Many-Body Entanglement Glossary

Entanglement is defined after choosing a state and a subsystem partition. The Many-Body Entanglement Overview explains how the definitions below connect to correlations and numerical representations. For a tensor-product bipartition $\mathcal H=\mathcal H_A\otimes\mathcal H_{\bar A}$,

$$\rho_A=\operatorname{Tr}_{\bar A}\rho.$$

Gauge constraints, identical particles, mode partitions, and algebraic subsystems may not admit this naive factorization and require their own conventions.

State and Partition Terms

Term	Compact meaning
Pure global state	$\rho=\lvert\psi\rangle\langle\psi\rvert$; then $S(A)=S(\bar A)$
Mixed global state	Classical uncertainty and quantum entanglement can both contribute to subsystem entropy
Schmidt decomposition	$\lvert\psi\rangle=\sum_i s_i\lvert i_A\rangle\lvert i_{\bar A}\rangle$, $s_i\ge0$, $\sum_i s_i^2=1$
Schmidt rank	Number of nonzero $s_i$; partition dependent
Spatial cut	Degrees of freedom assigned by region; geometry and boundary conditions matter
Mode or orbital cut	Partition by single-particle labels rather than real-space locality
Algebraic subsystem	A chosen observable subalgebra when tensor factorization is absent or unnatural

Measures and Spectra

Object	Definition or use
Purity	$\operatorname{Tr}\rho_A^2$; equals one only for a pure reduced state
von Neumann entropy	$S(A)=-\operatorname{Tr}(\rho_A\ln\rho_A)$
Rényi entropy	$S_n(A)=(1-n)^{-1}\ln\operatorname{Tr}\rho_A^n$, $n>0$, $n\ne1$; the $n\to1$ limit gives $S(A)$
Mutual information	$I(A:B)=S(A)+S(B)-S(AB)$; includes total correlations
Negativity	$\mathcal N=(\lVert\rho^{T_B}\rVert_1-1)/2$; a positive value detects entanglement, while zero does not exclude all entangled states
Entanglement spectrum	Eigenvalues of $\rho_A$, or levels $\epsilon_i=-\ln\lambda_i$
Discarded weight	$w=\sum_{i>\chi}s_i^2$; keeping and renormalizing the largest $\chi$ coefficients gives squared overlap $1-w$ with the original pure state
Operator entanglement	Entanglement of a normalized vectorized operator under a specified doubled-space cut; not generally its entangling power

Entropy of a subsystem of a mixed global state is not, by itself, an entanglement measure. Logarithm base sets the units: natural logarithms give nats; base two gives bits.

Scaling and Network Terms

Term	Meaning and qualification
Area law	Leading entropy scales with the boundary size; subleading terms may carry essential physics
Logarithmic violation	A multiplicative or additive logarithm modifies boundary scaling, as at Fermi surfaces or one-dimensional critical points
Volume law	Entropy scales with subsystem volume for a specified state family and subsystem fraction
Topological constant	A universal subleading combination after boundary terms are cancelled; not a raw single-fit intercept
Bond dimension $\chi$	Tensor-network virtual dimension; a cut crossing one MPS bond has $S\le\ln\chi$; a periodic interval usually crosses two, giving $S\le2\ln\chi$
Finite-entanglement scaling	A finite $\chi$ introduces an effective correlation length and can round critical behavior

What subsystem entropy does and does not measure

The Bell state $(\lvert00\rangle+\lvert11\rangle)/\sqrt2$ and the separable mixture $(\lvert00\rangle\langle00\rvert+\lvert11\rangle\langle11\rvert)/2$ both have $\rho_A=I/2$ and $S(A)=\ln2$. Their mutual information differs: $2\ln2$ for the Bell state, $\ln2$ for the mixture. Their negativities are respectively $1/2$ and zero. A reduced entropy alone therefore cannot distinguish these two kinds of correlation when the global state may be mixed.

Scaling also depends on the state family and the cut. An area law does not by itself establish an efficient simulation algorithm, and a volume law does not prove chaotic dynamics. See Area Laws, Entanglement Spectrum, and Matrix Product States for those distinctions.

References

• J. Eisert, M. Cramer, and M. B. Plenio, “Colloquium: Area laws for the entanglement entropy”, Reviews of Modern Physics 82, 277–306 (2010).
• G. Vidal and R. F. Werner, “Computable measure of entanglement”, Physical Review A 65, 032314 (2002).