In quantum mechanics, a matrix product state (MPS) is a quantum state of many particles (in N sites), written in the following form:
where are complex, square matrices of order (this dimension is called local dimension). Indices go over states in the computational basis. For qubits, it is . For qudits (d-level systems), it is .
It is particularly useful for dealing with ground states of one-dimensional quantum spin models (e.g. Heisenberg model (quantum)). The parameter 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 .
For states that are translationally symmetric, we can choose:
In general, every state can be written in the MPS form (with growing exponentially with the particle number N). However, MPS are practical when 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],[2] and.[3] In the context of finite automata see.[4] For emphasis placed on the graphical reasoning of tensor networks, see the introduction.[5]
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