DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Finite open MPS and injective translation invariant tensor

Definition

Let $N \geq 1$ and $d_{1}, \dots, d_{N}, D_{0}, \dots, D_{N} \geq 1$ be integers. Give each site $j$ the Hilbert space $H_{j} = C^{d_{j}}$ with standard inner product $⟨ x, y ⟩ : = \sum_{i = 1}^{d_{j}} x_{i} \overline{y_{i}}$ , linear in its first argument, and its fixed ordered standard basis $(∣ i_{j} ⟩)_{i_{j} = 1}^{d_{j}}$ . Tensor products use the ordered product basis and the product inner product. For every site and physical index choose a matrix $A_{j}^{i_{j}} \in M_{D_{j - 1} \times D_{j}} (C)$ , a boundary column $ℓ \in C^{D_{0}}$ , and a boundary column $r \in C^{D_{N}}$ . Here $M_{m \times n} (C)$ is the space of $m$ -by- $n$ complex matrices and $A^{*}$ denotes conjugate transpose. The associated finite open matrix product state (MPS) is the vector

$∣ ψ_{A} ⟩ : = \sum_{i_{1} = 1}^{d_{1}} \dots \sum_{i_{N} = 1}^{d_{N}} ℓ^{*} A_{1}^{i_{1}} \dots A_{N}^{i_{N}} r ∣ i_{1} ⟩ \otimes \dots \otimes ∣ i_{N} ⟩ \in H_{1} \otimes \dots \otimes H_{N} .$

The dimensions $D_{j}$ are the bond dimensions; the virtual indices between adjacent matrices are matrix indices summed in their product. The zero vector is allowed. A nonzero vector is normalized by division by its norm, and normalization is defined only when $⟨ ψ_{A}, ψ_{A} ⟩ > 0$ . Every vector in $H_{1} \otimes \dots \otimes H_{N}$ has an exact open MPS representation for some choice of bond dimensions, with $D_{0} = D_{N} = 1$ and $ℓ = r = (1)$ ; the internal dimensions are chosen in the construction, rather than prescribed arbitrarily.

For a translation invariant tensor, let $d, D \geq 1$ and use one family $A^{i} \in M_{D} (C)$ for every site. Its transfer map is the completely positive linear map

$E : M_{D} (C) \to M_{D} (C), E (X) : = \sum_{i = 1}^{d} A^{i} X A^{i *} .$

Here complete positivity means that for every integer $q \geq 1$ , the amplified map ${id}_{M_{q}} \otimes E$ sends positive semidefinite matrices to positive semidefinite matrices; a matrix is positive semidefinite when it is Hermitian and has nonnegative quadratic form on every vector.

The tensor is injective at length $L \geq 1$ when the matrices $A^{i_{1}} \dots A^{i_{L}}$ , over all $i_{1}, \dots, i_{L} \in {1, \dots, d}$ , span the full vector space $M_{D} (C)$ . No injectivity or transfer-spectrum condition is assumed by this definition.

For each block length $L \geq 1$ , define the blocked boundary map

$Γ_{L} : M_{D} (C) \to (C^{d})^{\otimes L}, Γ_{L} (X) : = \sum_{i_{1}, \dots, i_{L} = 1}^{d} tr (X A^{i_{1}} \dots A^{i_{L}}) ∣ i_{1} ⟩ \otimes \dots \otimes ∣ i_{L} ⟩,$

where $tr (Y) : = \sum_{a = 1}^{D} Y_{a a}$ . Let $P_{L}$ be the orthogonal projection onto $im Γ_{L}$ , set $h_{L} : = I - P_{L}$ , and, for $N \geq L$ , define the open-chain parent Hamiltonian

$H_{N, L} : = \sum_{j = 1}^{N - L + 1} I^{\otimes (j - 1)} \otimes h_{L} \otimes I^{\otimes (N - j - L + 1)} .$

For a positive semidefinite operator $K$ on a nonzero finite-dimensional inner-product space, define its ground energy by

$E_{0} (K) : = \inf {⟨ K v, v ⟩ : ∥ v ∥ = 1} .$

A ground-state vector of $K$ is a nonzero vector $ψ$ with $⟨ K ψ, ψ ⟩ / ∥ ψ ∥^{2} = E_{0} (K)$ . It is a zero-energy ground-state vector when $K ψ = 0$ ; positivity then gives $E_{0} (K) = 0$ .

Each local term is positive semidefinite with kernel $im Γ_{L}$ , and $H_{N, L}$ annihilates every finite open MPS formed from the same translation invariant tensor. Every nonzero such vector is a zero-energy ground-state vector of $H_{N, L}$ . This definition asserts no uniqueness of that ground state and no spectral-gap bound.

Definition references

Schmidt decomposition

Every finite-dimensional real or complex inner product space has an orthonormal basis

Facts & Assumptions

Given: Finite positive integers $N, d_{j}, D_{j}$ , the standard ordered bases of $H_{j} = C^{d_{j}}$ , site matrices of the displayed dimensions, and boundary vectors of the displayed dimensions.

[F1]

Schmidt decomposition: every vector in a finite-dimensional complex tensor product has a Schmidt expansion with positive singular coefficients and orthonormal factor families; the squared norm is the sum of their squares.

[F2]

Every finite-dimensional real or complex inner product space has an orthonormal basis: every finite-dimensional inner-product space has an orthonormal basis, including the zero-dimensional case with the empty basis.

Proof

1.1givenF1construct

For a nonzero vector, reshape its coefficient array as a matrix with row index $i_{1}$ and column index $(i_{2}, \dots, i_{N})$ . By [F1] this gives an exact finite sum of products of a first-site coefficient and a residual coefficient array. Reshape the residual array at site $2$ with row index $(a_{1}, i_{2})$ and remaining sites as column index, apply [F1] again, and store the resulting left coefficient array as the entries of $A_{2}^{i_{2}}$ with virtual indices $(a_{1}, a_{2})$ . Repeating this finite operation through site $N - 1$ leaves a final array defining $A_{N}^{i_{N}}$ ; every SVD factorization is exact, so the original coefficient of each ordered product-basis vector is the displayed matrix product. For $N = 1$ take the coefficients as scalar matrices. For the zero vector take all bond dimensions equal to $1$ and set the first site matrices to zero. For the nonzero construction choose each internal $D_{j}$ as the corresponding positive Schmidt rank; in both constructions use $D_{0} = D_{N} = 1$ with $ℓ = r = (1)$ . Thus all finite vectors admit a representation with suitably chosen bond dimensions.

1.2givenalgebra

For any integer $q \geq 1$ and positive semidefinite $Y \in M_{q} (C) \otimes M_{D} (C)$ , the amplification is $({i d}_{M_{q}} \otimes E) (Y) = \sum_{i = 1}^{d} (I_{q} \otimes A^{i}) Y (I_{q} \otimes A^{i *})$ . Each summand is positive semidefinite, so every amplification is positive and $E$ is completely positive. The displayed definition also fixes its exact input and output spaces and makes no trace-preserving or unitality assertion.

1.3givenF2constructalgebra

The map $Γ_{L}$ is linear and its image is a finite-dimensional subspace of $(C^{d})^{\otimes L}$ . By [F2], choose an orthonormal basis $u_{1}, \dots, u_{s}$ of that image and define $P_{L} v : = \sum_{a = 1}^{s} ⟨ v, u_{a} ⟩ u_{a}$ . Orthonormality gives $P_{L}^{*} = P_{L}$ and $P_{L}^{2} = P_{L}$ , and $v - P_{L} v$ is orthogonal to every $u_{a}$ . Thus $h_{L} = I - P_{L}$ is the orthogonal projection onto the complement, with $⟨ h_{L} v, v ⟩ = ∥ h_{L} v ∥^{2} \geq 0$ and kernel $im Γ_{L}$ .

2.1givenstep 1.3algebra∎

Fix a finite open MPS made from the same tensor and a block of $L$ consecutive sites. For each fixed physical-basis configuration outside that block, contraction of the outside tensors leaves a row $α$ on the left and a column $β$ on the right. With $X = β α$ , direct coordinates give $tr (X A^{i_{1}} \dots A^{i_{L}}) = \sum_{u, v} β_{u} α_{v} (A^{i_{1}} \dots A^{i_{L}})_{v u} = α A^{i_{1}} \dots A^{i_{L}} β$ , so that block slice is $Γ_{L} (X)$ . The outside basis configurations are orthonormal, so the full vector is a sum of outside basis vectors tensored with block vectors in $im Γ_{L}$ ; the corresponding $h_{L}$ therefore annihilates it. Each term in $H_{N, L}$ is positive semidefinite by step 1.3, and their sum annihilates the MPS. If that vector is nonzero, its normalized vector has quadratic form $0$ , while every unit vector has nonnegative quadratic form; hence $E_{0} (H_{N, L}) = 0$ , and it is a zero-energy ground-state vector by the Definition. The zero vector is annihilated but is excluded from ground-state vectors by definition. For example, if all tensor matrices vanish, then $Γ_{L} = 0$ , $h_{L} = I$ and $H_{N, L} = (N - L + 1) I$ , whose ground energy is positive. No uniqueness or positive-gap conclusion for a nonzero MPS follows from this argument.

Remarks

The diagram records the contracted virtual indices and the uncontracted physical basis legs in the finite open expression above. Horizontal edges carry bond spaces; each downward leg carries the indicated physical basis vector. The coefficient formula and proof are given algebraically above, so the diagram is explanatory only.

Finite open MPS and injective translation invariant tensor

Definition

Definition references

Facts & Assumptions

Proof

Remarks

Depends on

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