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

For two rebits, real product effects span a nine-dimensional hyperplane in the ten-

dimensional symmetric state carrier, leaving one product-invisible coordinate s(ρ) = tr[(J⊗J)ρ],

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

π

4 J⊗X

followed by one local population detector gives s= 1−2p∗. A single CNOT instead maps J⊗J to

X⊗Z, so one local X/Z parity statistic gives s= 2p(+)

XZ−1. The connected pulse compiles exactly

as CNOT–Ry(π/2)–CNOT; the two protocols are therefore Pareto alternatives rather than a unique

minimum. For continuous control, one generator K ∈so(4) gives full state observability exactly when

[K,J⊗J] = 0, 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.