Entanglement negativity in quantum field theory

Abstract

We develop a systematic method to extract the negativity in the ground state of a 1+1 dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose rho_A^{T_2} of the reduced density matrix of a subsystem A=A1 U A2, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity E=ln||ρ_A^{T_2}||. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result E\sim(c/4) ln(L1 L2/(L1+L2)) for the case of two adjacent intervals of lengths L1, L2 in an infinite system, where c is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.

4 pages, 5 figures

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• Entanglement negativity and entropy in non-equilibrium conformal field theory
• Conformal Blocks and Negativity at Large Central Charge
• Entanglement Entropy of Non Unitary Conformal Field Theory
• Random matrices and entanglement entropy of trapped Fermi gases
• Entanglement entropy and negativity of disjoint intervals in CFT: Some numerical extrapolations
• Universality in the geometric dependence of Renyi entropy
• Bose-Fermi duality and entanglement entropies
• On shape dependence of holographic mutual information in AdS4
• Macroscopic Quantum Entanglement of a Kondo Cloud at Finite Temperature
• Entanglement negativity in the multiverse
• Scaling of entanglement in $2+1$-dimensional scale-invariant field theories
• An Area Law for the Bipartite Entanglement of Disordered Oscillator Systems