Dependencies

Place a spin-\(\tfrac 12\) singlet on every bond, encoded by

and project the two spin-\(\tfrac 12\) particles at each site onto their joint spin-\(1\) subspace. With the physical states labelled by \(S_z=0,\pm 1\), the resulting tensor with \(d=3\) and \(D=2\) is

The three left factors are the spin-\(1\) states \(|\uparrow \uparrow \rangle \), \((|\uparrow \downarrow \rangle +|\downarrow \uparrow \rangle )/\sqrt2\), and \(|\downarrow \downarrow \rangle \) written as coefficient matrices. This is the tensor of [ 17 , Appendix A, “The AKLT state” ] , for the state of  [ 2 ] .

The CZX tensor of the review  [ 17 , Appendix A, “The CZX MPU” ] is the bond-two tensor \(A\), with the output index first:

and \(A^{00}=A^{11}=0\). Its entries are \(A^{ij}_{\alpha \beta }=\delta _{i,1-j}\delta _{\alpha ,i}(-1)^{i\beta }\). It is obtained from the bond-two tensor of Definition 31.3.1.1 by transposing every matrix and exchanging the two physical indices: \(A^{ij}=(M^{ji})^{\mathsf T}\). The bra-ket form of the same tensor in the review uses normalized virtual vectors and equals \(2^{-1/2}A\); its periodic operator is not unitary, and the matrices displayed above are taken as the source tensor  [ 180 ] .

Given bits \(q\) at torus plaquette corners, assign site \((x,y)\) the qubits \((q_{x,y+1},q_{x+1,y+1},q_{x+1,y},q_{x,y})\) and give each native bond the corresponding adjacent pair.

Let \(E_R\mu \) be the clockwise label sequence, with bottom and left pairs reversed. This is a bijective change of coordinates. For \(c\in \{ 0,1\} ^P\), set \(\iota (c)_i=(c_i,c_{i+1})\), define \(\mu (c)=E_R^{-1}\iota (c)\), and let \(F_R|c\rangle =A_R|\mu (c)\rangle \).

Number the crossing bonds clockwise, first across the top, then down the right, across the bottom, and up the left. The corresponding first plaquette corners are \((i,h)\), \((w,h-j)\), \((w-k,0)\), and \((0,\ell )\) on the four sides. Reverse the native pair order on the bottom and left.

For a native boundary configuration \(\mu \), let \(f_R\mu \) flip both bits on each crossing bond and let \(\phi _R(\mu )=\prod _{e\in \partial R}(-1)^{a_eb_e}\), where \(\mu _e=(a_e,b_e)\).

A PEPS tensor \(A\) is vertex-injective if, at every vertex \(v\), the family of physical vectors \((A_v(\eta ,\cdot )\in \mathbb {C}^d)_{\eta }\) indexed by virtual configurations \(\eta \) on the edges incident to \(v\) is linearly independent in \(\mathbb {C}^d\). Equivalently, extending \(A_v\) by linearity to a map from the free vector space on virtual configurations into the physical space \(\mathbb {C}^d\) gives a linear map with trivial kernel. This matches the injectivity formulation of [ 220 , Section 3 ] , where the local tensor is interpreted as a map from virtual configurations to the physical space.

  \(\ker A_v=\{ 0\} \).

The projector onto the symmetric subspace of \((\mathbb {C}^2)^{\otimes 4}\) is \(\Pi _{\mathrm{sym}}=\frac{1}{24}\sum _{\pi \in S_4}P_{\pi }\).

With the singlet matrix \(Y=\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)\), the AKLT tensor on the square lattice is \(A=\Pi _{\mathrm{sym}}(\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y)\), with physical index \(s\in \{ 0,1\} ^4\) (physical dimension \(16\)) and virtual indices ordered top, right, down, left [ 17 , Appendix A, “The AKLT model” ] .

On the torus a configuration \(\sigma \) has energy

each nearest-neighbour bond counted once. For symmetric \(h\) this is the sum of \(h\) over unordered bonds.

Let \(M=\sum _{i,j}e^{-\beta h(i,j)/2}|i)(j|\). The classical-model tensor is

the top and right legs copy the physical index \(s\), and the down and left legs carry \(M_{sb}\) and \(M_{sl}\) at bond indices \(b\) and \(l\) [ 17 , Appendix A, “PEPS from classical models” ] . The classical-model PEPS places this tensor at every site of the torus.

The cluster tensor is \(A=|0\rangle (00{+}{+}|+|1\rangle (11{-}{-}|\) with \((\pm |=[(0|\pm (1|]/\sqrt{2}\): the top and right legs copy the physical index \(s\), and the down and left legs carry \((-1)^{sb}/\sqrt{2}\) at bond index \(b\) [ 17 , Appendix A, “The cluster state” ] .

For \(N{\gt}0\) and effective spins \(c\in \{ 0,1\} ^N\), the closed chain of \(N\) boundary legs carrying \(c\) has the pair \((c_m,c_{m+1})\) on leg \(m\), indices modulo \(N\). The map \(\iota \) from \(N\) effective spins into the legs is

A virtual leg carries two qubits \((a,b)\). The operator \(V=(X\otimes X)\, CZ\) acts on a leg by \(V|ab\rangle =(-1)^{ab}|\bar a\bar b\rangle \).

The operator \(U_{CZX}=U_XU_{CZ}\) on the sixteen states of a site, and its product \(U_{CZX}^{\otimes V}\) over the sites \(V\) of the lattice, whose entry at \((\sigma ',\sigma )\) is \(\prod _v(U_{CZX})_{\sigma '_v\sigma _v}\) [ 11 ] .

Label the four qubits of a site \(|ijkl\rangle \), with \(i\) at the top left, \(j\) at the top right, \(k\) at the bottom right and \(l\) at the bottom left, and label the four basis vectors of a bond by pairs of qubits. The CZX tensor is

so each leg carries the pair of corner qubits on its side, listed clockwise around the site [ 17 , Appendix A, “The CZX model” ] . The CZX PEPS places this tensor at every site with its down and left labels read in reverse, so that each bond identifies the pair \((a,b)\) at one end with the pair \((b,a)\) at the other.

The GHZ tensor has \(D=d\) and \(A^i_{\alpha \beta \gamma \delta }=\delta _{i=\alpha =\beta =\gamma =\delta }\) [ 17 , Appendix A, “The GHZ state” ] .

Let a finite set of bonds have distinct endpoints \(a(e),b(e)\) in a finite vertex set \(V\). Parallel bonds are counted with multiplicity. The product of controlled-\(Z\) gates has diagonal coefficient \(g(\sigma )=\prod _e(-1)^{\sigma _{a(e)}\sigma _{b(e)}}\). The stabilizer \(K_v\) flips the qubit at \(v\) and applies \(Z\) at the opposite endpoint of every incident bond. Thus

On the square torus, take the horizontal and vertical positive bonds; the normalized circuit state has coefficients \(2^{-N/2}g(\sigma )\) [ 17 , Appendix A, “The cluster state” ] .

For a region \(R\) in a finite PEPS graph, let \(T_R(\mu ;\sigma )\) be the contraction of its site tensors with physical configuration \(\sigma \), fixing the crossing-bond labels \(\mu \) and summing only the internal-bond labels. It defines the linear map from the crossing virtual indices to the physical system of \(R\). No edge wholly outside \(R\) enters this contraction. This is the finite-region tensor used when blocking and cutting the PEPS in [ 235 , Section 6, proof of Theorem 6.9 ] .

Let \(\psi \) be the closed PEPS vector on a finite graph and let \(R\) be a region. Its unnormalized reduced density is \(\rho _R=\operatorname {tr}_{R^c}|\psi \rangle \langle \psi |\). If \(M_R(\sigma ,\tau )=\psi (\sigma ,\tau )\) is its coefficient matrix across the cut, then \(\rho _R=M_RM_R^\dagger \). This is the physical cut associated with the regional construction of [ 17 , Section IV.C.1 ] , local source lines 2003–2011.

The RVB tensor is \(A=P(\mathbb {1}\otimes \mathbb {1}\otimes Y\otimes Y)\) with \(Y=\left(\begin{smallmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right)\), virtual indices ordered top, right, down, left.

For a physical configuration \(\sigma \) on the vertices, the PEPS state coefficient is obtained by summing over all virtual configurations and, for each such configuration, multiplying the local tensor entries over all vertices.

A PEPS tensor on \(G\) with physical dimension \(d\) consists of a bond dimension \(D_e\) for every edge \(e\), together with a local tensor at each vertex \(v\) whose virtual indices are indexed by the edges incident to \(v\) and whose physical index lies in \(\{ 0,\ldots ,d-1\} \). The local tensor at \(v\) has one physical leg and one virtual leg for each incident edge:

For \(n,m{\gt}1\), the discrete \(n\times m\) torus has vertex set \(\mathbb {Z}/n\mathbb {Z}\times \mathbb {Z}/m\mathbb {Z}\). Its graph joins \((x,y)\) to \((x+1,y)\) and to \((x,y+1)\), with the coordinates read cyclically; every edge is of one of these two forms, so every edge is horizontal or vertical. Translation by \((a,b)\) is the graph automorphism \(\tau _{a,b}(x,y)=(x+a,y+b)\). The induced action on edges sends an edge with unordered endpoint pair \(\{ u,v\} \) to the edge with unordered endpoint pair \(\{ \tau _{a,b}(u),\tau _{a,b}(v)\} \); in particular it preserves the horizontal and vertical orientation classes. For a general graph isomorphism \(\phi :G\simeq G'\), the same endpoint action gives a bijection on edges and a bijection from the edges incident to \(v\) to the edges incident to \(\phi (v)\).

On the torus vertices, retain only consecutive coordinate values without wraparound. The result is the ordinary rectangular grid, a subgraph of the periodic square lattice.

Let \(a^i_{\alpha \beta \gamma \delta }\) be a tensor with physical dimension \(d\) and bond dimension \(D\). Its torus PEPS on the torus of width \(w\) and height \(h\) has bond dimension \(D\) on every edge and, at every site \(v\), the local tensor \(a\) with its four virtual indices read on the top, right, down and left legs of \(v\) [ 17 , Appendix A, “Two dimensions: PEPS” ] . The right leg of \(v\) is the left leg of \(v+(1,0)\), and the top leg of \(v\) is the down leg of \(v+(0,1)\).

A bond-dimension function on the torus is orientation-uniform with horizontal dimension \(D_h\) and vertical dimension \(D_v\) if \(D_{e_h}=D_h\) for every horizontal edge \(e_h\) and \(D_{e_v}=D_v\) for every vertical edge \(e_v\). The reference horizontal edge is the right edge at the origin, and the reference vertical edge is the up edge at the origin.

The entries of the AKLT tensor are

and the span of the vectors \(s\mapsto A^s_{\alpha \beta \gamma \delta }\) over all virtual indices is the range of \(\Pi _{\mathrm{sym}}\).

\(\Pi _{\mathrm{sym}}\) is a Hermitian idempotent commuting with \(U^{\otimes 4}\) for every \(2\times 2\) matrix \(U\). Its entries are \((\Pi _{\mathrm{sym}})_{s,t}=\binom {4}{|s|}^{-1}\) if \(|s|=|t|\) and \(0\) otherwise, and its range, the spin-\(2\) subspace, has dimension five.

\(A^{s}_{urdl}\neq 0\) if and only if exactly one of \(u,r,d,l\) differs from \(2\), that index being \(s\) if it is \(u\) or \(r\) and \(1-s\) if it is \(d\) or \(l\). The nonzero entries equal \(-1\) when the down or left index is \(1\) and \(+1\) otherwise.

Suppose the native square torus has periods \(W,H\geq 3\). Let \(R\) be the coordinate rectangle

with \(x_0+w\leq W\), \(y_0+h\leq H\), \(w{\lt}W\), and \(h{\lt}H\). Then there are \(2h\) horizontal and \(2w\) vertical crossing bonds, and hence

In particular a bounded square of side length \(L\) has \(4L\) crossing bonds. The coordinate intervals may start at zero or end at their torus periods; the formula includes the cyclic seam bonds in these cases. These are the actual lattice bonds used in the rectangular boundary construction of [ 235 , Theorem 6.9 ] . The period restriction comes from the finite simple-graph model; see [ 96 ] .

If a function \(f\) on the sites of the torus satisfies \(f(x+1,y)=f(x,y)\) and \(f(x,y+1)=f(x,y)\) for all \((x,y)\), then \(f\) is constant.

Let \(w,h\ge 3\). The right edges and the up edges of the sites are pairwise distinct and exhaust the edges of the torus, and the coefficient of the torus PEPS of \(a\) is

the sum running over all assignments of bond indices \(h_{(x,y)}\) to the horizontal edges from \((x,y)\) to \((x+1,y)\) and \(v_{(x,y)}\) to the vertical edges from \((x,y)\) to \((x,y+1)\).

Let \(a\) be a four-leg tensor and \(\alpha \mapsto X_\alpha \) a map from the bond indices to a monoid \(M\) with \(X_tX_r=X_lX_b\) whenever \(a^{s}_{trbl}\neq 0\). Fix a rectangle of \(m\) columns and \(n\) rows of sites of the torus. Consider the string from its upper left to its lower right corner that crosses the left legs of the leftmost column, from top to bottom, and then the down legs of the bottom row, from left to right, and the string that crosses the top legs of the top row and then the right legs of the rightmost column. On every labelling of the bonds whose term in the contraction of the torus PEPS of \(a\) is nonzero the two strings have the same product, and so, for a semiring \(M\) with a \(\mathbb {C}\)-module structure, the contracted network with the first string inserted equals the contracted network with the second string inserted. This is the pulling-through equation of a matrix product operator symmetry [ 17 , Section “MPO and PEPO” ] for strings that are diagonal in the bond basis.

For every \(N{\gt}0\) and every \(s,t\in \{ 0,1\} ^N\), the periodic contraction satisfies

In particular, its basis action is (17).

For every \(N{\gt}0\) and every \(s,t\in \{ 0,1\} ^N\), the periodic operator \(O_N(A)\) of the review’s tensor satisfies

Thus \(O_N(A)=D_NX^{\otimes N}\), and

where \(U_N\) is the periodic operator of Definition 31.3.1.1.

For any finite region, a physical configuration determines at most one nonzero incident-bond assignment. Its open-region coefficient is one precisely when this assignment exists and has the specified boundary; otherwise it is zero. Different boundary columns have disjoint physical supports, so the region Gram matrix is diagonal in the native boundary basis. No labels on edges wholly outside the region are summed.

Each local tensor and each actual region witness coefficient equals one. For every \(c\) there is a physical configuration \(\sigma _c\) with \(F_R(\sigma _c,c')=\delta _{c,c'}\). In particular, \(F_R\) is injective.

If a coefficient of \(A_R|\mu \rangle \) is nonzero, then \(E_R\mu =\iota (c)\) for some \(c\in \{ 0,1\} ^P\).

The trace of \(\rho _R\) is nonzero. Dividing by that trace preserves its range and rank \(2^P\).

For every effective boundary column there is a physical configuration on the complement whose closed-state slice is exactly that column. Consequently the actual unnormalized reduced density satisfies

On the faithful physical boundary coordinates,

This numbering is a bijection from \(\{ 0,\ldots ,P-1\} \) to the actual crossing bonds. Its cyclic successor is ordinary addition in \(\mathbb Z/P\mathbb Z\). Every bond has the specified endpoint in \(R\), the perimeter corners are distinct in the torus, and \(P\) is even. The assertions include rectangles meeting a coordinate seam.

The image of the actual region map is exactly the span of the columns of \(F_R\), and its dimension is \(2^P\).

The review’s printed CZX matrix product operator \(O(A)\) satisfies \(U_RF_R=F_RO(A)\) on the actual rectangle image.

Let \(U_R\) apply the CZX on-site symmetry at each site of \(R\). Then

This identity holds for the actual open contraction of every finite region. Reversal of a native bond pair preserves its phase and commutes with its simultaneous bit flip.

On a torus with \(w,h\ge 3\) the AKLT PEPS has coefficient

the sum running over the spin-\(\tfrac 12\) labels \(\tau _v\in \{ 0,1\} ^4\) of the four legs of every site. The factor \(Y_{b,a}\) is the coefficient of \(|a\rangle |b\rangle \) in the singlet \(|01\rangle -|10\rangle \), so the state is \(\bigotimes _v\Pi _{\mathrm{sym}}\) applied to a singlet on every edge.

Regard \(A\) as the map from the virtual space \((\mathbb {C}^2)^{\otimes 4}\) to the physical space. For every \(U\in SU(2)\),

For an arbitrary \(2\times 2\) matrix \(U\) the same identity holds with \(\bar U\) replaced by \((\operatorname {adj}U)^{T}\).

On a torus of width and height at least three the classical-model PEPS has coefficient \(\operatorname{Coeff}(\sigma )=e^{-\beta H(\sigma )/2}\), so \(|\operatorname{Coeff}(\sigma )|^{2}=e^{-\beta H(\sigma )}\) is the Gibbs weight of \(\sigma \) [ 173 ] .

Let the torus have width and height at least three and let \(O=\sum _{\sigma }O(\sigma )|\sigma \rangle \langle \sigma |\) be a diagonal observable. For the classical-model PEPS state \(|\psi \rangle =\sum _{\sigma }\operatorname{Coeff}(\sigma )|\sigma \rangle \),

that is, \(\langle \psi |O|\psi \rangle /\langle \psi | \psi \rangle \) is the expectation value of \(O\) in the Gibbs state \(e^{-\beta H}/Z\) [ 17 , Appendix A, “PEPS from classical models” ] .

On a torus with \(w,h\ge 3\) and \(N=wh\), the printed cluster PEPS is \(2^{-N/2}\) times the normalized controlled-\(Z\) circuit state and is nonzero. Its span is precisely the common \(+1\) eigenspace of

On a torus with \(w,h\ge 3\) and \(N=wh\) sites the cluster PEPS has coefficient

For every \(N{\gt}0\), the chain of boundary legs determines \(c\), so \(\iota \) is an isometry, \(\iota ^\dagger \iota =\mathbb {1}\). It satisfies \(V^{\otimes N}\iota =\iota \, U_N\) and \(\iota ^\dagger V^{\otimes N}\iota =U_N\), where \(U_N|c\rangle =(-1)^{\sum _mc_mc_{m+1}}|\bar c\rangle \) is the CZX matrix product unitary of Theorem 31.3.1.2; in terms of the review’s operator \(O(A)\) with the printed matrices, \(V^{\otimes N}\iota =(-1)^N\iota \, O(A)\) [ 17 , Appendix A, “The CZX MPU” ] , [ 11 ] . For actual positive proper bounded rectangles, the physical region image and this cyclic boundary action are identified in Section 25.5, together with the actual reduced-density support and rank, preserved by trace normalization. Nonzero spectral values, entropy, isometric normalization of the physical boundary map, and nonrectangular regions remain outside that result [ 164 ] .

On a torus with \(w,h\ge 3\) the CZX PEPS is invariant under \(U_{CZX}^{\otimes V}\) [ 17 , Appendix A, “The CZX model” ] , [ 11 ] .

On a torus with \(w,h\ge 3\) the CZX PEPS has coefficient

where \((i_v,j_v,k_v,l_v)\) are the four qubits of \(\sigma (v)\) and the factor at \(v\) is the coefficient of \(|0000\rangle +|1111\rangle \) on the four qubits around the plaquette at the upper right corner of \(v\) [ 17 , Appendix A, “The CZX model” ] , [ 11 ] .

Applying \(U_{CZX}\) to the physical index of the CZX tensor equals applying \(V\) to each of its four virtual legs [ 17 , Section III.B, “Non-injective PEPS: SPT phases” ] : writing \(A^s_{trbl}\) for the entry of the CZX tensor with physical index \(s\) and top, right, bottom and left legs \(t,r,b,l\),

For \(d\ge 2\), the four-leg GHZ tensor is not injective. For \(d\ge 2\) and \(w,h\ge 2\), the simple-graph GHZ PEPS is not vertex-injective.

Every permutation \(\pi \) of the \(d\) labels acts on the GHZ site tensor by

For every positive \(w,h\), the contraction in (1) is unchanged when \(\sigma \) is replaced by \(\pi \circ \sigma \). For \(w,h\ge 2\), the simple-graph GHZ PEPS therefore satisfies \(\operatorname{Coeff}(\pi \circ \sigma )=\operatorname{Coeff}(\sigma )\). In particular, for \(d\ge 1\) the on-site \(\mathbb {Z}_d\) shift is a symmetry, and its virtual action is the same shift on every bond.

On every torus with \(w,h\ge 1\), the GHZ tensor generates the unnormalized state \(\sum _{i=0}^{d-1}|i,i,\ldots ,i\rangle \) [ 17 , Appendix A, “The GHZ state” ] . With horizontal and vertical assignments of bond indices \(h_{(x,y)},v_{(x,y)}\) as in Lemma 25.1.3, its coefficient is

For \(w,h\ge 2\), the simple-graph GHZ PEPS has the same coefficients.

For the loop-free bond family of Definition 25.3.3, \(K_v\psi =\psi \) for every \(v\) if and only if \(\psi =cg\) for some \(c\in \mathbb {C}\).

For every finite PEPS graph, region \(R\), and pair of physical configurations \(\sigma ,\tau \) on \(R,R^c\), respectively, \(\psi (\sigma ,\tau )=\sum _\mu T_R(\mu ;\sigma )T_{R^c}(\mu ;\tau )\). This identity includes zero-dimensional virtual spaces.

For a region \(R\), let \(F_v\) be a physical matrix at each vertex, allowing different input and output dimensions at different vertices. Their product matrix is

Then \((FL)_R=F_RL_R\), \((F^\dagger )_R=F_R^\dagger \), and identity local maps give the identity on the physical space of the region. In particular, local isometries give an isometry \(F_R\).

Applying \(F_R\) to the actual open-region tensor is the contraction of the transformed site coefficients:

This is the physical operation on a block used in [ 235 , §6.1, accessible virtual coordinates ] .

The operator \(\rho _R\) is positive semidefinite, and \(\operatorname {ran}\rho _R=\operatorname {ran}M_R \subseteq \mathcal G_R\). No injectivity of the regional tensor is assumed.

Let \(w,h\ge 3\). A dimer covering of the torus is a set \(c\) of edges such that every site lies on exactly one edge of \(c\). The RVB PEPS has coefficient

the sum running over all dimer coverings, and each edge of \(c\) written as \((v,v+e)\) with \(e\in \{ (1,0),(0,1)\} \). The factor \(Y_{\sigma _{v+e},\sigma _v}\) is the coefficient of the singlet \(|01\rangle -|10\rangle \) on the two sites, so the state is the superposition of all nearest-neighbour singlet coverings with the translation-invariant orientation.