相对于一组测量的量子压缩
Quantum compression relative to a set of measurements
摘要：
在这项工作中，我们调查的可能性压缩的量子系统的一个较小的维度的方式，保持测量统计的一组给定的可观的。在这个过程中，我们允许任意数量的经典边信息。我们发现，后者可以是有界的，这意味着最小的压缩维数是稳定的，在这个意义上，它不能减少允许小的错误。各种界限上的最小压缩维数的证明，并提供了一个基于SDP的算法计算。结果基于两种独立的方法：使用Arveson不动点结果的算子代数方法和依赖于不可约多项式和Bézout定理的代数几何方法。后一种方法允许将结果从单个拷贝水平提升到多个拷贝的情况，并且从完全正映射提升到仅仅正映射。
英文摘要：
In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bézout's theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.
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