Conditional Gaussian filtering by Arthurs--Kelly readout in a three-mode cluster wire
Measurement readout programs the Gaussian operation implemented by a continuous-variable cluster wire. For the standard three-node wire, momentum homodyne readout gives an additive parity channel with finite-squeezing noise. We classify the pattern-resolved input--output covariance transformations generated by calibrated uncorrelated Arthurs--Kelly readout on one or both consumed nodes. Any finite Arthurs--Kelly position record produces a conditional Gaussian filter whose gain and residual covariance depend on the input covariance entries, rather than an input-independent additive Gaussian channel. Writing $\mathrm{A}$ for Arthurs--Kelly readout and $\mathrm{H}$ for homodyne readout, the readout pattern selects the filtered sector, with $\mathrm{AH}$ selecting position, $\mathrm{HA}$ selecting momentum, and $\mathrm{AA}$ retaining both. The hierarchy $0 < τ_{\mathrm{AA}} < \min \left\{ τ_{\mathrm{AH}}, τ_{\mathrm{HA}} \right\} < 1 = τ_{\mathrm{HH}}$ shows that finite Arthurs--Kelly position records contract the gain-transferred covariance area. Our results identify calibrated uncorrelated Arthurs--Kelly readout as a measurement-level method for covariance-sensitive Gaussian filtering inside a fixed cluster graph.