Ratio Negativity in Quantum Entanglement

• Ratio negativity is a family of negativity-based normalizations that reparameterize traditional negativity measures to yield bounded entanglement monotones in quantum systems.
• It provides operational insights in continuous-variable setups by ensuring multiplicative decay in entanglement swapping for chains of two-mode squeezed vacuum states.
• The framework extends to 4d CFT and fermionic systems, using universal term and Rényi negativity ratios to compare entanglement against entropy and thermal contributions.

Ratio negativity denotes a family of negativity-based normalizations rather than a single universally standardized invariant. In the most explicit usage, it is the bounded monotone $$\chi_{\mathcal N}(\rho)=\mathcal N(\rho)/(\mathcal N(\rho)+1)$$, introduced for continuous-variable quantum networks as a special $f$-negativity built from the standard bipartite negativity $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$ (Zhao et al., 2024). In adjacent literatures, closely related constructions include the ratio of the universal part of logarithmic negativity to the universal part of entanglement entropy in $4d$ conformal field theory, $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$, and the fermionic Rényi negativity ratios $R_r$ and $\tilde R_r$, defined by subtracting thermal Rényi entropy from untwisted and twisted Rényi negativities (Perlmutter et al., 2015, Wang et al., 10 Mar 2025). A separate but historically important usage concerns the dependence of negativity on geometric cross ratios for disjoint intervals in $1+1d$ quantum field theory, where the logarithmic negativity is a scale-invariant function of the harmonic ratio alone (Calabrese et al., 2012).

1. Terminological scope and main variants

The literature uses “ratio negativity” in several technically distinct ways. The clearest explicit instances can be organized as follows.

Context	Quantity	Role
Continuous-variable Gaussian networks	$$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$	Bounded entanglement monotone and characteristic swapping measure (Zhao et al., 2024)
$4d$ CFT universal terms	$f$0	Compares universal logarithmic negativity and universal entanglement entropy (Perlmutter et al., 2015)
Corner and cone singularities in CFT	$f$1	Compares universal negativity and entropy for smooth versus singular entangling regions (Kim et al., 2016)
Interacting fermionic many-body systems	$f$2, $f$3	Rényi negativity ratios used as practical proxies for logarithmic negativity (Wang et al., 10 Mar 2025)
Disjoint intervals in $f$4 QFT/CFT	$f$5	Cross-ratio-controlled negativity rather than a quotient normalization (Calabrese et al., 2012)

Several papers outside the continuous-variable setting explicitly state that they do not introduce a quantity literally named ratio negativity, but instead analyze the closest relevant normalized negativity observable in their own context (Perlmutter et al., 2015, Eltschka et al., 2015). This suggests that the term is context-dependent rather than canonical.

2. Ratio negativity as a bounded $f$6-negativity

In the continuous-variable network formulation, ratio negativity is defined from the standard negativity

$f$7

by choosing the $f$8-negativity with

$f$9

This yields

$\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$0

so $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$1 and is a strictly increasing reparametrization of $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$2 (Zhao et al., 2024). The bounded range is one of its defining advantages over both $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$3 and logarithmic negativity.

For a pure bipartite state $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$4 with Schmidt coefficients $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$5,

$\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$6

For pure Gaussian states brought by local Gaussian unitaries to a product of two-mode squeezed factors with squeezing parameters $\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$7, the same quantity becomes

$\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$8

For a two-mode squeezed vacuum state,

$\mathcal N(\rho)=\frac12(\|\rho^{T_A}\|_1-1)$9

one obtains the especially simple identity

$4d$0

For $4d$1-mode mixed Gaussian states with partially transposed covariance-matrix symplectic eigenvalue $4d$2,

$4d$3

(Zhao et al., 2024).

The same work places ratio negativity inside a broader $4d$4-family,

$4d$5

and proves that $4d$6 is an entanglement monotone whenever $4d$7 is concave. Since $4d$8 is concave on $4d$9, $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$0 is an entanglement monotone. At the same time, it is nonconvex: for

$$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$1

the corresponding $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$2-negativity is non-convex for

$$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$3

so ordinary ratio negativity, corresponding to $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$4, is a nonconvex entanglement monotone (Zhao et al., 2024).

3. Characteristic measure for Gaussian entanglement swapping

The principal operational significance of ratio negativity appears in continuous-variable quantum networks built from pure Gaussian links, especially chains of two-mode squeezed vacuum states. In that setting the relevant notion is a characteristic measure $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$5 satisfying

$$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$6

so that the end-to-end entanglement of a swapped link is multiplicative along the chain (Zhao et al., 2024).

For the optimal Gaussian deterministic swapping protocol on TMSVSs, ratio negativity obeys exactly this rule: $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$7 The same multiplicativity holds for the $$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$8-ratio negativity,

$$\mathcal X=\frac{C_{\rm univ}[\mathscr E(\rho)]}{C_{\rm univ}[S(\rho_A)]}$$9

For a chain of $R_r$0 identical links $R_r$1, the characteristic-measure equation gives

$R_r$2

hence, specifically,

$R_r$3

Because $R_r$4 for a TMSVS, the swapped entanglement decays exponentially as $R_r$5. The paper identifies this as the quantity that determines the exponential decay of optimal entanglement swapping on a chain of pure Gaussian states and therefore the characteristic length of entanglement distribution (Zhao et al., 2024).

This role places ratio negativity in the same structural position that concurrence and $R_r$6-concurrence occupy in discrete-variable swapping protocols. The comparison is explicit in the network-oriented discussion: ratio negativity is the characteristic measure for optimal Gaussian deterministic swapping of TMSVSs (Zhao et al., 2024).

4. Universal-term ratios in conformal field theory

A second major usage replaces the bounded monotone $R_r$7 by ratios of universal contributions to negativity and entropy in relativistic quantum field theory. In $R_r$8 CFT, the natural normalized object is

$R_r$9

where $\tilde R_r$0 denotes the cutoff-independent universal term and $\tilde R_r$1 is logarithmic negativity (Perlmutter et al., 2015). The interpretation is immediate: $\tilde R_r$2 means the universal part of logarithmic negativity exceeds that of entanglement entropy, $\tilde R_r$3 means equality, and $\tilde R_r$4 means negativity is smaller. The paper also uses

$\tilde R_r$5

and shows that there is no universal positivity theorem for $\tilde R_r$6.

For smooth entangling surfaces in $\tilde R_r$7 CFT, the universal entropy is

$\tilde R_r$8

and the negativity-to-entropy ratio becomes

$\tilde R_r$9

This makes $1+1d$0 topology-sensitive, shape-sensitive, and anomaly-sensitive. For the sphere, $1+1d$1 is positive. For the torus, positivity requires $1+1d$2, which the paper states holds in all known examples. For higher genus, the behavior depends sharply on whether $1+1d$3 or $1+1d$4; in $1+1d$5 theories, sufficiently high-genus Lawson surfaces can drive the universal entropy negative, and the ratio can become negative (Perlmutter et al., 2015).

A closely related construction appears in the study of corner and cone singularities, where the paper defines

$1+1d$6

for pure-state bipartitions, with $1+1d$7 in the situations considered (Kim et al., 2016). Near the smooth-corner regime the singular-surface ratio is

$1+1d$8

In most examples singular entangling surfaces increase $1+1d$9, but the paper also gives counterexamples: in $$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$0 the free scalar has $$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$1, in $$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$2 the free scalar has $$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$3, and in higher-derivative holographic models $$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$4 may either increase or decrease depending on couplings (Kim et al., 2016). In this CFT usage, ratio negativity is not a monotone rescaling of $$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$5, but a quotient of universal field-theoretic data.

5. Rényi negativity ratios in fermionic systems

In interacting fermionic many-body systems the relevant ratio construction is the Rényi negativity ratio. Because fermionic partial transpose is not unique at the level of moments, the paper distinguishes untwisted and twisted fermionic partial transposes and defines the corresponding Rényi negativities

$$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$6

The thermal Rényi entropy is

$$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$7

The Rényi negativity ratios are then

$$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$8

and

$$\chi_{\mathcal N}(\rho)=\frac{\|\rho^{T_A}\|_1-1}{\|\rho^{T_A}\|_1+1}$$9

(Wang et al., 10 Mar 2025).

The paper emphasizes that untwisted and twisted fermionic partial transposes yield the same logarithmic negativity, but generally different Rényi negativities and hence different Rényi negativity ratios. The twisted family is conceptually preferred because it is Hermitian and satisfies the replica-continuation relation

$4d$0

so it is directly tied to logarithmic negativity (Wang et al., 10 Mar 2025).

The rank-2 twisted case is trivial: $4d$1 Nontrivial twisted behavior therefore begins at higher rank, and the paper focuses especially on rank $4d$2. Its main numerical conclusion is that the rank-4 twisted Rényi negativity ratio obeys an area law and decreases monotonically with temperature, whereas higher-rank untwisted RNRs can be non-monotonic in temperature. This leads to the central interpretive claim that the twisted RNR, especially $4d$3, is the more faithful Rényi proxy for logarithmic negativity in fermionic systems (Wang et al., 10 Mar 2025).

6. Cross-ratio-controlled negativity for disjoint intervals

A different, geometric sense of ratio negativity arises in $4d$4 quantum field theory and critical many-body systems, where negativity for disjoint intervals is controlled by a conformal cross ratio rather than by a quotient of scalar entanglement measures. In the field-theoretic construction for

$4d$5

the logarithmic negativity is obtained from the even-replica continuation of

$4d$6

and for two disjoint intervals the final answer is a scale-invariant function only of the harmonic ratio/cross ratio (Calabrese et al., 2012, Calabrese et al., 2012). The same framework gives the adjacent-interval result

$4d$7

in an infinite system (Calabrese et al., 2012, Calabrese et al., 2012).

At large central charge the cross-ratio dependence near $4d$8 is controlled by a nontrivial conformal block, and the leading adjacent-limit singularity is

$4d$9

The large-$f$00 analysis also finds a nontrivial monodromy problem and branch structure, while for $f$01 the negativity is non-perturbatively small (Kulaxizi et al., 2014).

Outside equilibrium CFT ground states, the same cross-ratio language persists but with different asymptotics. At the measurement-driven entanglement transition in one-dimensional hybrid random circuits, the logarithmic negativity of two disjoint intervals obeys

$f$02

for small cross ratio $f$03, with fitted values

$f$04

The corresponding mutual information scales as

$f$05

The paper stresses that this power law is unlike the disjoint-interval negativity of any unitary CFT ground state, where the negativity decays faster than any power at small cross ratio (Shi et al., 2020). In this geometric usage, “ratio negativity” refers to cross-ratio dependence rather than to normalization by another correlation measure.

7. Related normalizations, comparison ratios, and conceptual limits

Ratio negativity is closely connected to other normalization schemes for negativity. A particularly important example is the dimension-aware modified negativity

$f$06

introduced to make the dimension dependence transparent. For a maximally entangled pure state of Schmidt rank $f$07,

$f$08

For axisymmetric states,

$f$09

whereas for arbitrary bipartite states one always has

$f$10

This is not called ratio negativity, but it is a closely related normalization of the same trace-norm data (Eltschka et al., 2013).

Another adjacent comparison problem concerns negativity versus concurrence. In $f$11 systems the paper on partial transpose and concurrence does not introduce a quantity explicitly called ratio negativity, but it derives pure-state comparison inequalities

$f$12

for Schmidt rank $f$13. These imply

$f$14

and, for pure states of two qubits,

$f$15

Again, the issue is not a named ratio negativity, but a ratio involving negativity that compares different entanglement monotones on equal footing (Eltschka et al., 2015).

Taken together, these constructions show that “ratio negativity” is best understood as an umbrella term for several normalization strategies built from negativity or logarithmic negativity. The bounded monotone $f$16, the CFT quotients $f$17 and $f$18, the fermionic Rényi negativity ratios $f$19 and $f$20, and the cross-ratio-controlled functions $f$21 are mathematically distinct objects. This suggests that any technical use of the term should specify whether the relevant “ratio” refers to a monotone reparametrization, a quotient of universal terms, a thermal subtraction, or a geometric cross ratio.