AI 中文总结
针对不可克隆加密问题，引入泡利编码方案，证明其获胜概率下界，分析自然方法的局限，求解半定规划松弛得到渐近上界，并证明其针对有界局域维敌手的强不可克隆安全。
AI 中文摘要
不可克隆比特问题探究量子加密能否防止敌手在密钥披露后生成两个均能揭示明文的系统。我们引入并研究泡利编码（Pauli Encodings），这是一类简单的单比特加密方案，其密文为泡利串的归一化本征空间投影算子。对每个含K个泡利串的泡利编码，我们证明最优纠缠一夫一妻制获胜概率的通用下界为1/2 + 1/(2√K)，并对若干结构化族得到更精确的界。随后我们确立不可克隆安全自然方法的两个局限：其一，若泡利串限制为长度n的X和Z串，则该编码不安全；其二，我们发现通用的3/4障碍，表明仅基于两两猜测边际的论证无法建立不可克隆不可区分安全。当泡利串两两反对易时，该协议成为[Quantum 10, 2157 (2026)]研究的协议，我们利用其对称性求解自然半定规划松弛的第三层，得到获胜概率约0.5556的渐近上界。最后，我们证明针对有界局域维敌手的强不可克隆不可区分安全，以及若干高效泡利族的强不可区分安全，第一级NPA计算为强不可克隆不可区分安全提供了额外数值证据。
英文摘要
The unclonable bit question asks whether quantum encryption can prevent an adversary from producing two systems that both reveal the plaintext once the key is disclosed. We introduce and study Pauli Encodings, a simple class of one-bit encryption schemes whose ciphertexts are normalized eigenspace projectors of Pauli strings. For every Pauli Encoding with K Pauli strings, we prove a universal lower bound $1/2+1/(2\sqrt{K})$ on the optimal monogamy-of-entanglement winning probability, together with sharper bounds for several structured families. We then establish two limitations of natural approaches to unclonable security. First, if the Pauli strings are restricted to strings of X and Z of length n, the encoding is not secure. Second, we identify a universal 3/4 obstruction showing that arguments based only on pairwise guessing marginals cannot establish unclonable-indistinguishable security. When the Pauli strings all pairwise anticommute, the protocol becomes the one studied in [Quantum 10, 2157 (2026)]. We exploit the symmetry of this protocol to solve the third level of the natural semidefinite programming relaxation obtaining an asymptotic upper bound of approximately 0.5556 on the winning probability. Finally, we prove strong unclonable-indistinguishable security against bounded-local-dimension adversaries and strong indistinguishability security for several efficient Pauli families. First-level NPA computations provide additional numerical evidence towards the strong unclonable-indistinguishable security.
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