Unambiguous state discrimination (USD) measurements are attractive because outcomes are either marked as conclusive (i.e., error free) or inconclusive (i.e., erased). We study affine filtering measurements, a structured variant of USD for decoding classical linear codes over pure-state classical-quantum channels, where a conclusive outcome identifies an affine subspace containing the transmitted codeword and an inconclusive outcome is treated as an erasure. For a group-covariant indexing of pure-state codewords, we show that the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. We use the resulting measurement to build a quantum decoding framework for local codes, and we demonstrate (via simulations on regular LDPC codes from Gallager ensembles using single parity check local constraints) that affine filtering based decoding can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding methods on i.i.d. pure-state channels. In an independent and concurrent work, Buzet and Chailloux study similar fine-grained USD measurements for symmetric families of states. Their focus is on the code-agnostic setting whereas our focus is on code-aware constructions and decoding.

翻译：无歧义状态判别（USD）测量因其结果要么被标记为确定性的（即无错误）要么不确定性的（即被擦除）而具有吸引力。我们研究了仿射滤波测量——一种用于解码纯态经典-量子信道上经典线性码的USD结构化变体，其中确定性结果可识别包含所传输码字的仿射子空间，而不确定性结果被视为擦除。针对群协变索引的纯态码字，我们证明仿射滤波测量的最优设计可表述为半定规划，并通过基于特征的对称化方法将其简化为线性规划。利用所得测量，我们为局部码构建了量子解码框架，并通过基于Gallager集成正则LDPC码（采用单校验局部约束）的仿真证明：在独立同分布纯态信道上，基于仿射滤波的解码性能优于逐符号USD和逐符号优效测量解码方法。在同期独立工作中，Buzet与Chailloux研究了针对对称态族群的类似细粒度USD测量，其研究重点在于码无关场景，而我们的工作侧重建码感知的构造与解码。