The existence of quantum error correcting codes is one of the most counterintuitive and potentially technologically important discoveries of quantum information theory. However, standard error correction refers to abstract quantum information, i.e., information that is independent of the physical incarnation of the systems used for storing the information. There are, however, other forms of information that are physical - one of the most ubiquitous being reference frame information. Here we analyze the problem of error correcting physical information. The basic question we seek to answer is whether or not such error correction is possible and, if so, what limitations govern the process. The main challenge is that the systems used for transmitting physical information, in addition to any actions applied to them, must necessarily obey these limitations. Encoding and decoding operations that obey a restrictive set of limitations need not exist a priori. We focus on the case of erasure errors, and we first show that the problem is equivalent to quantum error correction using group-covariant encodings. We prove a no-go theorem showing that that no finite dimensional, group-covariant quantum codes exist for Lie groups with an infinitesimal generator (e.g., U(1), SU(2), and SO(3)). We then explain how one can circumvent this no-go theorem using infinite dimensional codes, and we give an explicit example of a covariant quantum error correcting code using continuous variables for the group U(1). Finally, we demonstrate that all finite groups have finite dimensional codes, giving both an explicit construction and a randomized approximate construction with exponentially better parameters.
Error Correction in Lattice Quantum Electrodynamics with Quantum Reference Frames
ETH Zurich（苏黎世联邦理工学院）；Institute of Quantum Optics and Quantum Information Vienna (IQOQI-Vienna), Austrian Academy of Sciences（奥地利科学院维也纳量子光学与量子信息研究所）；University of Vienna（维也纳大学）
Quantum reference frames for general symmetry groups
一个完全关系的量子理论必然需要考虑量子参考系的变化，其中量子参考系是相对于其他系统来描述的量子系统。通过引入一个关系的形式主义，确定坐标系的对称群$G$的元素，我们定义了一个通用的操作员之间的量子参考系与一个组$G$可逆地改变。这推广了已知算子的平移和提升到任意有限和局部紧群，包括非阿贝尔群。我们展示了在何种条件下，可以唯一地分配坐标选择的物理系统（形成参考系），以及如何可逆地转换它们之间，提供坐标系之间的转换是“在叠加”的其他坐标系。从关系物理和参考系相干变化的原理出发，得到了量子参考系的变化。我们证明了一个定理，即量子参考系的变换与这些原理一致时是么正的当且仅当参考系带有$G$的左、右正则表示。在对称群$G$是半直积$G = N \rtimes P$或直积$G = N \times P$的情况下，我们还定义了经典和量子系统参考系的不可逆变化，提供了多个量子参考系可逆和不可逆变化的例子.最后，我们应用的关系形式主义和变化的参考框架，在这项工作中开发的维格纳的朋友的情况下，发现类似的结论，在关系量子力学使用明确的变化参考系，而不是间接推理使用测量算子。
