Real-Gate Readout of the Two-Rebit Tomographic Defect: Connected Pulses, CNOT Parity, and Copy-Depth Resources

Abstract

For two rebits, real product effects span a nine-dimensional hyperplane in the ten-dimensional symmetric state carrier, leaving one product-invisible coordinate $s(\rho)=\operatorname{tr}[(J\otimes J)\rho]$, whose magnitude is the established rebit concurrence. We formulate access to this coordinate as a copy-depth, control-topology, and orientation-resource problem. With an i.i.d. source but coherent copy depth one per round, arbitrary separated real-local processing with product ancillas, classical communication, postselection, and real terminal effects has no direct linear sensitivity to $s$; this gives nonidentifiability for unrestricted mixed states, though not under every state promise. After reunion, two exact direct meters are available. A connected pulse

$U_*=\exp\!\left(\frac{\pi}{4}J\otimes X\right)$

followed by one local population detector gives $s=1-2p_*$. A single CNOT instead maps $J\otimes J$ to $X\otimes Z$, so one local $X/Z$ parity statistic gives $s=2p_{XZ}^{(+)}-1$. The connected pulse compiles exactly as CNOT–$R_y(\pi/2)$–CNOT; the two protocols are therefore Pareto alternatives rather than a unique minimum. For continuous control, one generator $K\in\mathfrak{so}(4)$ gives full state observability exactly when $[K,J\otimes J]\neq0$, assuming access to the complete product-effect span. Two identical copies recover $|s|$ by separated local collective measurement, while any finite number of identical copies remains orientation-blind under separated real-local collective processing with no oriented shared resource; an oriented shared rebit reference supplies the missing sign standard. Exact detector-response, pulse-angle, and general implemented-gate calibration formulas are derived. These results concern controlled state readout within real quantum theory; they neither restore local tomography nor imply process tomography, empirical inequivalence with complex quantum theory, or a reconstruction of the complex formalism.

The resource hierarchy

The joint gate does not create the coordinate. It transduces an already existing global quantity into a population or parity signal — observability is a property of a state together with an interface, a control topology, a copy depth, and a reference structure.

Where it sits in the release

CFQF-Q1 takes its quantum-reconstruction context from CFQF-Q4 and supplies the two-rebit observability context to CFQF-Q3, whose countermodel uses the same hidden coordinate.

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