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Quantum Mechanics

Many-Body Entanglement Glossary

Canonical treatment: Many-Body Entanglement: Overview owns the explanations, derivations, computational workflows, exercises, and references. This page is a compact terminology and routing aid.

Entanglement is defined only after choosing a state and a subsystem question. For a tensor-product bipartition H=HA⊗HAˉ\mathcal H=\mathcal H_A\otimes\mathcal H_{\bar A},

ρA=Tr⁡Aˉρ.\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

Section titled “State and Partition Terms”
TermCompact meaning
Pure global stateρ=∣ψ⟩⟨ψ∣\rho=\lvert\psi\rangle\langle\psi\rvert; then S(A)=S(Aˉ)S(A)=S(\bar A)
Mixed global stateClassical uncertainty and quantum entanglement can both contribute to subsystem entropy
Schmidt decomposition∣ψ⟩=∑isi∣iA⟩∣iAˉ⟩\lvert\psi\rangle=\sum_i s_i\lvert i_A\rangle\lvert i_{\bar A}\rangle, si≥0s_i\ge0
Schmidt rankNumber of nonzero sis_i; partition dependent
Spatial cutDegrees of freedom assigned by region; geometry and boundary conditions matter
Mode or orbital cutPartition by single-particle labels rather than real-space locality
Algebraic subsystemA chosen observable subalgebra when tensor factorization is absent or unnatural

Measures and Spectra

Section titled “Measures and Spectra”
ObjectDefinition or use
PurityTr⁡ρA2\operatorname{Tr}\rho_A^2; equals one only for a pure reduced state
von Neumann entropyS(A)=−Tr⁡(ρAln⁡ρA)S(A)=-\operatorname{Tr}(\rho_A\ln\rho_A)
Rényi entropySn(A)=(1−n)−1ln⁡Tr⁡ρAnS_n(A)=(1-n)^{-1}\ln\operatorname{Tr}\rho_A^n
Mutual informationI(A:B)=S(A)+S(B)−S(AB)I(A:B)=S(A)+S(B)-S(AB); includes total correlations
NegativityMixed-state entanglement diagnostic based on partial transpose
Entanglement spectrumEigenvalues of ρA\rho_A, or levels ϵi=−ln⁡λi\epsilon_i=-\ln\lambda_i
Discarded weightSum of omitted Schmidt probabilities in a truncation
Operator entanglementEntanglement after treating an operator as a vector in operator space

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

Section titled “Scaling and Network Terms”
TermMeaning and qualification
Area lawLeading entropy scales with the boundary size; subleading terms may carry essential physics
Logarithmic violationA multiplicative or additive logarithm modifies boundary scaling, as at Fermi surfaces or one-dimensional critical points
Volume lawEntropy scales with subsystem volume for a specified state family and subsystem fraction
Topological constantA universal subleading combination after boundary terms are cancelled; not a raw single-fit intercept
Bond dimension χ\chiTensor-network virtual dimension; across one MPS bond, S≤ln⁡χS\le\ln\chi
Finite-entanglement scalingA finite χ\chi introduces an effective correlation length and can round critical behavior

Interpretation Checklist

Section titled “Interpretation Checklist”
  • Name the partition, geometry, boundary conditions, symmetry sector, and state class.
  • Distinguish physical energy spectra from entanglement spectra.
  • Separate ultraviolet-sensitive leading terms from universal combinations.
  • State the subsystem fraction and order of thermodynamic, continuum, and long-time limits.
  • For numerical results, report truncation, positivity, normalization, convergence, and statistical-error audits.
  • Do not infer efficient classical simulation from an area law alone, or chaos from a volume law alone.
Section titled “Cross-Links”