| logarithmic negativity | |
|---|---|
| Name | Logarithmic negativity |
| Unit | dimensionless |
| Symbols | E_N |
| Field | Quantum information science |
logarithmic negativity
Logarithmic negativity is an entanglement measure for mixed states in quantum mechanics and quantum information theory that quantifies the degree to which a bipartite density matrix fails to be positive under partial transposition. It is widely used because it is computable for many finite-dimensional systems and provides an upper bound on distillable entanglement, making it valuable in studies of entanglement in quantum optics, condensed matter physics, and quantum field theory.
The logarithmic negativity E_N of a bipartite state ρ_{AB} on Hilbert space H_A ⊗ H_B is defined in terms of the trace norm of the partial transpose ρ_{AB}^{T_B}. It captures nonseparability that is detected by the Peres–Horodecki criterion (positive partial transpose, PPT). Because E_N is monotone under local operations and classical communication (LOCC) in certain contexts and is nonincreasing under separable operations, it is used as an operational proxy for entanglement for states where other measures (e.g., entanglement of formation) are hard to compute. It plays a role in evaluating entanglement resources for protocols such as quantum teleportation and entanglement distillation.
For a density operator ρ_{AB}, the partial transpose with respect to subsystem B is defined by its matrix elements in product bases: ⟨i_A j_B|ρ^{T_B}|k_A l_B⟩ = ⟨i_A l_B|ρ|k_A j_B⟩. The trace norm ||X||_1 = Tr(√(X†X)) gives the negativity N(ρ) = (||ρ^{T_B}||_1 − 1)/2, which equals the absolute sum of negative eigenvalues of ρ^{T_B}. The logarithmic negativity is E_N(ρ) = log_2 ||ρ^{T_B}||_1 = log_2 (2N(ρ) + 1). This definition makes E_N additive on tensor products and yields nonzero values for PPT-violating states. The logarithm base is commonly 2 to express entanglement in bits; other bases may be used for natural units.
Logarithmic negativity satisfies several important properties: it is an entanglement monotone under deterministic LOCC for pure states and is convex on ensembles in many practical settings. It is an upper bound on the distillable entanglement E_D (E_D ≤ E_N), linking it to resource theory statements proven by authors such as Vlatko Vedral and the Horodecki family. E_N detects free entanglement not hidden by PPT constraints but cannot distinguish bound entangled states with PPT. For Gaussian states in continuous-variable systems, E_N has a direct relation to symplectic spectra and can be computed from covariance matrices; this connects it to work by Serafini, Adesso, and Illuminati on Gaussian entanglement. E_N is also connected to operational tasks: it bounds the fidelity of quantum teleportation and sets limits in one-shot entanglement manipulation protocols studied in the context of quantum Shannon theory and the Resource theory of entanglement.
Computing E_N requires diagonalization of ρ^{T_B} or evaluation of its trace norm. For finite-dimensional systems, numerical linear algebra (e.g., eigenvalue solvers) suffices; common toolchains include libraries used in computational physics and quantum chemistry at institutions such as CERN and major research groups. For two-qubit systems, analytical formulas exist, and E_N reduces to simple functions of concurrence for pure states studied by William K. Wootters. For continuous-variable Gaussian states, E_N is computed from the symplectic eigenvalues of the partially transposed covariance matrix using Williamson's theorem; this approach is central in experimental quantum optics settings such as squeezing and entanglement generation in optical parametric amplifiers and cavities used by groups at Caltech and Max Planck Institute for the Science of Light. Example calculations include entangled two-mode squeezed vacuum states, thermal states of harmonic lattices, and reduced density matrices extracted from matrix product states and density matrix renormalization group simulations in many-body models.
Logarithmic negativity is applied across several domains: in quantum information it quantifies resources for quantum cryptography and teleportation protocols implemented by experimental teams at institutions like IQOQI Vienna and IBM Quantum. In many-body physics and statistical mechanics, E_N is used to diagnose entanglement in ground states and thermal states of lattice models (e.g., Heisenberg model, Ising model), and to study entanglement scaling near quantum phase transitions investigated in works by John Cardy and others. In high-energy physics and holographic studies, logarithmic negativity has been proposed as a probe of entanglement structure across spatial bipartitions in conformal field theories (CFTs), connecting to calculations by Pablo Calabrese and collaborators. It is also used in non-equilibrium dynamics (quantum quenches), where time-evolution of E_N reveals spreading of entanglement and information.
Related measures include the negativity N(ρ), entanglement of formation, concurrence, relative entropy of entanglement, and Rényi entropies, each with different operational interpretations and computational properties. Extensions of logarithmic negativity address multipartite settings, fermionic systems (fermionic partial transpose), and continuum quantum field theories, where regularization and renormalization issues arise; such work links to authors like H. Casini and M. Huerta. Alternative PPT-based quantifiers and monotones (e.g., PPT entanglement cost) complement E_N in resource-theoretic frameworks developed by researchers at Perimeter Institute and University of Oxford. The interplay between computability and operational relevance continues to motivate comparative studies of entanglement measures in both theoretical and experimental quantum science.
Category:Quantum entanglement Category:Quantum information theory