The eavesdropper's deficit: entangled mediators, the (2n+1)² law, and the unbreakable revival window

Mustafa Boudjerada

Published Aug 25, 2026• 0 citations

Abstract

This note studies mediated quantum entanglement when the mediator is not only heated but also entangled with an unobserved environment. It investigates how hidden correlations affect the relationship between decoherence, information leakage, and the ability to recover which-path information. The first result is an unbreakable revival window: at specific revival times, the mediator interaction closes exactly, restoring the full entanglement between the target systems. The reduced state becomes identical to the ideal mediator case, with no remaining which-path information anywhere, even when the mediator is correlated with an external environment. The second result is the eavesdropper’s deficit: coherence loss and locally accessible information are not the same quantity when the environment is entangled. The paper derives an exact relation showing that the coherence lost can exceed the information available to any local observer by a factor of (2n̄+1)². The missing information is not destroyed; it is stored nonlocally in correlations between the mediator and its environment. Local measurements cannot recover this hidden record, while the complete joint system preserves the full information balance. These results demonstrate that entangled environments can separate decoherence from observable information leakage: a system may lose coherence strongly while leaving only limited locally accessible evidence about the cause. All results are derived in exactly solvable Gaussian models, verified through analytical and numerical checks, and accompanied by a reproducibility package.

DOI: 10.5281/zenodo.22096142

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