Entanglement, separability and correlation topology of quantum systems over parametric space of interaction potential

The standard understanding of formal quantum theory is based upon the belief that the state of two interacting quantum systems can jointly evolve as, either an entangled state, e.g. in case of measurement or decoherence, or a separable state, e.g. in case of gate operations, through two different processes, i.e. process-1 and process-2, respectively, as suggested by von Neumann, although nothing much is known about such processes in terms of physical interaction. The present work, exploring the correlation topologies of two interacting quantum systems in parametric space of their interaction potential, reveals that process-1 and process-2 are not different kinds of physical interactions but depend on the interaction parameters to result in either an entangled or a separable state. However, under energy conservation restriction, it is impossible to travel from one maximally-entangled state to another in the topological space by continuously varying such interaction parameters without crossing an intermediate separable state, and vice-versa. Nevertheless, a maximal entangled state is shown to be achieved by violating energy conservation utilizing the energy-time uncertainty or a catalytic separable ancillia. The work explores the nature of interaction potential needed to rotate a qubit state on the entire Bloch sphere, thereby revealing a novel method of measuring the qubit phase avoiding its state-collapse. Further, the manipulation of the degree of non-local entanglement of two space-like apart entangled-qubits in a controlled manner by local unitary operation on one has been illustrated. The generalization of process-1 and process-2 in terms of interaction potential to create entanglement or separability suggests a necessary revisit of the fundamental quantum paradoxes and several other quantum limitations including decoherence for advancing the field of quantum technology as a whole.

💡 Research Summary

The manuscript revisits the foundational distinction made by von Neumann between “process 1” (measurement‑type interactions that generate entanglement) and “process 2” (gate‑type interactions that keep subsystems separable) and asks a simple but profound question: are these two processes really different physical mechanisms, or can they be understood as different regimes of a single underlying interaction? By analysing two‑level systems that start in pure product states and evolve under a joint unitary generated by a Hamiltonian H = H₀ + V, the author shows that the answer lies in the structure of the interaction potential V and its parameters.

Two families of potentials are identified that always produce separable final states, irrespective of the interaction strength or duration. The first, V₁ = V₀(|χ⟩⟨χ| − |φ⟩⟨φ|)⊗I, merely adds a global phase η = V₀t to each qubit. This corresponds to a rotation about the azimuthal angle on the Bloch sphere and enables a non‑destructive measurement of the qubit phase without any external gate. The second, V₂ = V₀(|χ⟩⟨φ| + |φ⟩⟨χ|)⊗I, changes the polar angle of the first qubit while leaving the second untouched, effectively acting as a single‑qubit amplitude‑modulating gate. Both are classified as “process 2” because the joint state remains a product at all times.

A third class, V₃ = κ|χ⟩⟨χ|⊗|χ⟩⟨χ| + η|φ⟩⟨φ|⊗|φ⟩⟨φ|, preserves the computational basis of both qubits but couples their populations. The concurrence of the resulting state is C = |e^{-i(κ‑η)t} − 1|/2, reaching unity (maximal Bell‑state entanglement) when κ = η. This family embodies “process 1”.

A central topological result follows from imposing strict energy conservation (⟨V⟩ = 0). In the parameter space (κ, η), the set of maximally entangled points is disconnected by a “separable manifold” where C = 0. Consequently, one cannot continuously deform one maximally entangled state into another while remaining on the energy‑conserving surface; an intermediate separable state must be crossed. This establishes a topological obstruction: the entanglement manifold is not homotopic to the separable manifold under the conservation constraint.

To bypass this obstruction, the author invokes the energy‑time uncertainty relation (ΔE·Δt ≥ ħ/2). By allowing a brief violation of energy conservation, the system can traverse a shortcut through the forbidden region and reach a different maximally entangled state without passing through a separable point. An alternative is the use of a catalytic separable ancilla: a third system that remains unentangled but, through a carefully engineered three‑body interaction, mediates the required non‑conserving transition while the total system’s energy budget stays effectively unchanged.

Beyond the abstract topology, the paper presents a concrete protocol for locally controlling non‑local entanglement. If two distant qubits share a Bell pair, applying a local unitary U(θ, φ) on one side changes the overall concurrence according to a simple analytic expression derived from the same V₃‑type interaction. This demonstrates that, contrary to the common belief that local operations cannot alter the amount of shared entanglement, the underlying interaction potential can be tuned so that a purely local gate effectively “steers” the entanglement magnitude.

The author also discusses experimental relevance. In double‑quantum‑dot (DQD) charge qubits, the electrostatic coupling between the electron and the metal gate electrodes naturally realizes V₁‑type interactions, enabling phase‑gate operations without external control lines. Superconducting transmons or trapped‑ion platforms can be engineered to implement V₃‑type couplings via tunable capacitive or phononic mediators, providing a route to deterministic Bell‑state generation without dissipative reservoirs.

Finally, the manuscript connects its findings to the broader literature on correlation‑thermodynamic constraints, arguing that the “correlation‑thermodynamic” bound on steady‑state entanglement is a manifestation of the same topological limitation identified here. By exploiting energy‑time uncertainty or catalytic ancillae, these thermodynamic limits can be relaxed, opening new avenues for energy‑efficient entanglement generation in quantum technologies.

In summary, the work reframes entanglement versus separability as a continuous function of interaction‑potential parameters, uncovers a topological barrier imposed by energy conservation, and proposes concrete mechanisms—both theoretical and experimentally viable—to overcome this barrier. The insights have immediate implications for quantum gate design, entanglement distribution, decoherence mitigation, and the foundational interpretation of measurement in quantum mechanics.