---
title: Matrix Product States
url: https://www.emergentmind.com/topics/matrix-product-states
type: topic
---

# Matrix Product States

A matrix product state (MPS) is a structured, low-parameter ansatz for representing elements of tensor product spaces, most notably wavefunctions of quantum many-body systems on one-dimensional lattices. The formalism provides the foundation of modern tensor network methods in both physics and computer science, including the density matrix renormalization group, and admits extensions to stochastic processes, quantum information, algebraic geometry, and operator algebraic frameworks.

## 1. Definitions, Algebraic Structure, and Parameterization

For a chain of $N$ sites, each with local Hilbert space $\mathbb{C}^d$, an MPS of bond (auxiliary) dimension $D$ is the state
\[
|\psi_N\rangle = \sum_{s_1,\dots,s_N=1}^d \operatorname{Tr}\left[ A^{s_1} A^{s_2} \cdots A^{s_N} \right]\, |s_1 s_2 \cdots s_N\rangle,
\]
where the $A^s$ are $D \times D$ complex matrices, and the trace is over the auxiliary indices. With open boundary conditions, the trace is replaced by boundary vectors or matrices. The bond dimension $D$ controls the representational power, as the maximal Schmidt rank across any bipartition is at most $D$ [2411.04149].

In the language of tensor networks, the MPS is a 1D tensor contraction: each $A^{s_n}$ corresponds to a physical index $s_n$ and two virtual (bond) indices contracted along the chain. MPS with site-dependent tensors $A^{[n]s_n}$ remain within the class, with translation-invariant MPS corresponding to $A^{[n]} \equiv A$ for all $n$ [2212.13806], [1210.2812].

Algebraically, the image of the MPS parameterization forms a low-dimensional (typically nonlinear) algebraic variety within the exponentially large tensor space, governed by classical trace identities and invariant theory (see Section 6) [2212.13806], [1210.2812].

## 2. Canonical Form, Gauge Redundancy, and Geometry

MPS parameterizations are not unique. There exists a gauge freedom: for any invertible $X$, $A^s \mapsto X^{-1}A^s X$ leaves $|\psi_N\rangle$ invariant, up to possible changes in boundary conditions [1210.7710], [1211.3935]. This redundancy underlies a principal fiber bundle structure for the manifold of MPS: the parameter space is the total space, the physical states form the base manifold, and the structure group is $\mathrm{PGL}(D)$ [1210.7710].

Gauge fixing (canonical form) is implemented by imposing local orthogonality constraints:
- Left-canonical: $\sum_{s}(A^s)^\dagger A^s = I$.
- Right-canonical: $\sum_{s} A^s (A^s)^\dagger = I$.

The geometry of the MPS manifold is Kählerian, with a well-defined induced metric (the pullback of the Fubini–Study metric) and explicit formulas for tangent spaces, connections, and curvature tensors [1210.7710]. This structure is crucial for variational optimization (DMRG), time-dependent variational principle (TDVP), and systematic error control.

## 3. Physical Significance: Entanglement, Expressiveness, and Representational Power

The bond dimension $D$ quantifies the amount of entanglement an MPS can express: the entanglement entropy across any bipartition is at most $\log D$ [1803.08220], [2411.04149]. MPS naturally obey an area law, efficiently approximating ground states of gapped 1D local Hamiltonians and some critical systems (modulo logarithmic corrections). For generic states, exact representation requires $D$ scaling exponentially with system size, but most physical states of interest are efficiently approximable, as shown in lattice gauge theory applications [1310.4118]. 

The expressiveness of MPS is characterized algebraically: for translation-invariant $D=2$ MPS on $N$ qubits, the MPS variety saturates at dimension 5 once $N \gtrsim 5$, indicating that most quantum states cannot be represented as low-bond MPS [1210.2812], [2212.13806]. Explicit polynomial equations characterize which states admit an exact MPS representation.

## 4. Computational Methods and Applications

### 4.1 DMRG and Variational Algorithms

The density matrix renormalization group algorithm is the variational optimization of the MPS ansatz. The energy expectation value is minimized over the tensor parameters, typically via alternating least-squares (sweeping) [1310.4118]. The computational cost per sweep scales as $O(dD^3)$, where $d$ is the physical dimension.

Excited states and dynamics can also be addressed within the MPS manifold using tangent-space formalism, which allows systematic construction of time-evolved or excited wavefunctions [1210.7710], [1211.3935].

### 4.2 Infinite-Volume and Operator-Algebraic Framework

Consistency and projectivity conditions ensure the extension of finite MPS to states on the infinite quasi-local $C^*$-algebra. Specifically, the matrices must satisfy
\[
\sum_{s} (A^s)^\dagger A^s = I_D, \quad \sum_{s} (A^s)^\dagger \otimes A^s = \sum_{s} A^s \otimes (A^s)^\dagger,
\]
ensuring the existence and uniqueness of the limiting (infinite-volume) state and correspond to fixed-point equations of quantum channels [2411.04149].

### 4.3 Physical Models and Numerical Benchmarks

Applications include precise simulation of the Schwinger model (lattice QED) [1310.4118], efficient encoding of ground states, and computation of order parameters and excitation spectra. The approach extends to systems with chemical potential, out-of-equilibrium situations, and systems with non-Abelian or dynamically adapted symmetries [1905.01609].

## 5. Extensions: cMPS and Gaussian MPS

### 5.1 Continuous Matrix Product States (cMPS)

The cMPS formalism extends the tensor network paradigm to quantum field theories in one spatial dimension. A cMPS of bond dimension $D$ for bosonic or fermionic fields $\hat\psi(x)$ is
\[
|\chi\rangle = \operatorname{Tr}_{\mathrm{aux}} \left[ \mathcal{P}\exp\left( \int_0^L dx\, \big( Q(x) \otimes I + R(x) \otimes \hat\psi^\dagger(x) \big) \right) \right] | \Omega \rangle,
\]
where $Q(x), R(x)$ are $D\times D$ auxiliary matrices, and $\mathcal{P}$ denotes path-ordering [1002.1824], [1211.3935].

cMPS arise as exact continuum limits of discrete MPS under appropriate scaling, and the formalism enables direct variational optimization for continuum models (e.g., Lieb–Liniger gas), producing results indistinguishable from Bethe ansatz solutions for modest $D$ [1002.1824]. The area law remains valid for cMPS, and the bond dimension controls an effective intrinsic ultraviolet cutoff.

### 5.2 Gaussian MPS (GMPS)

Gaussian MPS generalize MPS to lattices of bosonic modes. Bonds correspond to projected infinitely squeezed EPR states, and onsite maps are Gaussian channels. Any pure translationally invariant Gaussian state can be approximated arbitrarily well by a GMPS of finite bond dimension [1201.3945]. GMPS arise as unique ground states of local quadratic Hamiltonians whose range is determined by the bond structure.

Correlation functions can be computed efficiently via Fourier-space techniques, and exponential decay of correlations is guaranteed in 1D gapped systems.

## 6. Algebraic Geometry, Invariant Theory, and Connections to Stochastic Models

The MPS varieties are low-dimensional algebraic subvarieties indexed by physical and bond dimensions and chain length. The defining ideals are generated by polynomial identities of matrices (e.g., Cayley–Hamilton, trace identities), and for $D=2$, $n=2$, explicit Hilbert series and dimension formulas for spans are known [2212.13806].

Parameter identifiability: For $D=d=2$ MPS, global state reconstruction up to phase is possible from 3- or 4-site reduced density matrices, depending on boundary conditions, and the corresponding rational maps are generically $N$-to-one [1210.2812].

There exists a direct connection between MPS and hidden Markov models (HMM): classical Markov processes map to MPS with positive tensors, and their algebraic parametrizations align, facilitating algorithms and rational parameterizations [1210.2812], [2502.12641]. Every MPS can be viewed as an "observed" process from a quantum hidden Markov chain (EHMM), with a dual construction connecting any MPS to a corresponding EHMM and quantifying the relative entropy divergence between observed and ideal amplitudes [2502.12641].

In classical probability, stochastic MPS (sMPS) represent steady states of non-equilibrium processes and encode "Schmidt-like" coefficients whose Shannon entropy bounds the necessary bond dimension [1003.2545]. The MPS formalism accurately captures steady-state statistics (e.g., asymmetric exclusion process) with manageable bond dimension except at criticality.

## 7. Generalizations, Limitations, and Current Directions

MPS efficiently represent a broad, yet fundamentally restricted, class of quantum states: those satisfying area laws in 1D and possessing only moderate entanglement or correlation length. The approach extends to infinite chains, stochastic and quantum processes, non-exclusion models with unbounded occupation numbers [1707.06123], and quantum systems with adaptive or emergent symmetries [1905.01609].

However, for highly entangled (volume-law) states or systems with long-range correlations, the required bond dimension becomes prohibitive. Algebraic geometry methods rigorously identify the locus of tensor states outside the MPS variety, and property-testing protocols reveal sharp sample complexity separations between product states and nontrivial MPS in quantum information tasks [2201.01824].

Recent work aims to systematize infinite-volume limits via $C^*$-algebraic projectivity conditions [2411.04149], to connect all MPS with quantum Markov models [2502.12641], and to clarify the role of invariants, parameter spaces, and identifiability for practical state and process tomography.

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**References**:  
- "Matrix Product States in Quantum Spin Chains" [2411.04149]  
- "Calculus of continuous matrix product states" [1211.3935]  
- "Continuous Matrix Product States for Quantum Fields" [1002.1824]  
- "Algebraic Geometry of Matrix Product States" [1210.2812]  
- "Matrix product states, geometry, and invariant theory" [2212.13806]  
- "Matrix Product States as Observations of Entangled Hidden Markov Models" [2502.12641]  
- "Stochastic Matrix Product States" [1003.2545]  
- "Matrix Product States for Interacting Particles without Hardcore Constraints" [1707.06123]  
- "Matrix Product States for Lattice Field Theories" [1310.4118]  
- "Gaussian Matrix Product States" [1201.3945]  
- "Matrix Product States: Entanglement, symmetries, and state transformations" [1901.07448]  
- "Testing matrix product states" [2201.01824]  
- "Matrix Product States for Quantum Stochastic Modelling" [1803.08220]  
- "Geometry of Matrix Product States: metric, parallel transport and curvature" [1210.7710]  
- "Matrix Product States with adaptive global symmetries" [1905.01609]  
- "Implementing Entangled States on a Quantum Computer" [1811.09833]  
- "Matrix Product States for Trial Quantum Hall States" [1211.3353]

Source: https://www.emergentmind.com/topics/matrix-product-states