No Definite Dark Direction for General Instruments

Abstract

A general no-click instrument has no definite dark direction exactly when its limit effect vanishes, exactly when the survival defect after d steps is positive definite, and exactly when Tr(rho F) = 0 for every density matrix rho.

Theorem 1.1 (Four equivalent forms of the absence of dark directions).

$$\begin{gathered} \sum _{a\in \alpha }{Q}_a^{\ast }{Q}_a+\sum _{i\in \iota }{L}_i^{\ast }{L}_i=I\wedge S_N\to F\Rightarrow \\ {D}_d=0⟺F=0⟺I-S_d>0⟺ \\ \forall \rho \ge 0\text{ with }\mathrm{Tr}\rho =1,\mathrm{Tr}(\rho F)=0. \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Measurement/GeneralInstrumentNoDarkDirection.no_dark_direction_tfae (✓ 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, D_d the stable dark layer and F the limit of S_N. The four conditions are: D_d is zero; F is zero; I - S_d is positive definite; Tr(rho F) = 0 for every density matrix rho.

Every vector of D_d satisfies S_N v = v from step d on, so F v = v; hence F = 0 forces D_d = 0. Conversely, if F is not zero, the Rayleigh quotient of F attains a largest value lambda > 0 on the unit sphere, so lambda I - F is positive semidefinite. On its kernel M, which contains the maximizer, the fixed-point equation gives <v, F v> = sum over a of <Q_a v, F Q_a v>, which is at most lambda times the sum of |Q_a v|^2, that is lambda (|v|^2 - sum of |L_i v|^2). Equality forces every L_i v to vanish and every Q_a v to lie in M, so M is a nonzero subspace annihilated by the click operators and invariant under the no-click operators, and it lies in D_d.

Since D_d is the kernel of the positive semidefinite operator I - S_d, it is zero exactly when I - S_d is positive definite. Testing F against the rank-one densities v v^* / |v|^2 shows that Tr(rho F) = 0 for all densities exactly when F = 0.

References

• Truth anchor: D5/S3/Quantum/Measurement/GeneralInstrumentNoDarkDirection.no_dark_direction_tfae
• Dependency: D5/S3/Quantum/Measurement/GeneralInstrumentSurvivalLimit