A New Definition of Quantum Superposition

Abstract

The usual description of the superposition of two (pure quantum) states is ambiguous, since the binary operation of addition in a Hilbert space does not pass down to the quotient projective space. Even though Dirac noted this as early as 1930, it is often assumed that superposition is a binary operation acting on two states with a value that is a unique state. The goal of this paper is to motivate a rigorous, geometrical definition of the superposition of states in the setting of complex projective space, which has been argued elsewhere to be the natural geometric phase space for quantum theory. The upshot is that the new definition of the superposition of two pure states, viewed as two distinct points in the projective space, is the unique (complex) line on which those two points lie. Next, a comparison is given between superposition and expansion in an orthonormal basis. Then the geometry of complex projective space is precisely defined using the Erlangen program. Finally, this approach is compared with another rigorous definition of superposition.

13 pages