AI 中文总结
本文以花态为研究对象，计算其各类纠缠度量，揭示非纠缠操作下存在已知最大纠缠不可逆性间隙，还得出其在局域操作和经典通信下的精确纠缠成本，相关结论利用了循环群不确定关系。
AI 中文摘要
纠缠作为最显著的精妙量子现象之一，其神秘性体现在复杂的操作结构中，存在可在不同有效程度上操控纠缠的自由操作类层级。本文使用由（偶数）局域维度2k参数化的“花态”，以揭示这一多样图景的若干方面。我们计算了花态的所有主要纠缠度量，发现所有形式的可蒸馏纠缠（均为1个ebit，与局域维度无关）与局域操作和经典通信（LOCC）下的纠缠成本（已知为log(2√k)）之间存在巨大差距。即使在严格更强大的非纠缠（NE）操作类下，我们证明其成本仍等于log(1+√k)，仅比LOCC的成本少约1个ebit。这一结果通过计算这些态新近引入的 tempered 纠缠负性得以证明，它展示了NE操作下已知最大的“不可逆性间隙”，即可蒸馏纠缠与纠缠成本的差值，等于Θ(1/2 log d)，其中d为局域维度。一个值得注意的推论是，著名的挤压纠缠在NE操作下并非单调量。最后，我们计算了花态在LOCC操作下的精确成本，其由施密特数给出，该施密特数在多份拷贝上具有可加性，等于min_{r|k} log(r + k/r)；当k为素数时，其简化为log(k+1)，约为标准LOCC成本的两倍。这些最新结果利用了Tao和Meshulam证明的循环群上的不确定关系。
英文摘要
The mysterious nature of entanglement, one of the most prominent exquisitely quantum phenomena, is reflected in its intricate operational structure, with a hierarchy of classes of free operations that enable its manipulation at different levels of effectiveness. Here we use the class of 'flower states', parametrised by their (even) local dimension $2k$, to shine light on some aspects of this varied landscape. We compute all the main entanglement measures for flower states, uncovering a large gap between all forms of distillable entanglement, equal to 1 ebit independently of the local dimension, and the entanglement cost under local operations and classical communication (LOCC), known to be equal to $\log\big(2\sqrt{k}\big)$. Even under the strictly more powerful class of non-entangling (NE) operations, we show that their cost is still equal to $\log\big(1+\sqrt{k}\big)$, only about an ebit less than for LOCCs. This result, which we prove by calculating the recently introduced tempered entanglement negativity for these states, demonstrates the largest known 'irreversibility gap', i.e. the difference between distillable entanglement and entanglement cost, under NE operations, equal to $Θ\big(\frac12 \log d\big)$, with $d$ being the local dimension. A notable consequence is that the celebrated squashed entanglement is not a monotone under NE operations. Finally, we compute the exact cost under LOCC operations for flower states; this is given by the Schmidt number, which turns out to be additive over multiple copies and equal to $\min_{r|k} \log\left( r + \frac{k}{r} \right)$; for prime $k$ this reduces to $\log(k+1)$, about twice the standard LOCC cost. These last results leverage the uncertainty relations over cyclic groups proved by Tao and Meshulam.
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