Data Processing Inequality for The Quantum Guesswork
Ilyass Mejdoub, Julien Béguinot, Olivier Rioul
TL;DR
The paper extends classical guesswork to the quantum regime by formalizing a quantum guessing framework with ensembles $\mathcal{E}=\{(\rho_x,p_X(x))\}$ and POVMs, defining the quantum guesswork $G(X|\mathcal{E})$ as a joint optimization over guessing strategies and measurements. It proves the equivalence of two prior definitions $G_1$ and $G_2$ with the operational $G$, and establishes core data-processing properties: unitary invariance, post-processing DPI under quantum channels, and pre-processing DPI via majorization, tying guesswork to entropy and Holevo-type bounds. The work provides a refined bounding methodology that relates $G(X|\mathcal{E})$ to classical quantities $H(X)$, $H(X|Y)$ and $\chi(\mathcal{E})$, including a Rioul-type lower bound and known bounds such as McEliece-Yu and Massey as special cases. These results advance the analysis of quantum state discrimination under guesswork criteria and have implications for security proofs and information-processing tasks where guesswork, rather than error probability, is the relevant performance metric.
Abstract
Non-orthogonal quantum states pose a fundamental challenge in quantum information processing, as they cannot be distinguished with absolute certainty. Conventionally, the focus has been on minimizing error probability in quantum state discrimination tasks. However, another criterion known as quantum guesswork has emerged as a crucial measure in assessing the distinguishability of non-orthogonal quantum states, when we are allowed to query a sequence of states. In this paper, we generalize well known properties in the classical setting that are relevant for the guessing problem. Specifically, we establish the pre and post Data Processing Inequalities. We also derive a more refined lower bound on quantum guesswork.