The curious case of "XOR repetition" of monogamy-of-entanglement games

Andrea Coladangelo; Qipeng Liu; Ziyi Xie

REVIEW 3 major objections 3 minor 14 references

The curious case of "XOR repetition" of monogamy-of-entanglement games

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that XOR repetition of the monogamy-of-entanglement game keeps optimal winning probability cos²(π/8) ≈ 0.85 for every n, contradicting the exponential-decay conjecture; it also gives an exponential-decay bound for a random-

desk verdict Main XOR result is a clean, surprising theorem; the GL section has a real but repairable counting error and a loose conjecture equivalence. read the letter →

arxiv 2509.01831 v1 pith:WNGLEAQB submitted 2025-09-01 quant-ph

Andrea Coladangelo , Qipeng Liu , Ziyi Xie This is my paper

classification quant-ph MSC 81P4594A60 PACS 03.67.-a03.67.Dd

keywords monogamy-of-entanglementgamesXORrepetitiondecisionvariantsquantumcryptographyunclonablebitGoldreich-Levingameparallel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether a decision version of the classic monogamy-of-entanglement game can have an advantage that decays exponentially with the number of qubits — a property that would yield an information-theoretically unclonable bit. It answers this question for two decision variants. The first, XOR repetition, is shown not to decay at all: for any n, the optimal winning probability is cos²(π/8) ≈ 0.85, the same value as the single-qubit game, and an optimal strategy needs no entanglement between Alice, Bob, and Charlie. This refutes the conjecture in [AGLL25] that the advantage over random guessing decays exponentially. The second, a Goldreich–Levin style variant with a random mask r, is shown to have advantage at most 0.93^n against semi-classical (entanglement-free) adversaries; the paper states a conjecture that would extend this to all adversaries, which would yield an unclonable bit.

What carries the argument

The load-bearing object is the two-dimensional subspace S = span{|0_Y⟩^{⊗n}, |1_Y⟩^{⊗n}}, spanned by the tensor powers of the ±1 eigenstates of Pauli Y. The parity observable X_θ, which measures the XOR of the n outcomes, maps S to itself and acts on it either as X or as Z, depending on |θ| mod 4 and y·θ. An explicit isomorphism from S to C² therefore maps the n-qubit optimal state to the optimal single-qubit state, reducing the whole game to the n=1 case; a separate simulating-referee reduction proves optimality in the reverse direction.

What would settle it

For n=2, compute the optimal winning probability of the XOR game by semidefinite programming over all strategies; a value above cos²(π/8) ≈ 0.8536 would refute Theorem 1. Independently, count the pairs (θ,r) outside S: for n=2 there are 9 of them, not 4; re-deriving Theorem 7 with the corrected complement probability (3/4)^n must still yield the stated 0.93^n bound, and any failure there would invalidate the bound as stated.

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Extended reading notes

Core claim

The central discovery is that the n-qubit XOR game is exactly as hard to win as the one-qubit game. For any basis-choice string θ, the parity observable X_θ preserves the two-dimensional subspace S spanned by the n-fold tensor powers of the ±1 eigenstates of Pauli Y; under an isomorphism from S to a single qubit, X_θ becomes either X or Z, and the state |φ_y⟩ maps to the optimal single-qubit state. This gives 2^{n+1} states, forming two orthonormal bases, that win with probability cos²(π/8) for every θ. Optimality follows by a reduction that simulates the referee's measurements on n−1 qubits, so any winning probability above cos²(π/8) would lift to a single-qubit violation of the optimality

Load-bearing premise

In Section 5.1, the proof bounds the effect of removing the 'bad' pairs (θ,r) by asserting the complement of S has size 2^n; the actual complement has size 3^n, and while the correct failure probability (3/4)^n is still exponentially small, all subsequent statements that use δ(n) = 2^{-n}/(1−2^{-n}) inherit this miscount as written.

Editorial extensions

If this is right

The XOR-repetition advantage over random guessing is cos²(π/8) − 1/2 ≈ 0.35 for every n, so this particular 'search-to-decision' reduction cannot yield exponential decay.
Because the optimal strategy is product over the three registers, the no-decay effect is intrinsic to the measurement structure, not to entanglement resources.
For the random-mask variant, entanglement-free adversaries are exponentially weak: winning probability ≤ 1/2 + 0.93^n, so decision-based monogamy games with negligible advantage remain viable candidates.
The 2^{n+1} optimal states form two orthonormal bases of 'parity-biased' states, giving a structured family that is robust to any basis choice.
Proving the paper's Conjecture 1 would extend the exponential decay to all quantum adversaries, yielding an information-theoretically secure unclonable bit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the subspace-invariance mechanism is the reason for no decay, then any n-fold game whose observable algebra preserves a two-dimensional subspace spanned by tensor powers of a single-qubit eigenstate will collapse to a constant-size game; testing three-basis or random-basis variants could map the boundary of this phenomenon.
The no-decay result suggests that XOR of all outcomes is too coarse a predicate for constructing unclonable bits; the random-mask predicate is the more promising direction, as the paper's semi-classical result already shows.
A testable sharpening: the paper states (Remark 6) that the 2^{n+1} states are the only optimizers but gives no proof; an explicit proof or a counterexample would settle whether the subspace S is forced.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know: the headline result is right and worth your time. Theorem 1—XOR repetition of the TFKW game has optimal winning probability cos²(π/8) for every n—is proved cleanly, and it refutes the Ananth et al. conjecture of exponential decay. The construction of the |φ_y⟩ states is explicit and the optimality argument reduces directly to the n=1 case, so no hidden assumptions or fitted parameters. That is a genuinely nice result.

The paper also gives a more direct proof of the semi-classical Goldreich-Levin bound previously obtained by Champion et al. and Çakan–Goyal. That part is not new, but the proof via Parseval is useful, and the explicit Conjecture 1 tying general security to an operator inequality is a reasonable framework.

Where the paper gets soft is Section 5.1. The complement of the conditioning event S has size 3ⁿ, not 2ⁿ: for each θ with k ones, there are 2^{n−k} strings r that vanish on H, so the complement probability is (3/4)ⁿ, not 2^{−n}. That makes δ(n) and Equations (15)–(16) wrong as written. The error is load-bearing for the proof of Theorem 7, but it is repairable—(3/4)ⁿ is still exponentially small, and the bound 0.93ⁿ survives after rerunning the Parseval calculation. There is also a second, separate issue: in going from Eq. (17) to W^(1), the conditional expectation over S is replaced by an unconditional expectation, so the claimed equivalence of Conjecture 1 to general security is not established as written. For semi-classical strategies the two coincide (P_b⊗Q_b = δ_{b=c} I), so Theorem 7 is unaffected. Remark 6 also claims uniqueness of the optimal states without proof; that is a minor unsupported assertion.

I looked for circularity or fabricated results and found none. The citation pattern is fair; the paper explicitly credits the prior semi-classical results.

Who should read it: anyone working on monogamy games, unclonable encryption, or search-to-decision reductions in quantum cryptography. The main theorem deserves referee time, and the flaws are identifiable and fixable. I would send it to peer review with a request to correct the counting and tighten the induction over W^(1).

Referee Report

3 major / 3 minor

Summary. The paper studies two decision variants of the Tomamichel–Fehr–Kaniewski–Wehner monogamy-of-entanglement game. Variant 1 ("XOR repetition") asks Bob and Charlie to guess only the parity of the referee's n-bit outcome. The paper claims that, contrary to a conjecture of Ananth et al., the optimal winning probability is exactly cos^2(pi/8) for every n, with no exponential decay. The lower bound is established by an explicit family of 2^{n+1} states that are cos^2(pi/8)-biased for every basis string theta (Theorem 3); optimality is argued by a reduction to the n=1 game (Section 3.2). Variant 2 ("Goldreich-Levin") adds a random string r and asks for the parity r·x. For semi-classical strategies (no entanglement between A and BC, deterministic answers), Theorem 7 claims an upper bound of 1/2 + 0.93^n. Section 5.1 rewrites the optimal winning probability and reduces the general problem to Conjecture 1, which is claimed equivalent to exponential decay for fully quantum strategies and would imply an unclonable bit.

Significance. If the main XOR result (Theorem 1) is correct, it is a crisp and surprising counterexample to the natural intuition that decision versions of this monogamy game should inherit exponential decay from the search game. The explicit optimal states are simple and the proof is direct, with no fitted parameters or numerical assumptions; it also gives a rare example where a search-to-decision reduction fails completely. The semi-classical security bound for Variant 2 and the explicit conjecture linking the general case to an unclonable bit are valuable and likely to stimulate further work. However, the paper currently contains a concrete counting error in the secondary Variant-2 analysis, a mismatch between conditional and unconditional expectations in the claimed equivalence with Conjecture 1, and a gap in the reduction proving optimality of the XOR game. These issues are repairable, and the central XOR theorem appears sound, but the manuscript needs careful revision.

major comments (3)

[Section 3.2] The reduction to XORMonogamy(1) is not a valid strategy under the game rules as stated. In the n=1 game Alice may only prepare the initial state and send registers; she cannot, after the referee has received A1, sample theta_2...theta_n, measure A2...An, and broadcast the resulting theta_2...theta_n and x_2...x_n to Bob and Charlie. Without those values both Bob and Charlie cannot compute x_2 xor ... xor x_n. The gap is repairable: since the assumed n-game success probability p > cos^2(pi/8) is averaged over theta_2...theta_n, there exists a fixing of theta_2...theta_n for which the conditional success probability is at least p; Alice can pre-measure A2...An in those fixed bases and send the outcomes along with B and C at setup. The proof should be rewritten in this form.
[Section 5.1, Eqs. (14)-(16); Section 5.3, delta(n)] The claim that the complement of S has size sum_{k=1}^n C(n,k) = 2^n is wrong. For a fixed theta with |H|=k>=1, the forbidden r are those with r_H = 0^H, giving 2^{n-k} choices; adding the k=0 case gives 2^n + sum_{k=1}^n C(n,k)2^{n-k} = 3^n. Hence Pr[(theta,r) not in S] = (3/4)^n, not 2^{-n}, and Eqs. (15)-(16) as well as delta(n)=2^{-n}/(1-2^{-n}) are incorrect as written. The mistake is repairable: delta becomes (3/4)^n/(1-(3/4)^n), still exponentially small, and the Parseval calculation yields ||W^(2)|| <= 1/2 (1+delta) ((2+sqrt(2))/4)^{n/2} <= 1/2 (1+delta) 0.92389^n, which preserves the advertised 0.93^n for non-vacuous n. But the current text contains a concrete counting error at a load-bearing point.
[Section 5.1, Eq. (17) and definition of W^(1)] The derivation gives \tilde{gamma}^{(1)}_{ijkl} with the conditional expectation E_{theta,r:(theta,r) in S}, but W^(1) is then defined as 1/2 I ⊗ E_{theta,r} sum_b P_b ⊗ Q_b, i.e. with the unconditional expectation. These two operators differ by the normalization 1/Pr[S] and by the restriction to S. Consequently Eq. (22) does not follow for general strategies, and the claimed equivalence between exponential decay for all strategies and Conjecture 1 is not established as written. For semi-classical strategies the two expressions do coincide, because P_b ⊗ Q_b = delta_{b=c(theta,r)} I makes the sum equal to I for every (theta,r), so Theorem 7 is unaffected. The general equivalence needs to be repaired.

minor comments (3)

[Theorem 3, statement for |psi_y>] In the paragraph defining |psi_y>, the text says 'over performing a measurement of |phi_y>' but should refer to |psi_y>.
[Section 5.3, Eq. (28)] The first equality in Eq. (28) is missing the sum over i: it should read E_{theta,r,b} sum_i <v_i| P_b^{theta,r} ⊗ Q_b^{theta,r} |v_i>, not E_{theta,r,b} <v_i| ... |v_i> without the sum.
[Remark 6] Remark 6 claims that the 2^{n+1} optimal states are the only optimal strategies but explicitly says no proof is provided. If this claim is kept, it should be formulated as a conjecture or proven; otherwise it is an unsupported assertion in an otherwise proof-driven paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the XOR-repetition result is derived from explicit states and reduced to the external TFKW single-qubit bound; Variant 2 uses standard Fourier/operator arguments and no fitted parameters.

full rationale

The paper's central claim, Theorem 1, is self-contained in the achievability direction: the family |φ_y> and |ψ_y> is explicit, and the cos^2(π/8) bias is proven by direct Pauli calculations (Eqs. (3)-(9)), with no parameter fitted to the target quantity. Optimality is reduced to the known n=1 optimality of the TFKW game, which is an external result, not an input of this paper and not a self-citation. The Variant 2 semi-classical bound uses Parseval's identity, Cauchy-Schwarz, and standard operator-norm inequalities; no fitted parameter is renamed as a prediction, and no equation reduces to its own input by construction. The paper's self-citations ([CLLZ21], [CV22], [AGLL25], [AKL+22], [AKL23]) are contextual or concern the conjectured statement, not load-bearing proof ingredients. Conjecture 1 is explicitly presented as an equivalence, not as a derived theorem. The reader's identified counting error in Section 5.1—the complement of S has size 3^n, not 2^n, making Eqs. (15)-(16) and δ(n) incorrect as written—is a correctness defect in a secondary rewriting step, but it is repairable and is not a form of circularity: it does not make any claimed prediction equal to its input. No circular step meeting the required evidentiary standard is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No numbers are fitted or introduced ad hoc; the optimal states |phi_y> and |psi_y> are explicit and the proof delivers exact bias. The only external input is the TFKW single-qubit optimality result. The subspace S is a mathematical construction, not a new physical entity.

assumptions (3)

standard math The optimal winning probability of the single-qubit TFKW monogamy game is cos^2(pi/8).
Used in Section 3.2 for the optimality reduction and in the n=1 strategy. It is an external theorem from Tomamichel et al., not proven in this paper.
domain assumption The referee's measurements are tensor products of single-qubit Z/X measurements, with eigenvectors |x_theta>.
This defines the game and is needed so that the parity observable X_theta has the claimed eigen-decomposition. It is inherited from the TFKW game definition.
domain assumption Without loss of generality, optimal strategies can be taken pure with projective measurements.
Section 4 assumes a pure state |v> and projectors P, Q. This is a standard purification argument, stated without proof.

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Cite this review

Pith. "Pith review of The curious case of "XOR repetition" of monogamy-of-entanglement games." pith.science (2026). https://pith.science/paper/WNGLEAQB

@misc{pith2026250901831,
  author       = {Pith},
  title        = {Pith review of: The curious case of "XOR repetition" of monogamy-of-entanglement games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNGLEAQB}},
  note         = {Machine review of arXiv:2509.01831}
}

abstract

In this work, we consider "decision" variants of a monogamy-of-entanglement game by Tomamichel, Fehr, Kaniewski, and Wehner [New Journal of Physics '13]. In its original "search" variant, Alice prepares a (possibly entangled) state on registers $\mathsf{ABC}$; register $\mathsf{A}$, consisting of $n$ qubits, is sent to a Referee, while $\mathsf{B}$ and $\mathsf{C}$ are sent to Bob and Charlie; the Referee then measures each qubit in the standard or Hadamard basis (chosen uniformly at random). The basis choices are sent to Bob and Charlie, whose goal is to simultaneously guess the Referee's $n$-bit outcome string $x$. Tomamichel et al. show that the optimal winning probability is $\cos^{2n} {(\frac{\pi}{8})}$, following a perfect parallel repetition theorem. We consider the following "decision" variants of this game: - Variant 1, "XOR repetition": Bob and Charlie's goal is to guess the XOR of all the bits of $x$. Ananth et al. [Asiacrypt '24] conjectured that the optimal advantage over random guessing decays exponentially in $n$. Surprisingly, we show that this conjecture is false, and, in fact, there is no decay at all: there exists a strategy that wins with probability $\cos^2{(\frac{\pi}{8})} \approx 0.85$ for any $n$. - Variant 2, "Goldreich-Levin": The Referee additionally samples a uniformly random $n$-bit string $r$ that is sent to Bob and Charlie along with the basis choices. Their goal is to guess the parity of $r\cdot x$. We show that the optimal advantage over random guessing decays exponentially in $n$ for the restricted class of adversaries that do not share entanglement. A similar result was already shown by Champion et al. and \c{C}akan et al.; we give a more direct proof. Additionally, we put forward a reasonably concrete conjecture that is equivalent to exponential decay for general adversaries.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Unclonable secret sharing

Prabhanjan Ananth, Vipul Goyal, Jiahui Liu, and Qipeng Liu. Unclonable secret sharing. In International Conference on the Theory and Application of Cryptology and Information Security , pages 129--157. Springer, 2025

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Prabhanjan Ananth, Fatih Kaleoglu, Xingjian Li, Qipeng Liu, and Mark Zhandry. On the feasibility of unclonable encryption, and more. In Annual International Cryptology Conference , pages 212--241. Springer, 2022

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Unclonable secret sharing

Prabhanjan Ananth, Vipul Goyal, Jiahui Liu, and Qipeng Liu. Unclonable secret sharing. In International Conference on the Theory and Application of Cryptology and Information Security , pages 129--157. Springer, 2025

work page 2025

[2] [2]

Unclonable encryption, revisited

Prabhanjan Ananth and Fatih Kaleoglu. Unclonable encryption, revisited. In Theory of Cryptography Conference , pages 299--329. Springer, 2021

work page 2021

[3] [3]

On the feasibility of unclonable encryption, and more

Prabhanjan Ananth, Fatih Kaleoglu, Xingjian Li, Qipeng Liu, and Mark Zhandry. On the feasibility of unclonable encryption, and more. In Annual International Cryptology Conference , pages 212--241. Springer, 2022

work page 2022

[4] [4]

Cloning games: A general framework for unclonable primitives

Prabhanjan Ananth, Fatih Kaleoglu, and Qipeng Liu. Cloning games: A general framework for unclonable primitives. In Annual International Cryptology Conference , pages 66--98. Springer, 2023

work page 2023

[5] [5]

Towards Unconditional Uncloneable Encryption

Pierre Botteron, Anne Broadbent, Eric Culf, Ion Nechita, Cl \'e ment Pellegrini, and Denis Rochette. Towards unconditional uncloneable encryption. arXiv preprint arXiv:2410.23064 , 2024

work page Pith review arXiv 2024

[6] [6]

Uncloneable Encryption from Decoupling

Archishna Bhattacharyya and Eric Culf. Uncloneable encryption from decoupling. arXiv preprint arXiv:2503.19125 , 2025

work page Pith review arXiv 2025

[7] [7]

Uncloneable quantum encryption via oracles

Anne Broadbent and S \'e bastien Lord. Uncloneable quantum encryption via oracles. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography , 2020

work page 2020

[8] [8]

Unbounded leakage-resilient encryption and signatures

Alper C akan and Vipul Goyal. Unbounded leakage-resilient encryption and signatures. Cryptology ePrint Archive , 2024

work page 2024