Incompatible measurements on quantum causal networks

The existence of incompatible measurements, epitomized by Heisenberg's uncertainty principle, is one of the distinctive features of quantum theory. So far, quantum incompatibility has been studied for measurements that test the preparation of physical systems. Here we extend the notion to measurements that test dynamical processes, possibly consisting of multiple time steps. Such measurements are known as testers and are implemented by interacting with the tested process through a sequence of state preparations, interactions, and measurements. Our first result is a characterization of the incompatibility of quantum testers, for which we provide necessary and sufficient conditions. Then we propose a quantitative measure of incompatibility. We call this measure the robustness of incompatibility and define it as the minimum amount of noise that has to be added to a set of testers in order to make them compatible. We show that (i) the robustness is lower bounded by the distinguishability of the sequence of interactions used by the tester and (ii) maximum robustness is attained when the interactions are perfectly distinguishable. The general results are illustrated in the concrete example of binary testers probing the time evolution of a single-photon polarization.

Figure 1

Pictorial representation of a setup testing a property of a quantum process $\mathcal{E}$. The input and output of the process are labeled as 0 and 1, respectively. The setup consists of the preparation of the input (and possibly an ancilla) into state $\mathrm{\Psi }$ and in the execution of a measurement (POVM) $\mathit{P}$ on the output. Any such setup can be described by a quantum tester, a suitable generalization of the notion of positive operator-valued measure.

Figure 2

Pictorial representation of a measurement setup testing a property of multitime quantum processes.

Figure 3

Diagrammatic representation of an ancilla-free quantum tester.

Figure 4

This figure illustrates the splitting of the parameter space of two-outcome factorized rank 1-testers into two regions $\mathsf{M}$ and $\mathsf{D}$ investigated separately. The robustness for testers is given by robustness of normalization states in the (shaded blue) region $\mathsf{M}$ (Proposition 19). In the remaining region $\mathsf{D}$ this is no longer the case as it is illustrated in Fig. 6.

Figure 5

Illustration of the optimal choice of the normalizations $\tilde{\rho }$ and $\tilde{\sigma }$ of the admixed testers ${\mathit{N}}^{\left(\mathit{A}\right)}$ and ${\mathit{N}}^{\left(\mathit{B}\right)}$ for the incompatibility of testers defined in Eq. (58).

Figure 6

Robustness of incompatibility of two qubit testers as defined in Eq. (58). (Top) Dependence on $\theta$ and situations for various choices of $\phi$. The solid line depicts the bound of Eq. (59). (Bottom) Dependence on $\phi$ for various choices of $\theta$. Dotted lines represent Eq. (59), which coincides with $R_{\mathrm{t}}(\mathit{A},\mathit{B})$ for $\theta \ge 0.6797$.

Figure 7

Illustration of quantum causal network, or $N$-comb (lower orange blocks), that can be “measured” by a quantum $N$-tester (upper blue blocks). Here ${\mathcal{H}}_i,i\in \{0,1,\cdots ,2N-1\}$ are the Hilbert spaces describing the inputs and outputs of the network (and outputs and inputs of the $N$-tester), ${\mathcal{K}}_j,j\in \{0,1,\cdots ,N-1\}$ are the Hilbert spaces of the internal memories of the network, and ${\mathcal{L}}_l,l\in \{0,1,\cdots ,N-1\}$ are the Hilbert spaces of the internal memories of the $N$-tester. $\mathrm{\Psi }$ is an input state to the $N$-tester (which can be chosen to be pure without loss of generality); ${\mathcal{U}}^{\left(l\right)}, l\in \{1,2,\cdots ,N-1\}$ are its quantum channels (which can be chosen to be unitary without loss of generality); and $\mathit{P}:={\left\{P_j\right\}}_{j\in \mathsf{S}}$ is a POVM, representing a measurement on the final output systems of the network of tested processes ${\mathcal{E}}^{\left(j\right)},j\in \{1,2,\cdots ,N\}$.

Figure 8

Diagrammatic representation of a tester with classical ancilla.

Figure 9

To obtain common normalization $I\otimes \omega$ for the same $\lambda$, lines connecting $\rho$ with $\sigma$ and $\tilde{\rho }$ with $\tilde{\sigma }$ must be parallel.

Figure 10

In qubit case the common normalization given by Bloch vector $\mathit{x}$ points at the intersection of lines connecting Bloch vectors $\mathit{r}$ with $\tilde{\mathit{r}}$ and $\mathit{s}$ with $\tilde{\mathit{s}}$, while the vectors $\mathit{r}-\mathit{s}$ and $\tilde{\mathit{r}}-\tilde{\mathit{s}}$ must be parallel.