费米子量子香农理论中的非可加性

中文摘要 AI 辅助

我们研究了在宇称超选择规则和由正则反对易关系支配的复合规则下，费米子系统中的集体态制备和信道编码。对于双模高斯态 $\tau$，我们得到了对于每个 $m\geq1$ 的精确纠缠形成代价，$E_{\mathrm F}^{\mathrm{phys}}(\tau^{\otimes m})=\lceil m/2\rceil\log2$。两份拷贝的形成值与一份相同，违反了可加性和强超可加性，而正则化将单份拷贝的值减半。对于相关联的高斯信道，我们确定了在每个块长度下的最小输出熵和Holevo信息，并证明了二者的非可加性。这些结果共同给出了与Shor和Hastings相关的四个可加性问题的费米子对应问题的精确答案。一次物理使用不携带经典信息，而两次使用完美传输一个比特。无辅助经典容量和零错误经典容量均等于每模式半比特。一个尖锐的有限块界限给出了强逆：在高于容量的每个速率下，解码成功概率消失。这些优化覆盖了所有物理纯态分解和信道输入，高斯系综和经典码达到最优。形成代价和最小输出熵公式对每个Rényi阶 $\alpha\in(0,\infty]$ 均成立。我们还确定了无辅助量子容量和零错误量子容量，两者均等于每模式半个逻辑量子比特，并证明了强逆：在高于容量的每个速率下，纠缠保真度消失。对于有噪声的高斯族，我们确定了单份和双份拷贝的形成值，并证明了满秩时的严格非可加性。对于 $\tau$，在局部宇称保持测量下的贝尔违背首次出现在三份拷贝时。

英文摘要

We study collective state preparation and channel encoding in fermionic systems under parity superselection and composition governed by the canonical anticommutation relations. For a two-mode Gaussian state $τ$, we obtain the exact entanglement of formation, $E_{\mathrm F}^{\mathrm{phys}}(τ^{\otimes m})=\lceil m/2\rceil\log2$ for every $m\geq1$. Two copies have the same formation value as one, violating additivity and strong superadditivity, while regularization halves the single-copy value. For an associated Gaussian channel, we determine the minimum output entropy and Holevo information at every blocklength and prove nonadditivity of both. Together, these results give exact answers to the fermionic counterparts of the four additivity questions associated with Shor and Hastings. One physical use carries no classical information, whereas two uses transmit one bit perfectly. The unassisted classical and zero-error classical capacities both equal one half bit per mode. A sharp finite-block bound gives a strong converse: the decoding success probability vanishes at every rate above capacity. These optimizations cover all physical pure-state decompositions and channel inputs, with Gaussian ensembles and classical codes attaining the optima. The formation and minimum output entropy formulas hold for every Rényi order $α\in(0,\infty]$. We also determine the unassisted quantum and zero-error quantum capacities, both equal to one half logical qubit per mode, and prove a strong converse: entanglement fidelity vanishes at every rate above capacity. For a noisy Gaussian family, we determine the one- and two-copy formation values and prove strict nonadditivity at full rank. For $τ$, Bell violation under local parity-preserving measurements first appears at three copies.

发表机构

• Perimeter Institute for Theoretical Physics（佩里尔研究所）
• Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
• Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）
• Department of Applied Mathematics, University of Waterloo（滑铁卢大学应用数学系）
• Department of Combinatorics and Optimization, University of Waterloo（滑铁卢大学组合优化系）

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