Researchers have generalized a measure of symmetry breaking, entanglement asymmetry, to account for quantum systems fracturing into exponentially many disconnected sectors. Lorenzo Gotta of the University of Geneva, Filiberto Ares of SISSA and INFN and Sara Murciano of Université Paris-Saclay demonstrate that while asymmetry typically has an upper bound, it can scale extensively in these fragmented systems.

This distinction allows researchers to differentiate between “classical” and “quantum” fragmentation, a key insight into the behavior of complex quantum states. The work qualitatively reproduces recent findings on asymmetry dynamics in random quantum circuits, suggesting a universal behavior for these U(1) charges in local ergodic systems.

Entanglement Asymmetry Measures Symmetry Breaking in Quantum States

Entanglement asymmetry, a quantitative measure of symmetry breaking in quantum states, typically remains bounded by a universal fraction of its maximum possible value, but this limitation does not hold true for systems exhibiting Hilbert-space fragmentation. Researchers discovered this distinction while investigating inhomogeneous U(1) charges, specifically dipole and multipole moments, revealing a unique behavior in fragmented systems. The study builds on a resurgence of interest in entanglement asymmetry, initially developed as a quantum resource theory distinguishing symmetry-respecting operations from those generating asymmetric states.

Unlike conventional symmetries where asymmetry grows logarithmically, asymmetry can scale extensively in fragmented quantum systems, enabling differentiation between “classical” and “quantum” fragmentation. This extensive scaling is particularly notable because it contrasts with previous findings concerning spatially homogeneous charges, which showed asymmetry limited to half its maximum allowed value.

The team’s work uses the commutant algebra formalism to generalize entanglement asymmetry, specifically to account for Hilbert-space fragmentation, a dynamic where a system splits into exponentially many disconnected sectors. Researchers identified states that saturate these upper bounds on asymmetry, providing a benchmark for understanding the limits of symmetry breaking in these complex systems.

The average entanglement asymmetry within a subsystem A, denoted as E[Δ $S_{A, Q_p}^{n}$ ], was analyzed using an ensemble of random matrix product states, with the bond dimension linked to an effective time parameter. “This asymmetry allows us to determine the maximal symmetry breaking for fragmented symmetries and to compare it with that of conventional symmetries,” the authors state in their report.

U(1) Charge Asymmetry Bounds in Conventional Systems

This fragmentation manifests as an exponential increase in disconnected sectors within the quantum system, altering the expected scaling of asymmetry. Researchers found that while conventional symmetries typically exhibit logarithmic growth in asymmetry, fragmented systems can display extensive scaling, a key distinction for understanding the nature of fragmentation itself. This approach allows for the analysis of systems where the number of disconnected sectors grows exponentially with size, a stark contrast to the polynomial growth seen in systems with conventional symmetries.

The work demonstrates that inhomogeneous U(1) charges, such as dipole and higher multipole moments, provide a natural setting for faster asymmetry growth due to the larger number of charge sectors they create. This framework extends beyond fragmented systems, encompassing standard U(1) symmetry groups and their extensions.

Commutant Algebra Formalism Generalizes Asymmetry to Fragmentation

This distinction allows for a differentiation between “classical” and “quantum” fragmentation, revealing fundamental differences in how symmetry is broken. Researchers have established a framework to quantify this behavior, deriving general upper bounds on asymmetry and identifying specific states that reach those limits. The team’s approach uses a commutant algebra formalism, a mathematical tool to generalize entanglement asymmetry to account for the unique dynamics of fragmented systems.

The framework defines asymmetry as the difference between the entropy of a symmetrized state and the original state, a measure applicable to any integer values. “This is a generalization of the entanglement asymmetry for U(1) groups,” the researchers write, noting the framework shares similar properties with existing measures.

The work demonstrates that the number of distinct sectors grows polynomially with system size for conventional symmetries, resulting in a logarithmic scaling of asymmetry. However, fragmented systems can exhibit a significantly larger number of sectors, allowing entanglement asymmetry to grow extensively with system size, reflecting the exponentially large number of dynamically disconnected sectors.

Random Matrix Product States Model Asymmetry Dynamics

Multipole charges, arising in systems experiencing Hilbert-space fragmentation, allow for asymmetry to scale extensively, a departure from the logarithmic growth observed in conventional symmetries. The ability to distinguish these fragmentation types stems from the way fragmented systems distribute weight across numerous charge sectors, amplifying entanglement asymmetry. Analysis of superpositions of frozen configurations, random matrix product states, and Haar-random states confirms this behavior, providing a broad validation of the findings.

The team employed random matrix product states (MPS) as a framework for analytical exploration, noting that increasing the bond dimension of these states mirrors increasing time in local chaotic circuits. This connection allows for the qualitative reproduction of asymmetry dynamics observed in both Haar-random brickwork circuits and Floquet experiments. The average entanglement asymmetry was calculated across an ensemble of MPS, generated by applying Haar random matrices to qubits initialized in a product state.

Results indicate that while asymmetry typically scales logarithmically with system size, the prefactor is larger in these inhomogeneous cases. “We compute both the maximal asymmetry one can achieve in this case, but also the typical that one would get,” the researchers write, highlighting the comprehensive nature of their analysis. States breaking these symmetries distribute weight across more charge sectors, further enhancing the entanglement asymmetry. The work identifies states that saturate the derived upper bounds, providing concrete examples of maximal symmetry breaking.

Extensive Asymmetry Distinguishes Classical from Quantum Fragmentation

This approach allows researchers to quantify symmetry breaking in a way that highlights the unique properties of these fragmented systems. A key finding is that the asymmetry provides a lower bound on the variance of a charge operator within a given state. The work suggests that understanding this asymmetry is important for characterizing the behavior of quantum systems in fragmented regimes and distinguishing between fundamentally different types of fragmentation.

Asymmetry Correlates with Fragmentation and Sector Count

The degree to which symmetry breaks down in quantum systems correlates directly with the extent of fragmentation within those systems, a relationship recently detailed in a theoretical analysis. While asymmetry, a measure of symmetry breaking, typically scales logarithmically with system size in conventional scenarios, it can grow extensively in systems exhibiting Hilbert-space fragmentation, where dynamics occur across an exponentially increasing number of disconnected sectors. Researchers derive general upper bounds for the asymmetry and identify states that saturate them.

Entanglement Asymmetry Links to Quantum Fisher Information (QFI)

Entanglement asymmetry, a measure of symmetry breaking, directly correlates with a system’s potential sensitivity to transformations, as larger asymmetry implies a larger quantum Fisher information (QFI). This connection positions fragmented quantum systems, those exhibiting exponentially many disconnected sectors, as potential resources for quantum technologies demanding enhanced sensitivity, such as quantum sensing.

By identifying the bond dimension with an effective time parameter, they qualitatively reproduce recent results on asymmetry dynamics in random, suggesting a universal behavior for the asymmetry of U(1) charges in local ergodic systems. This framework enables the identification of states that not only maximize asymmetry but also serve as promising resources for advanced quantum technologies.

👉 More information
🗞 Enhancing entanglement asymmetry in fragmented quantum systems
✍️ Lorenzo Gotta, Filiberto Ares and Sara Murciano
🧠 DOI: http://link.aps.org/doi/10.1103/lppm-48p3

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