Matrix product state
Matrix product state (MPS) is a pure quantum state of many particles, written in the following form:
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.
MPS decomposition is not unique.
One method to obtain MPS is to use Schmidt decomposition N − 1 times.
can be expressed as a Matrix Product State, up to normalization, with
or equivalently, using notation from:
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
Note that tensor product is not commutative.
In this particular example, a product of two A matrices is:
where the are Pauli matrices, or
Majumdar–Ghosh ground state can be written as MPS with
- Density matrix renormalization group
- Variational method (quantum mechanics)
- Markov chain
- Projected Entangled Pair States (PEPS)
- Multistate Landau–Zener Models
- State of Matrix Product States – Physics Stack Exchange
- A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
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