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：受经典信道确定性识别（其中编码器不允许使用随机化）的启发，我们重新审视了量子信道识别问题，但增加了额外限制：消息编码必须使用纯量子态而非一般的混合态。结合先前考虑的同时解码器与一般解码器的区分，这提出了一个二维的不同识别容量谱，其行为在理论上可能截然不同。我们证明了两个新结果作为主要发现：首先，我们表明所有四种组合（纯/混合编码器、同时/一般解码器）都具有双指数增长的码本规模，且相应的识别容量确实以一般量子信道的经典传输容量为下界，该容量由Holevo-Schumacher-Westmoreland定理给出。其次，我们证明量子信道的同时识别容量等于使用纯态编码的同时识别容量，从而留下三个线性排序的识别容量。通过考虑一些简单示例，我们最终证明这三者均不相同：一般识别容量可能大于纯态编码识别容量，而后者又可能大于纯态编码同时识别容量。