6 Wielandt Bound
Quantum Wielandt theory, established by Sanz et al. [ 233 ] , bounds the word length required for products of local MPS matrices to span the full matrix algebra \(M_{D}(\mathbb {C})\). For a primitive normalized tensor with bond dimension \(D\), the uniform vector-spreading index \(q(\mathcal{E}_A)\) and the injectivity length \(\iota (A)\) satisfy \(\iota (A)\le (D^2-\operatorname{kr}(A)+1)D^2\), with the cumulative bound \(T_{D^2}(A)=M_{D}(\mathbb {C})\) serving as an intermediate step.
6.1 Cumulative span
The exact-word identities used in this section are proved in Section C.1. The stabilization and spectral linear algebra behind the cumulative estimates are collected in Section C.2.
The word span at length \(n\) is the subspace
This is [ 233 , Section II ] .
The cumulative span at level \(n\) is \(T_n(A)=S_0(A)+S_1(A)+\cdots +S_n(A)\).
For an MPS tensor \(A\), its fixed-length vector span is the fixed-length vector span of the underlying finite matrix family.
6.1.1 Paper primitivity and indices
For the quantitative bounds, write \(d'=\dim S_1(A)\) for the number of linearly independent Kraus operators. The source definitions and Proposition 3 of [ 233 ] distinguish uniform vector spreading, eventual exact-word spanning, and spectral strong irreducibility.
The Kraus rank is \(\operatorname{kr}(A)=\dim S_1(A)\).
The Kraus family has eventually full Kraus rank if there is a positive integer \(i\) such that \(S_i(A)=M_{D}(\mathbb {C})\).
The transfer map is primitive in the sense of [ 233 ] if there is a positive integer \(q\) such that
for every nonzero \(\varphi \in \mathbb {C}^D\). The same \(q\) works for all nonzero vectors.
The full-Kraus-rank index is
If the set is empty, its value is defined to be zero.
The primitivity index is
If the set is empty, its value is defined to be zero.
An MPS tensor \(A\) is strongly irreducible if there exists a matrix \(\rho {\gt}0\) such that
and \(\mathcal{E}_A\) is irreducible. This formulation includes explicit irreducibility alongside the peripheral spectrum and fixed-point condition of Sanz et al. [ 233 , Proposition 3(c) ] ; see [ 138 ] .
The next seven results use the following notation. For nonnegative integers \(d\) and \(D\), let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices, and let \(\mathcal{E}_K\) be its Kraus map.
If \(\mathcal{E}_A(X)=\mu X\), then \(\mathcal{E}_A^n(X)=\mu ^nX\) for every \(n\geq 0\).
This is the corresponding theorem in the companion quantum-channel volume [ 216 , “Powers on an eigenspace” ] for the eigenspace of \(\mu \).
Let \(A\) be a normalized MPS tensor, and suppose \(\mathcal{E}_A(X)=\mu X\) with \(X\neq 0\), \(\mu \neq 1\), and \(\mu ^p=1\). Then there is a nonzero Hermitian matrix \(H\) such that
Moreover, \(H\) is not positive semidefinite.
Apply Theorem [ 216 , “Hermitian power-fixed point from a finite-order eigenmatrix” ] to the transfer channel. The resulting matrix is nonzero, Hermitian, trace zero, and fixed by \(\mathcal{E}_A^p\). A nonzero positive-semidefinite matrix has positive trace, so this matrix is not positive semidefinite.
Suppose there is a length \(q\geq 0\) such that \(H_q(A,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \in \mathbb {C}^D\). Let \(p{\gt}0\). If \(\rho \) and \(\sigma \) are nonzero positive-semidefinite fixed points of \(\mathcal{E}_A^p\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).
This is Theorem [ 216 , “Uniqueness of power-fixed positive matrices for a Kraus map” ] for the Kraus family formed by the matrices of the tensor.
Under normalization, primitivity in Wolf’s sense is equivalent to eventual full Kraus rank.
Under normalization, primitivity in Wolf’s sense is equivalent to the conjunction of eventual full Kraus rank, normality, and strong irreducibility.
Eventual full Kraus rank is equivalent to normality. The remaining pairwise primitivity equivalences are stated in Theorem 6.1.1.15.
If \(S_i(A)=M_{D}(\mathbb {C})\) for some positive \(i\), then \(H_i(A,\varphi )=\mathbb {C}^D\) for every \(\varphi \neq 0\), so \(A\) is primitive in the sense of Definition 6.1.1.3. No normalization is imposed on \(A\).
Applying every matrix of \(M_{D}(\mathbb {C})\) to a nonzero vector \(\varphi \) gives all of \(\mathbb {C}^D\), and \(S_i(A)\varphi \) is the span of the length-\(i\) words applied to \(\varphi \).
Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(A\) is primitive in the sense of Definition 6.1.1.3, then \(\mathcal{E}_A\) is strongly irreducible.
Regard the matrices of \(A\) as a finite Kraus family. Theorems [ 216 , “Positive-definite fixed point from vector spreading” ] and [ 216 , “Primitivity from vector spreading” ] give a positive-definite fixed point and show that \(1\) is the only peripheral eigenvalue, and Theorem [ 216 , “Irreducibility from vector spreading” ] gives irreducibility of \(\mathcal{E}_A\). Together these are the three conditions of Definition 6.1.1.6.
Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). The following are equivalent:
there is a positive \(q\) for which \(H_q(A,\varphi )=\mathbb {C}^D\) for every \(\varphi \neq 0\);
there is a positive \(i\) for which \(S_i(A)=M_{D}(\mathbb {C})\);
\(\mathcal{E}_A\) is strongly irreducible.
Proposition 3 of [ 233 ] states the same equivalence with item (3) replaced by the literal source condition: \(1\) is the only peripheral eigenvalue, and its one-dimensional eigenspace is generated by a positive-definite fixed point. Here item (3) includes the additional irreducibility condition.
Theorem 6.1.1.13 is (2)\(\Rightarrow \)(1), and Theorem 6.1.1.14 is (1)\(\Rightarrow \)(3).
For (3)\(\Rightarrow \)(2), write \(\mathcal{E}_A^m=P_\rho +(\mathcal{E}_A-P_\rho )^m\). The complementary powers tend to zero. If \(S_m(A)\neq M_{D}(\mathbb {C})\) for every \(m\), choose a nonzero matrix \(B\) orthogonal to \(S_m(A)\). The associated trace pairing vanishes on the left, while positive definiteness gives a lower bound \(c\lVert B\rVert ^2\) for the \(P_\rho \) term and the complementary term is \(o(\lVert B\rVert ^2)\). For large \(m\) this is a contradiction.
Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(A\) has eventually full Kraus rank, or equivalently is normal, then \(\mathcal{E}_A\) is strongly irreducible.
By Theorem 6.1.1.15, \(\iota (A)\) is a positive length with \(S_{\iota (A)}(A)=M_{D}(\mathbb {C})\). Therefore \(H_{\iota (A)}(A,\varphi )=\mathbb {C}^D\) for every \(\varphi \neq 0\), so the defining minimum for \(q(\mathcal{E}_A)\) is at most \(\iota (A)\).
Let \(D{\gt}0\) and suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If there is a positive \(n\) for which \(S_n(A)=M_{D}(\mathbb {C})\), then there is a unique positive definite density matrix \(\rho \in M_{D}(\mathbb {C})\) such that every fixed point of \(\mathcal{E}_A\) is a scalar multiple of \(\rho \).
This is [ 244 , Theorem 6.15 ] .
By Theorem 6.1.1.15, the span hypothesis makes \(\mathcal{E}_A\) strongly irreducible, giving a positive-definite fixed point \(\rho _0\) together with peripheral primitivity (unique peripheral eigenvalue \(1\)) and irreducibility of \(\mathcal{E}_A\). Peripheral primitivity and irreducibility of \(\mathcal{E}_A\) together give the complementary transfer-map gap at \(\rho _0\) (Lemma C.5.5): the restriction of \(\mathcal{E}_A\) off the fixed-point projection at \(\rho _0\) has spectral radius strictly less than one. This gap forces every fixed point of \(\mathcal{E}_A\) to be a scalar multiple of \(\rho _0\); rescaling \(\rho _0\) by the inverse of its (positive) trace gives the unique density-matrix representative \(\rho \).
The source establishes this by showing that \(\mathcal{E}_A^n(|\psi \rangle \! \langle \psi |)\) has full rank for each pure state and ruling out multidimensional fixed spaces via Corollary 6.5 of [ 244 ] . The present argument proceeds via the Proposition 3 equivalence (Theorem 6.1.1.15).
For every \(n\), one has \(T_n(A)\le T_{n+1}(A)\).
The span \(T_{n+1}\) includes all words of length at most \(n+1\) and therefore contains all words of length at most \(n\).
For every \(n\), one has \(\dim T_n(A)\le D^2\).
Since \(T_n(A)\subseteq M_{D}(\mathbb {C})\) and \(\dim M_{D}(\mathbb {C})=D^2\), the result follows.
If \(T_n(A)=T_{n+1}(A)\), then \(T_m(A)=T_n(A)\) for every \(m\ge n\).
If \(T_n=T_{n+1}\), then \(S_{n+1}\subseteq T_n\). Left-multiplying any element of \(S_{n+1}\) by \(A^i\) gives an element of \(S_{n+2}\), so \(S_{n+2}\subseteq T_{n+1}=T_n\). By induction, \(S_m\subseteq T_n\) for every \(m{\gt}n\), hence \(T_m=T_n\).
If \(T_n(A)\neq T_{n+1}(A)\), then \(\dim T_{n+1}(A){\gt}\dim T_n(A)\).
The inclusion \(T_n\le T_{n+1}\) from Lemma 6.1.1.19, together with \(T_n\neq T_{n+1}\), gives strict subspace inclusion and hence strict dimension growth.
One has \(S_L(A)=M_{D}(\mathbb {C})\) if and only if \(A\) is \(L\)-block injective.
Both conditions assert that the span of all products of length \(L\) equals \(M_{D}(\mathbb {C})\).
For every \(n\),
Consequently, if \(L{\gt}0\) and \(A\) is \(L\)-block injective, then \(A\) is \(m\)-block injective for every \(m\ge L\). This combines the block-injectivity discussion in [ 13 , Section II ] with [ 13 , Appendix C.3, Lemma L ] .
Every word of length \(n+1\) factors into its first letter and a word of length \(n\), which proves the displayed equality by bilinearity. Write \(L=n+1\). Since \(S_L(A)=M_{D}(\mathbb {C})\), one has \(S_n(A)\le S_L(A)\). Hence
Iterating this implication gives \(S_m(A)=M_{D}(\mathbb {C})\) for every \(m\ge L\).
6.2 Nonzero trace product
Let \(D{\gt}0\). If \(A\) is normal, then \(T_{D^2}(A)=M_{D}(\mathbb {C})\). This cumulative estimate corresponds to the intermediate step in the proof of [ 233 , Lemma 1 ] .
This is Theorem C.2.4.
Let \(D{\gt}0\). If \(A\) is normal, then \(T_{D^2-d'+1}(A)=M_{D}(\mathbb {C})\), where \(d'=\dim S_1(A)=\operatorname{kr}(A)\). This is the cumulative-span estimate used to prove the sharp trace conclusion in [ 233 , Lemma 1 ] .
If \(T_n(A)\neq T_{n+1}(A)\), then the dimension strictly increases. Since \(S_1(A)\subseteq T_1(A)\) and \(\dim S_1(A)=d'\), one has \(\dim T_1(A)\ge d'\). After at most \(D^2-d'\) additional strict-growth steps, the dimension reaches \(D^2=\dim M_{D}(\mathbb {C})\). Stabilisation before reaching \(M_{D}(\mathbb {C})\) contradicts normality. Hence \(T_{D^2-d'+1}(A)=M_{D}(\mathbb {C})\).
The conclusion of Theorem [ 216 , “Eigenvector extraction from a full cumulative span” ] holds for the matrix family of an MPS tensor whenever its cumulative span at level \(N\) is the full matrix algebra.
Apply the finite-family theorem to the matrices of the tensor.
Let \(D{\gt}0\), suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and assume that \(A\) is primitive in the sense of Definition 6.1.1.3. Then there is a word \(w\) with
This is the sharp trace conclusion of [ 233 , Lemma 1 ] .
Theorem 6.1.1.15 gives eventual full Kraus rank, hence normality. Put \(b=D^2-\operatorname{kr}(A)+1\) and let \(V_n\) be the span of \(A^w\) with \(1\le |w|\le n\). Then \(S_1(A)\subseteq V_1\), so \(\dim V_1\ge \operatorname{kr}(A)\). If \(V_n=V_{n+1}\) for some \(n\ge 1\), then \(V_n\) is closed under left multiplication by every \(A^i\). It therefore contains every positive-length word product; normality forces \(V_n=M_{D}(\mathbb {C})\). Thus, until the full matrix algebra is reached, the dimensions increase strictly, and the bound on \(\dim M_{D}(\mathbb {C})\) gives \(V_b=M_{D}(\mathbb {C})\). Express \(\mathbb {1}\) as a linear combination of the positive-length words defining \(V_b\). Since \(\operatorname{tr}(\mathbb {1})=D\neq 0\), at least one has nonzero trace.
By Theorem 6.2.2, the identity \(\mathbb {1}\) lies in \(T_{D^2-d'+1}(A)\) and is therefore a linear combination of word products of length at most \(D^2-d'+1\). At least one such product has nonzero trace.
If \(A\) is normal and \(D\ge 1\), then there exists a word \(w\) of positive length \(1\le |w|\le D^2-\operatorname{kr}(A)+1\) such that \(\operatorname{tr}(A^w)\neq 0\). This strengthens Theorem 6.2.5 by requiring \(|w|\ge 1\), a feature needed for the blocking argument in the general Wielandt bound below.
Define the positive-level cumulative span \(V_n=\operatorname{span}_{\mathbb {C}}\{ A^w:1\le |w|\le n\} \). The same stabilization-or-strict-growth dichotomy shows \(V_{D^2-d'+1}=M_{D}(\mathbb {C})\): if \(V_n\) stabilises and is not \(M_{D}(\mathbb {C})\), then \(V_n\) is closed under left multiplication by each \(A^i\), so \(S_{N}(A)\subseteq V_n\neq M_{D}(\mathbb {C})\) for every \(N\ge 1\), contradicting normality. Since \(V_{D^2-d'+1}=M_{D}(\mathbb {C})\), the identity \(\mathbb {1}\) lies in \(V_{D^2-d'+1}\) and is therefore a linear combination of word products of positive length. At least one such product has nonzero trace because \(\operatorname{tr}(\mathbb {1})=D\neq 0\) for \(D\ge 1\).
6.3 Eigenvector spreading
Given an MPS tensor \(A\) and a vector \(\varphi \in \mathbb {C}^D\), the cumulative vector span is
This is [ 233 , proof of Lemma 2(a) ] .
Let \(A\) be a normal MPS tensor and let \(A^{i_0}\varphi =\mu \varphi \), where \(\mu \neq 0\) and \(\varphi \neq 0\). Then \(K_{D-1}(A,\varphi )=\mathbb {C}^D\). This cumulative form supplies the dimension-growth step in [ 233 , Lemma 2(a) ] .
The vector span \(K_n\) is monotone and \(\dim K_n\le D\). If \(K_n=K_{n+1}\) for some \(n{\lt}D-1\), the span stabilises by Theorem C.2.7. But \(T_{D^2}(A)=M_{D}(\mathbb {C})\) by Theorem 6.2.1, so Lemma C.2.8 gives \(K_{D^2}(A,\varphi )=\mathbb {C}^D\), contradicting early stabilisation. Hence \(\dim K_n\) grows by at least \(1\) at each step, reaching \(D\) by step \(D-1\).
Proposition 3, in the form of Theorem 6.1.1.15, gives eventual full Kraus rank and hence normality. The cumulative spaces \(K_n(A,\varphi )\) cannot stabilise below \(\mathbb {C}^D\), because every matrix eventually lies in an exact word span. Their dimensions therefore reach \(D\) by \(n=D-1\), giving \(K_{D-1}(A,\varphi )=\mathbb {C}^D\). Finally, the eigenvector relation pads every shorter word to length \(D-1\) by a nonzero power of \(\mu \), so \(K_{D-1}(A,\varphi )=H_{D-1}(A,\varphi )\).
6.4 Quantum Wielandt bounds
Section C.3 proves the one-step augmentation facts used in cases (2) and (3). The blocking identities used in the general case are proved in Section C.4.
Theorem 6.2.1 gives the \(D^2\) cumulative-span bound. The full-rank index bound \(\iota (A)\le (D^2-d'+1)D^2\) ( [ 233 , Theorem 1, case (1) ] ) is proved in Theorem 6.4.8 below. The left-canonical normal blocked-injectivity estimate \(D^4\) is proved below in Theorem 6.5.5. A sharper \(3D^5\) canonical-form block-separation estimate, developed in the full blueprint, combines this \(D^4\) estimate with the direct-sum argument of [ 228 ] .
Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 6.1.1.3. If \(S_1(A)\) contains an invertible matrix \(X\), then
In the terminology of [ 233 ] , this is the case where the first application space \(S_1(A)\) contains an invertible operator. This is the exact word-span form of [ 233 , Theorem 1, case (2) ] .
Adjoin \(X\) as an extra first Kraus operator, obtaining \(\widetilde A\). Theorem 6.1.1.15 gives normality of \(A\), hence normality of \(\widetilde A\); the Kraus rank and all exact word spans are unchanged. The first Kraus operator of \(\widetilde A\) is invertible, so Theorem C.3.5 gives \(S_{D^2-\operatorname{kr}(\widetilde A)+1}(\widetilde A)=M_{D}(\mathbb {C})\). Transfer this equality back to \(A\).
Under the hypotheses of Theorem 6.4.2, one has \(\iota (A)\le D^2-\operatorname{kr}(A)+1\), where \(\iota (A)=\min \{ n\ge 1:S_n(A)=M_{D}(\mathbb {C})\} \).
The word-span equality (2) gives an admissible value in the defining minimum for \(\iota (A)\).
Let \(A\) be a normal MPS tensor with bond dimension \(D\ge 1\). Suppose that a Kraus operator \(A^{i_0}\) is non-invertible and that there exist a nonzero vector \(\varphi \in \mathbb {C}^D\) and a nonzero scalar \(\mu \) such that \(A^{i_0}\varphi =\mu \varphi \). Then, for every \(\psi \in \mathbb {C}^D\), \(|\varphi \rangle \! \langle \psi |\in S_{D^2-D+1}(A)\).
Let \(\mathcal L(M)=A^{i_0}M\), and choose \(r\) so that \(\mathcal L^r\) annihilates the generalized zero-eigenspace of \(\mathcal L\); equivalently, \(\operatorname{range}(\mathcal L^r)\) is its stabilized nonzero-spectral part. The dimension-growth argument gives a level \(n_0\) for which
Since \(A^{i_0}\varphi =\mu \varphi \) and \(\mu \neq 0\), iterating gives \((A^{i_0})^r\varphi =\mu ^r\varphi \) and hence \(\varphi =\mu ^{-r}(A^{i_0})^r\varphi \in \operatorname{range}((A^{i_0})^r)\). Therefore, for every \(\psi \in \mathbb {C}^D\), \(|\varphi \rangle \! \langle \psi |\in S_{r+n_0}(A)\). The corresponding Fitting-index estimate gives \(r+n_0\le D^2-D+1\). Since
one has \(|\varphi \rangle \! \langle \psi | =\mu ^{-1}A^{i_0}|\varphi \rangle \! \langle \psi |\). Thus \(|\varphi \rangle \! \langle \psi |\in S_k(A)\) implies \(|\varphi \rangle \! \langle \psi |\in S_{k+1}(A)\), because left multiplication by \(A^{i_0}\) maps \(S_k(A)\) into \(S_{k+1}(A)\). Iterating from \(k=r+n_0\) and using \(r+n_0\le D^2-D+1\) gives \(|\varphi \rangle \! \langle \psi |\in S_{D^2-D+1}(A)\).
Let \(D{\gt}0\), suppose \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and assume that \(A\) is primitive in the sense of Definition 6.1.1.3. Suppose \(A^{i_0}\) is not invertible and \(A^{i_0}\varphi =\mu \varphi \) with \(\varphi \neq 0\) and \(\mu \neq 0\). Then, for every \(\psi \in \mathbb {C}^D\),
This is the quantitative rank-one conclusion of [ 233 , Lemma 2(b) ] .
Theorem 6.1.1.15 first gives normality. Apply the Fitting decomposition to left multiplication \(\mathcal L(M)=A^{i_0}M\). Choose \(r\) so that \(\mathcal L^r\) kills its generalized zero-eigenspace. Dimension growth then gives \(n_0\) with
Since \((A^{i_0})^r\varphi =\mu ^r\varphi \) and \(\mu \neq 0\), the vector \(\varphi \) lies in the range of \((A^{i_0})^r\). Hence \(|\varphi \rangle \! \langle \psi |\in S_{r+n_0}(A)\) for every \(\psi \). The identity \(|\varphi \rangle \! \langle \psi |=\mu ^{-1}A^{i_0} |\varphi \rangle \! \langle \psi |\) pads this element to the exact length \(D^2-D+1\).
Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 6.1.1.3. If \(S_1(A)\) contains a non-invertible matrix \(X\) with a nonzero eigenvalue, then \(S_{D^2}(A)=M_{D}(\mathbb {C})\). More explicitly, it is enough to have \(X\varphi =\mu \varphi \) with \(\varphi \neq 0\) and \(\mu \neq 0\). In the terminology of [ 233 ] , this is the case where the first application space \(S_1(A)\) contains a non-invertible operator with a nonzero eigenvalue. This is the exact word-span form of [ 233 , Theorem 1, case (3) ] .
Theorem 6.1.1.15 gives normality. Adjoin \(X\) as an extra first Kraus operator. The augmented tensor remains normal and has the same exact word spans as \(A\). The eigenvector relation for \(X\) is now a one-generator eigenvector relation. The eigenvector-spreading argument gives full fixed-length vector span by length \(D-1\), and Lemma 6.4.4 gives the rank-one operators \(|\varphi \rangle \! \langle \psi |\) at length \(D^2-D+1\). Multiplying these two fixed-length spans gives \(S_{D^2}\) for the augmented tensor, hence for \(A\).
Under the hypotheses of Theorem 6.4.6, one has \(\iota (A)\le D^2\).
The equality \(S_{D^2}(A)=M_{D}(\mathbb {C})\) gives an admissible value in the defining minimum for \(\iota (A)\).
Let \(A\) be a normalized MPS tensor with \(D{\gt}0\), \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), that is primitive in the sense of Definition 6.1.1.3. Then
where \(\iota (A)=\min \{ n\ge 1:S_n(A)=M_{D}(\mathbb {C})\} \) is the full-Kraus-rank index. This is [ 233 , Theorem 1, case (1) ] .
Theorem 6.1.1.15 gives normality. By Theorem 6.2.4, there exists a positive-length word \(w\) with \(|w|\le D^2-d'+1\) and \(\operatorname{tr}(A^w)\neq 0\), where \(d'=\operatorname{kr}(A)\). The matrix \(A^w\) has nonzero trace, hence a nonzero eigenvalue \(\mu \) and eigenvector \(\varphi \) by Theorem [ 216 , “Nonzero trace implies eigenvector” ] . Set \(n=|w|\) and let \(B=A^{[n]}\) be the blocked tensor. Then \(B\) is normal by Theorem C.4.1, and the encoding of \(w\) as a single block index gives \(B^{i_0}=A^w\).
If \(A^w\) is invertible, then \(S_{D^2}(B)=M_{D}(\mathbb {C})\), whence \(S_{D^2n}(A)=M_{D}(\mathbb {C})\).
If \(A^w\) is non-invertible, then the eigenvector \(\varphi \) spans a proper subspace, and the rank-one extraction argument gives \(S_{D^2}(B)=M_{D}(\mathbb {C})\), whence \(S_{D^2n}(A)=M_{D}(\mathbb {C})\).
In the invertible case, the invertible Kraus operator \(B^{i_0}\) allows direct dimension growth, giving \(S_{D^2-k_B+1}(B)=M_{D}(\mathbb {C})\), where \(k_B=\operatorname{kr}(B)\). In the non-invertible case, the bound \(S_{D^2}(B)=M_{D}(\mathbb {C})\) follows from the rank-one extraction argument of [ 233 , Lemma 2(b) ] . In both cases, \(\iota (A)\le D^2n\le D^2(D^2-d'+1)\).
The quantitative bound \(\iota (A)\le (D^2-d'+1)D^2\) ( [ 233 , Theorem 1, case (1) ] ) gives an explicit upper bound on the injectivity length. For the Fundamental Theorem of MPS, the qualitative assertion that \(S_N(A)=M_{D}(\mathbb {C})\) for some finite \(N\) suffices.
6.5 Fixed-length matrix spanning
Eigenvector spreading establishes the cumulative vector span \(K_{D-1}(A,\varphi )=\mathbb {C}^D\) ( [ 233 , Lemma 2(a) ] ). To obtain block injectivity, this cumulative span is promoted to an exact fixed-length matrix span \(S_{n_0}(A)=M_{D}(\mathbb {C})\). Supporting word-product results appear in Section C.1, and the blocked fixed-length construction appears in Section C.4.
Suppose \(A^{i_0}\varphi =\mu \varphi \) with \(\mu \neq 0\). Then \(K_n(A,\varphi )=H_n(A,\varphi )\) for every \(n\). In particular, if \(K_n(A,\varphi )=\mathbb {C}^D\), then \(H_n(A,\varphi )=\mathbb {C}^D\).
Any word \(w\) with \(|w|\le n\) can be padded to exact length \(n\) by appending \(k=n-|w|\) copies of the index \(i_0\). The eigenvector relation gives \(A^{w\cdot i_0^k}\varphi =\mu ^kA^w\varphi \), so the padded vector is a nonzero scalar multiple of the original. Hence every generator of \(K_n\) lies in \(H_n\). The reverse inclusion \(H_n\le K_n\) is immediate.
Assume:
\(H_n(A,\varphi )=\mathbb {C}^D\) (length-\(n\) word products applied to \(\varphi \) span all of \(\mathbb {C}^D\)), and
for each standard basis vector \(e_j\), the rank-one operator \(|\varphi \rangle \! \langle e_j|\) lies in \(S_m(A)\).
Then \(S_{n+m}(A)=M_{D}(\mathbb {C})\).
Fix a matrix unit \(E_{ij}\). By hypothesis (1), there exists \(M_i\in S_n(A)\) with \(M_i\varphi =e_i\). By hypothesis (2), \(|\varphi \rangle \! \langle e_j|\in S_m(A)\). Their product satisfies
Lemma C.1.1 therefore gives \(E_{ij}\in S_{n+m}(A)\). Since the matrix units span \(M_{D}(\mathbb {C})\), \(S_{n+m}(A)=M_{D}(\mathbb {C})\).
A tensor \(A\) is \(L\)-block injective if and only if its \(L\)-blocked tensor \(A^{[L]}\) is injective.
The blocked physical alphabet is a reindexing of words of length \(L\). Decoding blocked indices therefore identifies the two spanning families, and hence the two injectivity conditions are equivalent.
If the blocked tensor \(A^{[L]}\) satisfies \(S_n(A^{[L]})=M_{D}(\mathbb {C})\), then \(S_{nL}(A)=M_{D}(\mathbb {C})\). In particular, \(A\) is \(L\)-block injective if and only if \(A^{[L]}\) is one-site injective.
By the blocked-tensor definition (18), each length-\(n\) word in the blocked alphabet \(\{ 0,\ldots ,d^L-1\} \) decodes to a length-\(nL\) word in the original alphabet \(\{ 0,\ldots ,d-1\} \), so \(S_n(A^{[L]})\subseteq S_{nL}(A)\). For \(n=1\), the \(d^L\) blocked letters are indexed by the length-\(L\) words of \(A\), giving \(S_1(A^{[L]})=S_L(A)\), which proves the stated equivalence.
Let \(A\) be a left-canonical normal MPS tensor with bond dimension \(D{\gt}0\). Then there is a positive blocking length \(L\le D^4\) such that the blocked tensor \(A^{[L]}\) is one-site injective.
This is the blocked-injectivity form of the quantum Wielandt input used in the Fundamental Theorem for translation-invariant tensors. It corresponds to the statement in [ 13 , Section II ] that every normal tensor becomes injective after at most \(D^4\) blockings, using the index estimate of [ 233 , Theorem 1 ] . The left-canonical/trace-preserving hypothesis is included because this is the canonical-form context in which the blocked-injectivity input is used.
Normality is eventual full Kraus rank, so Theorem 6.1.1.15 gives paper primitivity. Theorem 6.4.8 then yields
Trace preservation gives a nonzero Kraus operator, hence \(\operatorname{kr}(A)\ge 1\) and the right-hand side is at most \(D^4\). Set \(L=\iota (A)\). By definition, \(L{\gt}0\) and \(S_L(A)=M_{D}(\mathbb {C})\). Lemma 6.5.4 identifies the latter equality with one-site injectivity of \(A^{[L]}\).
Let \(A\) be a left-canonical normal MPS tensor with bond dimension \(D{\gt}0\). Then \(S_{D^4}(A)=M_{D}(\mathbb {C})\); equivalently, \(A\) is \(D^4\)-block injective.
Choose \(L\) with \(0{\lt}L\le D^4\) and \(S_L(A)=M_{D}(\mathbb {C})\), and write \(D^4=L+k\). Induction on \(k\), using positive-length propagation at each step, gives \(S_{L+k}(A)=M_{D}(\mathbb {C})\).
If \(A\) is \(N\)-block injective and \(m{\gt}0\), then \(A\) is \((mN)\)-block injective.
Induct on \(m\). Theorem C.1.2 gives \(S_{mN+N}(A)=S_{mN}(A)S_N(A)\), and the product of the full matrix algebra with itself is again the full matrix algebra.
Let \((A_j)_{j=1}^g\) be a finite family of positive-bond-dimension, left-canonical normal tensors. There is a positive integer \(L\) such that every \(A_j\) is \(L\)-block injective.
Apply Theorem 6.5.5 to each block. The product of the finitely many positive lengths is a common positive multiple, and exact block injectivity persists under positive multiples.
Let \((A_x)_{x\in X}\) be a finite separated normal-canonical family whose blocks have positive virtual bond dimension. Then there is a positive integer \(L\) such that every blocked tensor \(A_x^{[L]}\) is one-site injective. The same conclusion holds simultaneously for two finite separated normal-canonical families whose blocks have positive virtual bond dimension.
Here the family \((A_x)_{x\in X}\) is a basis of normal-canonical representatives (Definition 7.9.1.2) with distinct blocks; see [ 33 ] .
Apply Theorem 6.5.5 to each block. Taking the product of the finitely many resulting positive blocking lengths gives a common positive multiple, and injectivity persists under positive further blocking.
The quantitative rank-one result of Sanz et al. [ 233 , Lemma 2(b) ] is Theorem 6.4.5. Combined with fixed-length vector spreading, it yields matrix spanning via Theorem 6.5.2. Alternatively, the blocked construction in Theorem C.4.6 extracts a rank-one operator directly from normality without tracking explicit exponents.
The cumulative bound \(T_{D^2}(A)=M_{D}(\mathbb {C})\) (Theorem 6.2.1) yields exact-length spanning \(S_N(A)=M_{D}(\mathbb {C})\) of [ 233 , Theorem 1 ] under identity padding or blocked construction (Theorem C.4.6). This exact-length spanning is equivalent to injectivity of \(\Gamma _N\) and one-site injectivity of the blocked tensor \(A^{[N]}\) (Lemma 6.5.4).
The blocked normality and rank-one constructions used below are proved in Section C.4.
6.6 Primitive MPS tensors
Assume \(D{\gt}0\). A tensor \(A\) together with a matrix \(\rho \in M_{D}(\mathbb {C})\) is primitive if
where \(P\) is the fixed-point projection associated to \(\rho \). When the choice of \(\rho \) is irrelevant, we simply say that \(A\) is a primitive MPS tensor. This condition combines a complementary spectral gap with a nonzero This condition combines a complementary spectral gap with a nonzero positive semidefinite fixed point. When \(\rho {\gt}0\), this implies strong irreducibility and uniform vector spreading (Definition 6.1.1.3) by Theorem 6.1.1.15.
The fixed-point projection \(P\) depends only on the ray through \(\rho \): for every nonzero scalar \(c\) with \(\operatorname{tr}(c\rho )\neq 0\), the projection associated to \(c\rho \) equals the projection associated to \(\rho \). Write
for the normalized representative of the ray through a matrix of nonzero trace. Then \(\operatorname{tr}(\sigma )=1\); if \(\rho {\gt}0\) then also \(\sigma {\gt}0\); and if \(A\) is primitive with positive-definite fixed point \(\rho \), then \(A\) is primitive with fixed point \(\sigma \).
Both projections send \(X\) to \(\operatorname{tr}(X)\rho /\operatorname{tr}(\rho )\), so they agree. The trace of \(\sigma \) is \(\operatorname{tr}(\rho )/\operatorname{tr}(\rho )=1\), and a positive-definite matrix stays positive definite under multiplication by the positive real \(\operatorname{tr}(\rho )^{-1}\). For the gap certificate, the trace-preservation clause does not involve \(\rho \); positivity, nonvanishing and the fixed-point equation are preserved by a positive real rescaling; and the complementary spectral radius is unchanged because the two projections coincide.
6.7 Primitivity and normality
Throughout this section, primitivity means the condition of Definition 6.6.1: the tensor is in TP gauge, it has a nonzero PSD fixed point, and the complementary map has spectral radius \({\lt}1\).
6.7.1 From primitivity to normality
The primitivity condition of Definition 6.6.1 combines TP normalization, a nonzero PSD fixed point, and a complementary transfer-map gap. It is connected to normality through the following results.
When the fixed point \(\rho \) in Definition 6.6.1 is strictly positive definite, irreducibility holds and the peripheral spectrum is \(\{ 1\} \), so the tensor is strongly irreducible (Definition 6.1.1.6). By Theorem 6.1.1.15, this is equivalent to uniform vector spreading (Definition 6.1.1.3) and eventual full Kraus rank.
6.7.2 Primitive implies irreducible
By contrapositive. Assume \(A\) has an invariant projection \(P\) (\(P\neq 0\), \(P\neq \mathbb {1}\), \((\mathbb {1}-P)A^iP=0\) for all \(i\)). Set \(\sigma =P\rho P\).
Invariance of \(\sigma \) under \(\mathcal{E}_A^n\): the projection relation \((\mathbb {1}-P)A^iP=0\) implies \((\mathbb {1}-P)\mathcal{E}_A^n(\sigma )=0\) for all \(n\) (by induction on \(n\)).
Convergence: By Lemma C.5.6, \(\mathcal{E}_A^n(\sigma )\to \frac{\operatorname{tr}(\sigma )}{\operatorname{tr}(\rho )}\rho \). Hence \((\mathbb {1}-P)\frac{\operatorname{tr}(\sigma )}{\operatorname{tr}(\rho )}\rho =0\).
Nonzero scalar: Since \(P\neq 0\) and \(\rho \) is positive definite, \(\operatorname{tr}(P\rho P){\gt}0\), so \(\frac{\operatorname{tr}(\sigma )}{\operatorname{tr}(\rho )}\neq 0\). Therefore \((\mathbb {1}-P)\rho =0\).
Contradiction: Since \(\rho \) is positive definite, it is a unit in \(M_{D}(\mathbb {C})\). From \((\mathbb {1}-P)\rho =0\) we get \(\mathbb {1}-P=0\), i.e. \(P=\mathbb {1}\), contradicting \(P\neq \mathbb {1}\).
6.7.3 From irreducibility to full spanning via Burnside’s theorem
The irreducibility condition (Definition 7.1.1.1) is formulated in terms of invariant projections. For Burnside-style arguments it is more natural to work with invariant subspaces of \(\mathbb {C}^D\) under a finite matrix family.
For a finite matrix family \(A\), let \(\operatorname{alg}(A)\) be the unital \(\mathbb {C}\)-subalgebra of \(M_{D}(\mathbb {C})\) generated by the matrices \(\{ A^i\} _{i=0}^{d-1}\): \(\operatorname{alg}(A)=\mathbb {C}\langle A^i:i=0,\ldots ,d-1\rangle \subseteq M_{D}(\mathbb {C})\).
A subspace \(W\le \mathbb {C}^D\) is \(A\)-invariant if \(A^iW\subseteq W\) for every \(i\).
The matrices in \(A\) act irreducibly on \(\mathbb {C}^D\) if every \(A\)-invariant subspace is trivial, i.e. equal to \(0\) or to \(\mathbb {C}^D\).
Let \(W\le \mathbb {C}^D\) be an \(A\)-invariant subspace. In finite dimension the orthogonal projection \(P\) onto \(W\) exists. Invariance of \(W\) implies \((\mathbb {1}-P)A^iP=0\) for all \(i\). Thus \(P\) is a nontrivial invariant projection unless \(W=0\) or \(W=\mathbb {C}^D\), contradicting irreducibility of the tensor.
Assume \(D{\gt}0\). If \(A\) acts irreducibly on \(\mathbb {C}^D\), then the generated unital subalgebra is the full matrix algebra: \(\operatorname{alg}(A)=M_{D}(\mathbb {C})\). This is the complex finite-dimensional case of Burnside’s theorem (equivalently, Jacobson’s density theorem); see [ 206 ] .
Identify \(\operatorname{alg}(A)\) with its image under the algebra equivalence \(M_{D}(\mathbb {C})\simeq _{\mathbb {C}}\operatorname{End}_{\mathbb {C}}(\mathbb {C}^D)\). The irreducible-action hypothesis makes \(\mathbb {C}^D\) a simple module for this algebra. Over the algebraically closed field \(\mathbb {C}\), Schur’s lemma identifies the endomorphisms commuting with the action with scalars, and Jacobson density then forces the action map to be surjective onto all of \(\operatorname{End}_{\mathbb {C}}(\mathbb {C}^D)\).
Burnside’s theorem ensures that every irreducible subalgebra of \(M_{D}(\mathbb {C})\) coincides with the full matrix algebra \(\operatorname{End}_{\mathbb {C}}(\mathbb {C}^D)\), providing the algebraic foundation for matrix spanning.
6.7.4 From primitivity to strong irreducibility and normality
The complementary transfer-map gap of Definition 6.6.1 implies normality (eventual full Kraus rank) via the convergence of transfer-map iterates to the fixed-point projection, without requiring an identity element in \(S_1(A)\).
Let \(D{\gt}0\), let \((A,\rho )\) be primitive in the sense of Definition 6.6.1, and let \(\mathcal{E}_A\) be irreducible. Then \(\rho {\gt}0\).
Let \(P\) be the support projection of \(\rho \). Since \(\mathcal{E}_A\) is completely positive and fixes \(\rho \), the support of \(\mathcal{E}_A(X)\) lies in the support of \(\rho \) for every \(X\ge 0\) supported in \(P\), so \(\mathcal{E}_A\) maps the corner \(PM_{D}(\mathbb {C})P\) into itself. Irreducibility forces \(P=0\) or \(P=\mathbb {1}\), and \(P\neq 0\) because \(\rho \neq 0\); hence \(P=\mathbb {1}\) and \(\rho {\gt}0\).
Irreducibility of \(\mathcal{E}_A\) gives irreducibility of the tensor \(A\) by the last clause of Lemma D.4.9: if \(P\) were a nontrivial invariant projection of \(A\), then \(\mathcal{E}_A\) would map the corner \(PM_{D}(\mathbb {C})P\) into itself. Tensor irreducibility, the trace-preserving normalization, and the peripheral spectrum \(\{ 1\} \) then give primitive fixed-point data \((A,\rho )\) with \(\rho \ge 0\) nonzero by the first clause of the same lemma. Finally, Theorem 6.7.4.1 makes \(\rho \) positive definite because \(\mathcal{E}_A\) is irreducible.
If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then it satisfies the strengthened condition in Definition 6.1.1.6: \(\mathcal{E}_A\) is irreducible, the fixed point \(\rho \) is positive definite, and the peripheral spectrum of \(\mathcal{E}_A\) is \(\{ 1\} \). The source clause is [ 233 , Proposition 3(c) ] ; the definition used here records irreducibility separately.
Lemma C.5.7 gives irreducibility of \(\mathcal{E}_A\). The complementary-gap hypothesis gives peripheral primitivity directly: the only eigenvalue of modulus one is \(1\). Together with the positive-definite fixed point \(\rho \), these are exactly the three conditions in Definition 6.1.1.6. The complementary-gap consequences are proved in Section C.5.
If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then \(A\) is normal (eventually full Kraus rank).
6.8 One-step padding and exact word spans
Cumulative spanning \(T_N(A)=M_{D}(\mathbb {C})\) is weaker than exact-length spanning \(S_L(A)=M_{D}(\mathbb {C})\). When \(\mathbb {1}\in S_1(A)\), products of shorter words pad directly to a common length. Supporting proofs appear in Section C.6.
Suppose
and \(S_L(A)=M_{D}(\mathbb {C})\). Then \(S_m(A)=M_{D}(\mathbb {C})\) for every \(m\ge L\). Equivalently, \(L\)-block injectivity propagates to every larger homogeneous length.
If \(S_n(A)=M_{D}(\mathbb {C})\), then for every \(X\in M_{D}(\mathbb {C})\), substituting (4) gives
Indeed, \((A^i)^\dagger X\in S_n(A)\) because \(S_n(A)=M_{D}(\mathbb {C})\). Thus \(S_{n+1}(A)=M_{D}(\mathbb {C})\). Iterating this implication gives the result for every \(m\ge L\).
If \(\operatorname{alg}(A)=M_{D}(\mathbb {C})\) and \(\mathbb {1}\in S_1(A)\), then \(A\) is normal.
Since \(\operatorname{alg}(A)=M_{D}(\mathbb {C})\), the ascending chain condition for finite-dimensional subspaces gives an \(N\) such that \(T_N(A)=M_{D}(\mathbb {C})\). Apply Theorem C.6.5.
6.9 Block injectivity from TP/primitive/irreducible blocks
Let \(A\) be a left-canonical MPS tensor with \(D{\gt}0\). Suppose that \(A\) is tensor-irreducible and that its transfer map is primitive. Then there exists a positive blocking length \(L\) such that \(A^{[L]}\) is one-site injective.
If \(D=1\), trace preservation gives a nonzero one-site Kraus matrix, and every nonzero scalar matrix spans \(M_{1}(\mathbb {C})\). Hence \(A\) is one-site injective, so Lemma 2.5.4 gives the positive witness \(L=1\). Suppose now that \(D\ge 2\). Theorem 7.11.1.1 gives a block-injectivity witness \(L\). If \(L=0\), Corollary C.1.4 would force \(D=1\), a contradiction. Thus \(L{\gt}0\), and the blocked-chain equivalence identifies this \(L\)-block injectivity with one-site injectivity of the blocked tensor \(A^{[L]}\).