Abstract
Given a multipartite quantum system, what are the possible ways to impose mutual independence among some subsystems, and the presence of correlations among others, such that there exists a quantum state which satisfies these demands? This question and the related notion of a pattern of marginal independence (PMI) were introduced in [1], and then argued in [2] to be central in the derivation of the holographic entropy cone. Here we continue the general information theoretic analysis of the PMIs allowed by strong subadditivity (SSA) initiated in [1]. We show how the computation of these PMIs simplifies when SSA is replaced by a weaker constraint, dubbed Klein’s condition (KC), which follows from the necessary condition for the saturation of subadditivity (SA). Formulating KC in the language of partially ordered sets, we show that the set of PMIs compatible with KC forms a lattice, and we investigate several of its structural properties. One of our main results is the identification of a specific lower dimensional face of the SA cone that contains on its boundary all the extreme rays (beyond Bell pairs) that can possibly be realized by quantum states. We verify that for four or more parties, KC is strictly weaker than SSA, but nonetheless the PMIs compatible with SSA can easily be derived from the KC-compatible ones. For the special case of 1-dimensional PMIs, we conjecture that KC and SSA are in fact equivalent. To make the presentation self-contained, we review the key ingredients from lattice theory as needed.
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Acknowledgments
It is a pleasure to thank S. Hernández-Cuenca and M. Parisi for useful discussions. MR would also like to thank QMAP, the University of California Davis, and California Institute of Technology for their hospitality during various stages of this project. TH has been supported by funds from the University of California, the U.S. Department of Energy grant DE-SC0020360 under the HEP-QIS QuantISED program, the Heising-Simons Foundation “Observational Signatures of Quantum Gravity” collaboration grant 2021-2817, the U.S. Department of Energy grant DE-SC0011632, and the Walter Burke Institute for Theoretical Physics. VH has been supported in part by the U.S. Department of Energy grant DE-SC0009999 and funds from the University of California. MR has been supported by the University of Amsterdam, via the ERC Consolidator Grant QUANTIVIOL, and by the Stichting Nederlandse Wetenschappelijk Onderzoek Instituten (NWO-I).
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He, T., Hubeny, V.E. & Rota, M. On the relation between the subadditivity cone and the quantum entropy cone. J. High Energ. Phys. 2023, 18 (2023). https://doi.org/10.1007/JHEP08(2023)018
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DOI: https://doi.org/10.1007/JHEP08(2023)018