Stabilizer disentangling of conformal field theories

TL;DR

The paper introduces Stabilizer Disentangling via local Clifford gates (CAMPS) as a practical tool to cool entanglement in 1D lattice models hosting conformal field theories. It reveals two regimes of entanglement cooling—constant-gain and log-gain—governed by the sign of the mutual Stabilizer Renyi entropy and the amount of global magic, with the latter correlating with a reduced effective central charge. Through analytical treatment of cluster Ising models and extensive DMRG/MPS computations on XXZ and tricritical Ising models, the authors show that the disentangling efficiency hinges on nonlocal stabilizer information (negative mSRE) and that LCD states exhibit pronounced log-scaling of SMEE with system size. The work demonstrates that CAMPS can map certain CFTs to lower-entanglement descriptions, providing a path to more efficient variational representations (CAMPS) and highlighting rich connections between entanglement, magic, and stabilizer space. It also outlines avenues for extending CAMPS to other tensor-network formalisms and for deeper field-theoretic understanding of mSRE in continuum limits.

Abstract

Understanding how entanglement can be reduced through simple operations is crucial for both classical and quantum algorithms. We investigate the entanglement properties of lattice models hosting conformal field theories cooled via local Clifford operations, a procedure we refer to as stabilizer disentangling. We uncover two distinct regimes: a constant gain regime, where disentangling is volume-independent, and a log-gain regime, where disentanglement increases with volume, characterized by a reduced effective central charge. In both cases, disentangling efficiency correlates with the target state magic, with larger magic leading to more effective cooling. The dichotomy between the two cases stems from mutual stabilizer Renyi entropy, which influences the entanglement cooling process. We provide an analytical understanding of such effect in the context of cluster Ising models, that feature disentangling global Clifford operations. Our findings indicate that matrix product states possess subclasses based on the relationship between entanglement and magic, and clarifying the potential of new classes of variational states embedding Clifford dynamics within matrix product states.

Figures (11)

• Figure 1: a-b: Clifford augmented MPS. (a): schematics of a single sweep of local stabilizer disentangling protocol applied onto a matrix product state (MPS); (b): Hilbert space structure of many-body steates for a spin system. The manifold comprising MPS can be further split with respect to how much they can be disentangled by local Clifford: local-Clifford disentanglable (LCD; disentangling improves with size), non local-Clifford disentanglable (nLCD; disentangling does not improve with size), and fully disentanglable (fLCD). In the text, we elaborate examples of all three cases: tricritical Ising point, XXZ critical, and cluster Ising model. c-e: Illustration of the Stabilizer disentangling protocol (see Sec.\ref{['sec:dis_protocol']}). (c): Obtain the groundstate with DMRG and bring it to a right-normalized form; (d) Apply the gates in $\tilde{\mathcal{C}}_2$ (see \ref{['par:optimsearch']}) to the first bond, selecting the gate that minimizes the entanglement entropy on that bond; (e) Apply the chosen gate to the MPS, perform a Singular Value Decomposition, update the first two tensors in the MPS, and repeat step (d) on the next bond.
• Figure 2: Entropy gain $\Delta$ compared with magic observables in the ground states of the XXZ chain across various $J_z$ values within the critical phase, for a system size $L = 32$. The data are presented on a relative scale, with each panel scaled as follows: (a): $\Delta_{\text{max}} = 0.67$, $m_2^{\text{max}} = 0.27$, $m_2^{\chi = 2, \text{max}} = 0.30$; (b): $\Delta_{\text{max}} = 0.67$, $|L_{AB}^{\text{max}}| = 0.27$. (a) Comparison with the full state SRE density $m_2$ and the local magic $m_2^{\chi = 2}$. The computation of $m_2$ and $m_2^{\chi = 2}$ is performed via perfect Pauli sampling with $N_S = 10^3$ samples. (b) Comparison with mSRE $L_{AB}$ where $A$, $B$ are two partitions located at the boundary of the chain, each of length $\ell_A = \ell_B = L/4$. The computation of $L_{AB}$ is performed via Pauli-Markov sampling with $N_S = 10^4$ samples.
• Figure 3: Stabilizer disentangling within the cricical phase in XXZ model. For (a), (c) data are shown for a fixes length of the system $L = 512$, varying the size of the partitions $A$, $B$ ($\ell_A = l$, $\ell_B = L - l$) from $l = 8$ to $l = 256$. For (b), (d) data are presented for varying total system lengths $L$ with fixed, equally sized partitions $A$ and $B$ (where $\ell_A=\ell_B = L/2$). (a), (b): Scaling of Entanglement Entropy (EE) and Stabilizer-Minimized Entanglement Entropy (SMEE). Dashed lines represent the fitted curves, with both curves sharing the same slope. (c), (d) Scaling of entropy gain $\Delta$. It exhibits sublogarithmic scaling; however, due to the very small scale on the y-axis, it can effectively be considered saturated.
• Figure 4: Entropy gain $\Delta$ compared with magic observables in the ground states of the Tricritical Ising model across various $J_z$ values, for two system sizes: $L = 64$ ((a) and (b)) and $L = 128$ ((c) and (d)). The data are presented on a relative scale, with each panel scaled as follows: (a): $\Delta_{\text{max}} = 0.34$, $m_2^{\text{max}} = 0.29$, $m_2^{\chi = 2, \text{max}} = 0.29$; (b): $\Delta_{\text{max}} = 0.34$, $|L_{AB}^{\text{max}}| = 0.003$; (c): $\Delta_{\text{max}} = 0.34$, $m_2^{\text{max}} = 0.31$, $m_2^{\chi = 2, \text{max}} = 0.31$; (d): $\Delta_{\text{max}} = 0.34$, $|L_{AB}^{\text{max}}| = 0.002$. (a), (c): Comparison with the full state SRE density $m_2$ and the local magic $m_2^{\chi = 2}$. The computation of $m_2$ and $m_2^{\chi = 2}$ is performed via perfect Pauli sampling with $N_S = 10^3$ samples. (b), (d): Comparison with mSRE $L_{AB}$ where $A$, $B$ are two partitions located at the boundary of the chain, each of length $\ell_A = \ell_B = L/4$. The computation of $L_{AB}$ is performed via Pauli-Markov sampling with $N_S = 10^4$ samples.
• Figure 5: Entanglement entropy (EE) and Stabilizer-minimized Entanglement Entropy (SMEE) in Tricritical Ising point. For (a), (c) data are shown for a fixed length of the system $L = 512$, varying the size of the partitions $A$, $B$ ($\ell_A = l$, $\ell_B = L - l$) from $l = 8$ to $l = 256$. For (b), (d) data are presented for varying total system lengths $L$ with fixed, equally sized partitions $A$ and $B$ (where $\ell_A=\ell_B = L/2$). (a), (b): EE and CMEE scaling with size. While both follow the expected logarithmic trend, CMEE features a distinct prefactor. Dashed lines represent the linear fits. (b), (d): The gain $\Delta$ scales logarithmically with size at the TCI point. The dashed line represent the linear fit.