Effect Algebras

Effect algebras describe events that can be added when they are mutually exclusive outcomes of one measurement. They accommodate noisy and generalized measurements without assuming that all events are sharp propositions. Effects and Effect Algebras develops the definitions and operator-order results; the examples here explain their meaning for measurement.

Measurement events beyond projections

A quantum effect $E$ obeys $0\leq E\leq I$, and its probability is $\operatorname{Tr}(\rho E)$. Projections are the special sharp effects satisfying $E^2=E$. If a detector registers an ideal event $P$ with efficiency $\eta$, a simple model for the registered event is $E=\eta P$.

For $0<\eta<1$, a state in the range of $P$ produces a click only with probability $\eta$. The effect is not a proposition that is certainly true in that state. This example motivates a probability-event language broader than the lattice of sharp subspaces.

Coarse-graining uses a partial sum

For two outcomes of the same POVM, forgetting which occurred produces the effect

$$E\oplus F=E+F,\qquad E+F\leq I.$$

The upper bound ensures that the coarse-grained probability is never greater than one. A complement $I-E$ completes a binary measurement. The operation is partial because arbitrary effects need not describe exclusive alternatives.

Consider a detector with outcomes “red click,” “blue click,” and “no click,” described by $E_r=\eta P$, $E_b=(1-\eta)P$, and $E_0=I-P$. The first two effects overlap in support, yet their sum is $P$. Thus exclusivity of detector labels must not be confused with orthogonality of the operators representing unsharp events.

Likewise, joint measurability of two binary tests is not the same as being exclusive outcomes. Independent coin tosses are jointly measurable even when the two yes probabilities sum to more than one.

What event structure leaves unspecified

The same effect can arise from different instruments. After reporting a click, an apparatus may preserve a state within a subspace, reprepare a fixed state, or transfer information to an environment. The probability effect alone does not distinguish these mechanisms.

This matters for contextuality: operational equivalence of outcome probabilities and equivalence of full transformations are separate claims. It also matters for reconstruction, since an event algebra without composition and dynamics does not fix an entire physical theory.

Exercises

For $E=I/2$, can the binary measurement reveal anything about the input state? Can its instrument still disturb the state?

Solution

Both outcomes have probability $1/2$ for every input, so the classical outcome reveals no state information. An instrument can nevertheless reprepare a fixed state after each outcome and erase the input. Probability statistics alone do not specify disturbance.

References

• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — effects and instruments. Publisher.
• Foulis, David J., and Mary K. Bennett, “Effect algebras and unsharp quantum logics,” Foundations of Physics 24, 1331–1352 (1994). Paper.