Entanglement in systems with continuous degrees of freedom

Characterize and analyze entanglement in quantum systems that possess continuous degrees of freedom, developing frameworks that rigorously identify, describe, and handle entanglement in continuous-variable settings.

Background

The authors highlight that understanding entanglement for systems with continuous degrees of freedom remains unresolved. This refers to continuous-variable quantum systems, such as modes of the electromagnetic field, where entanglement criteria and operational tools are more complex than in finite-dimensional cases.

Progress on this problem is crucial for bridging theoretical insights with experimental platforms that naturally operate in continuous-variable regimes.

References

In spite of continuous progress, the current state of entanglement theory is still marked by a number of outstanding unresolved problems. These problems range from the complete classification of mixed-state bipartite entanglement to entanglement in systems with continuous degrees of freedom, and the classification and quantification of multipartite entanglement for arbitrary quantum states.

— Epistemic vs Ontic Classification of quantum entangled states?  (1206.2916 - Caponigro et al., 2012) in Section 'Systems and Partitions', opening paragraph

We have not characterized the entanglement generated by the free Dirac CTQW across the hierarchical levels of the p-adic tree G_{l}{3}, nor the entanglement between the four spinor (particle/antiparticle) channels at a fixed node.

— p-Adic Dirac Equations, Continuous-Time Quantum Walks, and Quantum Networks  (2609.04358 - Zúñiga-Galindo, 3 Sep 2026) in Section 6, subsection “Open problems,” item (5) (Entanglement structure)

The above definitions coincide if at least one of the marginal entropies of the state $\rho$ is finite Section 5, but the global coincidence of these definitions is an interesting open problem (it is clear that the r.h.s. of (\ref{EF-d}) is not less than the r.h.s. of (\ref{EF-c}) for any state $\rho$, but it is not clear how to prove its coincidence for any mixed states with infinite marginal entropies).

— The Moreau-Yosida approximation of the Entanglement of Formation: basic properties and accuracy estimates  (2609.30246 - Shirokov, 24 Sep 2026) in Section 1, Introduction