Entanglement cost for infinite-dimensional physical systems
arxiv(2024)
摘要
We prove that the entanglement cost equals the regularized entanglement of
formation for any infinite-dimensional quantum state ρ_AB with finite
quantum entropy on at least one of the subsystems A or B. This generalizes
a foundational result in quantum information theory that was previously
formulated only for operations and states on finite-dimensional systems. The
extension to infinite dimensions is nontrivial because the conventional tools
for establishing both the direct and converse bounds, i.e., strong typically,
monotonicity, and asymptotic continuity, are no longer directly applicable. To
address this problem, we construct a new entanglement dilution protocol for
infinite-dimensional states implementable by local operations and a finite
amount of one-way classical communication (one-way LOCC), using weak and strong
typicality multiple times. We also prove the optimality of this protocol among
all protocols even under infinite-dimensional separable operations by
developing an argument based on alternative forms of monotonicity and
asymptotic continuity of the entanglement of formation for infinite-dimensional
states. Along the way, we derive a new integral representation for the quantum
entropy of infinite-dimensional states, which we believe to be of independent
interest. Our results allow us to fully characterize an important operational
entanglement measure – the entanglement cost – for all infinite-dimensional
physical systems.
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