Concept record · edition 0.1.0
Matrix product state (MPS)
A one-dimensional chain of tensors in which each physical site carries one tensor joined to its neighbours by virtual bonds.
Why it matters
It is the ansatz underlying DMRG and the best-understood tensor network, with well-defined canonical forms and truncation control.
What this does not establish
Efficiency is not automatic: it depends on the entanglement across cuts staying bounded, which fails for critical systems and for long-time dynamics.
Claims using this concept
- tn-001 · Established resultA matrix product state represents a many-body state with a cost controlled by its bond dimension, which bounds the entanglement it can carry across any cut.
- tn-002 · Established resultThe one-dimensional area law proves that ground states of gapped one-dimensional local Hamiltonians have entanglement bounded independently of system size.
- tn-003 · Established resultDMRG is understood as variational optimization over the manifold of matrix product states.
Sources
review article · identifier verified
The density-matrix renormalization group in the age of matrix product states
Ulrich Schollwöck · 2010
arXiv:1008.3477
Standard reference for MPS canonical forms, SVD truncation, and the reformulation of DMRG in tensor-network language.
Source record →review article · identifier verified
A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
Román Orús · 2013
arXiv:1306.2164
Review covering tensor-network structure, entanglement scaling, and contraction strategy; the general reference for this atlas.
Source record →primary paper · identifier verified
Density matrix formulation for quantum renormalization groups
Steven R. White · 1992
DOI:10.1103/PhysRevLett.69.2863
Primary source for DMRG, the variational method later understood as optimization over matrix product states.
Source record →