Question

Let $G$ be a finite bipartite graph with vertex bipartition $(V,W)$. Suppose we want to find a matching $f\colon V\to W$, i.e., an injective function such that for all $v\in V$, the vertices $v$ and $f(v)$ are adjacent. We may not be sure what to choose for $f(v)$, so we choose all possible vertices at once as follows. Let $K$ be a field and $KS$ the vector space with basis $S$. Define a linear transformation $\varphi\colon KV\to KW$ by letting $\varphi(v)$ be the sum of all vertices in $W$ adjacent to $v$ (or more generally, some linear combination of such vertices). It is easy to show that if $\varphi$ is injective, then a matching $f\colon V\to W$ exists. We can think of an injective $\varphi$ as a "quantum matching" (since it is in all possible "states" $f(v)$ at once). Choosing a nonzero term in a certain determinant "collapses" the wave function $\varphi$ to the matching $f$. For further details and some nice applications, see Sections 4-6 of http://math.mit.edu/~rstan/algcomb.pdf.