# Matrix product state

A matrix product state of five particles, as drawn with Penrose graphical notation (also known as tensor diagram notation).

Matrix product state (MPS) is a pure quantum state of many particles, written in the following form:

${\displaystyle |\Psi \rangle =\sum _{\{s\}}{\text{Tr}}[A_{1}^{(s_{1})}A_{2}^{(s_{2})}\cdots A_{N}^{(s_{N})}]|s_{1}s_{2}\ldots s_{N}\rangle ,}$

where ${\displaystyle A_{i}^{(s_{i})}}$ are complex, square matrices of order ${\displaystyle \chi }$ (this dimension is called local dimension). Indices ${\displaystyle s_{i}}$ go over states in the computational basis. For qubits, it is ${\displaystyle s_{i}\in \{0,1\}}$. For qudits (d-level systems), it is ${\displaystyle s_{i}\in \{0,1,\ldots ,d-1\}}$.

It is particularly useful for dealing with ground states of one-dimensional quantum spin models (e.g. Heisenberg model (quantum)). The parameter ${\displaystyle \chi }$ is related to the entanglement between particles. In particular, if the state is a product state (i.e. not entangled at all), it can be described as a matrix product state with ${\displaystyle \chi =1}$.

For states that are translationally symmetric, we can choose:

${\displaystyle A_{1}^{(s)}=A_{2}^{(s)}=\cdots =A_{N}^{(s)}\equiv A^{(s)}.}$

In general, every state can be written in the MPS form (with ${\displaystyle \chi }$ growing exponentially with the particle number N). However, MPS are practical when ${\displaystyle \chi }$ is small – for example, does not depend on the particle number. Except for a small number of specific cases (some mentioned in the section Examples), such a thing is not possible, though in many cases it serves as a good approximation.

MPS decomposition is not unique.

Introductions in.[1] and.[2] In the context of finite automata:[3]

## Obtaining MPS

One method to obtain MPS is to use Schmidt decomposition N − 1 times.

## Examples

### Greenberger–Horne–Zeilinger state

Greenberger–Horne–Zeilinger state, which for N particles can be written as superposition of N zeros and N ones

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes N}+|1\rangle ^{\otimes N}}{\sqrt {2}}}}$

can be expressed as a Matrix Product State, up to normalization, with

${\displaystyle A^{(0)}={\begin{bmatrix}1&0\\0&0\end{bmatrix}}\quad A^{(1)}={\begin{bmatrix}0&0\\0&1\end{bmatrix}},}$

or equivalently, using notation from:[3]

${\displaystyle A={\begin{bmatrix}|0\rangle &0\\0&|1\rangle \end{bmatrix}}.}$

This notation uses matrices with entries being wave functions (instead of complex numbers), and when multiplying matrices using tensor product for its entries (instead of product of two complex numbers). Such matrix is constructed as

${\displaystyle A\equiv |0\rangle A^{(0)}+|1\rangle A^{(1)}+\ldots +|d-1\rangle A^{(d-1)}.}$

Note that tensor product is not commutative.

In this particular example, a product of two A matrices is:

${\displaystyle AA={\begin{bmatrix}|00\rangle &0\\0&|11\rangle \end{bmatrix}}.}$

### W state

W state, i.e. a being symmetric superposition of a single one among. Even through the state is permutation-symmetric, its simplest MPS representation is not.[1] For example:

${\displaystyle A_{1}={\begin{bmatrix}|0\rangle &0\\|0\rangle &|1\rangle \end{bmatrix}}\quad A_{2}={\begin{bmatrix}|0\rangle &|1\rangle \\0&|0\rangle \end{bmatrix}}\quad A_{3}={\begin{bmatrix}|1\rangle &0\\0&|0\rangle \end{bmatrix}}.}$

### AKLT model

The AKLT ground state wavefunction, which is the historical example of MPS approach:,[4] corresponds to the choice[5]

${\displaystyle A^{+}={\sqrt {\frac {2}{3}}}\ \sigma ^{+}={\begin{bmatrix}0&{\sqrt {2/3}}\\0&0\end{bmatrix}}}$
${\displaystyle A^{0}={\frac {-1}{\sqrt {3}}}\ \sigma ^{z}={\begin{bmatrix}-1/{\sqrt {3}}&0\\0&1/{\sqrt {3}}\end{bmatrix}}}$
${\displaystyle A^{-}=-{\sqrt {\frac {2}{3}}}\ \sigma ^{-}={\begin{bmatrix}0&0\\-{\sqrt {2/3}}&0\end{bmatrix}}}$

where the ${\displaystyle \sigma {\text{'s}}}$ are Pauli matrices, or

${\displaystyle A={\frac {1}{\sqrt {3}}}{\begin{bmatrix}-|0\rangle &{\sqrt {2}}|+\rangle \\-{\sqrt {2}}|-\rangle &|0\rangle \end{bmatrix}}.}$

### Majumdar–Ghosh model

Majumdar–Ghosh ground state can be written as MPS with

${\displaystyle A={\begin{bmatrix}0&|\uparrow \rangle &|\downarrow \rangle \\{\frac {-1}{\sqrt {2}}}|\downarrow \rangle &0&0\\{\frac {1}{\sqrt {2}}}|\uparrow \rangle &0&0\end{bmatrix}}.}$