Entanglement and tensor-network states

Divide a pure state into subsystems A and B and arrange its amplitudes into a matrix. Singular-value decomposition writes the state as a sum of paired orthonormal states, with nonnegative coefficients $\lambda_\alpha$. Their squares $p_\alpha$ sum to one. Tracing out B leaves A with eigenvalues $p_\alpha$. A global pure state can therefore give a mixed local state: the uncertainty comes from correlations with the omitted subsystem, not from assuming an unknown classical preparation of the whole system.

The von Neumann entropy of that reduced state measures bipartite entanglement for a globally pure state. A product state has one nonzero Schmidt weight and zero entropy. A maximally entangled pair of qubits has two equal weights and entropy $\log 2$ using natural logarithms. If the full state is already mixed, reduced-state entropy also contains classical and thermal uncertainty; it is no longer by itself a unique measure of entanglement between the parts. Always state the purity assumption before interpreting the number.

Repeated decompositions along a one-dimensional chain produce a matrix-product state, or MPS. For an open-chain MPS, a virtual bond index links neighboring tensors and carries correlations across that cut. If its dimension is $\chi$, the Schmidt rank across that bond is at most $\chi$ and its entropy is at most $\log\chi$. For finite open chains every state has an exact MPS representation if sufficiently large bond dimensions are permitted. The computational advantage comes only when a much smaller representation achieves the desired accuracy.

For one cut, retain the largest Schmidt weights and discard the rest. The discarded weight is the sum of the omitted probabilities. After normalizing the truncated state, its squared overlap with the original at that cut is one minus that weight. This is a precise local compression statement. A long sequence of truncations, a time-evolution algorithm, or an approximate variational optimizer needs additional error accounting; one small discarded weight is not a universal bound on every later observable or on a simulation's total error.

For one-dimensional chains with bounded finite-dimensional local spaces and bounded finite-range interactions, a unique ground state separated by a uniformly positive gap obeys an entanglement area law: entropy across a cut stays bounded as chain length grows. This motivates compact MPS approximations, though accuracy is governed by the full Schmidt tail rather than entropy alone. Critical chains can show logarithmically growing entanglement, and approaching small gaps can demand larger bonds. Density-matrix renormalization group optimizes an MPS variationally; it is a numerical method with convergence questions, not an exact solution merely because an area law is expected.

Generic highly excited states and long-time unitary dynamics can develop much more entanglement. A global quench often produces approximately linear growth before finite-size saturation, forcing the bond dimension needed for fixed accuracy to increase rapidly. Representing a two-dimensional system as a one-dimensional ordering also creates wider cuts and growing cost. Tensor-network variants exploit other geometries, but their contractions and optimization introduce further challenges. Report size, boundary conditions, retained bonds, discarded weights, and observable convergence rather than advertise compression without an error target.