Matrix product state

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

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

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.

The MPS decomposition is not unique. For introductions see and. In the context of finite automata see. For emphasis placed on the graphical reasoning of tensor networks, see the introduction.