Dependencies

Let \(T\) be a matrix product operator tensor and \(A_x\) a matrix product state tensor, and let \(T\cdot A_x\) be the tensor of the operator applied to the state, \((T\cdot A_x)^i=\sum _jT^{ij}\otimes A_x^j\). If the periodic operators of \(T\) map the periodic vectors of \(A_x\) to those of a normal tensor \(A_y\) of positive bond dimension at every positive length, then \(T\cdot A_x\) reduces to \(A_y\) with a nilpotent remainder. If, at every positive length, they map them to a finite weighted sum \(\sum _s\mu _s^N|V^{(N)}(A_s)\rangle \) over normal tensors of positive bond dimension with nonzero weights \(\mu _s\), then \(T\cdot A_x\) admits a simultaneous reduction to the weighted blocks \(\mu _sA_s\).

If, in the setting of Theorem 29.5.1, some block occurrence \(s\) of positive bond dimension admits no nonzero sitewise right intertwiner \(X\) with \(B^iX=XC_s^i\) for all \(i\), then every compression of \(B\) in that setting has a nonzero remainder, and the unital algebra generated by the \(B^i\) is not closed under conjugate transposition. In particular, the stacked product of the anomalous CZX symmetry with itself, for which both sitewise intertwiner spaces vanish, generates a unital algebra that is not closed under conjugate transposition.

If the stacked product of \(M_\alpha \) and \(M_\beta \) admits the block triangular form of Theorem 29.5.1 for a family \(\{ C_s\} _{s\in S}\) of operator tensors, then \(O_N(M_\alpha )O_N(M_\beta )=\sum _{s\in S}O_N(C_s)\) for every \(N\geq 1\). If the tensor \(T\cdot A_x\) admits the same form for a family \(\{ C_s\} _{s\in S}\) of state tensors, then \(O_N(T)|V^{(N)}(A_x)\rangle =\sum _{s\in S}|V^{(N)}(C_s)\rangle \) for every \(N\geq 1\).

Let \(\{ M_c\} _c\) be matrix product operator tensors of positive bond dimensions \(\chi _c\), each normal when its pair of physical indices is treated as one index, with \(O_N(M_a)O_N(M_b)=\sum _cN_{ab}^cO_N(M_c)\) for every \(N\geq 1\). Then the stacked product of \(M_a\) and \(M_b\) has fusion tensors \(V_{ab}^{c,\mu }\), \(W_{ab}^{c,\mu }\), \(\mu {\lt}N_{ab}^c\), as in Theorem 29.6.1.1, with nilpotency order at most \(\chi _a\chi _b\). Likewise, if \(O_N(T)|V^{(N)}(A_x)\rangle =\sum _yM_y|V^{(N)}(A_y)\rangle \) for every \(N\geq 1\) with normal \(A_y\) of positive bond dimension, then \(T\cdot A_x\) has action tensors onto \(M_y\) copies of each \(A_y\).

Let \((V,W)\) and \((V',W')\) be two reductions of the stacked product \(B\) of \(M_a\) and \(M_b\) onto the same normal tensor \(M_c\), each with a remainder all of whose products of at least \(K\) factors vanish. Then there is \(z\neq 0\) with \(VB^{w}=z\, V'B^{w}\) and \(B^{w}W=z^{-1}B^{w}W'\) for every word \(w\) of length greater than \(2K\). When \(O_N(M_a)O_N(M_b)=O_N(M_c)\) for every \(N\geq 1\), Corollary 29.6.1.2 provides such reductions with \(K=\chi _a\chi _b\).

Let \(\{ U_g\} _{g\in G}\) be a matrix product operator representation of a group \(G\), not necessarily unitary, by tensors \(T_g\) of positive bond dimension whose doubled-index tensors are normal, with \(U_gU_h=U_{gh}\) on periodic chains of every positive length. Injective tensors are normal, so this covers the injective representations of [ 199 ] ; neither unitarity, simplicity, nor \(U_e=\operatorname{id}\) is needed. Then for all \(g,h\in G\) there are fusion tensors \(V\) and \(W\) with \(WV=\operatorname{id}\) and \(W\, (T_gT_h)^{i_1j_1}\cdots (T_gT_h)^{i_Nj_N}\, V =T_{gh}^{i_1j_1}\cdots T_{gh}^{i_Nj_N}\) for every \(N\geq 0\), and the remainder is nilpotent of order at most \(D_gD_h-D_{gh}+1\).

Under the hypotheses of Theorem 29.8.1 there are maps \(V_s\) and \(W_s\) with \(W_sV_t=\delta _{st}\operatorname{id}\), \(W_sB^wV_s=C_s^w\) for every word \(w\), and the sitewise relations \(B^iV_s=V_sC_s^i\) and \(W_sB^i=C_s^iW_s\) for every physical index \(i\) and block occurrence \(s\).

Let \(A\) be a normal tensor (NT) of bond dimension \(D_A\geq 1\), let \(B\) be an arbitrary tensor of bond dimension \(D_B\), and suppose that \(|V^{(N)}(B)\rangle =|V^{(N)}(A)\rangle \) for every \(N\geq 1\). Then there are maps \(V:\mathbb {C}^{D_A}\to \mathbb {C}^{D_B}\) and \(W:\mathbb {C}^{D_B}\to \mathbb {C}^{D_A}\) with \(WV=\operatorname{id}\) such that, for every \(N\geq 0\) and every word of length \(N\),

and the remainder \(R^i=B^i-VA^iW\) generates a nilpotent non-unital algebra. Explicitly, for every word of length \(r=D_B-D_A+1\),

In particular \(D_B\geq D_A\).

Let \(T_a\) be a matrix product operator tensor, \(A_x\) a tensor, and \(A_y\) normal tensors of positive bond dimension that are pairwise not gauge related. Two families of action tensors that split \(T_a\cdot A_x\) over the blocks \(A_y\) with multiplicities \(M_{a,x}^y\), as in (13), differ by a unique \(X_{ax}^y\in \mathrm{GL}_{M_{a,x}^y}(\mathbb {C})\) for each \(y\) in the sense of (14). This is the gauge freedom of action tensors in [ 199 , Section 3.1 ] .

Let \(T_a\), \(T_b\) and \(T_c\) be matrix product operator tensors, with the \(T_c\) normal as tensors on the pair alphabet, of positive bond dimension, and pairwise not gauge related. Two families of fusion tensors that split the stacked product \(T_aT_b\) over the blocks \(T_c\) with multiplicities \(N_{ab}^c\), as in (13), differ by a unique \(Y_{ab}^c\in \mathrm{GL}_{N_{ab}^c}(\mathbb {C})\) for each \(c\) in the sense of (14).

For a multi-block compression with maps \(V_s\) and \(W_s\), let \(K=\bigoplus _{s\in S}\mathbb C^{D_s}\) and define \(L:\mathbb C^{D_B}\to K\) and \(R:K\to \mathbb C^{D_B}\) by \(Lx=(W_sx)_s\) and \(R(y_s)_s=\sum _sV_sy_s\).

Let \(M\) be a finite-dimensional \(F\)-module, \(S\) a finite set of target slots, and \(C_s\), \(s\in S\), a family of tensors. A labelled invariant flag of \(M\) over \((S,C)\) is an invariant chain \(0=H_0\leq H_1\leq \cdots \leq H_r=M\) together with a bijective labelling of the steps \(\{ 1,\ldots ,r\} \) by \(S\sqcup \{ 1,\ldots ,z\} \), for some \(z\geq 0\), such that a step labelled by \(s\in S\) has subquotient \(H_k/H_{k-1}\) isomorphic to the word-algebra module of \(C_s\), and a step labelled by an index in \(\{ 1,\ldots ,z\} \) has one-dimensional subquotient on which every generator acts as zero.

Tensors \(A\) and \(A'\) of bond dimensions \(D\) and \(D'\) are gauge related if there are matrices \(X\in \mathbb {C}^{D\times D'}\) and \(Y\in \mathbb {C}^{D'\times D}\) with \(XY=\operatorname{id}\), \(YX=\operatorname{id}\) and \(A^iX=XA'^i\) for every physical index \(i\), so that \(A'^i=YA^iX\). The two inverse relations force \(D=D'\), and then \(A'\) is gauge equivalent to \(A\) in the sense of Definition 2.4.1, with gauge matrix \(Y=X^{-1}\).

Fix \(d\geq 1\). The word algebra is the free unital associative algebra \(F=\mathbb {C}\langle x_1,\ldots ,x_d\rangle \), realized as the algebra of finite \(\mathbb {C}\)-linear combinations of words in the \(d\) letters under concatenation, so that the words form a basis. The augmentation \(\mathrm{aug}:F\to \mathbb {C}\) is the algebra map sending every generator to zero; on a basis word \(w\), \(\mathrm{aug}(w)=1\) if \(w\) is empty and \(0\) otherwise. The positive ideal is the two-sided ideal \(F_+=\ker (\mathrm{aug})\) of elements with zero constant term.

A tensor \(C=(C^i)_{i=0}^{d-1}\) on \(\mathbb {C}^D\) defines an \(F\)-module structure on \(\mathbb {C}^D\) through the representation \(\rho _C(x_i)=C^i\), extended multiplicatively so that a word \(w\) acts by \(\rho _C(w)=C^w\).

For a finite-dimensional \(F\)-module \(M\), the trace character is the linear functional \(p\mapsto \operatorname{tr}(\rho _M(p))\) on \(F\), and the word trace of a word \(w\) is \(\operatorname{tr}(\rho _M(w))\).

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

For a finite family of natural numbers \((N_i)_{i\in I}\), the common buffer length is

with \(N_i-1\) read as \(0\) when \(N_i=0\). It is positive, satisfies \(N_i\leq L+1\) for every \(i\in I\), and is the least positive integer with this property.

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

An MPS tensor \(A\) has invariant-projector closure if every orthogonal projection \(P\) satisfying \(PA^i=PA^iP\) for all \(i\) also satisfies \(A^iP=PA^iP\) for all \(i\). This is the projector hypothesis in the sufficient condition for canonical form of  [ 13 , Section 2 ] .

A tensor \(A\) is \(L\)-block injective if

An injective tensor is \(1\)-block injective.

The product tensor of two MPO tensors contracts the bra index of the first factor with the ket index of the second and takes the tensor product of the bond spaces:

In the vertical reading of [ 13 , Section 4.5 ] , this joins two vertically read tensors along an original horizontal bond. It is the tensor written \(M_\alpha M_\beta \) in [ 13 , Theorem 4.14(iii) ] .

The operator generated by \(M\) on \(N\) sites is the matrix \(\rho ^{(N)}(M)\) indexed by configurations \(\sigma ,\tau \in \{ 0,\ldots ,d{-}1\} ^N\) and given by

By combining the pair \((i,j)\) into a single physical index in \(\{ 0,\ldots ,d^2-1\} \), one obtains an MPS tensor with physical dimension \(d^2\) and bond dimension \(D\), with \(M^{\mathrm{MPS}}_{(i,j)}=M^{ij}\), identifying \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \) with \(\{ 0,\ldots ,d^2{-}1\} \) via the product encoding.

Let \(A\) and \(B\) have the same physical alphabet and respective bond dimensions \(D_A\) and \(D_B\). A reduction from \(B\) to \(A\) is a pair \(V\in \mathbb {C}^{D_A\times D_B}\) and \(W\in \mathbb {C}^{D_B\times D_A}\) such that

for every word \(\mathbf a\), including the empty word. This is the reduction of [ 221 , Proposition 20 and the following definition ] .

Let \(\widetilde A=(\widetilde A^i)_i\) be injective, with bond dimension \(D_A\), and choose the coefficient dual \(C=(C^i)_i\) from a right inverse to the linear-combination map of \(\widetilde A\). Thus \(C^i_{a'a}\) is the coefficient of \(\widetilde A^i\) in the chosen expansion of the matrix unit \(\lvert a\rangle \! \langle a'\rvert \). For a tensor \(\widetilde B\) of bond dimension \(D_B\), define the mixed-bond matrix on \(\mathbb {C}^{D_B}\otimes \mathbb {C}^{D_A}\) by

The superscript \(T\) is the ordinary transpose, not the conjugate transpose. This is the matrix drawn in [ 221 , proof of Proposition 20, lines 3817–3838 ] .

Let \((V,W)\) be a reduction from \(B\) to \(A\). A natural number \(m\) is an exterior buffer length for \((V,W)\) if

for every nonempty word \(\mathbf c\) and all words \(\mathbf p,\mathbf q\) with \(|\mathbf p|\geq m\) and \(|\mathbf q|\geq m\). Every \(m'\geq m\) is then also an exterior buffer length. This is the exterior fusion equation of [ 197 , lines 1409–1498 ] , with exterior length \(m\geq \ell \) and central length \(|\mathbf c|=n+1\), \(n\geq 0\).

Let \((V,W)\) be a reduction from \(B\) to \(A\). For each physical letter \(i\), define

The residual algebra \(\mathcal R(B,A;V,W)\) is the nonunital subalgebra of \(M_{D_B}(\mathbb {C})\) generated by the matrices \(N^i\). This is the residual algebra of Proposition 21 in [ 221 ] .

A natural number \(L\) is a residual-word nilpotency bound if every word \(N^{i_1}\cdots N^{i_L}\) of exactly \(L\) residual letters is zero. The residual nilpotency length \(\ell (B,A;V,W)\) is the infimum of the set of exact-length bounds. Definition 8 of [ 221 ] instead asks that every residual word of every length \(n\geq L\) vanish; the passage from an exact-length bound to this consequence is stated separately in Theorem 29.11.21.

Let \((V,W)\) be a reduction from \(B\) to \(A\). For a word \(\mathbf w\), set \(\mathcal I(\varnothing )=\mathcal W(\varnothing )=0\) and define recursively

For \(\mathbf i=(i_1,\ldots ,i_n)\), also set

The substitution \(s=r+m\) identifies the second indexing set with \(0\leq r{\lt}s\leq n\). Thus the central \(A\)-word is nonempty, and \(\widehat{\mathcal I}\) is the part with \(r=0\).

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

The matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector

Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write

The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

For tensors \(A\) and \(B\) of possibly different bond dimensions, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

Let \(U,V\) be finite-dimensional \(F\)-modules with \(\operatorname{tr}(\rho _U(p))=\operatorname{tr}(\rho _V(p))\) for every \(p\in F_+\). Then the semisimplifications of \(U\) and \(V\) contain every simple \(F\)-module with the same multiplicity, except possibly the one-dimensional module \(\mathbf0\) on which every generator acts as zero: there are integers \(a,b\geq 0\) and a semisimple module \(M\) without a \(\mathbf0\) summand such that \(U^{\mathrm{ss}}\cong M\oplus \mathbf0^{\oplus a}\) and \(V^{\mathrm{ss}}\cong M\oplus \mathbf0^{\oplus b}\).

Let \(B=(B^i)_i\) act on \(\mathbb {C}^{D_B}\), and choose a composition series \(0=H_0\subset H_1\subset \cdots \subset H_r=\mathbb {C}^{D_B}\) of its word-algebra module. There is \(G\in \mathrm{GL}_{D_B}(\mathbb {C})\) such that every \(T^i=GB^iG^{-1}\) is block upper triangular for the resulting decomposition into \(r\) blocks, and the \(k\)-th diagonal block of \(T^i\) is the action induced by \(B^i\) on the simple subquotient \(H_k/H_{k-1}\).

If the word traces of a finite family of normal tensors of positive bond dimension sum to zero on every nonempty word, the family is empty.

Two \(\mathbb {C}\)-linear functionals on \(F\) agreeing on every nonempty word agree on the positive ideal \(F_+\).

The trace of a length-\(N\) word in the paired indices of \(M\) is the matrix entry of \(O_N(M)\) at the ket and bra configurations specified by the word. Consequently, if two tensors \(M\) and \(\widetilde M\) have equal word traces at every positive length, then \(O_N(M)=O_N(\widetilde M)\) for every \(N\geq 1\). If instead, for a scalar \(c\), every length-\(N\) word trace of \(M\) equals \(c^N\) times the corresponding trace of \(\widetilde M\), then \(O_N(M)=c^NO_N(\widetilde M)\). Rescaling a tensor \(A\) by \(c\) multiplies each length-\(N\) word trace by \(c^N\).

Let \(A\) and \(A'\) be normal tensors of bond dimensions \(D\) and \(D'\), and let \(X\in \mathbb {C}^{D\times D'}\) satisfy \(A^iX=XA'^i\) for every physical index \(i\). Then \(X=0\), or \(A\) and \(A'\) are gauge related in the sense of Definition 29.8.1.1.

If \(C\) is normal on \(\mathbb {C}^D\) with \(D{\gt}0\), its word-algebra module is simple.

For a finite-dimensional \(F\)-module \(M\) and a submodule \(K\leq M\), the word trace of \(M\) is the sum of the word traces of \(K\) and of \(M/K\).

Let \(o\) be a finite index set with block sizes \(n_k\), \(k\in o\), and let \(T^1,\ldots ,T^d\) be block upper triangular for the resulting decomposition of \(M_{\sum _k n_k}(\mathbb {C})\). Write \(T^i_{kk}\) for the \(k\)-th diagonal block of \(T^i\). Then \(T^w\) is block upper triangular for every word \(w\), and its \(k\)-th diagonal block is the word evaluation of the diagonal blocks,

for \(w=(i_1,\ldots ,i_N)\). If \(P_k\) is the coordinate embedding of the \(k\)-th block and \(Q_k\) the coordinate projection onto it, then for every matrix \(M\) on the same space, \(Q_kMP_k=M_{kk}\), and

On the word-algebra module of a tensor \(C\), the word trace of \(w\) is the periodic coefficient \(\operatorname{tr}(C^w)\).

A finite-dimensional simple \(F\)-module on which every generator acts as zero is one-dimensional. It is the module \(\mathbf0\) of Lemma 29.3.1.

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for any word \(w\), \((\zeta A)^w = \zeta ^{|w|} A^w\), where \(|w|\) is the length of \(w\).

A tensor \(A\) is \(L\)-block injective if and only if its \(L\)-blocked tensor \(A^{[L]}\) is injective.

The coefficient dual obeys the exact matrix-unit identity

Consequently,

In particular, pairing \(\widetilde A\) with its own dual gives the rank-one matrix whose entry is \(\delta _{x,a}\delta _{y,a'}\). The reversed order \(C^{i_n}\cdots C^{i_1}\) is forced by the transpose and records the upper-leg orientation in [ 221 , lines 3817–3865 ] .

If \(V,W\) form a reduction from \(B\) to \(A\), then they also form a reduction from \(cB\) to \(cA\) for every \(c\in \mathbb {C}\). If instead they form a reduction from \(B\) to \(cA\), then

If the one-site matrices \(\{ A^i\} _{i=0}^{d-1}\) span \(M_{D}(\mathbb {C})\), then the length-one word products span \(M_{D}(\mathbb {C})\), so \(A\) is \(1\)-block injective.

Let \(M\) be a monoid and let \(S\subseteq M\). Every element \(x\in \langle S\rangle _{\mathrm{sg}}\) of the subsemigroup generated by \(S\) admits a factorization

Conversely, whenever a tensor \(B\) admits the block triangular form of Theorem 29.5.1, clauses (i)–(iii), for a family \((C_s)_{s\in S}\), its periodic data are those of the direct sum: \(\operatorname{tr}(B^w)=\sum _{s\in S}\operatorname{tr}(C_s^w)\) for every nonempty word \(w\), that is, \(|V^{(N)}(B)\rangle =|V^{(N)}(\bigoplus _sC_s)\rangle \) for every \(N\geq 1\). When the blocks are rescaled tensors \(C_s=\mu _sA_s\), this reads \(\operatorname{tr}(B^w)=\sum _{s\in S}\mu _s^{|w|}\operatorname{tr}(A_s^w)\).

Let the bond space of \(B\) be \(\mathbb {C}^m\otimes \mathbb {C}^{D}\) and let \(\Pi _1,\ldots \) be mutually orthogonal rank-one idempotents of \(\mathbb {C}^m\) cut out by an invertible matrix \(U\) (the idempotents \(U^{-1}E_{jj}U\)). If every letter has the form \(B^i=\sum _{s\in S}\mu _s\, \Pi _{j(s)}\otimes C_s^i\) with distinct positions \(j(s)\), then \(B\) admits the block triangular form of Theorem 29.5.1 onto the weighted blocks \(\mu _sC_s\), with \(z=(m-|S|)D\) zero factors, the remainder \(R^i=0\), and sitewise intertwiners \(B^iV_s=V_s\, \mu _sC_s^i\) and \(W_sB^i=\mu _sC_s^iW_s\) for every slot: the extension splits.

In the setting of Theorem 29.5.1, suppose that the remainder vanishes, \(R^i=0\) for every physical index \(i\). Then for every \(i\) and every \(s\in S\),

Every \(\mathbb {C}\)-linear endomorphism of a finite-dimensional simple \(F\)-module \(Q\) is the action of some element of \(F\).

Let \(A=\bigoplus _{s\in S}C_s\) be a finite direct sum of nonzero normal tensors, and suppose that \(V_N(B)=V_N(A)\) for every \(N\geq 1\). If \(\sum _i(B^i)^\dagger B^i=\operatorname{id}\) and \(\sum _iB^i\rho (B^i)^\dagger =\rho \) for some \(\rho {\gt}0\), then \(D_B=\sum _sD_s\) and there are inverse maps \(L,R\) with \(LB^iR=\bigoplus _sC_s^i\) for every physical index \(i\).

Let \(C_s\), \(s\in S\), be a finite family of normal tensors of positive bond dimension, and let \(M\) be a finite-dimensional \(F\)-module whose word traces agree, on every nonempty word, with the sum of the word traces of the \(C_s\). Then \(M\) carries a labelled invariant flag over \((S,C)\).

Let \(M\) be a finite-dimensional \(F\)-module whose word traces are, on every nonempty word, the sum of the word traces of a finite family of simple tensor modules \(C_s\), \(s\in S\). Let \(K\leq M\) have simple quotient \(M/K\) on which some generator acts nontrivially. Then \(M/K\) is isomorphic to \(C_s\) for some \(s\in S\).

Let \(\{ M_\gamma \} _\gamma \) be a finite family of matrix product operator tensors of positive bond dimension, each normal when its pair of physical indices is treated as one index. Suppose that nonzero scalars \(\lambda _\gamma \) satisfy \(O_N(B)=\sum _\gamma \lambda _\gamma ^NO_N(M_\gamma )\) for every \(N\geq 1\). Then for every \(\gamma \) there are maps \(V_\gamma \) and \(W_\gamma \) with \(W_\gamma V_\gamma =\operatorname{id}\) and \(W_\gamma B^{i_1j_1}\cdots B^{i_Nj_N}V_\gamma =\lambda _\gamma ^NM_\gamma ^{i_1j_1}\cdots M_\gamma ^{i_Nj_N}\) for every \(N\geq 1\) and every sequence of index pairs. The pairs may be chosen mutually biorthogonal, and the remainder after all matched sectors are removed is nilpotent.

Let \(A=\bigoplus _{s\in S}C_s\) be a finite direct sum of nonzero normal tensors \(C_s=(C_s^i)_i\) of bond dimension \(D_s\), with repetitions allowed, and let \(B=(B^i)_i\) be an arbitrary tensor of bond dimension \(D_B\). Assume that the periodic vectors of \(A\) and \(B\) agree at every positive length, \(|V^{(N)}(B)\rangle =|V^{(N)}(A)\rangle \) for every \(N{\gt}0\); equivalently, \(\operatorname{tr}(B^w)=\operatorname{tr}(A^w)\) for every nonempty word \(w\). Then there are an integer \(z\geq 0\), an integer \(r=|S|+z\), an invertible matrix \(G\in \mathrm{GL}_{D_B}(\mathbb {C})\), and maps \(V_s:\mathbb {C}^{D_s}\to \mathbb {C}^{D_B}\), \(W_s:\mathbb {C}^{D_B}\to \mathbb {C}^{D_s}\), with the following properties.

1. Every \(GB^iG^{-1}\) is block upper triangular for one common decomposition into \(r\) blocks.

2. Exactly \(|S|\) of the diagonal slots are matched with the target slots: writing \(j(s)\) for the position matched with \(s\), there is \(X_s\in \mathrm{GL}_{D_s}(\mathbb {C})\) with \(\bigl(GB^iG^{-1}\bigr)_{j(s),j(s)}=X_sC_s^iX_s^{-1}\).

3. The remaining \(z\) diagonal slots are one-dimensional zero tensors.

4. The compression pairs are mutually biorthogonal: \(W_sV_t= \operatorname{id}_{D_s}\) if \(s=t\) and \(0\) otherwise.

5. For every nonempty word \(w\), \(W_sB^wV_s=C_s^w\).

6. Writing \(\widehat C^i=\sum _{s\in S}V_sC_s^iW_s\) and \(R^i=B^i-\widehat C^i\), every product \(R^{i_1}R^{i_2}\cdots R^{i_r}=0\): the non-unital algebra generated by the \(R^i\) is nilpotent.

7. \(D_B=\sum _{s\in S}D_s+z\).

Let \(\{ C_c\} _c\) be a finite family of normal tensors of positive bond dimension and let \(N_c\geq 0\) be integers. If a tensor \(B\) of bond dimension \(D\) satisfies \(|V^{(N)}(B)\rangle =\sum _cN_c|V^{(N)}(C_c)\rangle \) for every \(N\geq 1\), then for every pair \((c,\mu )\) with \(\mu {\lt}N_c\) there are maps \(V_{c,\mu }\) and \(W_{c,\mu }\) such that \(V_{c,\mu }W_{d,\nu }=\delta _{cd}\delta _{\mu \nu }\operatorname{id}\), \(V_{c,\mu }B^{i_1}\cdots B^{i_N}W_{c,\mu }=C_c^{i_1}\cdots C_c^{i_N}\) for every \(N\geq 0\), and every product of at least \(D\) matrices of the remainder \(B^i-\sum _{c,\mu }W_{c,\mu }C_c^iV_{c,\mu }\) vanishes.

An endomorphism of a finite-dimensional complex vector space, all of whose positive powers have vanishing trace, is nilpotent.

Suppose that a multi-block compression has zero remainder and that \(\sum _i(B^i)^\dagger B^i=\operatorname{id}\). Then \(\sum _sV_sW_s=\operatorname{id}\), the assembled matrices satisfy \(LR=\operatorname{id}\) and \(RL=\operatorname{id}\), and \(LB^iR=\bigoplus _sC_s^i\) for every physical index \(i\). In particular, \(D_B=\sum _sD_s\) and there are no unmatched zero slots.

Let \(\{ M_\gamma \} _{\gamma \in \Gamma }\) be a finite family of MPO tensors of common physical dimension \(d\), with \(D_\gamma {\gt}0\). Regard the pair of physical indices \((i,j)\) as one MPS index. Assume these MPS tensors are normal and, for distinct labels of equal bond dimension, are not related by \(M_\delta ^{ij}=zXM_\gamma ^{ij}X^{-1}\) for all \(i,j\), with \(z\neq 0\) and \(X\) invertible. Fix \(\alpha ,\beta \in \Gamma \) and write \(O_L(M)\) for the periodic operator generated by \(M\). Suppose \(O_L(M_\alpha )O_L(M_\beta )\) belongs to \(\operatorname{span}_{\mathbb {C}}\{ O_L(M_\gamma ):\gamma \in \Gamma \} \) for every \(L{\gt}0\).

There are complex diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) of sizes \(n_\gamma \geq 0\), with nonzero diagonal entries, such that for every \(L{\gt}0\),

Their sizes satisfy \(\sum _{\gamma \in \Gamma }n_\gamma D_\gamma \leq D_\alpha D_\beta \). Beyond a positive threshold, any coefficients of this operator expansion equal \(c_{\alpha ,\beta ,\gamma }^{(L)} =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\). Any other finite families of nonzero diagonal entries giving such an expansion at all sufficiently large lengths agree target by target, including multiplicities.

This uses the structure-coefficient notation of Ref.  [ 13 , Section 4.5 ] .

Let \(Q\) be a finite-dimensional simple \(F\)-module on which some generator acts nonzero. There is \(u\in F_+\) acting as the identity on \(Q\). If \(T\) is a finite-dimensional simple module not isomorphic to \(Q\), there is \(p\in F_+\) acting as the identity on \(Q\) and as zero on \(T\). More generally, if \(C_s\), \(s\in S\), is a finite family of word-algebra modules of tensors, each simple and none isomorphic to \(Q\), there is \(p\in F_+\) acting as the identity on \(Q\) and as zero on every \(C_s\).

In the setting of Theorem 29.5.1, suppose that the unital algebra generated by the matrices \(B^i\) is closed under conjugate transposition, that is, \((B^i)^\dagger \) lies in the unital algebra generated by the \(B^j\) for every \(i\). Then the word-algebra module of \(B\) is semisimple, and \(B\) admits a block triangular form as in Theorem 29.5.1 whose remainder vanishes: the block triangular form is block diagonal.

Invariant-projector closure makes the word-algebra module semisimple. In particular, suppose that the source tensor in Theorem 29.5.1 satisfies

Then its compression may be chosen with zero remainder and with two-sided sitewise intertwiners as in Corollary 29.8.2. Normalization and faithful stationarity are sufficient hypotheses here; they are not inferred from periodic equality.

Let \(\Lambda \) and \(\Lambda '\) be finite multisets in \(\mathbb {C}\setminus \{ 0\} \). If their power sums agree for all sufficiently large integers \(N\), then \(\Lambda =\Lambda '\), including multiplicities. In particular, if the power sums of \(\Lambda \) vanish for all sufficiently large \(N\), then \(\Lambda \) is empty.

Let \(B\) be an MPS tensor of bond dimension \(D_B\), and let \(\{ M_\gamma \} _{\gamma \in \Gamma }\) be a finite family of normal MPS tensors on the same physical alphabet, with \(D_\gamma {\gt}0\). For distinct \(\gamma ,\delta \) of equal bond dimension, assume there are no \(X\in \mathrm{GL}(D_\gamma ,\mathbb {C})\) and \(z\in \mathbb {C}\setminus \{ 0\} \) such that \(M_\delta ^i=zXM_\gamma ^iX^{-1}\) for every physical index \(i\). Suppose that for every \(N{\gt}0\) there are coefficients \(c_\gamma ^{(N)}\) satisfying

Then there are integers \(n_\gamma \geq 0\) and nonzero complex weights \(\mu _{\gamma k}\), indexed by \(0\leq k{\lt}n_\gamma \), such that for every \(N{\gt}0\),

The multiplicities satisfy \(\sum _{\gamma \in \Gamma }n_\gamma D_\gamma \leq D_B\). There is a positive threshold \(N_0\) such that, for every \(N\geq N_0\), any coefficients satisfying (8) obey \(c_\gamma ^{(N)}=\sum _{k=0}^{n_\gamma -1}\mu _{\gamma k}^{N}\). The weight multisets are unique: any finite families of nonzero weights giving an expansion of the form (9) at all sufficiently large lengths have, for each \(\gamma \), exactly the same entries with the same multiplicities. Empty weight multisets are allowed.

Let \(A=\bigoplus _{k=1}^g\bigoplus _{q=1}^{n_k}\mu _kA_{(k)}\) be a canonical target with each \(A_{(k)}\) normal of bond dimension \(D_k\) and \(\mu _k\in \mathbb {C}^\times \), and suppose \(|V^{(N)}(B)\rangle =|V^{(N)}(A)\rangle \) for every \(N{\gt}0\). For every block label \(k\) and copy \(q\in \{ 1,\ldots ,n_k\} \), there are maps \(V_{k,q}:\mathbb {C}^{D_k}\to \mathbb {C}^{D_B}\) and \(W_{k,q}:\mathbb {C}^{D_B}\to \mathbb {C}^{D_k}\), mutually biorthogonal, with

for every \(N\geq 1\) and every word.

Let the blocks \(C_c\) be normal of positive bond dimension and pairwise not gauge related in the sense of Definition 29.8.1.1, and let \((V,W)\) and \((\widetilde V,\widetilde W)\) be two split compressions of \(B\) as in (13) with the same blocks and multiplicities. Then \(\widetilde V_{c,\mu }W_{e,\nu }=0\) for \(e\neq c\), and \(\widetilde V_{c,\mu }W_{c,\nu }=(Y_c)_{\mu \nu }\operatorname{id}\) for a matrix \(Y_c\in \mathrm{GL}_{N_c}(\mathbb {C})\) whose inverse is given by \(V_{c,\mu }\widetilde W_{c,\nu }=(Y_c^{-1})_{\mu \nu }\operatorname{id}\). The matrices \(Y_c\) are the unique ones with

for every physical index \(i\), so the two compressions differ by \(\bigoplus _c\mathrm{GL}_{N_c}(\mathbb {C})\) on the support of \(B\), with no dressing by words. This is the uniqueness of fusion tensors under the zipper condition in [ 8 ] .

If \(A\) is \(N\)-block injective and \(m{\gt}0\), then \(A\) is \((mN)\)-block injective.

Let \(A\) be left-canonical, and suppose its transfer map has a positive-definite fixed point \(\rho \). Every orthogonal projection \(P\) satisfying \((\mathbb {1}-P)A^iP=0\) for all \(i\) commutes with every \(A^i\). Consequently \(A\) has invariant-projector closure. This is a sufficient stationary-weight criterion, obtained from [ 244 , Proposition 6.11 and Theorem 6.13 ] .

Every MPS tensor \(A\) admits a block-diagonal tensor \(\bigoplus _{k=1}^r A_k\) with each \(A_k\) irreducible, \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _k A_k)\), and \(\sum _{k=1}^rD_k=D\).

For every length \(L\), the operator generated by the product tensor is the product of the generated operators: \(O_L(M\mathbin {\cdot }N)=O_L(M)O_L(N)\).

Let \(A\) and \(B\) have the same physical alphabet and positive target bond dimension \(D_A{\gt}0\). Suppose that \(|V^{(N)}(B)\rangle =|V^{(N)}(A)\rangle \) for every \(N{\gt}0\). If \(A\) is injective, then there exist \(V\in \mathbb {C}^{D_A\times D_B}\) and \(W\in \mathbb {C}^{D_B\times D_A}\) forming a reduction from \(B\) to \(A\). The same conclusion holds when \(A\) is normal.

This is the \(\lambda =1\) equality case of [ 221 , Proposition 20, lines 3815–3938 ] . The unrestricted proportional premise does not imply the unscaled conclusion [ 146 ] .

Let \(A\) and \(B\) have the same physical alphabet and positive target bond dimension \(D_A{\gt}0\). Let \(c\in \mathbb {C}\) be nonzero, and suppose that

If \(A\) is injective, then there exist \(V\in \mathbb {C}^{D_A\times D_B}\) and \(W\in \mathbb {C}^{D_B\times D_A}\) such that

The same conclusion holds when \(A\) is normal.

This is the corrected nonzero-scalar form of [ 221 , Proposition 20 ] . The unscaled conclusion printed there is false for unrestricted \(c\ne 1\), and the case \(c=0\) does not admit this rescaling argument [ 146 ] .

A rectangular reduction satisfies the retraction identity (9) and the word identity (10). Conversely, the all-word identity alone implies \(VW=1_{D_A}\) by taking the empty word. Moreover, the target bond dimension cannot exceed the source bond dimension: \(D_A\leq D_B\).

Let \((V,W)\) be a reduction from \(B\) to \(A\), and write \(\mathbf i=(i_1,\ldots ,i_n)\). Then

These identities require only the reduction. They are precursors to the bound-truncated formulas in Theorem 29.11.22.

Let \((V,W)\) be a reduction from \(B\) to \(A\). If \(N\) is a residual-word nilpotency bound, then \(N-1\) is an exterior buffer length; in particular, if \(\mathcal{V}^+(B)=\mathcal{V}^+(A)\), then \(\ell (B,A;V,W)-1\) is an exterior buffer length.

For every \(L\), the same pair \((V,W)\) is a reduction from \(B^{[L]}\) to \(A^{[L]}\), and for every map \(f\) between physical alphabets it is a reduction from \(B\circ f\) to \(A\circ f\). If \(m\) is an exterior buffer length for \((V,W)\), \(L{\gt}0\), and \(m\leq kL\), then \(k\) is an exterior buffer length for \((V,W)\) as a reduction from \(B^{[L]}\) to \(A^{[L]}\); reindexing both tensors by the same map \(f\) also preserves exterior buffer lengths.

Let \(A\) be normal, let \(B\) have the same physical alphabet, and let \((V,W)\) and \((\widetilde V,\widetilde W)\) be two reductions from \(B\) to \(A\). Suppose that \(N_0\in \mathbb N\) is a residual-word nilpotency bound for both reductions: every word of length \(N_0\) in either residual tensor vanishes. Then there exists a single scalar \(\lambda \in \mathbb {C}\) with \(\lambda \ne 0\) such that, for every word \(\mathbf a\) satisfying

one has

If \(A\) and \(B\) have the same positive-length closed-chain coefficients and the two least residual nilpotency lengths are at most \(N_0\), then \(N_0\) is such a common bound and the same conclusion follows. The direct-bound statement itself requires no equality of closed-chain coefficients.

Following [ 221 , Theorem 22 ] , these boundary-dressed identities hold for \(|\mathbf a|{\gt}2N_0\).

Let \((N_i)_{i\in I}\) be a finite family of natural numbers with common buffer length \(L\), and let \((V,W)\) be a reduction from \(B\) to \(A\) for which \(N_i\) is a residual-word nilpotency bound for some \(i\in I\). Then \(1\) is an exterior buffer length for \((V,W)\) as a reduction from \(B^{[L]}\) to \(A^{[L]}\): (41) holds for the blocked tensors whenever the blocked words \(\mathbf p,\mathbf c,\mathbf q\) are nonempty.

This formalizes the reduction of the buffer length to \(1\) by blocking in [ 197 ] and [ 200 , Theorem 1 footnote ] ; see [ 74 ] .

Let \((V,W)\) be a reduction from \(B\) to \(A\), let \(L\) be any residual-word nilpotency bound, and write \(\mathbf i=(i_1,\ldots ,i_n)\). Then

Here \(n-L\) denotes truncated subtraction, equivalently \(\max (0,n-L)\) in integer notation. The endpoints with a residual word of length exactly \(L\) are retained as zero terms. These are the two contracted formulas of [ 221 ] .

Suppose \(\widetilde A\) and \(\widetilde B\) have equal positive-length MPV coefficients and \(\widetilde A\) is injective. This equality hypothesis is the \(\lambda =1\) specialization of the proportional premise printed in Proposition 20; the unrestricted unscaled conclusion is false [ 146 ] . Equality transports the trace of every nonempty word. For every \(n{\gt}0\),

For arbitrary tensors \(B_0,C_0\), every \(n\geq 0\), and every boundary matrix \(X\), define

If \(B_0\) has bond dimension \(D_B\) and \(C_0\) has bond dimension \(D_A\), its exact cross-sum expansion is

For \(n{\gt}0\), if \(B_0=\widetilde A\), \(C_0=C\), and \(X\in \mathbb {C}^{D_A\times D_A}\), then

The same-alphabet open-boundary contraction is the specialization \(X=B_0^w\). Thus \(X=\widetilde A^w\) gives the target self-evaluation, while equality of the positive-length MPV coefficients gives

For the padded blocked contraction, suppose additionally that the original tensors \(A\) and \(B\) have equal positive-length MPV coefficients. If \(\widetilde A\) and \(\widetilde B\) are obtained by blocking \(p{\gt}0\) original sites, then for every unblocked lower tail word \(w\),

The trace-power identity is the closed contraction at [ 221 , proof of Proposition 20, lines 3839–3865 ] , and the open-boundary identity is the padded contraction at lines 3866–3887. The source next uses a rank-one rewrite and comparison to derive \(VB^{i_1}\cdots B^{i_m}W=A^{i_1}\cdots A^{i_m}\) at lines 3888–3938.

Let \((V,W)\) be a reduction from \(B\) to \(A\), and let \(L\) be any residual-word nilpotency bound. If the central word \(\mathbf c\) is nonempty and the exterior words \(\mathbf p,\mathbf q\) satisfy

then

No closed-chain equality is assumed: only the reduction and the supplied residual bound enter this identity. It follows from the corrected residual expansion by eliminating every term with a sufficiently long residual buffer.

Under the reduction hypothesis, every word satisfies

Combining this recursion with the explicit window sum below gives the corrected residual expansion.

Let \((V,W)\) be a reduction from \(B\) to \(A\), and assume \(\mathcal{V}^+(B)=\mathcal{V}^+(A)\). Then every nonempty residual word has trace zero, every element of \(\mathcal R(B,A;V,W)\) has trace zero, and every element of this algebra is nilpotent.

This establishes the nil-algebra step of [ 221 , Proposition 21 ] in the equality case \(\lambda =1\); see [ 146 ] .

If \(L\) is a residual-word nilpotency bound and \(|\mathbf i|\geq L\), then

This conclusion uses only the assumed bound; it does not require equality of closed chains.

Let \((V,W)\) be a reduction from \(B\) to \(A\), and assume \(\mathcal{V}^+(B)=\mathcal{V}^+(A)\). Then every mixed product of \(D_B\) elements of \(\mathcal R(B,A;V,W)\) vanishes. Consequently \(D_B\) is a residual-word nilpotency bound. The least bound exists, is itself a bound, and satisfies

In particular, \(N^{\mathbf i}=0\) whenever \(|\mathbf i|\geq D_B\). The resulting vanishing of all sufficiently long mixed products is the nilpotency conclusion of Proposition 21 in [ 221 ] . The specific numerical estimate by \(D_B\) follows from the finite-dimensional nil-matrix theorem.

Let \((V,W)\) be a reduction from \(B\) to \(A\). For every word \(\mathbf i=(i_1,\ldots ,i_n)\),

This statement requires only the reduction. The range \(0\leq r{\lt}s\leq n\) ensures nonempty central words; see [ 124 ] .

Let \((V,W)\) be a reduction from \(B\) to \(A\). For arbitrary exterior words \(\mathbf p,\mathbf q\) and every nonempty word \(\mathbf u\),

This strengthens the sandwiched residual identity below by allowing arbitrary exterior \(B\)-words.

For every physical letter \(i\), one has \(N^i\in \mathcal R(B,A;V,W)\).

Let \((V,W)\) be a reduction from \(B\) to \(A\). For every nonempty word \(\mathbf u\),

No equality of closed chains is required. This is the sandwiched residual identity of [ 221 ] .

Under the reduction hypothesis, for every word \(\mathbf i\),

Let \(A\) be a not necessarily unital associative subalgebra of \(n\times n\) matrices over a field. If every matrix in \(A\) is nilpotent, then every ordered product of \(n\) matrices from \(A\) vanishes.

Let \(V\) be a finite-dimensional complex vector space and let \(S\) be a semisimple endomorphism of \(V\). Suppose that \(c\in \mathbb C\) is nonzero and

for every integer \(k{\gt}1\). Then

If \(N\) is nilpotent and commutes with \(S\), then, for every positive integer \(k\),

There is an integer \(J{\gt}0\) such that, for every \(k\geq J\),

This is the generic Jordan–Chevalley step in [ 221 , Proposition 20, lines 3839–3866 ] .

Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^N)=0\) for every integer \(N{\gt}1\), then \(A\) is nilpotent and

The source obtains the all-positive trace hypothesis in [ 16 , proof of Proposition III.3 ] ; its specialization is the elementwise nilpotence assertion used before the nil-matrix theorem. The result above is a stronger generalization because it assumes vanishing only for \(N{\gt}1\). It is useful independently and requires no positivity, normality, or diagonalizability.

For a finite family \(\{ A_j\} _{j\in J}\) of pairwise non-repeated periodic tensors, in the sense that no two are related by a gauge transformation and a unit-modulus scalar, whose periods divide a common positive integer \(p\), there is \(N_0\) such that for every \(N \ge N_0\), the map

Following [ 191 , lines 589–609 ] , this establishes linear independence of the periodic vectors ( [ 93 ] ).

Let \(A\) and \(B\) be periodic tensors. Then at least one of the following alternatives holds:

1. \(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \to 0\) as \(N\to \infty \);

2. \(A\) and \(B\) are related by the repeated-block relation, i.e. they differ by a unit-modulus phase and an invertible gauge.

Let \(f\colon X\to Y\) be a linear map over a division ring \(K\) whose range has dimension one. Then there are a nonzero linear functional \(\varphi \colon X\to K\), a nonzero vector \(y\in Y\), and a vector \(x\in X\) such that

Consequently, if \(I\) and \(J\) are finite index sets, then every rank-one matrix \(M\in K^{I\times J}\) over a field is an outer product of two nonzero vectors: there are \(w\in K^I\) and \(v\in K^J\) such that

In particular, if both indices of \(M\) are the product \(D_B\times D_A\), then there are nonzero rectangular matrices \(W\in K^{D_B\times D_A}\) and \(V\in K^{D_A\times D_B}\) such that

This is the rank-one factorization used in [ 221 , proof of Proposition 20, lines 3866–3936 ] .

If two tensors have the same MPV coefficients at every positive length, then so do their blocked tensors after any positive blocking length.