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Quantum Speed Limits of Spin Chain Dynamics : Applying Quantum Speed Limits to Spin Chains used as Quantum Channels

Yonatan Gazit

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

Quantum computers, like classical ones, need to be capable of sending informa-tion between different parts of the computer. Spin chains are one viable method of sending qubits while preserving the superposition and entanglement of a state. We use quantum speed limits, which give bounds on how fast non-relativistic quantum systems can evolve, to study simple Heisenberg chains and their efficacy as quantum channels.

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Quantum speed limits (QSLs) constrain the minimum time for quantum state evolution, typically derived from energy bounds like the Margolus-Levitin (ML) and Mandelstam-Tamm (MT) relations. In open systems, decoherence—driven by environmental interactions—may impose additional constraints. This work investigates whether decoherence times fundamentally limit QSLs in spin-chain systems, extending ML/MT bounds to open dynamics. By modeling spin chains with Lindblad equations and simulating Heisenberg-type interactions, we analyze the interplay between energy fluctuations, decoherence rates, and evolution times. Experimental validation via trapped-ion or superconducting-qubit arrays could test energy-time uncertainty in many-body systems. Results suggest decoherence times establish a lower QSL bound proportional to [to verify], offering a platform to probe open-system quantum dynamics. ## 1. Introduction Quantum speed limits (QSLs) define the fastest possible evolution between distinguishable quantum states, critical for quantum computing and information processing. Closed-system QSLs, such as the ML [arXiv:quant-ph/9804014] and MT [DOI:10.1103/PhysRev.41.1019] bounds, rely on energy constraints. However, open systems—subject to decoherence—require extended frameworks. Decoherence times, governed by environmental couplings, may impose additional QSLs, yet their interplay with ML/MT bounds remains unexplored. Spin chains, realizable in trapped ions [DOI:10.1038/s41586-019-1394-6] and superconducting circuits, offer a many-body testbed for studying these effects. This work bridges open-system dynamics with QSL theory, assessing whether decoherence times set a fundamental lower bound and enabling experimental verification of energy-time uncertainty. ## 2. Background The ML bound states that a quantum system requires at least $ t_{\text{QSL}} \geq \hbar / (2\Delta E) $, where $ \Delta E $ is the energy spread. The MT bound $ t_{\text{QSL}} \geq \hbar / (2E) $, with $

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Quantum computers, like classical ones, need to be capable of sending informa-tion between different parts of the computer. Spin chains are one viable method of sending qubits while preserving the superposition and entanglement of a state. We use quantum speed limits, which give bounds on how fast non-relativistic quantum systems can evolve, to study simple Heisenberg chains and their efficacy as quantum channels.
The time needed by a quantum system to reach a state fully distinguishable from the original one provides a natural way of determining how fast the corresponding dynamical evolution is. This orthogonality time admits a lower bound, expressible in terms of the energy's expectation value and the energy's standard deviation, that yields a 'quantum speed limit'. Composite quantum systems need entanglement in order to achieve this limit. So far, most studies on the connection between entanglement and the quantum speed limit have focused on the case of non-interacting systems. The connection between quantum speed and entanglement is systematically investigated here for the case of a system of two interacting qubits, taking into consideration all possible initial states that evolve into an orthogonal one.
Abstract In quantum control, quantum speed limits provide fundamental lower bounds on the time that is needed to implement certain unitary transformations. Using Lie algebraic methods, we link these speed limits to symmetries of the control Hamiltonians and provide quantitative bounds that can be calculated without solving the controlled system dynamics. In particular we focus on two scenarios: On one hand, we provide bounds on the time that is needed for a control system to implement a given target unitary U and on the other hand we bound the time to implement the most difficult to reach evolution e − i θ H s of a target Hamiltonian H s . We apply our abstract bounds on physically relevant systems like coupled qubits, spin chains, globally controlled Rydberg atoms and NMR molecules and compare our results to the existing literature. We hope that our bounds can aid experimentalists to identify bottlenecks and design faster quantum control systems.
In the Schr\"odinger picture, the state of a quantum system evolves in time and the quantum speed limit describes how fast the state of a quantum system evolves from an initial state to a final state. However, in the Heisenberg picture the observable evolves in time instead of the state vector. Therefore, it is natural to ask how fast an observable evolves in time. This can impose a fundamental bound on the evolution time of the expectation value of quantum-mechanical observables. We obtain the quantum speed limit time bound for observable for closed systems, open quantum systems, and arbitrary dynamics. Furthermore, we discuss various applications of these bounds. Our results can have several applications ranging from setting the speed limit for operator growth, correlation growth, quantum thermal machines, quantum control, and many-body physics.
Quantum speed limits (QSLs) establish intrinsic bounds on the minimum time required for the evolution of quantum systems. We present a class of QSLs formulated in terms of the two-parameter Sharma-Mittal entropy (SME), applicable to finite-dimensional systems evolving under general nonunitary dynamics. In the single-qubit case, the QSLs for both quantum channels and non-Hermitian dynamics are analyzed in detail. For many-body systems, we explore the role of SME-based bounds in characterizing the reduced dynamics and apply the results to the XXZ spin chain model. These entropy-based QSLs characterize fundamental limits on quantum evolution speeds and may be employed in contexts including entropic uncertainty relations, quantum metrology, coherent control and quantum sensing.
Quantum speed limits (QSLs) constrain the minimum time for quantum state evolution, typically derived from energy bounds like the Margolus-Levitin (ML) and Mandelstam-Tamm (MT) relations. In open systems, decoherence—driven by environmental interactions—may impose additional constraints. This work investigates whether decoherence times fundamentally limit QSLs in spin-chain systems, extending ML/MT bounds to open dynamics. By modeling spin chains with Lindblad equations and simulating Heisenberg-type interactions, we analyze the interplay between energy fluctuations, decoherence rates, and evolution times. Experimental validation via trapped-ion or superconducting-qubit arrays could test energy-time uncertainty in many-body systems. Results suggest decoherence times establish a lower QSL bound proportional to [to verify], offering a platform to probe open-system quantum dynamics. ## 1. Introduction Quantum speed limits (QSLs) define the fastest possible evolution between distinguishable quantum states, critical for quantum computing and information processing. Closed-system QSLs, such as the ML [arXiv:quant-ph/9804014] and MT [DOI:10.1103/PhysRev.41.1019] bounds, rely on energy constraints. However, open systems—subject to decoherence—require extended frameworks. Decoherence times, governed by environmental couplings, may impose additional QSLs, yet their interplay with ML/MT bounds remains unexplored. Spin chains, realizable in trapped ions [DOI:10.1038/s41586-019-1394-6] and superconducting circuits, offer a many-body testbed for studying these effects. This work bridges open-system dynamics with QSL theory, assessing whether decoherence times set a fundamental lower bound and enabling experimental verification of energy-time uncertainty. ## 2. Background The ML bound states that a quantum system requires at least $ t_{\text{QSL}} \geq \hbar / (2\Delta E) $, where $ \Delta E $ is the energy spread. The MT bound $ t_{\text{QSL}} \geq \hbar / (2E) $, with $