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Shadow Theory

Technical Companion: Detector Interfaces and Retained Outputs

Continuum validation of a phase-grid design

Jeremy RodgersSection 7 of 12Original PDF ↗

Continuum validation of a phase-grid design

Proposition 6.1 (Continuum validation of a phase-grid reader). Let ρ±(θ) be the pure states of Uz(θ)(a|0⟩± b|1⟩), where a, b ≥0, a2 + b2 = 1, s = a2 −b2, and Uz is a rotation by angle θ about the Bloch z axis. Let a fixed reader and bounded nonnegative decision loss give operators 0 ≤C± ≤CmaxI and risk R(M, θ) = 1 2 Tr[C+(M)ρ+(θ)] + 1 2 Tr[C−(M)ρ−(θ)].
Technical passage, source page 18.
Searchable transcription
Proposition 6.1 (Continuum validation of a phase-grid reader). Let ρ±(θ) be the pure states of
Uz(θ)(a|0⟩± b|1⟩), where a, b ≥0, a2 + b2 = 1, s = a2 −b2, and Uz is a rotation by angle θ about the
Bloch z axis. Let a fixed reader and bounded nonnegative decision loss give operators 0 ≤C± ≤CmaxI
and risk
R(M, θ) = 1
2 Tr[C+(M)ρ+(θ)] + 1
2 Tr[C−(M)ρ−(θ)].
For phase differences at most π,
Source page 19.
Searchable transcription
For phase differences at most π,
|R(M, θ) −R(M, θ′)| ≤Cmax p 1 −s2 sin(|θ −θ′|/2).
Source page 19.
Searchable transcription
|R(M, θ) −R(M, θ′)| ≤Cmax
p
1 −s2 sin(|θ −θ′|/2).

If a finite grid G has covering radius h ≤π in a compact phase interval Θ, and the admissible reader class is the same for both optimizations, then

VG ≤VΘ ≤VG + Cmax p 1 −s2 sin(h/2), VA := inf M sup θ∈A R(M, θ).
Source page 19.
Searchable transcription
VG ≤VΘ ≤VG + Cmax
p
1 −s2 sin(h/2),
VA := inf
M sup
θ∈A
R(M, θ).

Proof. The pure-state trace distance is exactly

D(ρ±(θ), ρ±(θ′)) = p 1 −s2 | sin((θ −θ′)/2)|.
Source page 19.
Searchable transcription
D(ρ±(θ), ρ±(θ′)) =
p
1 −s2 | sin((θ −θ′)/2)|.

For trace-zero Hermitian X and 0 ≤C ≤CmaxI, | Tr(CX)| ≤Cmax∥X∥1/2 by the positive and negative parts of X. Apply this to each state difference, average, and choose a grid point within h of each phase. Take infima over the identical admissible reader class. If the grid infimum is not attained, use an arbitrarily close reader and pass to its limiting risk bound.

This is a standard trace-distance metric-net argument newly specialized to this phase reader, not a new general minimax theorem. If constraints themselves are imposed only on grid points, they need their own modulus margin. In particular, equality of an inconclusive rate at grid points is not proof of that equality between grid points. For an n-copy joint reader one may use the safe modulus Cmaxn √
Technical passage, source page 19.
Searchable transcription
This is a standard trace-distance metric-net argument newly specialized to this phase reader, not
a new general minimax theorem. If constraints themselves are imposed only on grid points, they
need their own modulus margin. In particular, equality of an inconclusive rate at grid points is not
proof of that equality between grid points. For an n-copy joint reader one may use the safe modulus
Cmaxn
√

1 −s2 sin(h/2), capped by Cmax, from telescoping trace distance; the one-copy formula must not be reused unchanged.

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