Topological boundary state counts without universal gravity claims

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.

For a finite fusion transfer count model, let $e_0\in {\mathbb{R}}^I$ be the coordinate vector equal to $1$ at label $0$ and $0$ elsewhere, and define $v_n:=N^n{e}_0$. Then $v_0=e_0$, $v_{n+1}=N{v}_n$, and the number of length-$n$ paths from $0$ to a label $a$ is $(v_n{)}_a=(N^n{)}_{a0}$. Whenever $D_n>0$, the operator ${\rho }_n=D_n^{-1}{I}_{{\mathbb{C}}^{D_n}}$ is a density matrix and has von Neumann entropy $s({\rho }_n)=\log D_n$ in nats. For the supplied two-label matrix $N=\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right)$, with labels ordered $0,1$, one has $D_0=1$, $D_1=0$, $D_2=1$, and $D_{n+2}=D_{n+1}+D_n$ for every $n\ge 0$.

Finite fusion transfer count model

Sector entropy and relative entropy identities

Direct-sum sector code entropy model

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

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

John Watrous, The Theory of Quantum Information

The diagram displays the two-label matrix only. An arrow $b\to a$ with label $m$ represents $N_{ab}=m$ transitions; the count proof below uses matrix multiplication and does not rely on the drawing.

Fix an integer $n\ge 1$ and a real loop parameter $\delta$. Define $[0{]}_{\delta }=0$, $[1{]}_{\delta }=1$, and $[r+1{]}_{\delta }=\delta [r{]}_{\delta }-[r-1{]}_{\delta }$ for $r\ge 1$. Assume $[k{]}_{\delta }\mathrm{\ne }0$ for $2\le k\le n$; this list is empty when $n=1$. The abstract generator algebra $P_n(\mathbb{C},\delta )$ is the unital complex algebra with generators $u_1,\dots ,u_{n-1}$ and relations $u_i^2=\delta u_i$, $u_i{u}_{i\pm 1}{u}_i=u_i$, and $u_i{u}_j=u_j{u}_i$ when $\mathrm{\mid }i-j\mathrm{\mid }>1$. The diagram algebra ${\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ has basis the noncrossing pairings of $n$ ordered bottom and $n$ ordered top points; its product stacks the left factor above the right, deletes each closed loop, and multiplies by $\delta$ per loop. Its elementary diagram $d_i$ has a cap joining top points $i,i+1$, a cup joining bottom points $i,i+1$, and vertical strands elsewhere. The isomorphism ${\mathrm{\Phi }}_n:P_n(\mathbb{C},\delta )\to {\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ sends $u_i$ to $d_i$.

For every $1\le i\le n-2$, the adjacent relations in the presentation mean both $u_i{u}_{i+1}{u}_i=u_i$ and $u_{i+1}{u}_i{u}_{i+1}=u_{i+1}$. For matching the imported recursion's notation, write $U_i:=d_i$.

Set $p_1=1_1$. For $1\le r<n$, put $Q_r=p_r\otimes 1_1$ by adjoining one rightmost vertical strand and define

$p_{r+1}:=Q_r-\frac{[r{]}_{\delta }}{[r+1{]}_{\delta }}{Q}_r{d}_r{Q}_r\mathrm{.}$

The denominator hypotheses make this recursion defined. For each $1\le r\le n$, let ${\widehat{p}}_r:={\mathrm{\Phi }}_r^{-1}(p_r)$. The cited Jones-Wenzl result gives $p_n^2=p_n$ under these unit-denominator hypotheses. For $1\le r\le n$, let ${\iota }_{r,n}:P_r(\mathbb{C},\delta )\to P_n(\mathbb{C},\delta )$ send each generator $u_i$ to the same-named generator, and let ${\mathrm{ȷ}}_{r,n}:{\mathrm{T}\mathrm{L}}_r(\mathbb{C},\delta )\to {\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ append $n-r$ rightmost vertical strands. The maps commute with $\mathrm{\Phi }$: on each generator, both composites send $u_i$ to $d_i$ with those added strands, and both send the unit to the identity diagram. Since generators and unit generate $P_r$, the composites agree on every element.

Let $H$ be a nonzero finite-dimensional complex inner-product space whose inner product is linear in its first argument and conjugate-linear in its second. A unital complex-algebra representation is a complex-linear map $\pi :P_n(\mathbb{C},\delta )\to {\mathrm{End}}_{\mathbb{C}}(H)$ such that $\pi (xy)=\pi (x)\pi (y)$ for all $x,y\in P_n(\mathbb{C},\delta )$ and $\pi (1)=I_H$. Here ${\mathrm{End}}_{\mathbb{C}}(H)$ is the algebra of complex-linear endomorphisms, with composition as multiplication and $I_H$ as unit. Require each generator image to be self-adjoint, meaning $⟨\pi (u_i)x,y⟩=⟨x,\pi (u_i)y⟩$ for all $x,y\in H$. Define $P:=\pi ({\widehat{p}}_n)$, define $\mathrm{rank}P:={\dim }_{\mathbb{C}}\mathrm{im}P$, and restrict to data with $\mathrm{rank}P>0$. The algebraic representation as a physical boundary state space is an additional model identification and is not part of this definition.

The abstract Temperley–Lieb generator presentation

Temperley–Lieb planar diagram data and endomorphism modules

The generator presentation is the planar diagram algebra over every commutative ring

Temperley–Lieb quantum integers and quantum characteristic

Jones–Wenzl recursion over unit quantum integers

Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure

Kernel and image of a linear map

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

Joshua Chen, The Temperley-Lieb Categories and Skein Modules

Zhenghan Wang, Topological Quantum Computation

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras

For $r<n$, the presentation tower map ${\iota }_{r,n}$ is well-defined because the defining relations are unchanged and it preserves the unit. For basis diagrams $E,D\in {\mathrm{T}\mathrm{L}}_r$, appending $n-r$ rightmost vertical strands before stacking gives the same matching as stacking $E$ above $D$ first and then appending those strands; the appended strands create no closed loops. Hence ${\mathrm{ȷ}}_{r,n}(E\circ D)={\mathrm{ȷ}}_{r,n}(E)\circ {\mathrm{ȷ}}_{r,n}(D)$, and ${\mathrm{ȷ}}_{r,n}(1_r)=1_n$. Bilinearity then makes ${\mathrm{ȷ}}_{r,n}$ a unital algebra homomorphism. The two unital algebra homomorphisms ${\mathrm{\Phi }}_n\circ {\iota }_{r,n}$ and ${\mathrm{ȷ}}_{r,n}\circ {\mathrm{\Phi }}_r$ agree on every generator and the unit, so they agree on all of $P_r$; this identity transfers the unit-denominator recursion on diagrams to the presentation used by $\pi$.

The box is notation for the three-strand algebra element $p_3$; strands are ordered from left to right and continue through the box. For general $n$, the same box notation has $n$ strands. The picture records no physical apparatus, and no proof below relies on it.

Only $[2{]}_{\delta },\dots ,[n{]}_{\delta }$ are required to be nonzero; $[n+1{]}_{\delta }$ may vanish. The item asserts no root-of-unity quotient, modular representation classification, trace-preserving realization, or physical area law.

Under the finite Hilbert representation data, let $r:=\mathrm{rank}P>0$ and $\rho :=P\mathrm{/}r$. An orthogonal projection on $H$ means a complex-linear endomorphism $Q$ satisfying $Q^2=Q$ and $⟨Qx,y⟩=⟨x,Qy⟩$ for all $x,y\in H$. Then $P$ is an orthogonal projection, $\rho$ is a density matrix with eigenvalue $1\mathrm{/}r$ of multiplicity $r$ and eigenvalue $0$ on $\ker P$, and its entropy is $s(\rho )=\log r$ in nats. Under the published closure convention, the unnormalized diagrammatic closure trace of the algebraic Jones-Wenzl element is ${\mathrm{Tr}}_n(p_n)=[n+1{]}_{\delta }$. The proposition records this separately from the ordinary Hilbert-space trace ${\mathrm{tr}}_H(P)=r$ and does not assert the two values agree. A sufficient additional hypothesis for equality on this element is that the representation preserve these traces, ${\mathrm{tr}}_H(\pi ({\mathrm{\Phi }}_n^{-1}(X)))={\mathrm{Tr}}_n(X)$ for every $X\in {\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$.

Finite Hilbert boundary representation of Temperley-Lieb

Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure

Direct-sum sector code entropy model

Sector entropy and relative entropy identities

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

Kernel and image of a linear map

Unnormalized diagram closure and partial trace

Jones–Wenzl recursion over unit quantum integers

Joshua Chen, The Temperley-Lieb Categories and Skein Modules

Zhenghan Wang, Topological Quantum Computation

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras

John Watrous, The Theory of Quantum Information