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 $\mathrm{A}\mathrm{H}$ selecting position, $\mathrm{H}\mathrm{A}$ selecting momentum, and $\mathrm{A}\mathrm{A}$ retaining both. The hierarchy $0<{\tau }_{\mathrm{A}\mathrm{A}}<\min \{{\tau }_{\mathrm{A}\mathrm{H}},{\tau }_{\mathrm{H}\mathrm{A}}\}<1={\tau }_{\mathrm{H}\mathrm{H}}$ 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.