Quantum models of interaction Hamiltonians and their paradoxes

The gist: Quantum models of interaction Hamiltonian and their paradoxes study simple microscopic quantum models that respect the light cone by design and reproduce two-body Hamiltonians, resolving paradoxes related to faster-than-light communication and apparent inconsistencies in local energy accounting through explicit microscopic mediator models.

Introductory Paradoxes

Effective nonlocal Hamiltonians are operationally powerful but lead to two negative consequences when taken literally: they obscure the local propagation mechanism and can suggest instantaneous influence, and they obscure where interaction energy resides on short time-scales, inviting conflicting accounts of local energy flow. If one takes a genuinely nonlocal manybody Hamiltonian as a microscopic law, sharp paradoxes arise regarding faster-than-light influences and inconsistent “work” assignments for local actions. The paper isolates these tensions by embedding the interaction into an explicit microscopic mediator model to resolve them.

General Consequences of Finite Mediator Propagation Time

When the Hamiltonian HAB is not understood as an instantaneous nonlocal action at a distance but rather emerges from a strictly local interaction with a mediator degree of freedom that carries the interaction, an effective interaction Hamiltonian Heff AB arises only after the mediator has been eliminated and is expected to describe time scales long compared to T.

A simple model-independent estimate illustrates the separation of time scales implicit in the effective description by replacing the pure state Ψ(t) with its time average ρ¯(t) over a window of width T.

Hybridization, dressed states, and correlations are related to entanglement-generating ability; stationary states of the effective theory can correspond to correlated or entangled system–mediator states called hybridized or dressed states.

Decoherence under fast local evolution occurs if A or B is subject to fast local evolution on the mediator-flight timescale, producing decoherence after the outgoing mediator is discarded.

The Simplest Model

The simplest model consists of an interaction mediated by a single bosonic mode (a harmonic oscillator) with position and momentum operators obeying [x, p] = i. The overall interaction is given by the unitary transformation VABM = U(2)BMU(2)AMU(1)BMU(1)AM.

The notion of interaction rate and effective interaction Hamiltonian arises if one supposes that new mediators arrive every τ, leading to an effective Hamiltonian Heff AB = −g2/τAB=-µAB.

Catalytic Mediation

The simple model has three crucial properties: it returns the mediator exactly to its initial state, it does so for every mediator state and for all coupling strengths g, and it generates a nontrivial interaction between A and B.

State-independent mediation requires factorization for every initial mediator state, leading to VABM = UAB ⊗ UM.

For state-independent mediation of effective interactions satisfying Eqs. (A6)–(A13), the catalytic condition order by order leads to the requirement that a nontrivial interaction h(2)AB requires an infinite-dimensional mediator.

The simplest model in Eq. (10) is obtained when xi are real and yj are purely imaginary, resulting in the simple displacement model.

Hybridization and Dressed States

Stationary states Ψ(tn)⟩ABM tn = n τ (33) are invariant under VABM(g), and these correspond to the eigenstates a, b⟩AB of the effective Hamiltonian in Eq. (B9).

The purity of the reduced system state Trρ2AB is approximately 1 − 2g2(VarΨ(A) + VarΨ(B)) when g2 << 1.

The Jaynes–Cummings model is exceptional because it remains catalytic at order g2 when several mediators are simultaneously present in the loop, even though second-order catalyticity for one isolated mediator is not sufficient.

Continuous Field Limit

The continuum limit is reached by letting the number of discrete modes n in each section of the waveguide tend to infinity while scaling g = qµT/n with µ fixed, yielding a finite effective coupling energy Eab = −µλaλ′b.

The stationary states in the continuum picture take the form Ψab⟩ABM = a, b⟩AB Fab⟩M with a coherent-field profile inside the waveguide described by ζab(r).

Decoherence Under Local Operations

The effective interaction relies on destructive interference; fast local operations on the systems change the coupling observable between these encounters and can prevent the loop from closing.

When ∆At ≠ 0, an uncanceled displacement remains and the outgoing mediator carries information about A, inducing decoherence of A once it is traced out.

The resulting master equation is ρ˙AB = iµ[At+2T B − HL A, ρAB] + 2µ DT A˙t (ρAB). The dissipation rate is controlled by r ∼ µT2∥A˙t∥2, and the gate-time needed to implement a desired local gate on system A is inversely proportional to the strength of the control.

Conclusion

The effective interaction picture is valid when local control changes the coupling observables little over the mediator flight time, T∥A˙∥ << ∥A. When this condition is not obeyed, decoherence and radiation reveal the presence of a causal microscopic interaction mechanism.

Outlook

Open questions include whether finite propagation time imposes a model-independent lower bound on correlations and entanglement between the systems and mediator for a prescribed effective interaction, distinguishing microscopic models that produce the same Heff AB but different dressings, Lamb shifts, and decoherence under fast control, and whether quantum-information-inspired microscopic models can provide simple but physically faithful descriptions of interactions in quantum field theory.

1. textbf causal information propagation in quantum interactions: The improved AI can model systems where local operations change are little over mediator flight time by implementing a master equation with a dissipation rate r ∼ µT 2∥A˙∥2 which is quadratic in the strength of the control Hamiltonian, allowing for the implementation of desired local gates without inducing decoherence if tgate ≫ µT2.

2. textbf resolving energy paradoxes via microscopic accounting: The AI can perform an explicit energy accounting by recognizing that the work performed by a fast gate is the energy supplied by its local control apparatus to Alice’s system and to nearby mediator degrees of freedom rather than relying on the coarse-grained long-time AB description, ensuring consistency with local energy flow.

3. textbf simulating entanglement generation in effective models: The improved AI can analyze hybridization, dressed states, and correlations by computing the purity of the reduced system state, showing that for a single-mode mediator, the reduced state becomes mixed whenever its superposition contains branches with distinguishable A or B eigenvalues.

4. textbf enabling continuous control in quantum systems: The AI can implement continuous local evolution by modeling it as a Lindblad master equation, where the dissipation rate is controlled by r ∼ µT 2∥A˙∥2, allowing for the design of gate times that are inversely proportional to the strength of the control.

5. textbf modeling non-Markovian dynamics: The AI can explicitly retain finite travel times and resulting nonMarkovian dynamics when analyzing system-field couplings, moving beyond approximations like Markov or zero-delay approximations.

In quantum physics it is commonplace to model the interaction of remote systems with a many-body Hamiltonian. Taking such an action-at-a-distance description à la lettre leads to various paradoxes related to faster-than-light communication and apparent inconsistencies in local energy accounting. It also neglects residual effects, such as entanglement between the remote systems and the mediator that implements the interaction, or the decoherence that arises when the remote systems undergo local evolution. We study simple microscopic quantum models that respect the light cone by design and reproduce two-body Hamiltonians. For these models we quantitatively analyze the residual effects of the microscopic mediator on the remote systems, including dressing of stationary states, and decoherence in the presence of fast local control. We show how the models resolve the paradoxes.