Matrix product state

In quantum mechanics, 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 ^[1] and.^[2] In the context of finite automata see.^[3] For emphasis placed on the graphical reasoning of tensor networks, see the introduction.^[4]