Why initial system-environment correlations do not imply the failure of complete positivity: A causal perspective

The common wisdom in the field of quantum information theory is that when a system is initially correlated with its environment, the map describing its evolution may fail to be completely positive. If true, this would have practical and foundational significance. We here demonstrate, however, that the common wisdom is mistaken. We trace the error to the standard proposal for how the evolution map ought to be defined. We summarize this standard proposal and then show that it sometimes fails to define a linear map or any map at all. Further, we show that these pathologies persist even in completely classical examples. Drawing inspiration from the framework of classical causal models, we argue that the correct definition of the evolution map is obtained by considering a counterfactual scenario wherein the system is reprepared independently of any systems in its causal past while the rest of the circuit remains the same, yielding a map that is always completely positive. In a postmortem of the standard proposal, we highlight two distinct mistakes that retrospectively become evident (in its application to completely classical examples): (i) The types of constraints to which it appealed are constraints on what one can infer about the final state of a system based on its initial state; however, such inferences reflect not just the cause-effect relation between them—which is the basis for defining the correct evolution map—but also the common-cause relation. (ii) In a (retrospectively unnecessary) attempt to introduce variability in the input state, it inadvertently introduced variability in the inference map itself, and then tried to fit the input-output pairs associated to these different maps with a single map.

Figure 1

The scenario of interest. An initially correlated state of system and environment, ${\rho }_{S_{1}E}$, evolves according to a quantum channel ${\mathcal{E}}_{S_2|S_{1}E}$.

Figure 2

The most general circuit considered in the standard proposal.

Figure 3

An example where the input-output relation corresponds to a map that is linear but not completely positive.

Figure 4

An example where the input-output relation corresponds to a map that is nonlinear.

Figure 5

An example where the input-output relation does not correspond to any map.

Figure 6

(a) The classical analog of the circuit in Fig. 1. (b) The classical analog of the circuit in Fig. 2.

Figure 7

(a) The classical analog of the circuit in Fig. 1. (b) The hypothetical circuit which aids in defining the evolution map for the situation in panel (a).

Figure 8

(a) Repeat of Fig. 1, for ease of reference. (b) The hypothetical circuit which aids in defining the evolution map for the situation in panel (a).

Figure 9

Composition of system-environment interactions, leading to non-Markovian evolution on the system.

Figure 10

(a) The subset of circuits considered in the standard proposal wherein the setting variable $J$ determines the measurement on the system and one postselects on the measurement outcome $K$. (b) The same operational scenario as in panel (a), but reconceptualized as an intervention with setting $J$ and outcome $K$ on the system, rather than a preparation of a set of states labeled by $J$ and $K$.

Figure 11

(a) A one-time pad. The decoder and eavesdropper have distinct information about the key ($E$).