Negativity, Purification Measures, and Proposed Mixed-State Bulk Duals

Logarithmic negativity, reflected entropy, and entanglement of purification are inequivalent mixed-state measures. Their bulk candidates can all involve entanglement-wedge cross sections or replica branes in special holographic limits, but that shared geometry does not make the measures interchangeable. A proposed negativity dual must reproduce the partial-transpose replica, vanish on every PPT state, obey its monotonicity properties, and carry the correct regulator dependence. This page compares two leading geometric candidates across a wedge-connectivity transition and treats disagreement as evidence about their domains rather than redefining the boundary measure.

Required background. Entanglement negativity in QFT defines the partial transpose and regulator, while entanglement-wedge nesting supplies the candidate bulk region.

Helpful background. Reflected entropy and canonical purification supplies a distinct measure; the reflected-entropy cross-section proposal gives a controlled comparison; and measure selection ties each quantity to its operational question.

The boundary measures do different jobs

For a bipartite density matrix,

$$\mathcal E_N(A:B)=\log\lVert\rho_{AB}^{T_B}\rVert_1$$

is logarithmic negativity. It detects nonpositive partial transpose (NPT) entanglement and is an entanglement monotone under the appropriate deterministic local operations. It vanishes on separable states and, more broadly, on PPT bound-entangled states, so it is not faithful to every kind of entanglement (Vidal and Werner 2002, §§II–III).

Reflected entropy $S_R(A:B)$ is an entropy in the canonical purification and responds to classical as well as quantum correlations. Entanglement of purification

$$E_P(A:B)=\min_{|\Psi\rangle_{ABA'B'}}S(AA')$$

minimizes over all purifications. Neither has the PPT-zero property of negativity. Consequently, an unqualified equality $\mathcal E_N\propto E_W$, $S_R=2E_W$, and $E_P=E_W$ cannot hold for arbitrary states and orders.

Two geometric negativity candidates

Consider two disjoint intervals in the vacuum of a large-$c$ CFT$_2$:

$$A=[x_1,x_2],\qquad C=[x_2,x_3],\qquad B=[x_3,x_4],$$

with lengths $a=x_{12}$, $c_0=x_{23}$, and $b=x_{34}$. The cross ratio is

$$\eta=\frac{ab}{(a+c_0)(b+c_0)}.$$

One early geodesic-combination proposal in the proximity channel is

$$\mathcal E_{\rm comb} =\frac34\left[ S(A\cup C)+S(B\cup C)-S(C)-S(A\cup B\cup C) \right].$$

Using vacuum interval entropies, the cutoffs cancel and

$$\mathcal E_{\rm comb} =\frac{c}{4} \log\!\left[ \frac{(a+c_0)(b+c_0)}{c_0(a+b+c_0)} \right] =\frac{c}{4}\log\!\left(\frac1{1-\eta}\right).$$

Such combinations were motivated by large-central-charge replica channels and geodesic lengths (Chaturvedi, Malvimat, and Sengupta 2016, §§4–5). Their use depends on the interval configuration and dominant conformal block.

A second proposal uses a backreacted minimal entanglement-wedge cross section. In AdS$_3$, its unbackreacted leading form is often written

$$\mathcal E_{\rm EW}=\frac32E_W =\frac{c}{4} \log\!\left(\frac{1+\sqrt\eta}{1-\sqrt\eta}\right)$$

in the connected phase, with the numerical coefficient understood as the AdS$_3$ value of a dimension- and brane-dependent factor. A controlled replica derivation generally requires a finite-tension brane and may break the naive replica symmetry; plain $3E_W/2$ is therefore a leading shorthand, not a universal functional (Kudler-Flam and Ryu 2019, §§2–4; Dong, Qi, and Walter 2021, §§2–3).

Connectivity transition as a comparison

At leading classical order, the two-interval entanglement wedge changes from connected to disconnected at the RT saddle crossing, $\eta=1/2$ in the vacuum symmetric channel. Approaching from the connected side,

$$\mathcal E_{\rm comb}\to\frac{c}{4}\log2,$$

whereas

$$\mathcal E_{\rm EW}\to \frac{c}{4}\log(3+2\sqrt2).$$

The candidates already differ at order $c$. On the disconnected side, the classical cross section is absent, so the second gives zero at leading order. The geodesic combination remains positive if its connected-channel expression is continued mechanically. It must instead change conformal-block/saddle branch or be declared outside its regime.

This is the requested comparison. It does not show that one simple expression is universally correct. It shows which additional evidence is needed: the actual partial-transpose replica saddle, its symmetry or symmetry breaking, finite-tension backreaction, and the dominant channel on each side of the transition. Any agreement in adjacent-interval or pure-state limits is a check, not a derivation for disjoint mixed states.

Separable and PPT adversarial controls

Take the separable but classically correlated state

$$\rho_{AB}=\frac12|00\rangle\langle00| +\frac12|11\rangle\langle11|.$$

Its partial transpose is itself and is positive, so

$$\mathcal E_N(A:B)=\log\operatorname{Tr}\rho_{AB}=0.$$

Its reflected entropy and entanglement of purification are nonzero because the shared classical bit must be represented in their purifications. A cross-section functional that measures total wedge correlation can therefore be nonzero while negativity must vanish. Calling that cross section “negativity” without deriving the partial-transpose brane fails the defining-zero test.

PPT bound-entangled states sharpen the point: they are entangled yet $\mathcal E_N=0$. A candidate that is positive whenever the wedge is connected is not a general negativity dual. Holographic code states may occupy a restricted subset where such adversarial states are absent, but that restriction must be demonstrated rather than assumed.

Further controls are mandatory:

• Monotonicity: a deterministic local channel cannot increase the claimed negativity; a geometric operation must be shown to represent that channel before an area monotonicity is relevant.
• Regulator: adjacent QFT regions have UV-divergent negativity, while separated regions can be finite. The brane and counterterms must reproduce the same dependence.
• Pure-state limit: when $\rho_{AB}$ is pure, $\mathcal E_N$ equals the Rényi entropy of order $1/2$, not the von Neumann entropy; finite brane backreaction matters.
• Order: an $O(c)$ area result does not predict an $O(c^0)$ negativity left after wedge disconnection.

Status and handoff

Replica-brane constructions provide substantial evidence for holographic negativity in symmetric large-$c$ regimes, while cross-section and geodesic formulas are useful candidate reductions. Purification cross sections have different replica gluing and different operational meanings. Higher-curvature, quantum, gauge-edge, and finite-$N$ corrections remain measure specific.

The safe conclusion is comparative: name the boundary measure first, derive its replica or optimization in the stated theory, solve every competing bulk saddle, and test its defining properties. Modular response and QEC chapters may use these quantities as diagnostics, but neither turns a proposal into a universal identity.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Chaturvedi, P., Malvimat, V., and Sengupta, G. (2016). “Holographic quantum entanglement negativity.” Journal of High Energy Physics 2016(5), 172. DOI.
• Dong, X., Qi, X.-L., and Walter, M. (2021). “Holographic entanglement negativity and replica symmetry breaking.” Journal of High Energy Physics 2021(6), 024. DOI.
• Kudler-Flam, J., and Ryu, S. (2019). “Entanglement negativity and minimal entanglement wedge cross sections in holographic theories.” Physical Review D 99, 106014. DOI.
• Vidal, G., and Werner, R. F. (2002). “Computable measure of entanglement.” Physical Review A 65, 032314. DOI.