Energy-limited quantum dynamics

TL;DR

This work develops a rigorous framework for energy-limited quantum dynamics by introducing energy constraints relative to a reference Hamiltonian $G$, and by defining energy-limited channels and dynamics that bound energy growth. It links energy-limitedness to a single Heisenberg-picture operator inequality for Markovian dynamics and introduces energy-constrained norms (ECD and ECO) that exhibit submultiplicativity and enable state-dependent continuity bounds. The paper provides concrete results for bosonic Gaussian channels, coherent-state quantization, quantum birth processes, and Lie-group representations, illustrating both unitary and open-system dynamics under energy constraints. These tools yield refined quantum speed limits, open-system Trotter convergence bounds, and a pathway to apply energy-scale reasoning to infinite-dimensional systems, with implications for quantum information processing in continuous-variable settings. The framework also identifies open problems, including energy-loss bounds, nonstandard generators, and extensions to von Neumann algebras, guiding future work in energy-aware quantum dynamics.

Abstract

We consider quantum systems with energy constraints relative to a reference Hamiltonian. In general, quantum channels and continuous-time dynamics need not satisfy energy conservation. Physically meaningful channels, however, only introduce a finite amount of energy to the system, and continuous-time dynamics only increase the energy gradually over time. We systematically study such "energy-limited" channels and dynamics. For Markovian dynamics, energy-limitedness is equivalent to a single operator inequality in the Heisenberg picture. We observe new submultiplicativity inequalities for the energy-constrained diamond and operator norms and use them to derive new state-dependent continuity bounds for quantum speed limits.

Figures (2)

• Figure 1: Numerical comparison of $\norm{e^{-itH_1}\psi-e^{-itH_2}\psi}$, the first order of our bound \ref{['eq:intro_qsl']}, and the operator norm bound $t\norm{H_1-H_2}$ for the quantum speed limit problem. The system is $\mathcal{H}=(\mathbb C^2)^{\otimes 7}$ with reference Hamiltonian $S_x^2+S_y^2+S_z^2$ minus its ground state energy, where $S_j = \sum_{k=1}^7 1^{\otimes k-1}\otimes \sigma_j\otimes1^{\otimes N-k}$. To obtain a state with relatively small energy, we take a weighted superposition $\psi = c(\Omega +\frac{1}{2}\phi)$ of the ground state $\Omega$ and a Haar randomly chosen state $\phi$ ($c$ is a normalizing constant). On the left, the Hamiltonians are $H_1=S_x+R_1$ and $H_2=S_y+R_2$, where $R_1$ and $R_2$ are random hermitian matrices of operator norm $\norm{R_i}=\tfrac{1}{2}$. These generate little energy, and we see that our bound \ref{['eq:intro_qsl']} is much better than the operator norm bound. On the right, we have $H_1=S_x$ and $H_2=R$ is a random hermitian matrix with $\norm{R}=\norm{S_x}=7$. Even though $H_2$ generates a lot of energy, our bound is still better than the operator norm bound, but the benefit is not that large.
• Figure 2: Visualization of the inequality $f(E)\le f(E')\le \frac{E'}{E}f(E)$, valid for concave nondecreasing functions $f:\mathbb R^+\to\mathbb R^+$ and $0<E<E'$. The diagonal has slope $f(E)/E$.

Theorems & Definitions (110)

• Lemma 1
• Theorem 2: Informal
• Theorem 3: Informal
• Lemma 2.1
• proof
• Lemma 2.2: Shirokov-Weis weis_extreme_2021
• Lemma 2.3: Holevo Holevo_2003
• Lemma 2.4
• proof
• Definition 2.5