---
tag: 2E9DMB7A6Q
kind: concept
name: Quantum logic
msc: [81P10, 03G12, 06C15]
sources:
  - url: https://en.wikipedia.org/wiki/Quantum_logic
same_as:
  wikidata: Q1206311
  wikipedia: Quantum logic
---

## Summary

Quantum logic is the study of the propositions about a quantum system and of the logical operations on them.
In the approach of Birkhoff and von Neumann, a proposition is a closed subspace of the [[Hilbert space|HM2HHE3J51]] of the system.
These propositions form an [[orthomodular lattice|6FPKEFX0X4]].
When the Hilbert space has dimension at least $2$, this lattice is not distributive, so the distributive law of classical logic fails.
Abstract quantum logics are studied as orthomodular lattices or orthomodular partially ordered sets.

## Explanations

### L1 human

In classical physics, a statement such as "the position is in this interval" is true or false of each state.
Such statements combine like subsets, with "and", "or" and "not".
In quantum mechanics, a yes-no question corresponds to a closed subspace of a Hilbert space, the states for which the answer is surely yes.
"And" is intersection, "not" is the orthogonal complement, and "or" is the smallest closed subspace containing both.
With these operations "$p$ and ($q$ or $r$)" can differ from "($p$ and $q$) or ($p$ and $r$)".
This reflects the fact that some quantum questions cannot be answered at the same time.