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.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.

quant-pharXiv:2608.22043Aug 25, 20260 citations

The real symplectic groups in quantum mechanics and optics

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Symplectic quantum mechanics

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HAMILTONIAN METHODS AND QUANTUM MECHANICS

P A M Dirac

Jan 1, 19649 citations

Maximal accuracy and minimal disturbance in the Arthurs-Kelly simultaneous measurement process

D. M. Appleby

The accuracy of the Arthurs-Kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by Braginsky and Khalili in the context of measurements of a single quantum observable. A distinction is made between the errors of retrodiction and prediction. It is shown that the distribution of measured values coincides with the initial-state Husimi function when the retrodictive accuracy is maximized (this being the result first obtained by Busch, using somewhat different methods). It is further shown that the distribution of measured values is related to the final-state anti-Husimi function (the P representation of quantum optics) when the predictive accuracy is maximized. The disturbance of the system by the measurement is also discussed. A class of minimally disturbing measurements is characterized. It is shown that the distribution of measured values then coincides with one of the smoothed Wigner functions described by Cartwright.The accuracy of the Arthurs-Kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by Braginsky and Khalili in the context of measurements of a single quantum observable. A distinction is made between the errors of retrodiction and prediction. It is shown that the distribution of measured values coincides with the initial-state Husimi function when the retrodictive accuracy is maximized (this being the result first obtained by Busch, using somewhat different methods). It is further shown that the distribution of measured values is related to the final-state anti-Husimi function (the P representation of quantum optics) when the predictive accuracy is maximized. The disturbance of the system by the measurement is also discussed. A class of minimally disturbing measurements is characterized. It is shown that the distribution of measured values then coincides with one of the smoothed Wigner functions described by Cartwright.

arXiv:quant-ph/9803047Journal of Physics A Mathematical and GeneralJul 31, 199826 citations

\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.