Uniform Detection Certificate for General Instruments

Abstract

When a general no-click instrument has no definite dark direction, the survival effects decay geometrically in blocks of d rounds, and the survival probabilities of every initial state have sum at most d / g.

Theorem 1.1 (Block decay of the survival effects).

$$\begin{gathered} \sum _{a\in \alpha }{Q}_a^{\ast }{Q}_a+\sum _{i\in \iota }{L}_i^{\ast }{L}_i=I\Rightarrow \\ (D_d=0\Rightarrow \exists g>0,gI\le I-S_d)\wedge \\ \forall g>0,gI\le I-S_d\Rightarrow (\forall m,S_{md}\le (1-g{)}^{m}I)\wedge \\ \forall \rho \ge 0\text{ with }\mathrm{Tr}\rho =1,(N\mapsto \mathrm{Re}\mathrm{Tr}(\rho S_N))\text{ summable }\wedge \sum _N\mathrm{Re}\mathrm{Tr}(\rho S_N)\le \frac{d}{g}. \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Measurement/GeneralInstrumentDetectionCertificate.detection_certificate (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let alpha and iota be finite, let Q_a be the no-click and L_i the click Kraus operators on the d-dimensional space with the completeness relation, let S_N be the survival effects and D_d the stable dark layer; the order is the Loewner order.

If D_d is zero, then I - S_d is positive definite, so its spectrum is positive and compact, and some g > 0 satisfies g I <= I - S_d. For any such g, S_d <= (1 - g) I; the dual no-click map is positive, monotone and homogeneous, so S_{(m+1)d} = A^d(S_{md}) <= (1 - g)^m A^d(I) = (1 - g)^m S_d <= (1 - g)^{m+1} I.

For a density matrix rho, the real parts of the traces Tr(rho S_N) are nonnegative and decrease in N; the block of d consecutive terms starting at md is at most d (1 - g)^m, and the geometric sum of these block bounds is at most d / g.

References

• Truth anchor: D5/S3/Quantum/Measurement/GeneralInstrumentDetectionCertificate.detection_certificate
• Dependency: D5/S3/Quantum/Measurement/GeneralInstrumentNoDarkDirection