Measures of correlations in infinite-dimensional quantum systems (Q2821894)

Measures for the correlations inherent to composite quantum systems are often formally defined as linear combinations of marginal entropies of the subsystems. In particular, the quantum mutual information of a bipartite system \(\omega_{AB}\) reads (Eq. 4.1) NEWLINE\[NEWLINE I(A:B)_{\omega} = H(\omega_A) + H(\omega_B) - H(\omega_{AB}), NEWLINE\]NEWLINE where \(\omega_X\) is the partial state of subsystem \(X\), and \(H(\omega)\) is the von Neumann-entropy of the state \(\omega\). As another prominent example, the conditional mutual information of a tripartite system \(\omega_{ABC}\) reads (Eq. 6.1) NEWLINE\[NEWLINE I(A:C|B)_{\omega} = H(\omega_{AB}) + H(\omega_{BC}) - H(\omega_{ABC}) - H(\omega_{B}) . NEWLINE\]NEWLINE When generalizing such notions from finite-dimensional to infinite-dimensional subsystems, there are several technical difficulties to be overcome. In particular, the divergence to \(+\infty\) or to \(-\infty\) of individual summands in the linear combinations leaves, a priori, almost all states in infinite-dimensional systems without well-defined information-theoretic quantities. The paper tackles this problem and first introduces the notion of ``faithful'' functions \(F\) based on the ``self-consistency or stability of \(F\) with respect to state truncation'', which yields a ``replacement for continuity in infinite dimensions''. This eventually allows the ``faithful extension of a linear combination of marginal entropies'', which answers the question to which extent and how a generalization to infinite dimensions is possible. The main results are formulated as two theorems, one concerning the quantum mutual information in a bipartite system, which is then generalized to multipartite systems. The second theorem concerns the quantum conditional mutual information in a tripartite system. Other information-theoretic measures that are generalized to infinite-dimensional systems are a ``quantum version of the interaction information'', and the unconditional and conditional secrecy monotones. Finally, ``applications of the results to the theory of infinite-dimensional quantum channels and their capacities are considered. The existence of a Fawzi-Renner recovery channel reproducing marginal states for all tri-partite states (including states with infinite marginal entropies) is shown.''