Physlib

Physlib.ProbabilisticTheory.Effect.Complement

Complementary effects

i. Overview

The complement of an effect `e` is the yes/no test that fires exactly when `e` doesn't: `1 - e`. Physically, a state's probability of "no" is always `1` minus its probability of "yes".

ii. Key results

  • `Effect.complement` : the complementary effect `1 - e`.
  • `Effect.complement_antitone` : the complement reverses order.
  • `Effect.complement_mix` : the complement of a mixture is the mixture of the complements.

iii. Table of contents

  • A. The complement
  • B. Monotonicity of the complement
  • C. The complement and mixtures

A. The complement

B. Monotonicity of the complement

C. The complement and mixtures

3 declarations

theorem

The Complement of an Effect ee is 1−e1 - e

For any effect ee in the order interval [0,1][0, 1] of a partially ordered type EE with zero and unit elements, the underlying value of the complement of ee in EE is 1−e1 - e.

theorem

The Effect Complement 1−e1 - e is Antitone

In an order unit space EE, the complement operation on effects, defined as e↦1−ee \mapsto 1 - e, is antitone. That is, for any two effects e1,e2e_1, e_2 in the order interval [0,1][0, 1] of EE, if e1≤e2e_1 \le e_2, then 1−e2≤1−e11 - e_2 \le 1 - e_1.

theorem

The complement of a mixture of effects is the mixture of their complements

For any effects ee and ff in an order unit space EE, and for any scalar tt in the unit interval [0,1][0, 1], the complement of the mixture of ee and ff is equal to the mixture of the complements of ee and ff. That is, 1−(te+(1−t)f)=t(1−e)+(1−t)(1−f) 1 - (t e + (1 - t) f) = t (1 - e) + (1 - t) (1 - f) where te+(1−t)ft e + (1 - t) f denotes the mixture of effects and 1−e1 - e denotes the complementary effect.