Manuscript XXII-C@1 · XXII-C.tex · section · b9a05a24

A compact formula for signalling strengths

Exact data and unconditional influence › A compact formula for signalling strengths · lines 412–441

XXII-C@1#eq:strength-definition

For distinct $a, b \in J$ , let $c$ be the remaining party. For $S, T \in C_{a}$ , a receiver instrument ${B_{z}}_{z}$ at $b$ , and $M_{c} \in C_{c}$ , compare

P_{b ∣ S} (z) = Tr [W_{e} (S_{a} \otimes B_{z}^{b} \otimes M_{c})], P_{b ∣ T} (z) = Tr [W_{e} (T_{a} \otimes B_{z}^{b} \otimes M_{c})] .

Factors are placed in the fixed tensor slots. The settings at $b, c$ are identical within the compared pair.

Define

Δ_{e} (a, b) = S, T, {B_{z}}, M_{c} sup d_{TV} (P_{b ∣ S}, P_{b ∣ T}), d_{TV} (P, Q) = \frac{1}{2} z \sum ∣ P (z) - Q (z) ∣.

This is the classical-record branch of the unconditional signalling test of XXII-B CheungXXIIB (def:signalling).

Displayed formulas in this passage

unlabeled · line 417
$P_{b ∣ S} (z) = Tr [W_{e} (S_{a} \otimes B_{z}^{b} \otimes M_{c})], P_{b ∣ T} (z) = Tr [W_{e} (T_{a} \otimes B_{z}^{b} \otimes M_{c})] .$
XXII-C@1#eq:strength-definition · line 430
$Δ_{e} (a, b) = S, T, {B_{z}}, M_{c} sup d_{TV} (P_{b ∣ S}, P_{b ∣ T}), d_{TV} (P, Q) = \frac{1}{2} z \sum ∣ P (z) - Q (z) ∣.$

A compact formula for signalling strengths

Displayed formulas in this passage

Nearby in this section