Entanglement Negativity in QFT

Logarithmic negativity detects mixed-state entanglement through a trace norm after a partial transformation. In QFT, that short phrase hides essential choices: the bipartite algebra, a split or regulator, the bosonic or fermionic transformation, region geometry, and—in replica calculations—the analytic continuation. Negativity is not a von Neumann entropy, and an ordinary bosonic transpose must not be silently applied to a graded fermionic algebra.

Required background. Regulated Subregion Entropy supplies the lattice and split-factor constructions needed to obtain density operators from continuum regions. Helpful background. Rényi Entropies and Analytic Continuation supplies the distinction between integer moments, an interpolating function, and the replica limit. The graded algebra and mode-ordering conventions used below are developed on Fermionic Subsystems, Parity, and Graded Tensor Products.

The shared structure map separates negativity from total-correlation and canonical-purification measures. Use the canonical comparison table before comparing results obtained with different algebras or transformations.

Bosonic partial transpose and its domain

For a finite-dimensional state on ${\cal H}_{A_1}\otimes{\cal H}_{A_2}$, choose a product basis and transpose matrix indices only in $A_2$. With natural logarithms,

$${\cal E}(A_1{:}A_2) =\log\left\lVert\rho_A^{T_2}\right\rVert_1, \qquad \lVert X\rVert_1 =\operatorname{Tr}\sqrt{X^\dagger X}.$$

Although the matrix representing $T_2$ changes under a local basis change, its singular values and therefore ${\cal E}$ do not. Since $\operatorname{Tr}\rho_A^{T_2}=1$, the trace norm is at least one and ${\cal E}\geq0$. Positive partial transpose implies ${\cal E}=0$, but PPT-entangled states show that zero negativity does not imply separability in general dimensions Vidal and Werner 2002, eqs. (1)–(4), pp. 1–2, and §II.C after eq. (21), p. 4.

Local QFT algebras are type III and do not intrinsically provide a density matrix or a tensor-product partial transpose. A lattice reduction or a declared split type-I factor defines the displayed quantity. Other algebraic replacements exist, but they are new definitions whose hypotheses must be stated. Adjacent regions retain a contact divergence; for separated regions a continuum limit can be finite, but only after fixing the algebras, centers, state, and limiting prescription.

Replica parity and analytic continuation

Let the Hermitian bosonic partial transpose have eigenvalues $\lambda_i$. Its integer moments split into two sequences:

$$\begin{aligned} \operatorname{Tr}(\rho_A^{T_2})^{n_e} &=\sum_{\lambda_i>0}|\lambda_i|^{n_e} +\sum_{\lambda_i<0}|\lambda_i|^{n_e}, &&n_e\ \text{even},\\ \operatorname{Tr}(\rho_A^{T_2})^{n_o} &=\sum_{\lambda_i>0}|\lambda_i|^{n_o} -\sum_{\lambda_i<0}|\lambda_i|^{n_o}, &&n_o\ \text{odd}. \end{aligned}$$

Consequently,

$$\lim_{n_e\to1}\operatorname{Tr}(\rho_A^{T_2})^{n_e} =\lVert\rho_A^{T_2}\rVert_1, \qquad \lim_{n_o\to1}\operatorname{Tr}(\rho_A^{T_2})^{n_o}=1.$$

These are different analytic continuations Calabrese, Cardy, and Tonni 2013, §2.2, eqs. (13)–(15). Even the even sequence does not determine a unique continuation without analyticity and growth assumptions or an independent benchmark. For two disjoint intervals, the original compact-boson calculation obtained the integer moments but not the $n_e\to1$ continuation Calabrese, Cardy, and Tonni 2013, abstract and §5.

Adjacent intervals in the harmonic chain

Take a periodic harmonic chain in lattice units,

$$H=\frac12\sum_{j=0}^{N-1} \left[p_j^2+m^2q_j^2+(q_{j+1}-q_j)^2\right], \qquad [q_j,p_k]=i\delta_{jk},$$

and let $A_1,A_2$ be adjacent blocks containing $N_1,N_2$ sites. Write $\ell_i=aN_i$ before taking a continuum limit. In the grouped phase-space order $(q_j)_{j\in A},(p_j)_{j\in A}$, restrict the vacuum covariance matrices

$$X_{jk}=\langle q_jq_k\rangle, \qquad P_{jk}=\langle p_jp_k\rangle$$

to $A=A_1\cup A_2$. Bosonic partial transpose reverses the momenta in $A_2$. If $R$ is $+1$ on $A_1$ and $-1$ on $A_2$, the transformed covariance matrix is

$$V^{T_2}=X\oplus(RPR).$$

Let $\widetilde\nu_k$ be its symplectic eigenvalues in the convention where the vacuum value is $1/2$. The covariance calculation is then

$${\cal E}_{\rm cov} =\sum_k\max\!\left\{0,-\log(2\widetilde\nu_k)\right\}.$$

An exact two-site check is already nontrivial. For

$$H_2=\frac12(p_1^2+p_2^2) +\frac{m^2}{2}(q_1^2+q_2^2) +\frac K2(q_1-q_2)^2,$$

take $m>0$ and $K\geq0$. The normal-mode frequencies are $\omega_+=m$ and $\omega_-=\sqrt{m^2+2K}$. The site-bipartition ground state is locally symplectically equivalent to a two-mode squeezed vacuum with $r=\tfrac14\log(\omega_-/\omega_+)\geq0$; the required local squeezers do not change the entanglement. The covariance formula therefore gives

$${\cal E}_{1|1}=2r =\frac12\log\frac{\omega_-}{\omega_+}.$$

It vanishes at $K=0$ and, at fixed $K>0$, diverges as the uniform zero mode softens when $m\to0$. These two limits independently check both the implementation and the need for a zero-mode prescription.

On an infinite line, for $a\ll\ell_1,\ell_2$ and $m\ell_i\ll1$, the continuum CFT prediction is

$${\cal E}_{\rm CFT} =\frac{c}{4}\log \frac{\ell_1\ell_2}{a(\ell_1+\ell_2)} +\kappa+o(1),$$

with $c=1$ for the critical harmonic chain and a nonuniversal constant $\kappa$. On a circle, replace each length by its chord distance

$$d_N(n)=\frac{N}{\pi}\sin\frac{\pi n}{N}.$$

For equal adjacent blocks, subtracting the result at $n=N/4$ therefore removes both $\kappa$ and the cutoff:

$$\begin{aligned} \Delta {\cal E}(n;N) &={\cal E}(n,n;N)-{\cal E}(N/4,N/4;N)\\ &=\frac{c}{4}\,x(n;N)+o(1),\\ x(n;N)&=\log\tan\frac{\pi n}{N}. \end{aligned}$$

The adjacent-interval replica result and its finite-circle form are derived in Calabrese, Cardy, and Tonni 2013, §§4.2–4.3, eqs. (47)–(51) and (56)–(57), pp. 13–14; harmonic-chain realization in §§7 and 7.1, eqs. (168) and (174)–(180), pp. 29–31; §7.3, eqs. (186)–(193), pp. 32–33; §7.5, eqs. (194)–(199), pp. 35–36, PDF. A periodic chain at $m=0$ has a zero mode, so a reproducible calculation either uses Dirichlet boundaries or works at declared $m>0$ with $mL\ll1$, checks regulator dependence, and extrapolates only if claiming regulator removal. In either case, it must track $mL$, $a/\ell_i$, and $\ell_i/L$ separately.

A constant-free harmonic-chain benchmark

The reproducible benchmark accompanying this page sets the lattice spacing $a$, oscillator mass $M$, and nearest-neighbor spring constant $K$ to one. It uses a periodic chain with $N=128,256,512$ and regulates the zero mode with $m=10^{-7}$. For each $N$ it evaluates five equal-block fractions $n/N=1/8,3/16,1/4,5/16,3/8$ directly from the partially transposed covariance matrix and fits $\Delta{\cal E}=s_Nx$ through the origin. Repeating the calculation at $m=10^{-6}$ tests the zero-mode prescription. The numerical values and every control are stored in the structured benchmark record and its flat CSV export.

The result should be read as a deterministic convergence test, not as sampled data. At $N=512$ the primary fit gives $s_{512}=0.2499959$. The conservative systematic estimate is $4.75\times10^{-5}$, dominated by the mass-regulator shift; an additive constant is never compared across regulator conventions.

Fermionic partial time reversal

A fermionic partial transformation must respect the graded tensor product and parity superselection rule. Fix all Majoranas of $A_1$ before those of $A_2$, and within site $j$ use

$$c_{2j-1}=f_j+f_j^\dagger, \qquad c_{2j}=i(f_j-f_j^\dagger).$$

Expand a parity-even reduced state in ordered Majorana monomials,

$$\rho_A =\sum_{\substack{p+q\\ \mathrm{even}}} w_{\boldsymbol\mu,\boldsymbol\nu} c_{\mu_1}\cdots c_{\mu_p} c_{\nu_1}\cdots c_{\nu_q},$$

where the first $p$ factors belong to $A_1$. The partial-time-reversal convention used here is the complex-linear rule

$$\rho_A^{R_1} =\sum_{\substack{p+q\\ \mathrm{even}}} i^p w_{\boldsymbol\mu,\boldsymbol\nu} c_{\mu_1}\cdots c_{\mu_p} c_{\nu_1}\cdots c_{\nu_q}.$$

It does not reverse the order inside a monomial. This is the Majorana form of Shapourian, Shiozaki, and Ryu 2017, §II.B.2, eqs. (11)–(15), pp. 3–4, PDF; changing the phase rule defines a different fermionic transformation.

For a fermionic Gaussian state, $\rho_A^{R_1}$ remains Gaussian but is generally non-Hermitian Shapourian, Shiozaki, and Ryu 2017, §III.A, eqs. (28)–(35), pp. 5–6, PDF. The corresponding singular-value measure is

$${\cal E}_R(A_1{:}A_2) =\log\operatorname{Tr} \sqrt{\rho_A^{R_1}(\rho_A^{R_1})^\dagger}.$$

Thus complex eigenvalue “signs” have no role; singular values replace absolute values of real eigenvalues Shapourian, Shiozaki, and Ryu 2017, §II.B.2, eq. (20), p. 4, PDF. For a quadratic Kitaev, SSH, or hopping chain, restrict the covariance matrix to $A_1\cup A_2$, apply this fixed phase rule, and diagonalize the positive Gaussian operator $\rho_A^{R_1}(\rho_A^{R_1})^\dagger$. At a clean critical point this calculation and the fermionic replica construction give

$${\cal E}_R =\frac{c}{4}\log \frac{\ell_1\ell_2}{a(\ell_1+\ell_2)} +\kappa_R+o(1),$$

with $c=1/2$ for a single critical Majorana chain and $c=1$ for a critical Dirac/SSH chain Shapourian, Shiozaki, and Ryu 2017, §V.A, eqs. (81), (83)–(84), and (86)–(87), pp. 12–13, PDF. The same universal slope does not make different fermionic transformations identical: finite constants, topological signals, and even the appropriate spectral language can change.

Covariance evaluation for a Dirac chain

For a number-conserving Gaussian state, let

$$C_{ij}=\langle f_i^\dagger f_j\rangle, \qquad \Gamma=I-2C_A =\begin{pmatrix} \Gamma_{11}&\Gamma_{12}\\ \Gamma_{21}&\Gamma_{22} \end{pmatrix}$$

after restriction to $A=A_1\cup A_2$. The two Gaussian operators associated with $\rho_A^{R_1}$ and its adjoint have one-body covariance matrices

$$\Gamma_\pm =\begin{pmatrix} -\Gamma_{11}&\pm i\Gamma_{12}\\ \pm i\Gamma_{21}&\Gamma_{22} \end{pmatrix}, \qquad \Gamma_\times =(I+\Gamma_+\Gamma_-)^{-1}(\Gamma_++\Gamma_-).$$

If $\nu_j^\times$ are the eigenvalues of $\Gamma_\times$ and $\zeta_j$ those of $C_A$, the zero-charge-counting phase—the ordinary, uncharged negativity—gives

$$\begin{aligned} {\cal E}_R ={}&\sum_j\log\!\left[ \sqrt{\frac{1-\nu_j^\times}{2}} +\sqrt{\frac{1+\nu_j^\times}{2}} \right]\\ &+\frac12\sum_j \log\!\left[\zeta_j^2+(1-\zeta_j)^2\right]. \end{aligned}$$

The second line restores the normalization of $\rho_A^{R_1}(\rho_A^{R_1})^\dagger$; omitting it gives a wrong trace norm. This stable $O(|A|^3)$ calculation follows Murciano, Bonsignori, and Calabrese 2021, Appendix A, eqs. (140)–(145), printed pp. 36–37, PDF.

The benchmark uses the half-filled hopping ring

$$H_F =-\sum_{j=0}^{N-2}(f_{j+1}^\dagger f_j+f_j^\dagger f_{j+1}) +(f_0^\dagger f_{N-1}+f_{N-1}^\dagger f_0),$$

so $f_{j+N}=-f_j$. The allowed momenta are $k=2\pi(r+1/2)/N$; filling every mode with $-2\cos k<0$ removes a Fermi-level degeneracy Shapourian, Shiozaki, and Ryu 2017, §V.A, eq. (63) and the momentum-quantization paragraph after eq. (66), p. 11, PDF. For $N=128,256,512$, the calculation uses eight equal adjacent-block fractions from $\ell/N=1/32$ through $3/16$ and fits

$${\cal E}_R=b_N+s_N\log\tan\frac{\pi\ell}{N}.$$

The three lattice slopes approach the Dirac prediction $c/4=0.25$. A linear extrapolation in $1/N$ gives $s_\infty=0.249935$; varying the interval window and replacing the $1/N$ extrapolation by $1/N^2$ gives a conservative deterministic systematic envelope of $\pm0.0013$. Ordinary least-squares errors are recorded only as fit diagnostics because the covariance data are not random samples.

The same fit coordinate makes the finite-size trend easy to compare:

$N$	harmonic-chain $s_N$, $m=10^{-7}$	Dirac-chain $s_N$
128	0.249995	0.245428
256	0.249997	0.247666
512	0.249996	0.248816
reported estimate	$0.249996\pm0.000048$	$0.249935\pm0.001289$
CFT value	0.250000	0.250000

The scalar result is already close at each displayed size because reference subtraction cancels its large zero-mode-dependent offset. The fermionic ultraviolet correction is visibly larger but decreases regularly under size doubling. These different convergence rates are why one should report controls rather than only a visually persuasive straight line.

For a smallest-chain normalization check, the one-particle ground state of a two-site hopping dimer is

$$|\psi\rangle =\frac{f_1^\dagger+f_2^\dagger}{\sqrt2}|0\rangle, \qquad C=\frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix}.$$

It has Schmidt weights $1/2,1/2$ across the adjacent sites. For a pure state, fermionic partial-time-reversal negativity equals the Rényi entropy of order $1/2$, so ${\cal E}_R=\log2$ Shapourian, Shiozaki, and Ryu 2017, §III.A, eqs. (36)–(37) and the following paragraph, p. 6, PDF. The numerical implementation also returns zero for a product Slater determinant and for the infinite-temperature state, and is invariant under translating the interval pair or exchanging $A_1$ and $A_2$. An independently diagonalized single-particle Hamiltonian reproduces the analytic Fourier projector.

Convention-switch failure test

The requested adversarial switch produces the following outcomes.

Construction	Spectral object	What remains true	What must be withdrawn after a switch
Bosonic/type-I $T_2$	Hermitian $\rho^{T_2}$ with real eigenvalues	${\cal E}=\log\lVert\rho^{T_2}\rVert_1\geq0$; the even/odd formulas above apply	Any claim imported to a graded fermionic algebra without a Jordan–Wigner ordering
Fermionic matrix transpose	A Hermitian sum of two Gaussian operators in the standard construction	A trace norm is definable after fixing ordering and grading	Efficient one-Gaussian covariance formulas and convention-independent finite terms
Fermionic partial time reversal $R_1$	Generally non-Hermitian $\rho^{R_1}$; use singular values	${\cal E}_R\geq0$ and Gaussian evaluation for parity-even Gaussian states	Statements about negative eigenvalues or the bosonic even/odd eigenvalue branches

The matrix-transpose Gaussian decomposition is derived in Eisler and Zimborás 2015, §3.2, eqs. (24)–(27), p. 8; the pure-state contrast with partial time reversal is explicit in Shapourian, Shiozaki, and Ryu 2017, §III.A, eqs. (36)–(37) and the following paragraph, p. 6, PDF. Changing an algebraic center additionally changes the reduced state by changing which boundary observables are retained. Positivity of the chosen trace norm survives, but its value and continuum constant need not. The strongest cross-convention statement is therefore the conditional one: after fixing the graded algebra and transformation, the critical adjacent-interval calculations above reproduce the stated $c/4$ slope in the tested chain classes.

For the ordinary type-I measure, logarithmic negativity is nonincreasing on average under completely positive PPT-preserving instruments, a class containing LOCC Plenio 2005, Lemma and eqs. (7)–(8), p. 3, PDF. A fermionic operational statement must instead restrict to parity-preserving local operations and the same graded tensor product used in the definition.

Report the logarithm base, state, interval geometry, lattice or split factor, center, bosonic or fermionic transformation, replica sequence, zero-mode treatment, and extrapolation errors. The shared validity map shows why none of these entries is optional.

Exercises

1. Bell-pair normalization

For $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$, compute the spectrum of $\rho^{T_2}$ and the logarithmic negativity.

Solution

The density matrix contains diagonal terms $|00\rangle\langle00|/2$ and $|11\rangle\langle11|/2$ and off-diagonal terms $|00\rangle\langle11|/2$ and its adjoint. Partial transpose on the second factor sends the off-diagonal terms to $|01\rangle\langle10|/2$ and its adjoint. Hence

$$\operatorname{spec}(\rho^{T_2}) =\left\{\frac12,\frac12,\frac12,-\frac12\right\}.$$

The trace norm is $2$, so ${\cal E}=\log2$. The negative eigenvalue detects the entanglement, while the trace remains one.

2. Derive the replica split

Starting from real eigenvalues $\lambda_i$ of a Hermitian partial transpose, derive the even and odd moment formulas and their formal limits at one.

Solution

For even integers, both positive and negative eigenvalues contribute with a plus sign; for odd integers, a negative eigenvalue contributes with a minus sign. Continue those two real branches separately as

$$F_e(n)=\sum_{\lambda_i>0}|\lambda_i|^n +\sum_{\lambda_i<0}|\lambda_i|^n, \qquad F_o(n)=\sum_{\lambda_i>0}|\lambda_i|^n -\sum_{\lambda_i<0}|\lambda_i|^n.$$

Their formal limits at one are

$$\lim_{n\to1}F_e(n) =\sum_i|\lambda_i| =\lVert\rho^{T_2}\rVert_1, \qquad \lim_{n\to1}F_o(n) =\sum_i\lambda_i =\operatorname{Tr}\rho^{T_2}=1.$$

Writing the continuation with $|\lambda_i|^n$ is essential: $\lambda_i^n$ for a negative eigenvalue is not a single-valued real function at noninteger $n$. The calculation identifies the desired branches but does not establish a unique analytic continuation from integer data.

3. A two-mode covariance check

In the order $(q_1,p_1,q_2,p_2)$, a two-mode squeezed vacuum with $r\geq0$ has

$$V(r)=\frac12 \begin{pmatrix} \cosh2r\,\mathbb I_2 & \sinh2r\,Z\\ \sinh2r\,Z & \cosh2r\,\mathbb I_2 \end{pmatrix}, \qquad Z=\operatorname{diag}(1,-1).$$

Show that partial transpose on mode 2 gives symplectic eigenvalues $e^{-2r}/2$ and $e^{2r}/2$, and find ${\cal E}$.

Solution

Partial transpose reverses $p_2$, converting the correlation block $\sinh(2r)Z/2$ into $\sinh(2r)\mathbb I_2/2$. Passing to symmetric and antisymmetric quadratures diagonalizes the covariance matrix. Their canonical variances pair as $e^{\pm2r}/2$, giving

$$\widetilde\nu_- =\frac12e^{-2r}, \qquad \widetilde\nu_+ =\frac12e^{2r}.$$

Only $\widetilde\nu_-$ lies below $1/2$, so

$${\cal E}=-\log(2\widetilde\nu_-)=2r.$$

This is an exact covariance benchmark for any harmonic-chain implementation.

4. Inject a fermionic convention change

A calculation reports negative eigenvalues of $\rho^{R_1}$ after fermionic partial time reversal and inserts them into the bosonic even/odd formulas. Diagnose the failure. Which conclusions can still be retained?

Solution

Fermionic $\rho^{R_1}$ is generally non-Hermitian, so its eigenvalues can be complex and have no invariant positive/negative split. The bosonic parity formulas therefore do not apply. One must instead compute the singular values through

$$\sqrt{\rho^{R_1}(\rho^{R_1})^\dagger}.$$

After fixing a parity-even state, graded algebra, and the $R_1$ rule, the singular-value definition and its nonnegativity remain valid. In the clean critical Kitaev and SSH classes, the $c/4$ logarithmic slope also survives. Claims about eigenvalue signs, additive constants shared with a matrix transpose, or convention-independent topological signals must be withdrawn.

5. Read a central charge from the lattice fit

For equal adjacent intervals on a circle, derive

$$\log\frac{d_N(n)^2/d_N(2n)} {d_N(N/4)^2/d_N(N/2)} =\log\tan\frac{\pi n}{N}.$$

The fermionic size extrapolation gives $s_\infty=0.249935\pm0.0013$ for the slope multiplying the right-hand side. Find the corresponding effective central charge and decide whether it is consistent with one massless Dirac fermion.

Solution

Using $d_N(r)=(N/\pi)\sin(\pi r/N)$ and $\sin(2x)=2\sin x\cos x$,

$$\frac{d_N(n)^2}{d_N(2n)} =\frac{N}{2\pi}\tan\frac{\pi n}{N}.$$

At $n=N/4$, the tangent is one, so division by the reference removes $N/(2\pi)$ and leaves the stated logarithm. Since $s=c/4$,

$$c_{\rm eff}=4s_\infty =0.999740\pm0.0052.$$

The expected Dirac value $c=1$ lies comfortably inside this systematic envelope. The quoted uncertainty comes from finite-size and fit-window alternatives, not from statistical sampling.

References

• Calabrese, Pasquale, John Cardy, and Erik Tonni. “Entanglement Negativity in Extended Systems: A Field Theoretical Approach.” Journal of Statistical Mechanics: Theory and Experiment (2013): P02008. DOI. Open preprint.
• Eisler, Viktor, and Zoltán Zimborás. “On the Partial Transpose of Fermionic Gaussian States.” New Journal of Physics 17 (2015): 053048. DOI. Open preprint.
• Murciano, Sara, Riccarda Bonsignori, and Pasquale Calabrese. “Symmetry Decomposition of Negativity of Massless Free Fermions.” SciPost Physics 10 (2021): 111. DOI. Open PDF.
• Plenio, Martin B. “Logarithmic Negativity: A Full Entanglement Monotone That Is Not Convex.” Physical Review Letters 95 (2005): 090503. DOI. Open preprint.
• Shapourian, Hassan, Ken Shiozaki, and Shinsei Ryu. “Partial Time-Reversal Transformation and Entanglement Negativity in Fermionic Systems.” Physical Review B 95 (2017): 165101. DOI. Open preprint.
• Vidal, Guifré, and Reinhard F. Werner. “A Computable Measure of Entanglement.” Physical Review A 65 (2002): 032314. DOI. Open preprint.