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arXiv:2504.02926 · Physical Review Letters

Optimal Bound on Long-Range Distillable Entanglement

Jonah Kudler-Flam, Vladimir Narovlansky, Nikita Sopenko

Published Aug 6, 2025• 0 citations
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Abstract

We prove an upper bound on long-range distillable entanglement in D spatial dimensions. Namely, it must decay faster than 1/r, where r is the distance between entangled regions. For states that are asymptotically rotationally invariant, the bound is strengthened to 1/r^{D}. We then find explicit examples of quantum states with decay arbitrarily close to the bound. In one dimension, we construct free fermion Hamiltonians with nearest neighbor couplings that have these states as ground states. Curiously, states in conformal field theory are far from saturation, with distillable entanglement decaying faster than any polynomial.

DOI
10.1103/d9tn-jfg1

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We prove an upper bound on long-range distillable entanglement in D spatial dimensions. Namely, it must decay faster than 1/r, where r is the distance between entangled regions. For states that are asymptotically rotationally invariant, the bound is strengthened to 1/r^{D}. We then find explicit examples of quantum states with decay arbitrarily close to the bound. In one dimension, we construct free fermion Hamiltonians with nearest neighbor couplings that have these states as ground states. Curiously, states in conformal field theory are far from saturation, with distillable entanglement decaying faster than any polynomial.
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We prove continuity of distillable entanglement, the optimal asymptotic rate at which copies of a bipartite state can be converted into maximally entangled pairs by local operations and classical communication (LOCC) with vanishing error. We establish an explicit asymptotic continuity bound that holds for arbitrary bipartite states and depends logarithmically on the system dimension. This bound has the same structure as known continuity bounds for conditional entropy, entanglement cost, and relative entropy of entanglement. Our proof constructs an explicit LOCC protocol that converts either of two arbitrary bipartite states into the other at an ebit cost that vanishes as their trace distance tends to zero.
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