arXiv subjects
Publications and source records attributed to Carla Rubiliani.
We review recent progress on state-dependent Lieb--Robinson bounds for Bose--Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb--Robinson velocity is bounded by $t^{d-1}$ for large times, where $d$ denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound $t^{d+\epsilon}$.
In this note, we study transport properties of the dynamics generated by translation-invariant and possibly long-ranged Hamiltonians of Bose-Hubbard type. For translation-invariant initial states with controlled boson density, we improve the known bound on the local repulsive energy at time $t$ from $\langle n^2_x\rangle_t\lesssim t^{2d}$ to $\langle n^2_x\rangle_t\lesssim t^d$. This shows that bosonic on-site energies accumulate at most ballistically. Extending the result to higher moments would have powerful implications for bosonic Lieb-Robinson bounds. While previous approaches focused on controlling particle transport, our proof develops novel ASTLOs (adiabatic space-time localization observables) that are able to track the growth of local boson-boson correlations.
How the detailed structure of quantum complexity emerges from quantum dynamics remains a fundamental challenge highlighted by advances in quantum simulators and information processing. The celebrated Small-Incremental-Entangling (SIE) theorem provides a universal constraint on the rate of entanglement generation, yet it leaves open the problem of fully characterizing fine entanglement structures. Here we introduce the concept of Spectral-Entangling strength, which captures the structural entangling power of an operator, and establish a spectral SIE theorem: a universal speed limit for R'enyi entanglement growth at $\alpha \ge 1/2$, revealing a robust $1/s^2$ decay threshold in the entanglement spectrum. Remarkably, our bound at $\alpha=1/2$ is both qualitatively and quantitatively optimal, defining the universal threshold beyond which entanglement growth becomes unbounded. This exposes the detailed structure of Schmidt coefficients and enables rigorous truncation-based error control, linking entanglement structure to computational complexity. Building on this, we derive a generalized entanglement area law under an adiabatic-path condition, extending a central principle of quantum many-body physics to general interactions. As a concrete application, we show that one-dimensional long-range interacting systems admit polynomial bond-dimension approximations for ground, time-evolved, and thermal states, thereby closing the long-standing quasi-polynomial gap and demonstrating that such systems can be simulated efficiently with tensor-network methods. By explicitly controlling R'enyi entanglement, we obtain a rigorous, a priori error guarantee for the time-dependent density-matrix renormalization-group algorithm. Overall, our results extend the SIE theorem to the spectral domain and establish a unified framework that unveils the detailed and universal structure underlying quantum complexity.
We study the quantum dynamics generated by Bose-Hubbard Hamiltonians with long-ranged (power law) terms. We prove two ballistic propagation bounds for suitable initial states: (i) A bound on all moments of the local particle number for all power law exponents $\alpha>d+1$ in $d$ dimensions, the sharp condition. (ii) The first thermodynamically stable Lieb-Robinson bound (LRB) for these Hamiltonians. To handle the long-ranged and unbounded terms, we further develop the multiscale ASTLO (adiabatic space time localization observables) method introduced in our recent work [arXiv:2310.14896].
We consider the time-dependent Schr\"odinger equation that is generated on the bosonic Fock space by a long-range quantum many-body Hamiltonian. We derive the first bound on the maximal speed of particle transport in these systems that is thermodynamically stable and holds all the way down to microscopic length scales. For this, we develop a novel multiscale rendition of the ASTLO (adiabatic spacetime localization observables) method. Our result opens the door to deriving the first thermodynamically stable Lieb-Robinson bounds on general local operators for these long-range interacting bosonic systems.
We study general lattice bosons with long-range hopping and long-range interactions decaying as $|x-y|^{-\alpha} $ with $\alpha\in (d+2,2d+1)$. We find a linear light cone for the information propagation starting from suitable initial states. We apply these bounds to estimate the minimal time needed for quantum messaging, for the propagation of quantum correlations, and for quantum state control. The proofs are based on the ASTLO method (adiabatic spacetime localization observables). Our results pose previously unforeseen limitations on the applicability of fast-transfer and entanglement-generation protocols developed for breaking linear light cones in long-range and/or bosonic systems.