parallel repetition of the GHZ game comes from the field of quantum information. The GHZ game, first introduced by Greenberger, Horne and Zeilinger, is a central game in the study of quantum entanglement and has been studied in numerous works. For example, it is used for testing quantum entanglement and for device-independent quantum cryptography. In such applications a game is typically repeated to reduce the probability of error, and hence bounds on the value of the parallel repetition of the game may be useful.
We obtain quantitative estimates on the decay of the multiplayer optimal value under parallel repetition. In comparison to a previous work of the author in 2025 (arXiv: 2508.09380) which sought to generalize dependency-breaking and anchoring variables from two-player Quantum games, being able to establish quantitative estimates on the decay of the optimal value of a multiplayer game under parallel repetition is of interest to establish under different assumptions. Specifically, independently of the dependency-breaking and anchoring variables that have previously been employed to remove correlations from entangled information shared between Alice and Bob (hence removing dependencies), monotonic concave functions can be used in place of such variables to obtain rates of decay on the optimal value. The game-theoretic setting with two players was first analyzed with monotonic concave functions by Lanzenberger and Maurer. For qi,xi>0q_i , x_i > 0qi​,xi​>0 ∀1≤i≤N\forall 1 \leq i \leq N∀1≤i≤N where $N > 0 $ is the total number of players we adddress an open question raised in their work regarding potential generalizations of two-player monotonic concave functions, through amplification functions of the form ΨMult≡Ψ=N−∏1≤i≤Nexp[−qixi]Ψ_{\textit{Mult}} \equiv Ψ= N - \underset{1 \leq i \leq N}{\prod} \mathrm{exp} \big[ - q_i x_i \big]ΨMult​≡Ψ=N−1≤i≤N∏​exp[−qi​xi​], which in the multiplayer game-theoretic setting have more intricate combinatorial structures.
The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game G repeated n times in parallel is at most c_G*n^{-1/4}*log(n) for a constant c_G depending on G, provided that the entangled value of G is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.