Entanglement cost for infinite-dimensional physical systems
arXiv:2401.09554 · doi:10.1007/s00220-025-05431-1
Abstract
We prove that the entanglement cost equals the regularized entanglement of formation for any infinite-dimensional quantum state with finite quantum entropy on at least one of the subsystems or . 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-dimensional systems is nontrivial because the conventional tools for establishing both the direct and converse bounds, i.e., strong typicality, 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.
37+8 pages, no figures. v2 is close to the published version: main result unchanged; new definition of entanglement of formation using the continuous convex-roof construction; monotonicity strengthened from countably separable to general separable operations, and converse proof revised accordingly
References in corpus (48)
- Bell nonlocality
- Advances in Quantum Cryptography
- General Benchmarks for Quantum Repeaters
- Fundamental Limits of Repeaterless Quantum Communications
- Superadditivity of communication capacity using entangled inputs
- From Classical to Quantum Shannon Theory
- Fundamental rate-loss tradeoff for optical quantum key distribution
- Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
- Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
- Direct and Reverse Secret-Key Capacities of a Quantum Channel
- Quantum Capacities of Bosonic Channels
- Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture
- Limits and Security of Free-Space Quantum Communications
- Converse bounds for private communication over quantum channels
- Irreversibility for all bound entangled states
- A solution of the Gaussian optimizer conjecture
- Entanglement measures and their properties in quantum field theory
- Narrow Bounds for the Quantum Capacity of Thermal Attenuators
- Energy-constrained diamond norms and their use in quantum information theory
- Entanglement cost in practical scenarios
- Quantum trade-off coding for bosonic communication
- Entanglement cost and quantum channel simulation
- Separable Operations on Pure States
- Operational quantification of continuous-variable quantum resources
- Entropic characteristics of subsets of states
- Framework for resource quantification in infinite-dimensional general probabilistic theories
- Rényi squashed entanglement, discord, and relative entropy differences
- Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose
- Von Neumann entropy and majorization
- Exact solution for the quantum and private capacities of bosonic dephasing channels
- General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures
- Enhanced energy-constrained quantum communication over bosonic Gaussian channels
- Asymptotic state transformations of continuous variable resources
- Squashed entanglement in infinite dimensions
- Quantum Rényi and -divergences from integral representations
- On properties of the space of quantum states and their application to construction of entanglement monotones
- Reversibility of quantum resources through probabilistic protocols
- Optimal uniform continuity bound for conditional entropy of classical--quantum states
- Integral formula for quantum relative entropy implies data processing inequality
- Entanglement convertibility for infinite dimensional pure bipartite states
- Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions
- Continuity of characteristics of composite quantum systems
- Recoverability of quantum channels via hypothesis testing
- Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
- Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique
- Computable lower bounds on the entanglement cost of quantum channels
- On entropic quantities related to the classical capacity of infinite dimensional quantum channels
- Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication