Motivated by deterministic identification via (classical) channels, where the encoder is not allowed to use randomization, we revisit the problem of identification via quantum channels but now with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new results as our main findings: first, we show that all four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher-Westmoreland Theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples, we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity, which in turn can be larger than pure-state-encoded simultaneous identification capacity.

翻译：受经典信道确定性辨识（编码器不得使用随机化）的启发，本文重新审视量子信道中的辨识问题，但附加限制：消息编码必须使用纯量子态而非一般混合态。结合先前对同时解码器与一般解码器的区分，这形成了不同辨识容量的二维谱系，其行为在理论上可能差异显著。我们提出两个主要新结果：首先，证明所有四种组合（纯/混合编码器、同时/一般解码器）均具有双指数增长的码本规模，且相应辨识容量的下界由经典传输容量（由Holevo-Schumacher-Westmoreland定理给出）所界定。其次，证明量子信道的同步辨识容量等于纯态编码下的同步辨识容量，从而仅保留三种线性有序的辨识容量。通过考虑若干简单实例，最终证明三者互不相同：一般辨识容量可大于纯态编码辨识容量，而后者又可大于纯态编码同步辨识容量。