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This research establishes a rigorous framework for non-asymptotic entanglement manipulation, deriving a fundamental "Second Law" that quantifies how efficiently accessible quantum resources degrade under noise and probabilistic mixing. By bridging computational complexity with information-theoretic bounds, the work provides essential tools for optimizing entanglement distribution in practical quantum devices where unlimited processing power is unavailable.
Efficient Entanglement Manipulation Cookbook: From Mixing to a Computational Second Law
arXiv:2509.21988v2 Announce Type: replace Abstract: Quantum entanglement is an essential resource for implementing practical quantum devices. However, for the resource to be useful in large-scale applications, it must be accessible to parties with limited computational resources. Computational entanglement measures quantify the usefulness of entanglement in the presence of limited computational resources in the practical setting of non-asymptotic manipulations. In this paper, we build a framework of non-asymptotic efficient entanglement manipulations by systematically analyzing a wide range of properties of two recently introduced computational entanglement measures: the computational one-shot distillable entanglement and cost. To do so, we adapt definitions of some common properties of scalar entanglement measures to the case in which only lower or upper function bounds are available. Using the developed framework, we then proceed to investigate the mathematical behavior of the newly introduced entanglement measures under probabilistic state mixing and tensor-product operations. Furthermore, we analyze how the values of both measures change under local unitary and LOCC transformations. For this, we derive an explicit quantitative connection between the geometry of projective-net-generated families of states and their computational one-shot distillable entanglement, and use this connection to establish an unconditional separation between computational and information-theoretic one-shot distillable entanglement. Furthermore, we investigate the robustness of efficiently accessible entanglement against noise and derive the first quantitative continuity bound on computational distillable entanglement. Finally, as an application of the developed "dictionary" of properties, we establish a fundamental relation between computational entanglement measures that we call the "Second Law of Efficient Entanglement Manipulation".
Abstract:Quantum entanglement is an essential resource for implementing practical quantum devices. However, for the resource to be useful in large-scale applications, it must be accessible to parties with limited computational resources. Computational entanglement measures quantify the usefulness of entanglement in the presence of limited computational resources in the practical setting of non-asymptotic manipulations. In this paper, we build a framework of non-asymptotic efficient entanglement manipulations by systematically analyzing a wide range of properties of two recently introduced computational entanglement measures: the computational one-shot distillable entanglement and cost. To do so, we adapt definitions of some common properties of scalar entanglement measures to the case in which only lower or upper function bounds are available. Using the developed framework, we then proceed to investigate the mathematical behavior of the newly introduced entanglement measures under probabilistic state mixing and tensor-product operations. Furthermore, we analyze how the values of both measures change under local unitary and LOCC transformations. For this, we derive an explicit quantitative connection between the geometry of projective-net-generated families of states and their computational one-shot distillable entanglement, and use this connection to establish an unconditional separation between computational and information-theoretic one-shot distillable entanglement. Furthermore, we investigate the robustness of efficiently accessible entanglement against noise and derive the first quantitative continuity bound on computational distillable entanglement. Finally, as an application of the developed "dictionary" of properties, we establish a fundamental relation between computational entanglement measures that we call the "Second Law of Efficient Entanglement Manipulation".
60 pages, 4 figures. This version substantially extends and refines the original manuscript by adding a continuity bound, a computational second-law relation, and projective-net-based constructions yielding quantitative entanglement separations. The introduction and related discussions have also been revised to clarify the contributions and their relation to prior work
arXiv:2509.21988 [quant-ph]
(or arXiv:2509.21988v2 [quant-ph] for this version)
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