Fundamental Irreversibility from Discrete Time

• URL: http://arxiv.org/abs/2606.22187v1
• Date: Sat, 20 Jun 2026 18:33:13 GMT
• Title: Fundamental Irreversibility from Discrete Time
• Abstract summary: In 1964, Yu. A. Gol'fand proposed an extension of quantum mechanics to discrete time, predicting intrinsic non-unitarity and entropy increase.<n>We rigorously recast Gol'fand's discrete evolution equation as a Completely Positive Trace-Preserving quantum channel.<n>We quantify the loss of quantum coherence via fidelity decay and purity loss, establishing exact constraints for fault-tolerant quantum computing.
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• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: In 1964, Yu. A. Gol'fand proposed an extension of quantum mechanics to discrete time, predicting intrinsic non-unitarity and entropy increase. While historically significant, this formalism predates the modern theory of open quantum systems. In this work, we rigorously recast Gol'fand's discrete evolution equation as a Completely Positive Trace-Preserving (CPTP) quantum channel and derive its continuous-time coarse-grained limit. We demonstrate that the dynamics converge to a specific Lindblad master equation characterized by a fundamental time scale $τ$, which induces decoherence in both the energy basis and a fundamental operator basis $W$. We analyze the thermodynamic implications using Spohn's entropy production formalism, proving that the discrete time step induces a strictly positive entropy production rate driven by the decay of quantum coherences, thereby providing a microscopic foundation for the arrow of time independent of environmental coupling. Furthermore, we quantify the loss of quantum coherence via fidelity decay and purity loss, establishing exact constraints for fault-tolerant quantum computing. We further investigate the impact of this intrinsic decoherence on Discrete Time Crystals (DTCs), showing that Gol'fand dynamics impose a fundamental lifetime limit on time-translation symmetry breaking phases. Finally, we utilize precision data from optical lattice clocks, matter-wave interferometry, and neutrino oscillations to place stringent upper bounds on $τ$. Our results constrain the fundamental time discretization to $τ\lesssim 10^{-26}$ s, significantly tightening previous limits and offering a testable framework for quantum gravity phenomenology.

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