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 $I_m=(4/9)\,3^{-m}$, 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.