Entanglement Negativity in QFT

Logarithmic negativity detects mixed-state entanglement through a partial-transpose construction. In QFT it requires a declared bipartite algebra, regulator, bosonic or fermionic transpose convention, and replica continuation; it is not a von Neumann entropy and does not inherit entropy formulas by substitution.

Required background. Use regulated subregion entropy. Helpful background. Rényi analytic continuation supplies the continuation cautions.

Partial transpose and trace norm

For a finite-dimensional bipartite state $\rho_{AB}$, choose a basis on $B$ and define

$$\mathcal E(A{:}B) =\log\left\lVert\rho_{AB}^{T_B}\right\rVert_1.$$

Although the matrix operation depends on the chosen basis, the trace norm is invariant under local basis changes. A positive partial transpose gives $\mathcal E=0$, but the converse between zero negativity and separability fails in general dimensions.

In continuum QFT, one first supplies a type-I regulator, split factor, or algebraic replacement. Fermionic systems require a graded construction: ordinary matrix partial transpose, partial time reversal, and fermionic partial transpose are not interchangeable.

The structure diagram locates negativity on the mixed-state branch.

Logarithmic negativity uses a trace norm after partial transpose and can detect mixed-state entanglement. Its continuum value depends on the bipartite algebra, regulator, and bosonic or fermionic convention. Schematic.

Replica parity

Replica calculations evaluate moments $\operatorname{Tr}[(\rho^{T_B})^{n_e}]$ for even integers $n_e$ and analytically continue the even sequence to $n_e\to1$. Continuing the odd sequence computes a different object. This parity distinction underlies the twist-field construction of Calabrese, Cardy, and Tonni 2013, §§3–4.

For adjacent intervals in a harmonic chain, compute the covariance-matrix negativity at several lattice spacings and compare with the even-replica prediction. For a fermionic chain, state the graded tensor product and transpose prescription before comparing results. Boundary contact divergences can remain for adjacent regions; disjoint-region negativity has a different continuum structure.

Negativity is monotone under an appropriate class of local operations, but a QFT operational claim must state which local algebra and allowed operations are used. It does not equal entanglement entropy for a generic mixed state.

Convention failures

The validity map highlights choices that cannot be suppressed.

Bosonic and fermionic transpose constructions, even and odd replica sequences, adjacent and disjoint regions, and center choices define different quantities. Omitting any one can turn a valid calculation into a comparison of unlike measures. Schematic.

Report whether logarithms are natural, the partial-transpose convention, regulator and extrapolation, interval geometry, state, and replica sequence.

References

• Calabrese, Pasquale, John Cardy, and Erik Tonni. “Entanglement Negativity in Extended Systems: A Field Theoretical Approach.” Journal of Statistical Mechanics (2013): P02008. DOI.

Further reading

• Calabrese, Pasquale, John Cardy, and Erik Tonni. “Entanglement Negativity in Quantum Field Theory.” Physical Review Letters 109 (2012): 130502. DOI. Open preprint.