Estimation of Quantum Umlaut Information

Abstract

Quantifying correlations is important for understanding how quantum states and channels support information processing. Quantum umlaut information $U$ connects such correlations to reliability in hypothesis testing and certain communication settings, but its known formulas require classical descriptions of the underlying states. To estimate correlations without these descriptions, we introduce measured quantum umlaut information $U^M$ by replacing Umegaki relative entropy with measured relative entropy. Its variational formulation leads to a quantum generative adversarial framework for estimating $U^M$ from measurement statistics of unknown bipartite states and classical-quantum channels. This framework optimizes reference states and measurements without reconstructing the target states or channel outputs. For classical-quantum channels, it also optimizes the input probabilities. We further analyze the relation between $U^M$ and $U$, deriving a state-dependent gap bound and characterizing equality for full-rank bipartite states. Numerical simulations show trained estimates close on average to independently computed numerical references for $U^M$ across the tested states and channels. These estimates could help assess how well quantum memories preserve correlations and how noise or drift affects input-output dependence in quantum transmitters and state-preparation devices.

The paper asks how to estimate quantum umlaut information, $U(A;B)_\rho=\min_{\sigma_B}D(\rho_A\otimes\sigma_B\Vert\rho_{AB})$, when states are available through repeated preparations rather than classical descriptions. It introduces measured umlaut information $U^M$ by replacing Umegaki relative entropy with measured relative entropy, and extends this construction to classical–quantum (CQ) channels with optimized input probabilities. A variational formula expresses $U^M$ as an adversarial optimization: a generator minimizes over the reference state $\sigma_B$, while a discriminator maximizes over measurement bases and outcome scores. Both objective terms are accessible through single-copy measurement statistics without tomography. For CQ channels, one reference state is shared across labels, with an outer optimization over their probabilities. The theoretical results establish faithfulness, local-channel monotonicity, identical finiteness conditions to $U$, and $0\le U-U^M\le\log|\operatorname{spec}(\rho_{AB})|$. For full-rank states, a sharper state-dependent bound follows from a variational witness and Golden–Thompson inequality; equality holds exactly when the optimal product reference for $U$ commutes with $\rho_{AB}$. Smooth perturbations around equality states have an $O(\varepsilon^2)$ gap. Simulations use exact outcome probabilities for two-, four-, and six-qubit states and CQ label–output registers. Across 30 targets per size and 10 initializations, mean absolute relative errors against independently bounded numerical references are approximately $0.000016\%$, $0.67\%$, and $0.13\%$ for states, and $0.0077\%$, $6.0\%$, and $0.028\%$ for channels. Finite-shot training and device noise are not tested. Restricted circuits and incomplete optimization prevent certified bounds from trained values; discriminator score parameters scale exponentially.