Local Unitary Transformations in Quantum Systems

• Local unitary transformations are quantum operations where independent unitary operators act on spatially localized subsystems, establishing an equivalence relation among quantum states.
• They remove short-range entanglement to expose universal properties and enable the classification of distinct quantum phases via tensor network renormalization.
• LU transformations underpin experimental techniques in quantum control and photonics by facilitating high-fidelity, reconfigurable local operations.

A local unitary transformation is a quantum operation in which independent unitary operators act on separate, spatially localized subsystems of a many-body quantum system. Local unitary (LU) transformations are central to the theory of quantum phases, entanglement classification, and tensor network representations of quantum states. They provide the operational notion for distinguishing universal properties insensitive to local (short-range) quantum correlations and have become an organizing principle in condensed matter, quantum information science, and tensor network theory.

1. Definition and General Properties

A local unitary transformation acts as $U = \bigotimes_{i=1}^n U_i$, where each $U_i$ is a unitary operator acting on subsystem $i$ in a multipartite Hilbert space $H_1 \otimes H_2 \otimes \cdots \otimes H_n$. For systems with local degrees of freedom on a lattice, this generalizes to finite-depth quantum circuits, which may include contiguous layers of spatially local (often nearest-neighbor) unitaries.

Key properties include:

• Equivalence Relation: LU transformations define an equivalence relation: two states are LU-equivalent if a local unitary connects them.
• Phase Classification: In condensed matter, two gapped ground states are in the same phase if and only if an LU transformation exists connecting them, i.e., $$|\Phi(1)\rangle = \mathcal{T}\left[e^{-i\int_0^1 dg\, \widetilde{H}(g)}\right] |\Phi(0)\rangle$$, where $\widetilde{H}(g)$ is a sum of strictly local Hermitian operators (Chen et al., 2010).
• Removal of Short-Range Entanglement: LU transformations can remove all local (nonuniversal) entanglement but leave long-range entanglement untouched. LU-invariant properties therefore characterize topological order.

Constant-depth quantum circuits are operationally equivalent to LU transformations in the context of gapped quantum phases: any sequence of non-overlapping local unitaries applied in a bounded number of layers.

2. LU Transformations and Quantum Phases

LU transformations delineate the boundary between universal, long-range many-body entanglement (defining topological order) and transient, short-range entanglement.

• Classification Scheme: Gapped ground states are partitioned into universality classes—phases—according to LU equivalence. This is formalized by the operational notion that if $|\Phi(1)\rangle = U_{\rm LU}|\Phi(0)\rangle$ for some LU $U_{\rm LU}$, then $|\Phi(0)\rangle$ and $|\Phi(1)\rangle$ are in the same phase (Chen et al., 2010).
• Wave Function Renormalization: By repeated application of LU transformations, a generic state (e.g., a tensor-product state representing a symmetric or topologically ordered phase) is renormalized to a fixed-point wave function, which is characterized by its long-range entanglement pattern. For instance, the fixed-point tensors of topologically ordered states such as the toric code obey strict algebraic constraints (e.g., the pentagon identity for $F$-tensors).
• LU Transformation Algorithm: The wave function renormalization procedure consists of two moves:
  • F-move: A local unitary that reconnects parts of the tensor network graph, implemented via an $F$-tensor satisfying unitarity and pentagon-like identities:

  $$\sum_{n,\chi,\delta} F^{ijm',\alpha'\beta'}_{kln,\chi\delta}(F^{ijm,\alpha\beta}_{kln,\chi\delta})^* = \delta_{m\alpha\beta,\,m'\alpha'\beta'}$$

  and

  $$\sum_{t,\eta,\varphi,\kappa}F^{ijm,\alpha\beta}_{knt,\eta\varphi} F^{itn,\varphi\chi}_{lps,\kappa\gamma} F^{jkt,\eta\kappa}_{lsq,\delta\phi} = e^{i\theta_F}\sum_{\epsilon} F^{mkn,\beta\chi}_{lpq,\delta\epsilon} F^{ijm,\alpha\epsilon}_{qps,\phi\gamma}$$ - P-move: A projection onto the support space of a subsystem's reduced density matrix, thereby trimming redundant short-range structures.

States at the RG fixed point are fully specified by algebraic data $(N, F, P, A)$, which uniquely label distinct symmetry-breaking and topologically ordered phases (Chen et al., 2010).

3. LU Invariants and Quantum State Classification

The action of LU transformations preserves certain invariants, which are the basis for classification of quantum states with respect to their nonlocal properties.

• Bipartite Mixed States: LU invariants can be explicitly constructed from the spectral decomposition and matrix representations of eigenvectors. Two states $\rho$ and $\rho'$ are LU-equivalent if and only if all trace invariants built from products of their eigenvector matrices agree:

$$\left\{\text{Tr}\left[\left(A_{i_1}A_{i_1}^\dagger\right)\cdots\left(A_{i_k}A_{i_k}^\dagger\right)\right]\right\}$$

together with moments of the density matrix, provide a complete invariant set (Zhou et al., 2012).

• Multipartite Systems: For multipartite pure states, the singular values of coefficient matrices under all bipartitions provide a complete set of spectral LU invariants, while for mixed states, traces of products of these matrices across eigenstates give operational invariants for LU classification (Zhang et al., 2013). For full-rank mixed states with nondegenerate spectrum, these invariants are necessary and sufficient (Zhang et al., 2013).

• Reduced Density Matrix Method: Invariants constructed from the characteristic polynomials (and hence spectra) of matrices built from reduced density matrices of the quantum state allow efficient distinction of LU inequivalence, particularly valuable in degenerate cases (Wang et al., 2014).

• Structural Approaches: For two- and three-qubit states, LU-equivalence is determined by a finite set of polynomial invariants based on the Bloch representation vectors and tensors, and their inner/outer products. For two-qubit mixed states, only 14 invariants suffice for complete distinction (Sun et al., 2017).

4. LU Transformations in Tensor Networks and Operator Representations

LU transformations play dual roles in tensor network theory, both as transformations on tensor network states and as elements in tensor network representations of unitary operators.

• Classifying Tensor Product States: LU-based wavefunction renormalization “flows” general tensor product states to fixed points in their respective phase. Fixed-point tensors obey nonlinear algebraic equations dictated by the fusion rules, $F$-move constraints, and pentagon identities (Chen et al., 2010).

• Matrix Product and Unitary Network Representations: 1D locality-preserving unitaries admit a matrix product operator (MPO) representation, where local tensors are constrained to be unitary (Şahinoğlu et al., 2017). In this representation, the global unitary defined by the MPO preserves locality if and only if each tensor satisfies "fixed-point" factorization and isometry conditions.

  • Index Theory: The GNVW index, which classifies locality-preserving unitaries and quantum cellular automata, can be extracted from the rank ratio of the local tensor's singular value decompositions in the MPO (Şahinoğlu et al., 2017).
  • General Unitary Networks: Recent developments generalize MPOs to “unitary networks” (UN), where each local tensor is unitary under suitable reshaping and global unitarity is ordered via acyclic tensor contraction. The net "quantum information flow" through a UN matches the GNVW index for 1D quantum cellular automata and can encode non-trivial, topologically protected structure (Xie et al., 23 Aug 2025).

5. LU Operations in Entanglement Theory and State Transformations

LU transformations are instrumental in the operational classification and manipulation of quantum entanglement:

• Universality and Resource Conversion: Entanglement classes are partitioned into LU-equivalence classes, with further sub-classification determined by stochastic local operations and classical communication (SLOCC) for convertible (resource-theoretic) transformations (Neven et al., 2020).
• Implications for Entanglement Catalysis and Multi-State LOCC: Applying LUs across multiple copies of a multipartite state can non-trivially change the SLOCC class of the individual states, enable catalysis, and facilitate conversions not possible for a single copy. Joint LUs can redistribute entanglement, reflecting the non-additivity of entanglement measures in the multi-copy regime (Neven et al., 2020).
• Majorization Conditions and Symmetries: The theory provides explicit criteria (e.g., majorization relations) for when LU transformations effect meaningful conversions in multipartite GHZ-like states, and it specifies all LU symmetries of generalized GHZ states as products of local diagonal unitaries and local permutations (Neven et al., 2020).

6. LU Transformations in Physical and Experimental Systems

Local unitaries are directly implemented in various experimental contexts:

• Quantum Control and Optimal Gate Synthesis: The quantum optimal control problem can often be cast as identifying the control field $\varepsilon(t)$ that generates a prescribed local unitary propagator, with analytical results showing that the landscape is trap-free for controllable systems but convergence resources may scale unfavorably for large Hilbert space dimensions depending on the Hamiltonian's connectivity and dynamical Lie algebra depth (Moore et al., 2010).
• Photonics and Optical Platforms: In systems such as multi-plane light converters (MPLC), arbitrary local unitary transformations can be realized on spatial modes of light beams by combining programmable phase masks and free space propagation. Experimental results demonstrate high-fidelity implementation of arbitrary $2\times2$ unitaries with average transformation fidelity up to $0.85\pm0.03$, validating practical, reconfigurable LU transformations in photonic arrays (Martinez-Becerril et al., 9 Jul 2024).

7. LU Transformations and Topological Order

The role of LU transformations is foundational in distinguishing short-range entangled (symmetry-breaking) phases from long-range entangled (topologically ordered) phases.

• Invariant Matrices and Topological Signatures: The classification of topological phases is operationalized by invariants (e.g., the state-dependent $\tilde{S}$-matrix) that remain unchanged under LU transformations of shallow/finite circuit depth but differ for states in distinct topological phases (Haah, 2014). These invariants are constructed by algebraic operations on local regions and use the properties of projectors associated with anyonic sectors (fusion, braiding statistics, etc.).
• Circuit Depth Bounds: LU transformations of insufficient depth cannot connect ground states with different topological order: local invariants such as the $\tilde{S}$-matrix impose lower bounds on the depth required to map between such states, yielding a criterion based on locality and topological data (Haah, 2014).
• Many-body Entanglement Witnesses: The formalism of locally invisible operators introduced in this context serves as a witness for the existence of long-range entanglement that cannot be generated from a product state via shallow LU circuits (Haah, 2014).

---

In summary, local unitary transformations provide the theoretical and operational scaffold for classifying quantum phases, distinguishing entanglement structures, and implementing quantum information processing tasks across condensed matter, quantum information, and experimental platforms. Their invariance properties underpin both the identification of universal topological signatures and the design of efficient numerical and experimental schemes for quantum state manipulation, renormalization, and operator simulation.