Quantum Transfer Entropy of Incoherent Channels and Causal Interpretation

Transfer entropy plays an important role in causal inference and causal discovery in time series analysis. As an information-theoretic measure, transfer entropy can detect nonlinear causal structures without any prior model assumptions. The transfer-entropy-based causal analysis scenario thus has a broader range of applications than the model-based methods like Granger causality tests. This study generalizes the definition of transfer entropy to time-parameterized quantum processes. We define a basis-dependent operational measure of one-way information flow through incoherent channels, called quantum transfer entropy. Quantum transfer entropy provides a new characteristic of quantum channels. A nonzero quantum transfer entropy indicates the directed causal influence from the source to the target. In addition, we study controlled-Z channels and find that the quantum transfer entropy is a linear function of the total information of the control qubit—including both classical Shannon information and coherence in a given reference basis —thus providing an operational interpretation of information flow through quantum control channels. We further quantify the effect of a third party by studying the extreme case of the Toffoli gate, where the two control qubits are equivalent and symmetric with respect to the target qubit. This effect is characterized by the quantum transfer entropy conditioned on the third party (or the environment). We illustrate several bounds on the conditional quantum transfer entropy for the Toffoli gate in terms of local systems. These bounds allow us to estimate the conditional transfer entropy without any knowledge of the environment qubit. Finally, we employ the proposed quantum transfer entropy for causal discovery in both synthetic and real quantum time-series data.