Finite fusion transfer count model

A finite fusion transfer count model consists of a finite nonempty label set $I$, a distinguished label $0\in I$, and an explicitly supplied matrix $N=(N_{ab}{)}_{a,b\in I}$ whose entries lie in $\{0,1,2,\dots \}$. The entry $N_{ab}$ is the multiplicity of elementary transitions from label $b$ to label $a$; formally, the transition copies from $b$ to $a$ are the triples $(b,a,j)$ with integer $1\le j\le N_{ab}$. A length-$n$ path is a sequence of $n+1$ labels together with a chosen transition copy between each consecutive pair. View each natural-number entry as a real number under the usual inclusion $\mathbb{N}\subset \mathbb{R}$. For a finite set $J$ and $F=\mathbb{R}$ or $\mathbb{C}$, $F^J$ means the functions $J\to F$, with pointwise addition $(x+y{)}_j=x_j+y_j$ and scalar multiplication $(\lambda x{)}_j=\lambda x_j$. For real matrices $A,B$ indexed by $I$, define $(A+B{)}_{ab}:=A_{ab}+B_{ab}$. Their action on $x\in {\mathbb{R}}^I$ is $(Ax{)}_a:={\sum }_{b\in I}{A}_{ab}{x}_b$. Matrix multiplication is $(AB{)}_{ac}:={\sum }_{b\in I}{A}_{ab}{B}_{bc}$; it is associative by regrouping finite sums. These definitions are compatible because $((AB)x{)}_a={\sum }_{b,c\in I}{A}_{ab}{B}_{bc}{x}_c={\sum }_{b\in I}{A}_{ab}(Bx{)}_b=(A(Bx){)}_a$, and $((A+B)C{)}_{ac}={\sum }_b(A_{ab}+B_{ab})C_{bc}=(AC{)}_{ac}+(BC{)}_{ac}$. Thus the natural-entry powers can also act on real vectors and the addition used below distributes over matrix multiplication. Let $I_I$ be the identity matrix with entries $(I_I{)}_{ab}=1$ if $a=b$ and $0$ otherwise, and use the power convention $N^0:=I_I$ and $N^{n+1}:=N{N}^n$ for $n\ge 0$. Define the vacuum-return count $D_n:=(N^n{)}_{00}$. If $D_n>0$, set $H_n:={\mathbb{C}}^{D_n}$, with coordinates $x=(x_1,\dots ,x_{D_n})$ and inner product $⟨x,y⟩:={\sum }_{j=1}^{D_n}{x}_j\overline{y_j}$, linear in the first argument. This pairing satisfies the inner-product axioms of Real and complex inner product spaces, with the inner product linear in the first argument: finite distributivity gives $⟨\alpha x+\beta z,y⟩=\alpha ⟨x,y⟩+\beta ⟨z,y⟩$, and conjugating the finite sum gives $\overline{⟨y,x⟩}={\sum }_j\overline{y_j}{x}_j=⟨x,y⟩$. Writing $x_j=a_j+i{b}_j$ with real $a_j,b_j$ gives $⟨x,x⟩={\sum }_j(a_j^2+b_j^2)\ge 0$; if this sum is zero, each nonnegative summand is zero, hence every $a_j=b_j=0$ and $x=0$. Conversely $x=0$ gives zero. The conjugation identities used here follow directly from $\overline{a+ib}=a-ib$ and finite distributivity. Therefore $H_n$ is a finite-dimensional complex inner-product space; its coordinate unit vectors form a basis since every $x$ equals their linear combination with coefficients $x_j$, and a zero combination forces every coefficient to vanish. Let $I_{H_n}$ be its identity endomorphism and define ${\rho }_n:=D_n^{-1}{I}_{H_n}$. If $D_n=0$, no normalized operator ${\rho }_n$ is assigned. The matrix $N$ is model data; no fusion category, quantum group, or representation is presumed to realize it.

Real and complex inner product spaces, with the inner product linear in the first argument

Daniel Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction

John Watrous, The Theory of Quantum Information

The definition records a finite matrix count only. It does not assert the existence of a fusion category or identify $D_n$ with a gravitational state count.