Entanglement in Many-Body Physics

In a lattice system, choosing a region means choosing which site factors to keep when taking a partial trace. The resulting entropy tells us how a pure state connects that region to its surroundings. Its dependence on region size can reveal structure that is invisible in a single two-qubit example. For a mixed global state, the same entropy can instead include ordinary thermal mixing.

Helpful background. Entanglement entropy and the Schmidt decomposition define the quantities used below. Many-Body Entanglement Overview develops the broader setting of partitions, state classes, and scaling.

Entropy across a spatial cut

Consider a pure chain of spin-$1/2$ sites, arranged into independent Bell pairs. If exactly $k$ pairs cross a chosen cut, the reduced state has $2^k$ equal nonzero eigenvalues $2^{-k}$. Therefore

$$S(A)=k\quad\text{bits}.$$

Pairs wholly inside $A$ contribute pure factors and no entropy across this cut. Thus the location of entanglement matters, not just the total number of entangled pairs.

For nearest-neighbor dimers, a contiguous block cuts at most two pairs, so its entropy stays bounded as the block grows. This is a simple illustration of boundary scaling, or an area law. If instead every site in the left half of a chain is paired with a site in the right half, the half-chain entropy is $N/2$ bits: a volume law. The second construction is a possible state, not a claim about the ground state of a short-range gapped Hamiltonian.

In one dimension, a uniform spectral gap above the ground state of a local finite-dimensional spin chain gives an area-law theorem under the usual bounded-interaction assumptions (Hastings, 2007). Critical chains can have logarithmic growth. These are statements with hypotheses, not rules for every state or Hamiltonian. Area laws, volume laws, and many-body entanglement entropy develop their scope and finite-size tests.

The same local entropy can have different origins

The infinite-temperature state of $N$ qubits is

$$\rho=\frac{I}{2^N}=\bigotimes_{j=1}^N\frac{I_j}{2}.$$

A block of $\ell$ sites has entropy $\ell$ bits even though the entire state is product and unentangled. Two disjoint blocks also have zero mutual information. This differs sharply from the pure long-range Bell-pair construction, where the same volume scaling counts entanglement across the cut.

For a mixed state, mutual information measures total correlations, while negativity or other mixed-state measures address entanglement. Thermal entropy versus entanglement entropy explains when the same entropy expression answers different physical questions.

Schmidt spectra and tensor-network cost

Across a single bond of an open-chain matrix product state with bond dimension $\chi$, the Schmidt rank is at most $\chi$. Hence

$$S(A)\le\log_2\chi.$$

This gives a lower bound on the bond dimension required for an exact representation. Approximation depends on the full Schmidt probabilities: retaining the largest $\chi$ discards weight $\delta=\sum_{j>\chi}p_j$, and the normalized truncated vector has squared overlap $1-\delta$ with the original vector for that cut.

One small entropy value does not uniquely determine the tail or the error. For instance, $p=(1-\epsilon,\epsilon/(R-1),\ldots,\epsilon/(R-1))$ has rank $R$ for every $\epsilon>0$, but truncation to one coefficient loses only weight $\epsilon$. Several cuts and numerical convergence must be checked when approximating an entire chain. The entanglement spectrum keeps the information that a single entropy compresses; Schollwöck’s review develops the matrix-product-state algorithms.

Connecting entropy with other observables

Entanglement helps characterize many-body states, but an area law alone does not show that a state is product or topologically trivial. Likewise, an entropy maximum in a finite system does not by itself prove a phase transition. Compare size dependence, symmetry, correlation functions, and the relevant gaps or order parameters. Quantum phase transitions develops those comparisons.

A spatial cut, an orbital cut, and a formal particle partition need not give the same result. Continuum limits introduce ultraviolet and algebraic issues discussed in entanglement in QFT.

Exercises

1. A six-site state consists of Bell pairs on $(1,2)$, $(3,4)$, and $(5,6)$. Find the entropy of $A=\{2,3,4,5\}$ and the Schmidt rank across $A:\bar A$.

Solution

Pairs $(1,2)$ and $(5,6)$ cross the cut; pair $(3,4)$ lies entirely in $A$. Thus $S(A)=2$ bits and the Schmidt rank is four. The four nonzero reduced eigenvalues are all $1/4$. The pure internal Bell pair contributes no entropy across this cut.

2. A normalized pure-state Schmidt spectrum is $(0.9,0.05,0.03,0.02)$. Find the squared overlap after retaining the two largest coefficients and normalizing. Why is this not a complete error estimate for a long matrix product state calculation?

Solution

The discarded weight is $0.03+0.02=0.05$, so the squared overlap for this single truncation is $0.95$. A chain calculation can truncate many cuts and repeat truncations during evolution or optimization; errors can accumulate and observables need their own convergence checks.

References

• L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems,” Reviews of Modern Physics 80, 517–576 (2008), doi:10.1103/RevModPhys.80.517.
• J. Eisert, M. Cramer, and M. B. Plenio, “Area laws for the entanglement entropy,” Reviews of Modern Physics 82, 277–306 (2010), doi:10.1103/RevModPhys.82.277.
• M. B. Hastings, “An area law for one-dimensional quantum systems,” Journal of Statistical Mechanics P08024 (2007), doi:10.1088/1742-5468/2007/08/P08024.
• U. Schollwöck, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), doi:10.1016/j.aop.2010.09.012.