Compatibility of quantum instruments

Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the notion of parallel compatibility and show that these two notions are inequivalent. We then argue that the notion of traditional compatibility is conceptually incomplete and prove that, while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not capture channel compatibility. Hence, we propose parallel compatibility as the conceptually complete definition of the compatibility of quantum instruments.

Figure 1

Schematic representation of joint instruments for (a) traditionally compatible and (b) parallel compatible instruments. (a) Traditional compatibility: Schematic representation of Definition t2. Recovering the quantum output of either ${\mathcal{I}}_1$ or ${\mathcal{I}}_2$ can be done by first implementing the joint instrument $\mathcal{I}$ on the state $\rho$ and then performing the postprocessing of outcomes, i.e., taking the marginal over either $x$ or $y$. The downward arrows represent quantum systems. Clearly, in this case there is only one output quantum system. (b) Parallel compatibility: An example of parallel simultaneous implementation of two instruments (according to Definition t4), corresponding to Example t7. The simultaneous implementation of ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ can be done through the following steps: (i) implementing the channel $\mathrm{\Lambda }$ on the state $\rho$ which is the joint channel of the compatible channels ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ [where ${\mathrm{\Lambda }}_1\left(\rho \right)$ and ${\mathrm{\Lambda }}_2\left(\rho \right)$ can be considered as the approximate unequal clones (unless ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2$) of the state $\rho$, in general, and therefore, it can be considered as approximate asymmetric cloning], and then (ii) applying the instruments ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ on ${\mathrm{\Lambda }}_1\left(\rho \right)$ and ${\mathrm{\Lambda }}_2\left(\rho \right)$, respectively, such that ${\mathcal{J}}_1\circ {\mathrm{\Lambda }}_1={\mathcal{I}}_1$ and ${\mathcal{J}}_2\circ {\mathrm{\Lambda }}_2={\mathcal{I}}_2$. The existence of such a channel $\mathrm{\Lambda }$ and such instruments ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ implies the parallel compatibility of the instruments ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$, as explained in Example t7. The downward arrows represent quantum systems. Clearly, in this case there are two output quantum systems.