Quantum Coherence Limits Entanglement Spread in Complex Systems

Swati Choudhary and colleagues at Harish-Chandra Research Institute present quantitative bounds linking the spreading of quantum states in Krylov space to both entanglement and quantum coherence. The work clarifies how measures of this spreading, including spread complexity and the inverse participation ratio, relate to the entanglement of evolved states and the coherence of initial states. These bounds apply to bipartite and multipartite systems, and the researchers specifically connect initial state coherence and spread complexity for qubit and qutrit systems, offering key constraints linking Krylov-space complexity growth to underlying quantum properties. Krylov-space dynamics now provide quantifiable upper bounds for bipartite entanglement Entanglement measures now possess definitive upper bounds, a significant improvement over previous ambiguity where any level of entanglement remained unquantifiable relative to Krylov-space dynamics. Previously, assessing entanglement necessitated direct measurement of the quantum state, a process that can be destructive and challenging, particularly for complex systems.

This research demonstrates that the entanglement of an evolved state in bipartite systems is limited by both the entanglement within the Krylov basis vectors, states generated by repeatedly applying the Hamiltonian operator to an initial state, and the spread complexity, a measure of how far a quantum state propagates within this mathematical space. This establishes a key threshold. Previously, direct measurement was required to assess entanglement, but now it can be indirectly inferred from Krylov-space properties, opening avenues for studying complex quantum systems where direct measurement is impractical or impossible. The Krylov subspace, formed by applying powers of the Hamiltonian to the initial state, effectively captures the relevant degrees of freedom for the time evolution, allowing for a reduced description of the system’s dynamics. For systems comprising multiple particles, entanglement generated during evolution is constrained not only by the initial entanglement within the Krylov basis vectors, states generated by repeated application of the Hamiltonian, but also by the spread complexity, which measures how widely a quantum state extends within this mathematical space. This is particularly relevant in many-body quantum systems where entanglement can grow rapidly with system size, making its characterisation computationally expensive. The bounds established by Choudhary et al. provide a means to estimate the maximum possible entanglement achievable given the initial state and the Hamiltonian. Investigations into qubit and qutrit systems reveal connections between initial quantum coherence, specifically in the energy eigenbasis, and spread complexity, a relationship that holds true regardless of the Hamiltonian used. This suggests a fundamental link between the system’s ability to maintain superposition, its coherence, and its capacity for complexity growth within Krylov space. Furthermore, the inverse participation ratio, a measure of state delocalization, is also bounded by geometric measures in multipartite systems, providing a new way to assess complex quantum states. A lower inverse participation ratio indicates a more delocalized state, meaning the quantum information is spread across multiple basis states, while a higher value suggests localization. These geometric measures offer a complementary perspective to traditional entanglement measures. Krylov space complexity reveals entanglement and coherence boundaries Researchers are increasingly focused on quantifying quantum complexity, a vital step towards harnessing the power of quantum systems for computation and communication. Quantum complexity, in this context, refers to the resources required to prepare or simulate a particular quantum state. Understanding and controlling complexity is crucial for developing efficient quantum algorithms and building robust quantum technologies. Clear limits on how entanglement, a uniquely quantum correlation, and coherence, the ability of a quantum system to exist in multiple states simultaneously, are linked to the spreading of quantum states within Krylov space have been established. The work deliberately focuses on upper bounds, revealing a fundamental tension between expansion and limitation. While quantum systems can, in principle, explore vast Hilbert spaces, their evolution is ultimately constrained by the available quantum resources. Defining what limits complexity growth is as important as understanding how it expands, providing a benchmark against which new quantum technologies can be measured. Without such limits, assessing the potential of a quantum system becomes exceedingly difficult. Linking Krylov space, a tool for modelling quantum behaviour, to entanglement and coherence offers a pathway to assess the resources needed for practical quantum computation and communication. The Krylov subspace provides a computationally tractable framework for analysing the dynamics of quantum states, allowing researchers to focus on the most relevant degrees of freedom. Establishing definitive relationships between quantum complexity and fundamental properties like entanglement and coherence offers a new perspective through which to view quantum systems. A quantum state’s expansion within Krylov space, a mathematical framework used to model quantum behaviour, is bounded by inherent quantum resources; specifically, initial entanglement and system coherence constrain how far a state can spread, providing quantifiable limits on complexity growth.

This research provides a theoretical foundation for understanding the interplay between these factors and could ultimately lead to the development of more efficient and robust quantum technologies. The implications extend to areas such as quantum error correction, where understanding the limits of entanglement and coherence is crucial for protecting quantum information from decoherence. The bounds derived in this study are general and apply to a wide range of Hamiltonians and initial states. However, future research could focus on exploring these bounds for specific physical systems, such as those relevant to condensed matter physics or quantum chemistry. Furthermore, investigating the relationship between Krylov-space complexity and other measures of quantum complexity, such as circuit depth, could provide a more complete understanding of the resources required for quantum computation. The research demonstrated that the expansion of a quantum state within Krylov space is limited by its initial entanglement and coherence. This matters because it establishes quantifiable constraints on quantum complexity growth, offering a benchmark against which to evaluate new quantum technologies. These findings, applicable to qubit and qutrit systems, link a modelling tool, Krylov space, to fundamental quantum resources, aiding assessment of systems for practical computation and communication. Future work could explore these limits in specific physical systems and connect Krylov-space complexity to measures like circuit depth, potentially leading to more efficient quantum technologies. 👉 More information 🗞 Entanglement and Quantum Coherence in Krylov Space Dynamics 🧠 ArXiv: https://arxiv.org/abs/2603.26619 Tags: