AI 中文总结
研究海登 - 普雷斯基尔协议中，当解码器不知输出量子比特微观标识仅知相对顺序时的情况。通过分析保序删除信道，得出不同条件下最优纠缠保真度及恢复尺度，还考虑部分位置信息对恢复尺度的影响，揭示经典信息可改变量子输出需求。
AI 中文摘要
在海登 - 普雷斯基尔协议中，通常假设解码器知道收集到的输出量子比特的微观标识。我们研究当这些标签不可用时，仅保留接收量子比特的相对顺序会发生什么。由此产生的保序删除信道将一个\(n\)量子比特的加扰寄存器映射到长度为\(\ell\)的子序列。对于固定大小\(k\)的日志，我们证明当\(\ell = o(n^{2/3})\)时，最优纠缠保真度收敛到无输出值\(4^{-k}\)，对加扰酉矩阵一致成立。对于哈尔随机加扰酉矩阵，当\(\ell=\omega(n^{2/3})\)且\(\ell = o(n)\)时，它收敛到\(1\)。单调性给出固定误差恢复尺度\(\ell_{\mathrm{rec}}=\Theta(n^{2/3})\)。我们还考虑通过将寄存器划分为\(B\)个连续块并揭示每个接收量子比特的块起源而获得的部分位置信息。对于\(B\leq\sqrt{n}\)，恢复尺度变为\(\ell_{\mathrm{rec}}(n,B)\asymp n^{2/3}B^{-1/3}\)，对于\(B\geq\sqrt{n}\)，\(\ell_{\mathrm{rec}}(n,B)\asymp n/B\)。指数\(2/3\)可追溯到随机子序列之间的秩对齐巧合，它控制了逆命题和恢复论证。因此，即使是关于接收子系统起源的纯经典信息也可以改变海登 - 普雷斯基尔恢复所需的量子输出量。
英文摘要
In the Hayden--Preskill protocol, the decoder is usually assumed to know the microscopic identity of the collected output qubits. We study what happens when these labels are unavailable and only the relative order of the received qubits is preserved. The resulting order-preserving deletion channel maps an $n$-qubit scrambled register to a subsequence of length $\ell$. For a diary of fixed size $k$, we prove that the optimal entanglement fidelity converges to the no-output value $4^{-k}$ when $\ell=o(n^{2/3})$, uniformly over the scrambling unitary. For a Haar-random scrambling unitary, it converges to one when $\ell=ω(n^{2/3})$ and $\ell=o(n)$. Monotonicity then gives the fixed-error recovery scale $\ell_{\mathrm{rec}}=Θ(n^{2/3})$. We also consider partial position information obtained by dividing the register into $B$ consecutive blocks and revealing the block of origin of each received qubit. The recovery scale becomes $\ell_{\mathrm{rec}}(n,B)\asymp n^{2/3}B^{-1/3}$ for $B\leq\sqrt n$ and $\ell_{\mathrm{rec}}(n,B)\asymp n/B$ for $B\geq\sqrt n$. The exponent $2/3$ is traced to rank-aligned coincidences between random subsequences, which control both the converse and the recovery argument. Thus, even purely classical information about the origin of the received subsystems can change the amount of quantum output required for Hayden--Preskill recovery.
