Symmetry in quantum systems tied to hidden information

Yu-Jie Liu of Technical University of Munich and colleagues have linked the emergence of 1-form symmetries in quantum systems to the preservation of information about a system’s underlying structure. The work demonstrates that these symmetries, which act on sub-dimensional spatial manifolds, can appear even when explicitly broken at the microscopic level, presenting a challenge to traditional symmetry analysis.

Researchers analytically determined the conditions for 1-form symmetry emergence in product states on one- and two-dimensional lattices, and developed a method to detect these symmetries using a quantum error correction decoder. The findings suggest a connection between symmetry and information retention, revealing an information-theoretic transition detectable through long-range entanglement.

Emergent 1-Form Symmetries and Information Preservation

The existence of an emergent 1-form symmetry can be determined by examining the recoverable information about a microscopic, or “bare”, symmetry within a quantum system. This work proposes that these symmetries, which manifest at low energies despite being broken at the fundamental level, are fundamentally linked to information preservation, not just the system’s inherent properties.

Researchers established a criterion for identifying a 1-form symmetry in quantum states on lattice systems, even when the symmetry is emergent, by determining when information about a bare symmetry remains recoverable. This framework allows prediction of symmetry emergence independent of microscopic details, offering a new approach to characterizing these elusive phenomena.

The team demonstrated the utility of this approach by employing the Minimum Weight Perfect Matching (MWPM) algorithm, though any quantum error correction decoder could be used, to detect emergent 1-form symmetries and topological phase transitions in perturbed two-dimensional toric code models. Extending this detection protocol to planar geometries, the analysis reveals connections between emergent 1-form symmetries and quantum error correction, providing a physical basis for using QEC-based methods to identify quantum phases.

Further investigation into Z_2 lattice gauge theories suggests that particle condensation, including the Higgs and confinement regimes, is sharply defined by these information-theoretic transitions. “The connection between emergent 1-form symmetries and QEC provides a physical foundation for employing QEC-based approaches to detect quantum phases,” the paper states, highlighting the practical implications of this theoretical framework. In spin systems, these 1-form symmetry charges are interpretable as domain walls created by the endpoints of X-string operators, revealing a deeper connection between symmetry and topological order.

Higher-Form Symmetries: Robustness Beyond Microscopic Origins

This emergent behavior isn’t merely a quirk of low-energy physics, but is fundamentally linked to the preservation of information about the original, “bare” symmetry present in the system’s microscopic details. This connection offers a new lens through which to understand the emergence of order in complex quantum systems. Beyond these simplified cases, the team developed a method to efficiently detect 1-form symmetries using a global quantum error correction (QEC) decoder, allowing for analysis in analytically intractable regimes.

The research indicates that the boundaries between Higgs and confining regimes are sharply defined from an information-theoretic perspective. While 1-form symmetries are generally unstable at finite temperature, the study notes that 2-form symmetries in systems with more than one spatial dimension can remain thermally stable if creating their charges requires a finite energy cost proportional to the perimeter of the charge-creating operator. This suggests a deeper connection between symmetry, dimensionality and the energetic cost of creating topological defects, offering new avenues for controlling and characterizing quantum phases of matter.

Form Symmetries Defined by Decodability of String Operators

The decodability of string operators directly reveals the presence of emergent 1-form symmetries within quantum systems, according to new work linking symmetry to information retention. This connection establishes that a system’s ability to preserve information about a fundamental, or “bare,” 1-form symmetry dictates whether a larger, emergent symmetry will appear, even if the microscopic details obscure it.

The research demonstrates that losing this emergent symmetry coincides with an information-theoretic transition, detectable through analysis of long-range entanglement within the system’s post-measurement states. This framework decomposes the quantum state, viewing it as a 1-form symmetric state acted upon by charged string operators, allowing for a geometric interpretation of the symmetry.

The analysis, detailed in accompanying appendices, reveals that a particular choice of these charged string operators for each charge configuration is important for deforming the 1-form symmetry and ensuring its effective commutation with the charged strings. A key condition for the symmetry’s persistence in the thermodynamic limit is that the maximum overlap between the system’s state and specific symmetry-related operators must equal one. The loss of the 1-form symmetry, however, signals a transition where this condition fails, indicating a fundamental shift in the system’s properties.

The presence of the symmetry is a bulk property, independent of boundary conditions; even with open boundaries where non-contractible operators are undefined, the disappearance of the symmetry implies that the same charge configuration can be generated by drastically different string operators. This finding provides a new way to characterize topological order and offers a direct framework for its detection in increasingly complex quantum systems.

Information-Theoretic Transition as a Symmetry Loss Indicator

The emergence of 1-form symmetries, symmetries acting on sub-dimensional spatial manifolds, does not always signal a traditional quantum phase transition with observable singularities; instead, the loss of such a symmetry often represents an information-theoretic transition detectable through changes in long-range entanglement. Researchers found that the transition point, where the symmetry disappears, manifests as a shift in the entanglement characteristics of post-measurement states, moving from short-range to long-range entanglement.

This information-theoretic nature of 1-form symmetries is revealed by analyzing the post-measurement states resulting from measuring the symmetry’s charges, specifically the configuration of these charges denoted as 𝒎 = 𝑚𝑝 = ±1. As a practical application of their framework, a topological quantum phase transition characterized by the spontaneous breaking of the 1-form symmetry can be accurately determined by a disorder parameter.

Detecting 1-Form Symmetries with Quantum Error Correction

This approach extends to a broad class of two-dimensional abelian topological states, offering a pathway to characterize these symmetries in complex quantum systems. The work specifically examined the toric code on a square lattice subjected to an external field, a system where the presence or absence of symmetry can be linked to the efficiency of error correction. The detection method relies on monitoring changes in long-range entanglement within post-measurement states, a characteristic revealed by the loss of a 1-form symmetry.

While quantifying this entanglement directly presents challenges for both experiments and simulations, the researchers found a connection to QEC that allows for efficient detection. By applying the Minimum Weight Perfect Matching (MWPM) decoder, they could approximate the location of the information-theoretic transition marking the symmetry’s disappearance.

This parallels the behavior of the toric code as a transverse field increases, ultimately leading to the loss of its emergent Z symmetry at the Ising critical point. “In this work, we provide an information-theoretic perspective and a precise criterion for the emergence of 1-form symmetries, emphasizing its close connection to the decoding problem in quantum error correction,” the paper states. The researchers suggest this approach offers a new lens for understanding topological order and provides a means to identify these transitions with greater precision.

Z_2 Gauge Theory and Emergent 1-Form Symmetry Framework

This connection extends beyond simple observation; the loss of this emergent symmetry is fundamentally an information-theoretic transition, revealed by examining long-range entanglement. The framework applies to systems beyond the strictly topological, with implications for understanding phase transitions in one dimension. They further argue that their proposed theory for emergent 1-form symmetries offers new perspectives on particle condensation and suggests sharp information-theoretic phase boundaries between Higgs and confining regimes in the $Z_2$ lattice gauge theory.

Disorder Parameter Identifies Spontaneous Symmetry Breaking

Researchers have established a method for pinpointing topological quantum phase transitions by identifying a disorder parameter linked to the spontaneous breaking of 1-form symmetries. This approach moves beyond simply detecting the presence of these symmetries, instead focusing on their loss as a signal for a change in quantum state, using global quantum error correction (QEC) decoders to efficiently analyze complex systems. The work demonstrates that once a 1-form symmetry is detected, the transition to a broken state can be accurately determined, offering a new tool for characterizing these phases of matter.

The protocol detailed in the work uses QEC decoders to detect 1-form symmetries within subsystems of a quantum state, a significant departure from standard QEC settings that typically require specific boundary conditions. Applying QEC to restore a bare 1-form symmetry allows for efficient measurement of disorder parameters, which then reveal the topological quantum phase transitions characterized by symmetry breaking.

This is achieved by examining the expectation values of bare 1-form symmetry string operators, which function as a disorder parameter for the corresponding bare symmetry. A nonzero value indicates a 1-form symmetric phase, while a vanishing value signals spontaneous symmetry breaking. The research addresses a key question: can a low-energy theory exhibit a common emergent higher-form symmetry even when the microscopic level displays explicit breaking of the bare symmetry?

The existence of such an emergent symmetry constrains the low-energy effective Hamiltonian and guides the definition of parameters used to characterize distinct phases. This is not merely about observing symmetry, but understanding its role in defining order and disorder within a quantum system. The researchers write, highlighting the functional equivalence between inherent and emergent symmetry.

Form Symmetries Link Particle Condensation and Confinement

The emergence of 1-form symmetries offers new ways to understand the boundary between Higgs and confining regimes within Z₂ lattice gauge theory, according to new analysis of information retention within quantum systems. This work establishes a connection between the preservation of a microscopic 1-form symmetry and the appearance of its larger-scale counterpart, suggesting that symmetry isn’t solely a property of the system, but is linked to its origins.

Researchers found this connection allows for a sharper definition of phase transitions than previously possible, even when conventional diagnostic criteria fail. It provides a means to accurately pinpoint the moment of symmetry breaking using a disorder parameter.

Long-Range Entanglement Reveals Information-Theoretic Transitions

This challenges conventional symmetry analysis by suggesting that these symmetries aren’t simply inherent properties of a quantum system, but are tied to retained information about its origins. This transition manifests as an abrupt shift in the entanglement structure of states following measurements designed to project the system into specific 1-form symmetry charge configurations. A state initially exhibiting only short-range entanglement can become long-range entangled after these measurements, establishing a direct connection between measurement-induced entanglement and higher-form symmetries.

The work reveals that this information-theoretic transition is not necessarily a quantum phase transition with singularities in physical observables, but rather a singularity in non-linear functions of the system’s states. However, quantifying this entanglement directly can be computationally difficult, both in experiments and simulations.

To overcome this, they formulated a theory linking emergent 1-form symmetries to quantum error correction (QEC), offering a means to efficiently detect these symmetries and pinpoint the location of the information-theoretic transition. “The transition of a 1-form symmetry can generally be detected by monitoring the change in long-range entanglement of the post-measurement states,” the paper states.

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