Entanglement and Bond Dimension

The bond dimension needed for an MPS calculation is determined by the Schmidt spectrum at the accuracy requested. Entropy is a useful warning about resources, but it does not replace the discarded spectral tail. This page turns entanglement diagnostics into numerical choices; physical definitions and area-law statements belong to entanglement entropy and many-body area laws.

Required background. MPS compression explains how canonical bond singular values are obtained.

Rank, entropy, and retained probability

At one cut write the normalized weights as $p_a=\lambda_a^2$, ordered decreasingly. An exact rank-$D$ MPS obeys

$$S=-\sum_a p_a\log p_a\leq\log D.$$

Thus $D\geq e^S$ is a necessary exact-representation condition, not a sufficient approximation criterion. For a truncation tolerance $\epsilon_{\rm target}$, choose the smallest $D$ satisfying

$$\epsilon_D=\sum_{a>D}p_a\leq\epsilon_{\rm target}.$$

The entropy and this tail answer different questions. A large number of individually small weights may carry enough total probability to spoil the requested accuracy even when the leading singular values look converged.

A spectrum with a computable resource estimate

For a geometric spectrum $p_a=(1-q)q^{a-1}$, $0<q<1$, the tail is $\epsilon_D=q^D$. Consequently,

$$D\geq \frac{\log\epsilon_{\rm target}}{\log q}.$$

With $q=1/2$ and tolerance $10^{-6}$, $D=20$ is sufficient because $2^{-20}<10^{-6}$. With $q=0.9$, the same tolerance needs $D=132$. The entropy is

$$S=-\log(1-q)-\frac{q}{1-q}\log q.$$

For $q=1/2$, $S=2\log2$, so the entropy bound says only $D\geq4$. It misses the much larger dimension needed for a one-part-per-million tail.

This example is also a compression test: generate a diagonal coefficient matrix with the chosen Schmidt values, compress it, and compare the observed squared fidelity with $1-q^D$, allowing for the finite spectrum’s initial normalization.

System size, geometry, and real time

For a chain ordering of a two-dimensional cylinder, a cut crosses many physical bonds. The entanglement across the circumference can force $D$ to grow rapidly even when the Hamiltonian is local. Reordering sites changes which bipartitions the MPS must represent, so site order is an algorithmic choice with physical consequences.

After a global quench, entropy often grows approximately linearly over an intermediate time window. If $S(t)\approx st$, the rank bound already suggests $D\gtrsim e^{st}$. This is a diagnostic scaling estimate, not a universal law: localized systems, integrable protocols, finite size, and special initial states can behave differently.

An injective finite-$D$ MPS has a finite correlation scale even when approximating an infinite critical chain. Noninjective states, such as GHZ states, can retain nondecaying correlations; finite $D$ alone does not rule out long-range order. Converge transfer-operator correlation lengths alongside energy and local observables; a stable energy alone can hide inaccurate long-distance correlations.

A practical adaptive rule

Set both a maximum dimension and a discarded-weight tolerance. Increase the bond until the tail criterion passes or the dimension cap is reached. Record which bonds hit the cap and at what times. Converge the observable again with a larger cap; a run that repeatedly reaches its maximum is not automatically inaccurate, but its tolerance has ceased to be enforced.

Symmetry sectors constrain which multiplets may be retained. Splitting a required multiplet to meet a nominal dimension can break the intended symmetry. Report the actual dimension and the sector-resolved spectrum.

Exercises

A flat spectrum has $K$ equal weights. How many terms must be retained for squared fidelity at least $1-\epsilon$?

Solution

Every retained term contributes $1/K$, so the fidelity is $D/K$. One needs $D\geq\lceil(1-\epsilon)K\rceil$. Here $S=\log K$, and nearly the full exact rank remains necessary at small error. There is no rapidly decaying tail for SVD compression to exploit.

References

• Orús, Román, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014), arXiv:1306.2164 — representations and approximate contractions.
• Schollwöck, Ulrich, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), arXiv:1008.3477 — canonical gauges, contractions, and finite-chain algorithms.