LU-Invariant Network States

• Local-unitary-invariant network states are quantum states that remain unchanged under independent local unitary operations across multipartite systems.
• They are defined via tensor-network structures using techniques like singular value decomposition, Gram matrices, and hyperdeterminants to capture invariant properties.
• These invariants underpin effective algorithms for classifying entanglement, detecting quantum phases, and characterizing topological order in complex systems.

A local-unitary-invariant (LU-invariant) network state is a quantum state—often structured as a multipartite tensor network or graph—characterized and classified by quantities invariant under the independent action of local unitary (LU) transformations at each constituent subsystem. Invariant theory for quantum networks aims to provide both necessary and sufficient algebraic conditions for local unitary equivalence, thereby underpinning the operational and information-theoretic classification of entanglement and phases in complex quantum systems.

1. Definition and Structure of LU-Invariant Network States

Consider an $N$-partite quantum system with Hilbert space $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$. A pure quantum state $|\psi\rangle \in \mathcal{H}$ or a mixed state $\rho$ is said to be local-unitary-invariant if, for every tuple of local unitaries $U_1 \otimes \cdots \otimes U_N$, the quantities $\mathcal{J}(|\psi\rangle)$ or $\mathcal{J}(\rho)$ of interest remain unchanged, i.e.,

$$\mathcal{J}((U_1 \otimes \cdots \otimes U_N)|\psi\rangle) = \mathcal{J}(|\psi\rangle), \quad \mathcal{J}((U_1 \otimes \cdots \otimes U_N) \rho (U_1 \otimes \cdots \otimes U_N)^\dagger) = \mathcal{J}(\rho).$$

For network-structured states—such as tensor network states, graph states, matrix product states (MPS), or projected entangled pair states (PEPS)—the connectivity dictates natural multipartitions and the associated construction of invariants. The fundamental aim is to classify all physically indistinguishable states under LU actions and to provide algorithms for distinguishing non-equivalent classes (Zhang et al., 2013).

2. Construction of LU Invariants for Pure and Mixed States

Pure-State Invariants

For a pure state $|\psi\rangle$ expanded in the computational basis,

$$|\psi\rangle = \sum_{s_1, \ldots, s_N} \psi_{s_1\cdots s_N} |s_1 \cdots s_N\rangle,$$

one constructs the “flattened” coefficient matrix for a bipartition $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$0,

$\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$1

with $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$2 and $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$3. Under LU, $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$4 transforms via local unitaries on the respective blocks, preserving its singular values. Thus, the singular values of $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$5 (or, equivalently, the spectra of $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$6) are complete LU invariants for pure states. The set

$\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$7

fully characterizes LU-equivalence (Zhang et al., 2013).

Mixed-State Invariants

Letting $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$8 be a mixed state with spectral decomposition $\mathcal{H} = \bigotimes_{j=1}^N \mathbb{C}^{d_j}$9, one generalizes the above procedure by forming, for each bipartition $|\psi\rangle \in \mathcal{H}$0,

$|\psi\rangle \in \mathcal{H}$1

which transforms under LU as $|\psi\rangle \in \mathcal{H}$2. LU invariants are then obtained as

$|\psi\rangle \in \mathcal{H}$3

and products of mixed-component matrices,

$|\psi\rangle \in \mathcal{H}$4

Under full-rank, nondegenerate $|\psi\rangle \in \mathcal{H}$5, these invariants together with the eigenvalues of $|\psi\rangle \in \mathcal{H}$6 form a complete set: $|\psi\rangle \in \mathcal{H}$7 and $|\psi\rangle \in \mathcal{H}$8 are LU-equivalent if and only if their spectra and all traces above coincide for all bipartitions and all $|\psi\rangle \in \mathcal{H}$9 (Zhang et al., 2013).

3. LU Invariants in Tensor-Network and Graph States

The application to quantum networks proceeds by replacing the standard “first $\rho$0 parties vs. rest” cuts by network-specific partitions: one selects connected subsets (nodes or minimal edge-cuts), flattens tensors accordingly to obtain coefficient matrices, and computes singular values (pure states) or matrix traces (mixed states). This method provides a prescription for generating LU invariants in MPS, PEPS, and generic network geometries.

Examples

• GHZ Graph State: For $\rho$1, flattening over $\rho$2 yields a coefficient matrix whose singular values, $\rho$3, are invariant under local unitaries. Full classification requires checking all network cuts (Zhang et al., 2013).
• Matrix Product State: For a 4-site MPS, each bond (partition) is flattened, and the resulting singular values provide the corresponding LU invariants. One iterates over all network cuts to obtain a complete invariant set.

This network-oriented construction ensures that LU invariants are compatible with the underlying connectivity of the tensor network, which is essential for classifying network-generated entanglement (Zhang et al., 2013).

4. Algebraic and Tensor-Network Methods for Invariant Computation

Tensor network methods, such as diagrammatic tensor calculus and SVD decomposition, enable efficient computation of polynomial LU invariants. All polynomial invariants can be written as

$\rho$4

where $\rho$5 is a permutation in the commutant algebra generated by symmetric group actions (Brauer–Procesi theorem) (Biamonte et al., 2012). This approach naturally incorporates Schmidt reductions across network partitions, yielding explicit closed-form invariants in terms of singular values and allowing the direct expression of measures such as Rényi entropies: $\rho$6 where $\rho$7 is itself an LU invariant associated with the chosen network bipartition (Biamonte et al., 2012).

The graphical calculus further facilitates structural insight, enabling simplification via Penrose's string diagrams and diagrammatic SVD, which isolates singular value “loops” and encodes invariance under arbitrary local basis changes (Biamonte et al., 2012).

5. Alternative Invariant Constructions: Gram Matrices and Hyperdeterminants

Alternative algebraic constructions include invariants from Gram matrices and hyperdeterminants that do not depend on a particular pure-state decomposition of $\rho$8 (Zhang et al., 2013, Wang et al., 2014). The Gram-matrix approach constructs, for reduced density matrices $\rho$9 of a subset $U_1 \otimes \cdots \otimes U_N$0: $U_1 \otimes \cdots \otimes U_N$1 where $U_1 \otimes \cdots \otimes U_N$2 are matrix unfoldings of the eigenvectors of $U_1 \otimes \cdots \otimes U_N$3. The coefficients in the characteristic polynomial of $U_1 \otimes \cdots \otimes U_N$4 are LU invariants: $U_1 \otimes \cdots \otimes U_N$5 These invariants retain sensitivity to eigenvector orientation, resolving cases with spectral degeneracies where spectra alone are insufficient (Wang et al., 2014).

For general multipartite mixed states, hyperdeterminant-based invariants provide a decomposition-independent framework. One constructs associated Gram-type tensors $U_1 \otimes \cdots \otimes U_N$6 (for chosen order $U_1 \otimes \cdots \otimes U_N$7), and their hyperdeterminants or characteristic coefficients $U_1 \otimes \cdots \otimes U_N$8, which are invariants under all local unitaries and independent of any particular basis or decomposition (Zhang et al., 2013).

6. Completeness, Sufficiency, and Detectability

For full-rank, nondegenerate mixed states with the same spectrum, equality of all partition-wise invariants (singular values of coefficient matrices or traces of Gram matrix products) is necessary and sufficient for LU equivalence (Zhang et al., 2013). For pure states or mixed states with degeneracies, a complete set may require additional invariants encoding finer structural data, accomplished in practice by including Gram matrix coefficients and hyperdeterminants (Wang et al., 2014, Zhang et al., 2013).

In the context of two- and three-qubit systems, minimal complete sets of invariants can be defined in terms of Bloch vector inner products and triple scalar products derived from correlation data; this approach yields efficient decision criteria for LU equivalence in those dimensions (Sun et al., 2017).

7. Significance for Quantum Phases and Topological Order

Bulk LU invariants serve as tools for phase classification and entanglement detection. For example, in two-dimensional systems with topological order, one can construct an LU-invariant “$U_1 \otimes \cdots \otimes U_N$9-matrix” from a ground state wave function of a commuting-projector Hamiltonian, which coincides (up to normalization) with the modular $\mathcal{J}(|\psi\rangle)$0-matrix of the underlying anyonic theory (Haah, 2014). This matrix is insensitive to finite-depth quantum circuits and thus captures intrinsic topological features of the network state. LU invariants also play a central role in detecting non-trivial many-body entanglement via locally invisible operators, with direct applications to circuit-depth lower bounds for state preparation (Haah, 2014).

Summary Table of LU-Invariant Construction Approaches

Technique	Applicable To	Algebraic Object/Invariants
Coeff. matrix singular values	Pure states	$\mathcal{J}(|\psi\rangle)$1
Traces of mixed-state coefficient matrices	Mixed/full-rank states	$\mathcal{J}(|\psi\rangle)$2, products
Gram matrix characteristic coefficients	(Reduced) density ops	$\mathcal{J}(|\psi\rangle)$3 via $\mathcal{J}(|\psi\rangle)$4
Hyperdeterminants	Arbitrary mixed states	$\mathcal{J}(|\psi\rangle)$5 from Gram hypermatrices
Tensor network contractions	Network-structured	Polynomial invariants via diagrammatic SVD, COPY-fusion
Topological S-matrix invariants	2D topological states	$\mathcal{J}(|\psi\rangle)$6 entry expectation values

These approaches collectively provide a unified, network-aware, decomposition-independent basis for characterizing quantum states up to local unitary equivalence, directly supporting modern quantum information, condensed matter, and computational frameworks (Zhang et al., 2013, Biamonte et al., 2012, Wang et al., 2014, Zhang et al., 2013, Haah, 2014, Sun et al., 2017).