平衡差承诺下的最优纠缠辅助信源编码
Optimal entanglement-assisted source coding under a balanced-difference promise
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中文总结 AI 辅助
研究零误差信源编码中纠缠辅助通信的最小成本，针对平衡差承诺的候选词对，证明偶校验时需n条消息、奇校验时需2条消息，结合傅里叶分析与组合计数确定相关图最小特征值，解决Cao等猜想，并在Lean中形式化验证。
中文摘要 AI 辅助
纠缠可以减少编码任务所需的通信量，但确定最小可实现成本对于理解其极限至关重要。我们在一个零误差信源编码任务中探讨此问题：Alice 接收一个单词，Bob 知道包含该单词的一个无序候选对。Alice 不知道这个对，必须利用共享纠缠和一条经典消息使 Bob 无误地识别她的单词。候选对满足平衡差承诺：对于 $\mathbb{Z}_q^n$ 中的单词，其中 $n=q\ell$，每个模 $q$ 的余数在其坐标差中恰好出现 $\ell$ 次。对于所有整数 $q\geq2$ 和 $\ell\geq1$，我们证明当 $(q-1)\ell$ 为偶数时该任务恰好需要 $n$ 条消息，当其为奇数时需要两条消息。这些最小值允许任意有限维共享态（独立于输入）和任意局部测量。在偶校验情况下，这确立了现有纠缠辅助协议的最优性。在奇校验情况下，一个显式确定性协议无需纠缠即可实现一位的最优。我们的证明结合傅里叶分析与组合计数论证，以确定相关图的最小特征值。在偶校验情况下，这解决了 Cao 等人猜想 6.3 对平衡循环广义 Hadamard 图的谱断言。结合一个显式的奇校验二分，这确定了量子色数在偶校验时为 $n$，在奇校验时为 $2$，而经典色数也为 $2$。所有引理、定理和推论均在 Lean 中形式化并验证。
英文摘要
Entanglement can reduce the communication required for coding tasks, but establishing the minimum achievable cost is essential to understanding its limits. We address this question in a zero-error source-coding task where Alice receives a word and Bob knows an unordered pair of candidates containing it. Alice does not know the pair and must enable Bob to identify her word without error using shared entanglement and one classical message. The candidates satisfy a balanced-difference promise: for words in $\mathbb{Z}_q^n$ with $n=q\ell$, each residue modulo $q$ occurs exactly $\ell$ times in their coordinatewise difference. For all integers $q\geq2$ and $\ell\geq1$, we prove that the task requires exactly $n$ messages when $(q-1)\ell$ is even and two messages when it is odd. These minima allow arbitrary finite-dimensional shared states independent of the inputs and arbitrary local measurements. In even parity, this establishes optimality of an existing entanglement-assisted protocol. In odd parity, an explicit deterministic protocol achieves the optimum of one bit without entanglement. Our proof combines Fourier analysis with a combinatorial counting argument to determine the smallest eigenvalue of the associated graphs. In even parity, this resolves the spectral assertion of Cao et al.'s Conjecture 6.3 for balanced cyclic generalized Hadamard graphs. Together with an explicit odd-parity bipartition, this determines the quantum chromatic number as $n$ in even parity and $2$ in odd parity, where the classical chromatic number is also $2$. All lemmas, theorems, and corollaries are formalized and verified in Lean.
