Logarithmic Negativity in Quantum Systems

• Logarithmic negativity is a computable measure of bipartite entanglement defined via the trace norm of the partial transpose of the density matrix, distinguishing quantum correlations in mixed states.
• It is widely applied in quantum many-body systems, field theories, and lattice models, with methods such as correlation matrix techniques, replica tricks, and cumulant expansions.
• Its sensitivity to area laws, topology, and dynamical changes makes it a vital tool for probing critical, disordered, and out-of-equilibrium quantum systems.

Logarithmic negativity is a computable measure of bipartite entanglement in quantum many-body systems, quantum field theory, and quantum information theory, particularly suited for mixed states where standard entanglement entropy fails to detect quantum correlations. It is defined as the logarithm of the trace norm of the partial transpose of the reduced density matrix, and its mathematical properties and physical applications span lattice models, continuum field theories, strongly interacting and random systems, holography, and dynamical or out-of-equilibrium settings.

1. Formal Definition and Mathematical Properties

Given a bipartite quantum system with density matrix $\rho_{AB}$ on Hilbert space $\mathcal{H}_{A} \otimes \mathcal{H}_{B}$, the logarithmic negativity is

$$\mathcal{E}(\rho_{AB}) \equiv \log \left\| \rho_{AB}^{T_B} \right\|_1 = \log \sum_{k} |\lambda_k|$$

where $\rho_{AB}^{T_B}$ is the partial transpose with respect to subsystem $B$ and $\{\lambda_k\}$ are its eigenvalues. This measure is an entanglement monotone, is efficiently computable in many settings, and serves as an upper bound on the distillable entanglement (Wang et al., 2019). For pure states, and in several quantum field theories, the logarithmic negativity is closely related to the Rényi entropy with index $1/2$: $\mathcal{E} = S_A^{(1/2)}$ For mixed states, logarithmic negativity distinguishes quantum from classical correlations.

The logarithmic negativity can be generalized into a family of $\alpha$-logarithmic negativities, interpolating between the standard logarithmic negativity ($\alpha=1$) and the $\kappa$-entanglement ($\alpha=\infty$), with important monotonicity, normalization, faithfulness, and subadditivity properties (Wang et al., 2019).

2. Field-Theoretic and Lattice Implementations

In quantum many-body systems (spin chains, harmonic lattices, free fermions), the logarithmic negativity can be computed via correlation matrix techniques, path integral replica tricks, or (for random states) diagrammatic and random matrix methods (Blondeau-Fournier et al., 2015, Nobili et al., 2016, Shapourian et al., 2020). In conformal field theory (CFT) or massive QFT, negativity calculations are performed using four-point functions of "branch-point twist fields," with replica analytic continuation in even replica index $n_e \to 1$ (Blondeau-Fournier et al., 2015). In free-fermion models, negativity can be expanded systematically as a charge cumulant (FCS) expansion in connected correlators, facilitating practical calculations and revealing connections to topological features such as the Euler characteristic of the Fermi sea (Tang, 6 Feb 2024).

Table: Representative Calculation Approaches

System	Method	Key Feature
Free Bosons/Fermions	Correlation matrix, FCS	Cumulant expansions, rapid convergence (Tang, 6 Feb 2024)
Lattice Harmonic Models	Replica + Correlators	Area/corner scaling, vertex corrections (Nobili et al., 2016)
CFT / Massive QFT	Twist field CFT/replica	Universal scaling governed by mass spectrum (Blondeau-Fournier et al., 2015)
Random Mixed States	Diagrammatic/Moment expansions	Phase diagrams, semicircle spectrum (Shapourian et al., 2020)

For bosonic Gaussian states, the partial transpose is implemented by time-reversal on a subset of momenta; for fermionic systems, a careful definition respecting parity and anticommutation is needed (sometimes called "twisted" partial transpose) (Parez et al., 2023, Wang et al., 10 Mar 2025).

3. Universal and Geometry-Dependent Scaling Laws

• Area Law: In higher-dimensional lattice models (e.g., 2D harmonic lattices), the leading term of the logarithmic negativity between adjacent domains is proportional to the length of the shared boundary (the area law), with a universal subleading correction when the boundary has corners (Nobili et al., 2016).
• Corner Contributions: In 2D, when the interface contains vertices, the negativity receives logarithmic corrections determined by universal corner functions, which for explementary angles are numerically equal to those for the Rényi entropy of order $1/2$ (Nobili et al., 2016).
• Genus and Topological Dependence in CFT: In 4D conformal field theories, universal contributions to both logarithmic negativity and entanglement entropy are controlled by the central charges $a, c$ and the topology and geometry of the entangling surface. When $a > c$, nontrivial genus surfaces can yield arbitrarily negative universal entanglement entropy, and the negativity can drop below the entropy, violating standard finite-dimensional quantum mechanical bounds (Perlmutter et al., 2015).

Key Structural Formulas

• Universal negativity ratio for 4D CFTs:

$$X = \frac{(a-c)R_{\mathcal{A}} + 2c W_{\mathcal{A}}}{2cW_{\mathcal{A}} + (a-c)R_{\mathcal{A}}}$$

where $R_\mathcal{A}$ is the curvature integral and $W_\mathcal{A}$ is the WiLLMore energy of the surface (Perlmutter et al., 2015).

• In 1+1D massive QFT, large-distance corrections to negativity decay exponentially with distance, with rates determined solely by the mass spectrum, indicating universality:

$$\mathcal{E}^{\perp} \sim -\frac{c}{4}\log(m_1) + \mathcal{E}_{sat} - \frac{2}{3\sqrt{3}\pi} \sum_{\alpha} K_0(\sqrt{3} m_{\alpha} r)$$

where $K_0$ is the modified Bessel function, $m_{\alpha}$ are particle masses (Blondeau-Fournier et al., 2015).

4. Dynamics, Out-of-Equilibrium Scaling, and Disorder

Logarithmic negativity is a sensitive probe of quantum correlations in non-equilibrium and disordered systems.

• Quantum Quenches and Dynamics: After quenches, logarithmic negativity evolves according to a quasi-particle interpretation: initially zero, it shows linear growth delayed by half the inverse temperature ($\beta/2$) for adjacent intervals, or by half the separation for disjoint intervals; later saturates or vanishes depending on subsystem geometry (Fujita et al., 2018, Ghasemi et al., 2021). In dissipative or open systems, the negativity reflects both ballistic propagation and exponential decay due to dissipation, with negativity content per quasi-particle deviating from simple Rényi entropy forms as in the unitary case (Alba et al., 2022).
• Disorder and Random Systems: In the random singlet phase of disordered spin chains, the disorder-averaged negativity is set by the number of shared singlets, scaling logarithmically with subsystem size, and decaying algebraically for disjoint intervals (unlike exponential decay in translation-invariant systems) (Ruggiero et al., 2016). In random mixed states, negativity exhibits a phase diagram delicately dependent on system-bath partitioning, with features such as an entanglement plateau and transitions to PPT (separable) phases as the bath size dominates (Shapourian et al., 2020).
• Many-Body Localization (MBL) Transitions: At the critical point, logarithmic negativity between blocks exhibits scale-invariant exponential decay with distance-to-size ratio, while mutual information decays polynomially, suggesting distinct scaling for quantum versus classical correlations and hinting at possible fractal eigenstate organization (Gray et al., 2019).

5. Fermionic Systems and Computational Proxies

Direct computation of LN in interacting fermionic systems is challenging due to nontrivial fermionic statistics.

• Fermionic Partial Transpose and Proxies: Definitions of fermionic partial transpose ("untwisted" and "twisted") yield different Rényi negativities (RN); but only the twisted RN and its ratio ("RNR") are found to obey area-law scaling and decrease monotonically with temperature, matching expectations from LN in bosonic systems (Wang et al., 10 Mar 2025). This establishes twisted RNR as a robust proxy for LN in large-scale QMC studies. Careful stabilization and incremental algorithms are required to address the large variance of Grover determinants arising for high-rank RNs.
• Free Fermion Charge Correlator Expansion: For translationally invariant free-fermion systems, logarithmic negativity can be expressed as a rapidly convergent expansion in charge cumulants, offering practical and accurate evaluation in large systems. The leading term reproduces universal CFT scaling, and the expansion connects entanglement structure to charge fluctuations and Fermi surface topology (Tang, 6 Feb 2024).
• Fine Structure and Non-Universalities: For well-separated intervals in free fermion models, strict negativity of PTDM eigenvalues need not occur; rather, eigenvalues develop small imaginary parts, yielding non-vanishing negativity that scales as $(\log N)^{-1}$ for large intervals at half-filling, but with a highly non-universal, parity-sensitive profile (Bettelheim, 2023). The field theory continuum limit may smooth these details.

6. Critical Systems, Universality, and Holography

• Critical Scaling: In CFTs and quantum critical lattice models, logarithmic negativity at short separation displays area or boundary laws with universal coefficients dependent on CFT data (Parez et al., 2023). At large separation, for bosons, negativity can decay faster than any power or even vanish beyond a sudden-death distance (for single-spin subsystems), demonstrating that quantum critical ground states do not generally possess long-range bipartite distillable entanglement—despite the existence of long-range correlations or mutual information (Parez et al., 2023). For fermionic systems (with parity-sensitive definition of LN), negativity decays algebraically, controlled by the dimension of the leading fermionic operator.
• Holographic Duality: In AdS/CFT, logarithmic negativity is geometrized as the minimal area of a tensioned cosmic brane anchored to the entanglement wedge cross-section, possibly modified by quantum corrections from bulk negativity (Kudler-Flam et al., 2018, Kusuki et al., 2019). In large-$c$ 2D CFTs, for symmetric regions, $\mathcal{E} = (3/2)\,E_W$, where $E_W$ is the entanglement wedge cross-section area. Through a connection to the reflected entropy, $\mathcal{E} = (S_R^{(1/2)})/2$, matching CFT twist-operator calculations and the gravitational dual (Kusuki et al., 2019). Zamolodchikov's recursion for Virasoro blocks efficiently encodes the universal terms in LN (Kudler-Flam et al., 2018).

7. Excited States, Multiparty, and Topological Phases

• Excited States: For low-density/few-particle excited states in QFT, increments in Rényi and logarithmic negativity depend only on the overall size of the involved regions, independent of connectivity—demonstrated both analytically (via qubit pictures and twist field form factors) and numerically (Castro-Alvaredo et al., 2019).
• Area laws in random, topological, and higher-dimensional models: Negativity robustly detects the topology and geometry of Fermi seas, realizes area scaling in 2D systems, and serves as a diagnostic of entanglement structure in random, topological, and critical phases (Nobili et al., 2016, Tang, 6 Feb 2024).

8. Outlook, Computational Methods, and Physical Significance

Logarithmic negativity has emerged as a versatile quantum correlation measure, with practical computational schemes—correlation matrix methods, cumulant expansions, stochastic Monte Carlo approaches, and diagrammatic techniques—adapting to the needs of interacting, random, or free systems across platforms. It reveals distinctly quantum, sometimes topology-sensitive or geometry-sensitive information inaccessible to entanglement entropy or mutual information. Its modifications for fermionic systems and the operational meaning in higher-dimensional field theory (especially in the presence of topology, curvature, and symmetry constraints) remain active areas of research, including for holographic dualities and resource theories of quantum channels (Wang et al., 2019).

The landscape of logarithmic negativity thus interconnects quantum information, condensed matter, and high-energy theory, with developments in computational methods paralleling deep insights into the structure and universality of quantum entanglement.