Quantum channel marginal problem

Given a set of local dynamics, are they compatible with a global dynamics? We systematically formulate these questions as quantum channel marginal problems. These problems are strongly connected to the generalization of the no-signaling conditions to quantized inputs and outputs and can be understood as a general toolkit to study notions of quantum incompatibility. In fact, they include as special cases channel broadcasting, channel extendibility, measurement compatibility, and state marginal problems. After defining the notion of compatibility between global and local dynamics, we provide a solution to the channel marginal problem that takes the form of a semidefinite program. Using this formulation, we construct channel incompatibility witnesses, discuss their operational interpretation in terms of an advantage for a state-discrimination task, prove a gap between classical and quantum dynamical marginal problems, and show that the latter is irreducible to state marginal problems.

Figure 1

(a) A state marginal problem is asking whether, for a given set of local states (e.g., ${\rho }_{\mathrm{AB}},{\rho }_{\mathrm{BC}}$), there is a global state (${\rho }_{\mathrm{ABC}}$) that has them as its marginal states. (b) As a dynamical generalization of the state marginal problem, a channel marginal problem is asking whether, for a given set of local channels (e.g., ${\mathcal{E}}_{\mathrm{AB}},{\mathcal{E}}_{\mathrm{BC}}$, where ${\mathcal{E}}_{\mathrm{X}}={\mathcal{E}}_{\mathrm{X}|{\mathrm{X}}^{\prime }}$ when $X=X^{\prime }$), there is a global channel (${\mathcal{E}}_{\mathrm{ABC}}$) that has them as its marginal channels. (c) The commutation diagram used to define the marginal channel in a subsystem (Definition 2). For a local input coming from part of a global input, ${\mathrm{tr}}_{\mathrm{B}}\left({\rho }_{\mathrm{AB}}\right)$, the marginal channel ${\mathrm{Tr}}_{\mathrm{B}|\mathrm{B}}\left({\mathcal{E}}_{\mathrm{AB}}\right)$ of a global channel ${\mathcal{E}}_{\mathrm{AB}}$ should map it to the marginal state of the global output, ${\mathrm{tr}}_{\mathrm{B}}\left[{\mathcal{E}}_{\mathrm{AB}}\left({\rho }_{\mathrm{AB}}\right)\right]$. Unlike the case for states, the marginal of a channel is not always well defined.