Born rule

The Born rule states that an observable, measured in a system with normalized wave function ${\displaystyle |\psi \rangle }$ (see Bra–ket notation), corresponds to a self-adjoint operator ${\displaystyle A}$ whose spectrum is discrete if:


In the case where the spectrum of ${\displaystyle A}$ is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure (PVM) ${\displaystyle Q}$, the spectral measure of ${\displaystyle A}$. In this case:


A wave function ${\displaystyle \psi }$ for a single structureless particle in space position ${\displaystyle (x,y,z)}$ implies that the probability density function ${\displaystyle p}$ for a measurement of the particles's position at time ${\displaystyle t_{0}}$ is:


In some applications, this treatment of the Born rule is generalized using positive-operator-valued measures (POVM). A POVM is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of von Neumann measurements and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurements described by self-adjoint observables. In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics and can also be used in quantum field theory.^[2] They are extensively used in the field of quantum information.


In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices ${\displaystyle \{F_{i}\}}$ on a Hilbert space ${\displaystyle {\mathcal {H}}}$ that sum to the identity matrix,:^[3]: 90


The POVM element ${\displaystyle F_{i}}$ is associated with the measurement outcome ${\displaystyle i}$, such that the probability of obtaining it when making a measurement on the quantum state ${\displaystyle \rho }$ is given by:


where ${\displaystyle \operatorname {tr} }$ is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state ${\displaystyle |\psi \rangle }$ this formula reduces to:


The Born rule, together with the unitarity of the time evolution operator ${\displaystyle e^{-i{\hat {H}}t}}$ (or, equivalently, the Hamiltonian ${\displaystyle {\hat {H}}}$ being Hermitian), implies the unitarity of the theory, which is considered required for consistency. For example, unitarity ensures that the probabilities of all possible outcomes sum to 1 (though it is not the only option to get this particular requirement).