量子仪器的兼容性

中文摘要 AI 辅助

测量兼容性探讨的是：一族量子测量能否由一个父测量随后进行经典后处理来模拟。我们将这种测量兼容性扩展为测量、信道和仪器的 \(d\)-兼容性概念，其中父测量通过一个 \(d\) 维量子系统并将无限经典信息传递给后处理。我们将 \(d\)-兼容性表述为算子空间和算子系统的公共线性分解，并证明 \(d\)-兼容族构成闭集，即使具有无限多个设置和可分离的无限维输入也是如此。为了认证 \(d\)-不兼容性，我们定义了兼容性泛函，即通过 \(d\) 维寄存器公共分解的完全有界范数的最小乘积。利用算子空间的分解理论，我们证明了基于互不偏基定义的测量-制备信道和基于海森堡-外尔算子定义的收缩信道在特定 \(d\) 范围内是 \(d\)-不兼容的。我们还利用信道-态对偶性将多仪器的 \(d\)-兼容性与 Choi 集合的可制备性联系起来，并利用这一联系证明：对于二分态，仪器的（不）兼容性可用于认证 Schmidt 数以及仪器通过单向 LOCC 协议的可实现性。最后，我们的兼容性泛函还限制了兼容性的广义鲁棒性，我们通过一个内存受限的非瞬态制备博弈来解释这一点。

英文摘要

Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{$d$-compatibility} for measurements, channels, and instruments whose parent passes a $d$-dimensional quantum system and unlimited classical information to the post-processing. We formulate $d$-compatibility as a common linear factorization of operator spaces and operator systems and show that $d$-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify $d$-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a $d$-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are $d$-incompatible in certain regimes of $d$. We also link $d$-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.

发表机构

• University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
• University of Waterloo（滑铁卢大学）
• Perimeter Institute for Theoreticla Physics（基础物理研究所）

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