Negativity Spectrum in Mixed-State Entanglement

• Negativity spectrum is the signed eigenvalue distribution of a partially transposed density matrix that refines logarithmic negativity by revealing edge behavior, sign structure, and degeneracies.
• It emerges in diverse contexts—from bosonic and conformal systems to fermionic and random matrix models—using techniques such as moment analysis, Stieltjes transforms, and replica methods.
• Practical insights include quantifying entangled dimensions, detecting phase transitions in mixed states, and linking spectral properties to operational measures like logarithmic negativity.

The negativity spectrum is the spectrum, or eigenvalue distribution, of the partially transposed reduced density matrix $\rho_A^{T_2}$. In bosonic settings this operator is Hermitian but not positive semidefinite, so its spectrum generally contains both positive and negative eigenvalues; the logarithmic negativity is then $\mathcal E=\ln\|\rho_A^{T_2}\|_1$, obtained from the absolute values of those eigenvalues. As a result, the negativity spectrum refines logarithmic negativity: it resolves edge behavior, sign structure, degeneracies, and, in several fermionic formulations, even complex spectral branches. Across conformal field theory, gapped one-dimensional phases, disordered systems, free fermions, random matrix models, random tensor networks, and holography, it has emerged as a distinct spectral object rather than a mere byproduct of a scalar entanglement measure (Ruggiero et al., 2016, Kudler-Flam et al., 2021, Shapourian et al., 2019).

1. Definition and basic spectral structure

A standard representation of the negativity spectrum is the spectral density

$P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$

where $\{\lambda_i\}$ are the eigenvalues of $\rho_A^{T_2}$. Equivalently, one may characterize the spectrum through the moments

$R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$

and reconstruct $P(\lambda)$ by Stieltjes-transform or resolvent methods. This moment-based viewpoint is central in conformal, random-matrix, and tensor-network treatments because it translates partial transpose into replica data, permutation sums, or diagrammatic expansions (Ruggiero et al., 2016, Kudler-Flam et al., 2021).

For pure bipartitions, the negativity spectrum is not independent of the entanglement spectrum. If $\mu_i$ are the eigenvalues of the reduced density matrix $\rho_{A_1}$, then the eigenvalues of $\rho_A^{T_2}$ are

$\mathcal E=\ln\|\rho_A^{T_2}\|_1$0

This relation shows that the negativity spectrum is obtained from pairwise geometric means of entanglement eigenvalues together with a sign structure imposed by the partial transpose. In that sense, the pure-state negativity spectrum is a signed reorganization of the entanglement spectrum rather than an unrelated object (Ruggiero et al., 2016).

The same reduction becomes particularly explicit in low-rank examples. For a pure bipartite state in a generic $\mathcal E=\ln\|\rho_A^{T_2}\|_1$1 Hilbert space, the partial transpose has only four non-zero eigenvalues,

$\mathcal E=\ln\|\rho_A^{T_2}\|_1$2

where $\mathcal E=\ln\|\rho_A^{T_2}\|_1$3 for the reduced qubit state. Only one of these eigenvalues is negative, and the negativity is $\mathcal E=\ln\|\rho_A^{T_2}\|_1$4. This makes the negativity spectrum a fully explicit spectral compression of mixed-state entanglement data in the simplest qubit–qudit pure setting (Agrawal et al., 27 May 2025).

2. Universal one-dimensional results: conformal and gapped systems

In one-dimensional conformal field theories, the negativity spectrum for adjacent intervals is universal and depends only on the central charge $\mathcal E=\ln\|\rho_A^{T_2}\|_1$5, similarly to the entanglement spectrum. For the ground state of a critical chain, the sign dependence of the spectrum is weak for bulk eigenvalues and strong at the spectrum edges. In both pure and mixed adjacent-interval geometries, the support is $\mathcal E=\ln\|\rho_A^{T_2}\|_1$6, there is a delta peak at the largest positive eigenvalue $\mathcal E=\ln\|\rho_A^{T_2}\|_1$7, and the smallest negative eigenvalue approaches $\mathcal E=\ln\|\rho_A^{T_2}\|_1$8 only with very large scaling corrections. DMRG checks for the critical Ising and Heisenberg chains, together with exact harmonic-chain calculations, show that the bulk distribution follows the CFT prediction well, while the smallest eigenvalue is numerically delicate because the edge corrections are logarithmically slow (Ruggiero et al., 2016).

For pure complementary intervals, the CFT distribution can be written in terms of

$\mathcal E=\ln\|\rho_A^{T_2}\|_1$9

with

$P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$0

For adjacent intervals in a mixed state, the form changes quantitatively but preserves the same basic features: a bounded signed spectrum, edge asymmetry, and asymptotically equal numbers of positive and negative eigenvalues (Ruggiero et al., 2016).

A complementary one-dimensional result concerns gapped phases. In the limit of large adjacent regions in gapped one-dimensional models, the negativity spectrum was shown to be entirely reconstructed from the entanglement spectrum of the bipartite system. In the XXZ spin chain, where the entanglement spectrum is known from the corner transfer matrix, the negativity spectrum levels were found to be equally spaced, with spacing half that in the entanglement spectrum, and the degeneracy was described by combinatorial formulas related to integer partitions. The same work reported an asymptotic distribution of the negativity spectrum, exact results for the logarithmic negativity and moments of the partial transpose, and unusual scaling corrections in the limit $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$1 with the same correction exponent as the Rényi entropies (Mbeng et al., 2016).

Massive $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$2-dimensional QFT supplies a related, though moment-based, perspective. For adjacent regions the logarithmic negativity saturates to a finite value as the length scale $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$3, while for separated semi-infinite regions it vanishes. In both cases the leading large-$P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$4 corrections are universal modified-Bessel decays controlled only by the mass spectrum and not by the scattering matrix, and in the adjacent case the subleading structure detects bound states. This suggests that, beyond the strictly critical regime, partial-transpose data retain universal spectral information about massive excitations (Blondeau-Fournier et al., 2015).

3. Disordered phases and valence-bond states

In the random singlet phase, the negativity spectrum is naturally organized through the disorder-averaged moments of $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$5. Strong-disorder renormalization group analysis expresses these moments in terms of the numbers of singlets connecting $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$6 to $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$7 and $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$8 to $P(\lambda)=\sum_i \delta(\lambda-\lambda_i),$9. A central distinction is between

$\{\lambda_i\}$0

which are not equivalent. Accordingly, negativity and logarithmic negativity are not trivially related after disorder averaging, even though for a fixed state $\{\lambda_i\}$1. The disorder-averaged moments show universal logarithmic scaling with subsystem size, and these predictions were checked both by numerical SDRG and by exact free-fermion computations in the random XX chain (Turkeshi et al., 2019).

The generalized valence bond solid state provides an exact lattice realization in which the partial-transpose spectrum can be diagonalized through $\{\lambda_i\}$2 recoupling theory. For two blocks $\{\lambda_i\}$3 and $\{\lambda_i\}$4, the reduced density matrix and its partial transpose are written as tensor-network objects built from Clebsch–Gordan coefficients, $\{\lambda_i\}$5-moves, and $\{\lambda_i\}$6-symbols. In the adjacent-block case the spectrum of $\{\lambda_i\}$7 can be obtained exactly; negative eigenvalues arise from antisymmetric sectors under exchange of block-spin labels, yielding a closed expression for the negativity. The same analysis found that the negativity is non-zero only for adjacent subsystems, and it conjectured vanishing negativity for separated blocks for all integer-spin VBS states (Santos et al., 2016).

These two settings illustrate two distinct departures from clean CFT behavior. In the random singlet phase, averaging changes the relation between moments and entanglement measures; in the VBS case, the spectrum is controlled by finite-dimensional representation theory and exhibits an adjacency criterion rather than long-distance critical scaling. Both cases show that the negativity spectrum is sensitive not only to locality and dimensionality but also to the mechanism generating entanglement.

4. Fermionic partial transpose and complex spectra

For fermionic systems, the negativity spectrum is more subtle because there is a phase ambiguity in the definition of partial transpose. In one-dimensional free fermions, two natural versions were identified. The untwisted fermionic partial transpose $\{\lambda_i\}$8 is pseudo-Hermitian and can have complex eigenvalues, while the twisted partial transpose $\{\lambda_i\}$9 is Hermitian and therefore has a real spectrum. For adjacent intervals, the untwisted spectrum lies on six rays,

$\rho_A^{T_2}$0

whereas the twisted spectrum has only positive and negative real branches. In both cases the leading moments were computed by path-integral methods, and the resulting branch-resolved spectral distributions were written in terms of modified and ordinary Bessel functions (Shapourian et al., 2019).

A sharper nonstandard phenomenon appears for well-separated intervals in free fermionic systems. In that regime, none of the eigenvalues of the deformed reduced density matrix become negative; instead, they develop a small imaginary value, and those imaginary parts generate a non-zero logarithmic negativity. At half filling and for equal intervals, the resulting negativity is of order $\rho_A^{T_2}$1, and the analysis emphasizes that the effect is non-universal, depending non-smoothly on the Fermi level and on interval lengths measured in lattice units. A plausible implication is that, in fermionic problems, the phrase “negativity spectrum” need not refer to a signed real spectrum at all, but can denote a complexified spectral problem whose entanglement content is encoded in tiny departures from the real axis (Bettelheim, 2023).

The Schwinger model shows the same theme in a two-mode setting. There the fermionic partial transpose of the reduced density matrix has eigenvalues

$\rho_A^{T_2}$2

with $\rho_A^{T_2}$3, and the negativity is non-zero only when $\rho_A^{T_2}$4 are complex conjugates. In the massless model, the two-fermion negativity becomes proportional to the square of the fermion propagator, $\rho_A^{T_2}$5, so the distance dependence of the spectrum detects the crossover from algebraic fermionic decay at short distances to exponential bosonic decay at long distances, with crossover scale $\rho_A^{T_2}$6. In the massive model, tensor-network simulations show the same algebraic-to-exponential crossover governed by the first excited-state mass (Florio, 2023).

These fermionic results correct a common oversimplification. Negative real eigenvalues remain central in bosonic partial transpose, but fermionic negativity spectra can be pseudo-Hermitian, branch-valued, and intrinsically complex. The operationally relevant structure is then the full spectrum of the deformed operator, not only the sign of its real eigenvalues.

5. Random matrices, chaotic eigenstates, tensor networks, and holography

Random induced mixed states admit an analytically tractable negativity spectrum through a double-line diagrammatic expansion adapted to partial transpose. In the regime where both subsystems $\rho_A^{T_2}$7 and $\rho_A^{T_2}$8 are smaller than half the total system, the negativity spectrum is semicircular: $\rho_A^{T_2}$9 This immediately yields a PPT/NPT transition at $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$0: for $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$1, the support is entirely positive and the logarithmic negativity vanishes. The same framework identifies a plateau regime in which $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$2, a maximally entangled regime with $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$3, and non-GUE higher-order corrections despite the leading semicircle law (Shapourian et al., 2020).

Chaotic eigenstates furnish a related spectral problem through ETH and Wick-contraction techniques. Moments of $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$4 are organized by permutations on the geodesic $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$5, leading to hypergeometric expressions and a block-transposed Wishart resolvent for the averaged negativity spectrum. The transition of interest occurs when

$R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$6

or, in volume fractions, around $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$7. Near this transition the spectrum exhibits enhanced corrections analogous to those appearing in Rényi-entropy transitions, with different scaling depending on the replica continuation and on whether one studies logarithmic negativity or partially transposed entropy (McBride et al., 2023).

Random tensor networks and holographic fixed-area states extend these random-matrix ideas to geometric settings. A diagrammatic large-bond-dimension expansion, combined with a modified Ford–Fulkerson algorithm, identifies the dominant permutation assignments controlling $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$8. In pure states the negativity spectrum is simply determined by the entanglement spectrum, but in mixed-state phases the same methods produce semicircle, Marchenko–Pastur, Fuss–Catalan, and Motzkin-type spectra. In holography these spectra map onto connected and disconnected phases, phase transitions, multiboundary wormholes, and fixed-area states with bulk matter. The smallest random tensor network is the same as a micro-canonical version of Jackiw–Teitelboim gravity decorated with end-of-the-world branes, and in the Hawking-radiation problem the semiclassical negativity requires island contributions, verified directly in the JT model through Euclidean replica wormholes (Kudler-Flam et al., 2021).

6. Interpretation, bounds, and conceptual cautions

The negativity spectrum supports several operational and structural interpretations. One is as a witness of effective entangled dimension. For a bipartite state $R_n^{T_2}=\mathrm{Tr}(\rho_A^{T_2})^n,$9, the modified quantity

$P(\lambda)$0

has the property that $P(\lambda)$1 is a lower bound on the number of entangled local dimensions, and for axisymmetric states it equals the Schmidt number. In this sense, the partial-transpose spectrum can be read not only as an entanglement witness but as a counter of how many local levels participate in the entanglement (Eltschka et al., 2013).

A second perspective reverses the logic: the ordinary density-matrix spectrum constrains how much negativity can be produced by any global unitary. This was formalized through sets $P(\lambda)$2, consisting of states whose negativity never exceeds $P(\lambda)$3 along the entire unitary orbit. The resulting criteria are explicit inequalities on ordered eigenvalues and often require only a small subset of the spectrum. This suggests that the relation between entanglement spectrum and negativity spectrum is bidirectional in a broad sense: partial-transpose spectra diagnose entanglement, but the original density spectrum can also bound attainable negativity (Abellanet-Vidal et al., 2 Apr 2026).

A third interpretation is genuinely operational. For large random induced mixed states, logarithmic negativity typically equals the exact entanglement cost under positive-partial-transpose-preserving operations. In the PPT phase, $P(\lambda)$4 and both quantities vanish; in the maximally entangled phase, $P(\lambda)$5 at leading order; and in the entanglement-saturation phase the discrepancy is only $P(\lambda)$6, becoming negligible compared with $P(\lambda)$7. Here the spectrum of $P(\lambda)$8, together with the binegativity operator $P(\lambda)$9, supplies an operational interpretation of a quantity often treated purely as a computable entanglement monotone (Ouyang et al., 1 Jul 2026).

At the same time, positivity intuition must be used carefully. Perlmutter, Rangamani, and Rota showed that in $\mu_i$0 CFTs the universal part of entanglement entropy can become arbitrarily negative if and only if $\mu_i$1, and that in the same regime the logarithmic negativity does not always exceed the entanglement entropy. This does not concern the negativity spectrum directly, but it does show that finite-dimensional intuitions about positivity and monotonic dominance do not transfer straightforwardly to quantum field theory, especially when topology and anomaly coefficients enter (Perlmutter et al., 2015).

Taken together, these developments define the negativity spectrum as a spectral framework for mixed-state entanglement rather than a single technique. In pure states it is often reducible to the entanglement spectrum; in critical one-dimensional systems it obeys universal central-charge-controlled laws; in gapped, disordered, and valence-bond systems it reflects the underlying entanglement mechanism; in fermionic systems it may become complex; and in random, tensor-network, and holographic settings it undergoes spectral phase transitions. The recurring lesson is that the scalar logarithmic negativity captures only the coarsest summary of a much richer spectral object.