The incompatibility of quantum channels in general probabilistic theories

Abstract: In quantum theory, there exist sets of operations that cannot be performed simultaneously. These sets of operations are referred to as incompatible. While this definition of incompatibility extends to general probabilistic theories (GPTs), the dependency of the set of compatible sets on the definition of composite systems has not been thoroughly investigated. For quantum channels, compatibility is defined using the tensor product of Hilbert spaces, based on the conventional composite system. However, in GPTs, composite systems are not uniquely defined, and the set of states can vary from the minimal tensor to the maximal tensor. In this paper, in addition to the usual quantum compatibility, we introduce min-tensor-compatibility using the minimal tensor on the composite system of effect spaces and investigate their relationship employing noisy identity channels on qubits. As a result, we found that the set of min-tensor-compatible channel pairs is strictly broader than the set of quantum-compatible channel pairs. Furthermore, we introduce the concept of almost quantum compatible pairs of channels from an operational perspective. This concept corresponds to cases where the correlation functions in the verification of compatibility can be realized through a channel and local reinterpretation of effects. We demonstrate that the set of all almost quantum compatible channel pairs is strictly narrower than the set of all min-tensor-compatible channel pairs.