One method to obtain an MPS representation of a quantum state is to use Schmidt decomposition N − 1 times. Alternatively if the quantum circuit which prepares the many body state is known, one could first try to obtain a matrix product operator representation of the circuit. The local tensors in the matrix product operator will be four index tensors. The local MPS tensor is obtained by contracting one physical index of the local MPO tensor with the state which is injected into the quantum circuit at that site.

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 state vectors (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, i.e., the superposition of all the computational basis states of Hamming weight one.

${\displaystyle |\mathrm {W} \rangle ={\frac {1}{\sqrt {3}}}(|001\rangle +|010\rangle +|100\rangle )}$ 

Even though the state is permutation-symmetric, its simplest MPS representation is not.^[1] For example:

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

The AKLT ground state wavefunction, which is the historical example of MPS approach:,^[5] corresponds to the choice^[6]

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

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

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

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 ground state can be written as MPS with

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

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