The question of how quantities, like entanglement and coherence, depend on the number of copies of a given state $\rho$ is addressed. This is a hard problem, often involving optimizations over Hilbert spaces of large dimensions. Here, we propose a way to circumvent the direct evaluation of such quantities, provided that the employed measures satisfy a self-similarity property. We say that a quantity ${\cal E}(\rho^{\otimes N})$ is {} if it can be described as a function of the variables $\{{\cal E}(\rho^{\otimes i_1}),\dots,{\cal E}(\rho^{\otimes i_q}); N\}$ for $N>i_j$, while, preserving the tensor-product structure. If analyticity is assumed, recursive relations can be derived for the Maclaurin series of ${\cal E}(\rho^{\otimes N})$, which enable us to determine its possible functional forms (in terms of the mentioned variables). In particular, we find that if ${\cal E}(\rho^{\otimes 2^n})$ depends only on ${\cal E}(\rho)$, ${\cal E}(\rho^{\otimes 2})$, and $n$, then it is completely determined by Fibonacci polynomials, to leading order. We show that the one-shot distillable (OSD) entanglement is well described as a scalable measure for several families of states. For a particular two-qutrit state $𝜚$, we determine the OSD entanglement for ${𝜚}^{\otimes 96}$ from smaller tensorings, with an accuracy of $97\%$ and no extra computational effort. Finally, we show that superactivation of non-additivity may occur in this context.

中文翻译：

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表征量子资源的可扩展度量

数量（如纠缠和连贯性）如何取决于给定状态的副本数的问题 $\rho$已解决。这是一个难题，通常涉及对大尺寸希尔伯特空间的优化。在这里，我们提出了一种避免这种数量的直接评估的方法，只要所采用的措施满足自相似性即可。我们说数量$ {\ cal E}（\ rho ^ {\ otimes N}）$如果可以描述为变量$ \ {{\ cal E}（\ rho ^ {\ otimes i_1}），\ dots，{\ cal E}（\ rho ^ {\ otimes i_q}）; N \} $的$ñ>{\mathrm{一世}}_{Ĵ}$，同时保留张量积结构。如果假设具有分析性，则可以为$ {\ cal E}（\ rho ^ {\ otimes N}）$的Maclaurin系列推导递归关系，这使我们能够确定其可能的函数形式（根据所提及的变量） 。特别是，我们发现如果$ {\ cal E}（\ rho ^ {\ otimes 2 ^ n}）$仅取决于$ {\ cal E}（\ rho）$，则$ {\ cal E}（\ rho ^ {\ otimes 2}）$和$ñ$，则完全由Fibonacci多项式确定，直至领先。我们表明，一次蒸馏可蒸馏（OSD）纠缠被很好地描述为几个状态族的可扩展度量。对于特定的二态状态$𝜚$，我们确定OSD纠缠为 ${𝜚}^{\otimes 96}$ 来自较小的张量，精度为 $97％$无需额外的计算工作。最后，我们证明了在这种情况下可能发生非加性的超活化。