Theorem: General mixed-target interval

If $d_{\diamond }({\Lambda }_{\star },{\Lambda }_0)\le b$ on the good calibration and transfer event, set

Then

The result applies to subsystem and algebra targets (eq:subchoi)–(eq:algchoi).

Proof. For independent outcomes in $[-1,1]$, Hoeffding's inequality Hoeffding gives

The union bound and calibration imply, simultaneously,

Orthonormality gives the squared Hilbert–Schmidt norm as the sum of squared coefficient errors. The inequality ${\Vert X\Vert }_1\le \sqrt{D}{\Vert X\Vert }_2$ therefore gives ${\Vert \hat{J}-J({\Lambda }_0)\Vert }_1\le r$.

The trace-norm triangle inequality bounds the true Choi discrepancy between $\max \{0,a-r\}$ and $a+r$. Lemma lem:choi and the transfer triangle give (eq:generalinterval). Union with the calibration/transfer failure event proves the claim.