The Born Rule Is Not Optional
The Born rule is the link between quantum mathematics and experimental reality. It says: the probability of measuring outcome A is the squared amplitude of the state vector projected onto A.
\[P(A) = |\langle A | \psi \rangle|^2\]Every quantum prediction depends on this rule. But where does it come from?
Textbooks typically present it as a postulate, one of the axioms you accept to do quantum mechanics. Why squared amplitudes? Why not amplitudes directly, or some other function?
The LRT answer: the Born rule is forced by the requirement that measurement outcomes be distinguishable.
The Gleason Derivation
Andrew Gleason proved in 1957 that the Born rule is the only way to assign probabilities in Hilbert space while respecting the structure of the space.
Here’s the setup. You have:
- A quantum system living in Hilbert space (at least 3 dimensions)
- A set of possible outcomes, represented by orthogonal projectors
- A probability measure assigning values to those outcomes
What rules must this probability measure obey?
Non-negativity: Probabilities can’t be negative.
Normalization: Probabilities for a complete set of exclusive outcomes must sum to 1.
Frame function: The probability assigned to an outcome must be the same regardless of which other outcomes you’re considering alongside it.
This last requirement is crucial. If measuring “spin up in the z-direction” gives probability 0.5, that value can’t depend on whether you’re also considering spin down, or some arbitrary direction, or nothing else at all.
This is vehicle-invariance applied to probabilities. The probability of A is a fact about A.
Gleason’s theorem: The only probability measures satisfying these conditions have the form:
\[P(A) = \text{Tr}(\rho \cdot P_A)\]| For pure states, this reduces to the familiar Born rule: $P(A) = | \langle A | \psi \rangle | ^2$. |
Why This Matters for LRT
The standard reading of Gleason’s theorem: if you’re working in Hilbert space, then the Born rule is forced.
LRT adds the prior step: why are we working in Hilbert space?
The previous article explained how distinguishability constraints (local tomography, etc.) force complex Hilbert space as the arena for physical theories. Gleason then forces the probability rule within that arena.
The full chain:
- $L_3$ constraint: outcomes must be determinately distinguishable
- Local tomography: compound systems are distinguishable from local data
- Complex Hilbert space: the unique structure respecting (1) and (2)
- Gleason’s theorem: the unique probability measure on that structure
- Born rule: the operational form of (4)
The Born rule is a theorem, derived from deeper requirements about what it means for outcomes to exist determinately.
The Non-Contextuality Connection
Another way to see this: the Born rule ensures non-contextuality of probabilities.
Contextuality means the probability of an outcome depends on what other measurements you might perform. If I measure A in context {A, B, C}, do I get the same probability as measuring A in context {A, D, E}?
Physical outcomes with determinate identity must be non-contextual in this sense.
Gleason’s theorem proves that non-contextual probability assignments over Hilbert space must take the Born rule form. Any other assignment would be contextual: the probability of A would secretly depend on the measurement context.
Non-contextuality = vehicle-invariance = determinate identity = $L_3$.
The terminologies differ across fields, but they’re expressing the same constraint: what’s true about a physical outcome can’t depend on how you frame your question about it.
Why Squared?
Here’s an intuitive way to see why amplitudes get squared rather than used directly.
Quantum amplitudes can be negative or complex. Probabilities must be non-negative real numbers between 0 and 1.
| The simplest way to get a non-negative real from a complex number: take its squared magnitude, $ | z | ^2 = z \cdot z^*$. |
| This is necessary. Gleason proves there’s no other function that works. If you tried to use $ | \langle A | \psi \rangle | $ (without squaring), or $ | \langle A | \psi \rangle | ^4$, or any other power, you’d violate the frame function requirement. The probabilities would become contextual. |
The squaring in the Born rule is the unique operation that preserves non-contextuality while converting amplitudes to probabilities.
What About Many-Worlds?
In the many-worlds interpretation, all branches of the wavefunction are equally real. What do Born rule “probabilities” even mean?
A genuine interpretive puzzle, but it doesn’t affect the LRT derivation.
LRT operates at the level of mathematical structure. Whatever the Born rule represents (objective probabilities, rational betting weights, branch counting measures), Gleason’s theorem says it must have that specific mathematical form.
Many-worlds advocates can interpret the derivation as showing why branch weights must follow $\lvert\psi\rvert^2$. Collapse advocates can interpret it as deriving objective transition probabilities. The structural constraint is interpretation-independent.
Finite-Dimensional Restriction
Gleason’s theorem requires at least 3 dimensions. For a 2-dimensional system (a single qubit), there are other mathematically possible probability assignments that aren’t Born rule form.
No physical system is truly isolated. A qubit interacts with its environment, embedding it in a larger space. The 2D case is an idealization.
When you treat the qubit-plus-environment system (which is high-dimensional), Gleason applies. The Born rule for the qubit emerges from restricting the full-system Born rule probabilities.
The 3D restriction isn’t physically limiting. Any system capable of having distinguishable interactions with other systems is embedded in a space where Gleason forces the Born rule.
The Big Picture
Here’s what the Born rule derivation demonstrates about LRT:
The quantum formalism isn’t arbitrary. Physicists didn’t try various rules until the Born rule happened to fit the data. The Born rule is the unique probability assignment compatible with outcomes having determinate identity.
Constraints generate structure. Classical physics assumed anything mathematically consistent could be physical. LRT says only $L_3$-admissible configurations can be instantiated. That constraint forces precisely the structures we observe.
Logic reaches into physics. The three classical laws (Identity, Non-Contradiction, Excluded Middle) constrain what structures can exist through the instantiation barrier. The Born rule is what happens when you ask: how must probability work for outcomes that genuinely exist?
Next in the Series
The final article, “Common Objections to Logic Realism,” addresses skeptical responses: Isn’t this just philosophy? What about alternative logics? Can $L_3$ really have physical implications?
For the full technical treatment, see the Position Paper. For concept-by-concept breakdowns, explore Topics.
Logic Realism Theory was developed by James (JD) Longmire. This article is part of the LRT documentation at jdlongmire.github.io/logic-realism-theory.