[Paper Review] Hilbert Space Fragmentation in Open Quantum Systems

Yahui Li, Pablo Sala+1 more|arXiv (Cornell University)|May 5, 2023

Quantum Information and Cryptography95 references4 citations

TL;DR

This paper investigates Hilbert space fragmentation (HSF) in open quantum systems, showing that quantum HSF—fragmentation in an entangled basis—can stabilize highly entangled steady states when coupled to a tailored dissipative bath. It demonstrates that dephasing noise reduces quantum HSF to classical fragmentation, leading to separable steady states with classical correlations, while a fine-tuned dissipative coupling preserves quantum HSF, yielding a highly entangled steady state with strong memory of the initial state, as quantified by logarithmic negativity.

ABSTRACT

We investigate the phenomenon of Hilbert space fragmentation (HSF) in open quantum systems and find that it can stabilize highly entangled steady states. For concreteness, we consider the Temperley-Lieb model, which exhibits quantum HSF in an entangled basis, and investigate the Lindblad dynamics under two different couplings. First, we couple the system to a dephasing bath that reduces quantum fragmentation to a classical one with the resulting stationary state being separable. We observe that despite vanishing quantum correlations, classical correlations develop due to fluctuations of the remaining conserved quantities, which we show can be captured by a classical stochastic circuit evolution. Second, we use a coupling that preserves the quantum fragmentation structure. We derive a general expression for the steady state, which has a strong coherent memory of the initial state due to the extensive number of non-commuting conserved quantities. We show that it is highly entangled as quantified by the logarithmic negativity.

Motivation & Objective

• To understand how Hilbert space fragmentation (HSF) in open quantum systems is affected by coupling to a dissipative bath.
• To investigate whether quantum HSF—fragmentation in an entangled basis—can survive in the presence of dissipation.
• To determine if a dissipative environment can be engineered to preserve or enhance quantum correlations in fragmented systems.
• To develop a framework for identifying quantum HSF via the structure of the steady state under specific Lindblad dynamics.

Proposed method

• Uses the commutant and bond algebra formalism to characterize conserved quantities in quantum HSF systems.
• Analyzes Lindblad dynamics under two distinct couplings: dephasing (local $L_j = S_j^z$) and two-site dissipative couplings.
• Derives an effective Markov generator $\mathbb{W}_{\text{eff}} = \sum_i \frac{J_i^2}{\gamma} M_{i,i+1}$ that maps the dynamics to a classical stochastic process under dephasing.
• Maps the quantum dynamics under dephasing to a classical random circuit with two-site gates permuting configurations $\bm{\sigma} \in \{+, 0, -\}^N$, enabling large-scale simulations.
• Constructs the steady state as a projection onto the Krylov subspaces of the Temperley-Lieb model, using vectorization to relate mixed states to pure superpositions.
• Quantifies entanglement via number entropy $S_{\text{num}}$ and symmetry-resolved entanglement $S_{\text{res}}$, comparing their time evolution and saturation.

Experimental results

Research questions

• RQ1 Can quantum Hilbert space fragmentation survive in open quantum systems when coupled to a dissipative bath?
• RQ2 How does dephasing noise affect the quantum nature of HSF, and can it be mapped to classical dynamics?
• RQ3 What type of dissipative coupling preserves the quantum fragmentation structure and leads to non-trivial steady states?
• RQ4 Can the steady state of a quantum HSF system retain memory of the initial state, and how is this quantified?
• RQ5 Is there a protocol to experimentally distinguish quantum HSF from classical HSF using dissipative engineering?

Key findings

• Dephasing noise reduces quantum HSF to classical HSF, mapping the effective dynamics to a classical Markov process with the same block-diagonal structure as the pair-flip model.
• Despite vanishing quantum correlations, classical correlations emerge due to fluctuations of conserved quantities, captured by a classical stochastic circuit evolution.
• The steady state under dephasing is separable and corresponds to a uniform superposition over fully-paired configurations, with number entanglement scaling as $S \sim O(\sqrt{N})$ for large $N$.
• Symmetry-resolved entanglement $S_{\text{res}}$ saturates to zero in the steady state, confirming the absence of quantum correlations.
• A fine-tuned two-site dissipative coupling preserves the quantum fragmentation structure, leading to a steady state with strong coherent memory of the initial state.
• The steady state under this coupling is highly entangled, as confirmed by a non-zero logarithmic negativity, demonstrating that quantum HSF can stabilize long-lived entanglement in open systems.