Conditional Information Flow and Neural Representability of One-Dimensional Rotated Cluster States
Abstract
I study how measurement outcomes can remain conditionally dependent over long distances while the underlying state carries only a small internal memory, using one-dimensional rotated cluster states, whose exact matrix product state (MPS) bond dimension is D=2 at every rotation angle. Using exact MPS contraction, I compute the conditional mutual information (CMI) of two readout protocols: the entanglement-swapping endpoint CMI decreases from one bit at θ=0 to zero at θ=π/2, whereas a stationary bulk readout of the locally equivalent conventional cluster state vanishes at both limits. As an exact control, the spin-1 AKLT chain obeys Im=(4/9) 3−m, so its CMI length equals its two-point correlation length, for an identifiable but non-generic reason. I then compare learners on the same N=64, θ=0.35 target. A positive autoregressive GRU learns the computational-basis Born distribution from a few thousand samples, but an exactly computable Bell-basis readout shows that this distribution does not fix the phases. A complex RBM reaches the N=8 ground state under an exact objective, while a short generic recurrent variational Monte Carlo pilot is still improving when its runs end. As a positive control, an MPS-induced recurrent model whose hidden state is the bond vector reaches the ground state to near machine precision at χ=2, whereas a χ=1 control fails. These results show that long CMI does not require a large bond dimension, and that building the small virtual memory into the architecture greatly eases optimization.
Related
- Renormalization group and long-range conditional mutual information in hierarchical models
- Log-Euclidean Rényi Conditional Mutual Information
- Bosonic Quantum Conditional Mutual Information and Bose-Einstein Condensation
- Estimating Spectrum-Free Information Bounds for Generative Surrogates Beyond Conditional Collapse
- Universal Scaling of Conditional Mutual Information in Classical Stochastic Dynamics
BibTeX
@misc{xu2026conditional,
title = {{Conditional Information Flow and Neural Representability of One-Dimensional Rotated Cluster States}},
author = {Ziqi Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39392},
eprint = {2610.39392},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39392},
}
RIS
TY - UNPB
TI - Conditional Information Flow and Neural Representability of One-Dimensional Rotated Cluster States
AU - Ziqi Xu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39392
UR - https://arxiv.org/abs/2610.39392
AB - I study how measurement outcomes can remain conditionally dependent over long distances while the underlying state carries only a small internal memory, using one-dimensional rotated cluster states, whose exact matrix product state (MPS) bond dimension is D=2 at every rotation angle. Using exact MPS contraction, I compute the conditional mutual information (CMI) of two readout protocols: the entanglement-swapping endpoint CMI decreases from one bit at θ=0 to zero at θ=π/2, whereas a stationary bulk readout of the locally equivalent conventional cluster state vanishes at both limits. As an exact control, the spin-1 AKLT chain obeys I_m=(4/9) 3^−m, so its CMI length equals its two-point correlation length, for an identifiable but non-generic reason. I then compare learners on the same N=64, θ=0.35 target. A positive autoregressive GRU learns the computational-basis Born distribution from a few thousand samples, but an exactly computable Bell-basis readout shows that this distribution does not fix the phases. A complex RBM reaches the N=8 ground state under an exact objective, while a short generic recurrent variational Monte Carlo pilot is still improving when its runs end. As a positive control, an MPS-induced recurrent model whose hidden state is the bond vector reaches the ground state to near machine precision at χ=2, whereas a χ=1 control fails. These results show that long CMI does not require a large bond dimension, and that building the small virtual memory into the architecture greatly eases optimization.
ER -