# QM probability density function without Born's rule, invariant to wave-function phase

The QM probability density as a function of the wave function is given by Born's rule as a postulate. This leads to the probability density being invariant to the phase of the wave function.

Suppose instead that I leave out Born's rule, and instead more generally postulate that there exists a probability density as a function of the wave function, but invariant to the phase of the wave-function. How does this restrict the form of this probability density function?

• The probability density is already invariant to the phase of the wave function... – Jake Rose Jul 31 '19 at 0:22
• @JakeRose as defined by Born's rule, yes. I'm asking about postulating a general probability density function not limited by Born's rule to begin with. – Physiks lover Jul 31 '19 at 0:34
• @Physikslover Well, does this prospective probability density need to satisfy any additional requirements? (say, things like "be local with respect to the wavefunction".) Or are you asking about arbitrary functionals $\rho = \mathcal F[\psi]$ subject only to global-phase invariance? – Emilio Pisanty Jul 31 '19 at 1:22
• @EmilioPisanty I'm after a probability density function satisfying the usual Dirac–von Neumann axioms minus the Born rule plus only global-phase invariance. But it would add to my question helpfully if it could also be shown how the probability density is further restricted by the usual properties required of the wave-function as a consequence of Born's rule. – Physiks lover Jul 31 '19 at 2:20

Theorem. Suppose $$H$$ is a separable Hilbert space. A measure on $$H$$ is a function $$f$$ that assigns a nonnegative real number to each closed subspace of $$H$$ in such a way that, if $$\{ A_i \}$$ is a countable collection of mutually orthogonal subspaces of $$H$$, and the closed linear span of this collection is B, then $$f(B)=\sum_i f(A_i)$$. If the Hilbert space $$H$$ has dimension at least three, then every measure $$f$$ can be written in the form $$f(A)=\mathrm{Tr}(WP_A)$$, where $$W$$ is a positive semidefinite trace class operator and $$P_A$$ is the orthogonal projection onto $$A$$.
In other words, in dimension $$d>2$$, any probability measure on a Hilbert space is generated by a quantum state following the Born rule, so even if you came up with a different measure it would be possible to express it as the Born rule.