Quantum Fragmentation: Frozen Entangled States

A key mechanism driving quantum Hilbert space fragmentation has been revealed, a phenomenon where a quantum system breaks down into disconnected pieces unable to interact with each other. Zihan Zhou and colleagues at Princeton University show that this fragmentation arises from deficiencies in the local Hamiltonian, leading to the creation of entangled frozen states, complex entangled states that remain static even when the system evolves. The research, conducted across four models with increasing complexity, links classical fragmentation with its quantum counterpart, introducing concepts of weak and strong quantum fragmentation to categorise the behaviour of these systems. By analysing the resulting mobile quantum Krylov subspace, the team characterised the statistical properties of the energy spectrum, providing insights into the nature of quantum chaos and ergodicity within fragmented systems. Hilbert space fragmentation represents a significant departure from the conventional expectation of ergodicity in quantum systems, where one anticipates that a system will explore all accessible states given sufficient time; fragmentation creates isolated regions within this space.

Identifying Entangled Frozen States via Krylov Sector Analysis and Spectral Decomposition Spectral decomposition dissected the Hamiltonian, the mathematical description of the system’s energy, and identified instances of rank deficiency, a shortfall in the fundamental properties needed to fully define the system’s behaviour. The Hamiltonian operator, in this context, describes the total energy of the quantum system and governs its time evolution. Rank deficiency implies that the Hamiltonian does not span the entire Hilbert space, leading to the existence of zero-energy modes or ‘null directions’. These null directions are crucial as they define the space within which the entangled frozen states reside. Enabling the pinpointing of entangled frozen states, stable configurations of entangled particles that do not evolve with time, the analysis also allowed for understanding how these states compartmentalise the quantum space. The Krylov subspace, generated by applying powers of the Hamiltonian to an initial state, provides a means to explore the accessible states of the system. By analysing the dimensions and multiplicities of irreducible blocks within this Krylov subspace, the researchers could characterise the fragmentation. Investigations across four models, an asymmetric qubit projector, a $\mathbb{Z}$2-symmetric GHZ projector, a $\mathbb{Z}$3-symmetric cyclic qutrit projector, and the Temperley-Lieb model, explored how quantum systems break down into isolated parts. The asymmetric qubit projector serves as a simple example of a fragmented system, while the GHZ and cyclic qutrit projectors introduce symmetry considerations. The Temperley-Lieb model, a more complex model originating from statistical mechanics, allows for the investigation of fragmentation in systems with many interacting degrees of freedom. Analyses across these models revealed that the asymmetric and GHZ models yield precise calculations of irreducible Krylov dimensions and multiplicities, facilitating detailed characterisation of the fragmented space. The Temperley-Lieb model exhibited a growing number of irreducible blocks as system size increased, converging towards a Poisson distribution, indicating a higher degree of fragmentation and a loss of ergodicity. By focusing on the properties of these fragmented spaces, the approach bypassed complex simulations, offering a computationally efficient method for studying fragmentation. Entangled frozen states and the scaling of irreducible blocks define strong quantum fragmentation A number of irreducible blocks in strongly fragmented quantum systems grows as system size, L, approaches infinity, a threshold previously unattainable for characterising these complex behaviours. This exponential growth signifies that the number of disconnected pieces within the system increases dramatically with size, leading to a highly fragmented Hilbert space. Weakly fragmented systems, in contrast, maintain a constant number of these blocks, indicating a limited degree of fragmentation and the presence of some residual connectivity. This distinction offers a new way to categorise quantum fragmentation, moving beyond simple observation to quantitative classification. Entangled frozen states split mobile classical sectors into quantum and entangled components, fundamentally altering the system’s dynamics. Classical sectors represent regions of the Hilbert space that can be described by classical physics, while the quantum and entangled components represent deviations from this classical behaviour. This splitting effectively creates ‘islands’ of quantum coherence within a sea of classicality. Further research is needed to explore the scaling behaviour and control of these states, potentially establishing definitive links to practical applications like strong quantum error correction. In quantum error correction, the ability to isolate and protect quantum information is paramount; fragmented systems, with their inherent isolation, could potentially provide a natural framework for such protection. The origin of this fragmentation lies in rank deficiency within the local Hamiltonian, generating these embedded states. Understanding the precise relationship between the rank deficiency and the resulting fragmentation is crucial for designing systems with desired fragmentation properties. Rank deficiency explains Hilbert space fragmentation across quantum systems Quantum Hilbert space fragmentation, where complex systems divide into isolated components, has been pinpointed as originating from rank deficiency. This rank deficiency, as previously discussed, creates null directions in the Hamiltonian, allowing for the existence of entangled frozen states. However, the precise conditions determining whether fragmentation remains weak, with a few large isolated parts, or becomes strong, with exponentially growing complexity, are still unclear. Factors such as the specific form of the local Hamiltonian, the symmetry of the system, and the presence of disorder could all play a role in determining the degree of fragmentation. This ambiguity hinders efforts to predict and control fragmentation, limiting potential applications in areas like quantum error correction, where isolated components could be both a benefit and a hindrance. While isolation can protect quantum information, excessive fragmentation could render the system unusable by preventing information transfer between different components. Even without full predictive control, understanding how these fragmented states arise provides important insights for using their potential in quantum technologies, particularly in building more durable quantum computers. The ability to create and manipulate entangled frozen states could lead to new approaches to quantum computation and information processing. This work identifies how deficiencies in the interactions between components generate entangled frozen states, stable configurations of entangled particles that do not change over time. These states are ‘frozen’ in the sense that they do not evolve under the Hamiltonian dynamics, providing a degree of stability that is rare in quantum systems. Compartmentalising the system, these frozen states split conventionally understood classical sectors into quantum and entangled components, fundamentally altering its dynamics.

The team demonstrated this mechanism across four models, revealing that symmetry is not a prerequisite for fragmentation, though it can influence the resulting quantum spaces. The absence of a symmetry requirement broadens the scope of systems where fragmentation can occur, increasing the potential for practical applications. A quantum system breaks down into disconnected, non-interacting parts during the process, representing a fundamental shift in its behaviour and opening up new avenues for exploring the limits of quantum mechanics. The research demonstrated that a deficiency in the local Hamiltonian of a model is key to quantum Hilbert space fragmentation. This rank deficiency generates stable, entangled frozen states within the system, effectively splitting classical sectors into quantum and entangled components.

The team showed this fragmentation occurs in four different models, and importantly, does not require symmetry. After removing these frozen states, the remaining quantum system decomposes into irreducible blocks, exhibiting statistical properties consistent with Gaussian Orthogonal Ensemble level statistics. 👉 More information 🗞 Quantum Hilbert Space Fragmentation and Entangled Frozen States 🧠 ArXiv: https://arxiv.org/abs/2604.05218 Tags: