Non-Local Search-to-Decision Reduction over F2

Abstract: Non-local search-to-decision asks whether two noncommunicating parties, given the two shares of a bipartite encoding of a uniformly random string $x\in \mathbb{F}_2<sup>n$, can both predict the same random parity $\langle r,x\rangle$ without there also being local measurements with which both parties recover $x$. We prove that if their optimal probability of both recovering $x$ by local measurements is $p$, then their probability of both answering a common parity challenge correctly is at most $\min{1,\frac{1}{2}+5p<sup>{1/22}}$. The result is motivated by applications to unclonable encryption and quantum copy-protection. The proof is information-theoretic and does not provide an efficient extractor. The proof and the exposition were developed with assistance from ChatGPT using GPT-5.6 Sol Pro and Codex in the Ultra reasoning mode.

• The paper proves that any joint prediction advantage beyond the guessing baseline in non-local search scenarios implies a polynomially related probability of joint full-string recovery.
• The theorem applies to arbitrary bipartite ensembles under a uniform prior, with unrestricted local quantum measurements and concerns a joint evolving event with identical challenges.
• The findings are used for a common-challenge compiler to generate unclonable encryption schemes with information-theoretic search security.

The problem and the main result

The paper studies a non-local variant of the Goldreich–Levin search-to-decision reduction. A uniformly random string $x \in \mathbb{F}_2^n$ is encoded by an arbitrary quantum channel $\Phi$ into a bipartite state $\rho_x$, with one share given to Bob and one to Charlie, who cannot communicate. In the search experiment, both parties attempt to recover $x$ using local POVMs; let $p_{\mathrm{srch}}$ denote the optimal joint success probability, which lies in $[2^{-n}, 1]$. In the decision experiment, both parties receive the same challenge vector $r$ drawn uniformly from $\mathbb{F}_2^n$ and must each output $\langle r, x\rangle$; let $p_{\mathrm{pred}}$ be the probability that both are correct simultaneously. Since the shared challenge can be ignored (e.g., output the first coordinate of $\Phi$0), the trivial baseline is exactly $\Phi$1.

The main theorem (2608.19091) proves:

$\Phi$2

Equivalently, any joint prediction advantage $\Phi$3 over the guessing baseline forces local full-string extraction POVMs succeeding with probability at least $\Phi$4. Consequently, negligible optimal local recovery probability implies negligible joint prediction advantage over $\Phi$5. Two aspects of this statement deserve emphasis: it applies to arbitrary bipartite ensembles under the uniform prior, with unrestricted local quantum measurements; and it concerns the joint success event of both parties on an identical challenge, a setting in which prior non-local Goldreich–Levin results required independent challenges or imposed structural restrictions on the strategies.

The exponent $\Phi$6 is weak but sufficient for asymptotic purposes, and the proof is entirely information-theoretic. This is also its principal limitation: the argument establishes existence of witness extraction POVMs but does not yield a uniform efficient procedure for constructing them from efficient prediction measurements. Hence the theorem transfers information-theoretic search security, not computational search security.

Position relative to prior work

Prior non-local simultaneous Goldreich–Levin results split along several axes: the field, the joint distribution of hidden strings and challenges, and whether acceptance is exact or relational. Ananth, Kaleoglu, and Liu proved a shared-secret $\Phi$7 theorem with independent challenges and an efficient extractor [AKL23]; Kundu and Tan allowed arbitrary joint distributions over the two secrets, again with independent challenges [KT25]; Broadbent, Karvonen, and Lord applied the independent-challenge principle to uncloneable advice [BKL24]; Coladangelo and Gunn gave many-sample extensions [CG24]; and Ananth and Behera stated identical- and independent-sample conjectures over large prime fields, where only binary distinction tasks are treated [AB24]. Most relevantly, Coladangelo, Liu, and Xie analyzed an identical-challenge monogamy game but only for semi-classical strategies with an unentangled referee register, leaving general fully quantum strategies conjectural [CLX26].

The present result closes a specific gap: it handles a single challenge vector copied verbatim to both parties, against arbitrary entangled local quantum strategies. The author reserves the term "non-local search-to-decision" rather than "simultaneous Goldreich–Levin" precisely because no constructive efficient extraction procedure is provided—a deliberate terminological concession that mirrors the proof's actual content.

The main cryptographic corollary is a compiler for unclonable encryption: any one-time secret-key scheme for unclonable encryption of uniform messages with information-theoretic search security yields a one-bit-message scheme whose identical-challenge distinguishing advantage is negligible. This compiler structure appears in [AKL23] for independent challenges; the new theorem supplies the common-challenge analysis, building on the equivalence of Georgiou and Zhandry between unclonable encryption and selectively secure single-decryptor encryption [GZ20]. Because the compiler leaves the quantum part of the ciphertext untouched, the BB84 construction [BL20] and coset-state constructions retain their state structure, yielding an alternative information-theoretically secure plain-model construction for one-bit messages—though with weaker quantitative bounds than recent dedicated constructions [AS26], [Rag26], [BBC26].

Proof architecture

The proof proceeds through six stages, organized around a purification $\Phi$8 of each $\Phi$9 and the operator

$\rho_x$0

where $\rho_x$1 and $\rho_x$2 are the Fourier coefficients of Bob's and Charlie's difference observables at character $\rho_x$3, and $\rho_x$4 is the common-challenge correlation operator, so that $\rho_x$5.

Spectral decomposition. For each $\rho_x$6, Bob's space is split according to whether eigenvalues of $\rho_x$7 have absolute value at least $\rho_x$8, giving high/low projectors $\rho_x$9; Charlie's analogously via $x$0 with $x$1. This induces four orthogonal sectors $x$2 of the purification.

Pruning. Parseval's identity ($x$3) shows that $x$4: the $x$5 component itself defines valid local sub-POVMs whose normalization is bounded by search security. The entire $x$6 family plus any other sector of average weight below $x$7 is deleted from the algebraic estimate; Cauchy–Schwarz bounds the resulting perturbation of $x$8 by $x$9 where $p_{\mathrm{srch}}$0. If either one-high sector vanishes, a marginal bound immediately gives the target. The hard case has both $p_{\mathrm{srch}}$1.

Schur complement. The slack operator $p_{\mathrm{srch}}$2 has margin $p_{\mathrm{srch}}$3 on any subspace where either party is low. Rather than bounding the $p_{\mathrm{srch}}$4–$p_{\mathrm{srch}}$5 and $p_{\mathrm{srch}}$6–$p_{\mathrm{srch}}$7 cross terms individually—which would discard positivity and would require controlling half-extraction paths that search security alone cannot bound—the proof completes the square in the $p_{\mathrm{srch}}$8 component, equivalently taking the Schur complement of the positive $p_{\mathrm{srch}}$9 block. This reduces everything to a single averaged scalar $[2^{-n}, 1]$0, combining the direct $[2^{-n}, 1]$1 path with an $[2^{-n}, 1]$2 mediated path:

$[2^{-n}, 1]$3

Neumann expansion. On the $[2^{-n}, 1]$4 subspace, writing $[2^{-n}, 1]$5 with $[2^{-n}, 1]$6 and $[2^{-n}, 1]$7, a resolvent identity controls the replacement of $[2^{-n}, 1]$8 by $[2^{-n}, 1]$9 with error at most $r$0; the latter expands as a convergent Neumann series in $r$1, decomposing $r$2 into contributions $r$3, each carrying $r$4 copies of the common-challenge operator $r$5.

Reduction to search. Each $r$6, expanded over auxiliary challenge words $r$7 fixed independently of $r$8, factors into products $r$9 whose normalized adjoint-products form local sub-POVMs indexed by candidate strings. The key normalizability estimate is a telescoping identity plus Minkowski's inequality:

$\mathbb{F}_2^n$0

with the analogous Charlie-side bound costing nothing extra since his endpoint projector $\mathbb{F}_2^n$1 already normalizes the contractions. Search security then gives $\mathbb{F}_2^n$2, and summing the geometrically weighted series yields $\mathbb{F}_2^n$3, comfortably below the required threshold $\mathbb{F}_2^n$4. Choosing $\mathbb{F}_2^n$5 completes the proof, with the numerical constants (e.g., $\mathbb{F}_2^n$6) absorbing the deletion cost.

A notable feature of the architecture is that the low-weight test precedes the delicate block analysis, preserving the quantitative smallness of genuinely tiny sectors rather than replacing them by worst-case bounds; and that the square completion accounts exactly for the most unfavorable possible effect of the $\mathbb{F}_2^n$7 interactions rather than bounding them term-by-term.

Limitations and open problems

Two limitations are explicit. First, the extraction POVMs produced by the proof are existential witnesses hardwired to fixed challenge words; there is no uniform polynomial-time construction from the prediction measurements, so computational search security cannot be transferred. A candidate efficient reduction targeting the weaker conclusion $\mathbb{F}_2^n$8 was suggested during the work's development but remains unverified. Second, the result is specific to $\mathbb{F}_2^n$9; an analogue over larger finite fields covering arbitrary ensembles and the full negligible-advantage regime remains open, though candidate arguments exist in restricted structured regimes. It should also be noted that the quantitative bound degrades rapidly in the advantage-to-recovery direction ($\langle r, x\rangle$0), so the compiler's security loss is substantial even when applicable.

Conclusion

This paper resolves the shared-secret, identical-challenge, exact-output non-local search-to-decision problem over $\langle r, x\rangle$1 for unrestricted local quantum strategies, proving that joint prediction advantage over the $\langle r, x\rangle$2 baseline implies joint full-string recovery with polynomially related probability. The proof combines spectral sectoring, Schur-completion of the lowest sector, and a Neumann-series decomposition into finite-length common-challenge paths, each reducible to search security via Parseval-normalized witness measurements. The result supplies the missing analysis for the common-challenge case of unclonable-encryption compilers, while leaving open efficient extraction and extensions beyond $\langle r, x\rangle$3.