Entanglement negativity in quantum field theory (1206.3092v2)

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||\rho_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.

• The paper develops a replica approach to compute entanglement negativity via the partial transpose of the reduced density matrix.
• It applies the method to conformal field theories, deriving a universal logarithmic formula involving the central charge and interval lengths.
• Numerical simulations in harmonic chain models validate the predictions, offering insights for quantum computing and information processing.

Entanglement Negativity in Quantum Field Theory

The paper "Entanglement negativity in quantum field theory" by Pasquale Calabrese, John Cardy, and Erik Tonni addresses a critical aspect of quantum information science, focusing on the quantification of entanglement between non-complementary subsystems within quantum field theories (QFT). Utilizing a path integral formalism coupled with a replica approach, the authors develop a systematic method to extract entanglement negativity, a valuable entanglement measure particularly in mixed states.

Overview of Methodology

The entanglement negativity is defined through the partial transpose of the reduced density matrix of a subsystem. Specifically, for a subsystem $A = A_1 \cup A_2$, the partial transpose $\rho_A^{T_2}$ is computed, with the logarithmic negativity given by $\mathcal{E} = \ln ||\rho_A^{T_2}||$. The authors first validate their approach for the well-understood case of a pure state where standard results are reproduced, thereby establishing a baseline for their method's correctness. They then extend their framework to conformal field theories (CFTs), identifying universal entanglement characteristics near quantum critical points.

Conformal Field Theory Applications

In CFT, the authors derive a formula for the logarithmic negativity for two adjacent intervals within an infinite system. The result, $$\mathcal{E} \sim (c/4) \ln(\ell_1\ell_2/(\ell_1+\ell_2))$$, where $c$ is the central charge and $\ell_1, \ell_2$ are the lengths of the intervals, highlights the scale-invariant nature of the negativity. This function depends solely on the harmonic ratio of the intervals' endpoints in the case of disjoint intervals, demonstrating a scale-invariant property not previously rigorously established.

The paper also elaborates on a novel replica approach, constructing replicas for $\rho_A^{T_2}$ and analyzing the analytic continuation of the replicated trace norms to provide a quantitative measure of entanglement negativity.

Theoretical and Practical Implications

From a theoretical standpoint, this work enriches the understanding of entanglement properties in quantum many-body systems, especially near critical points where standard measures like entanglement entropy may not provide granular insights into non-complementary subsystem entanglements. It also broadens the applicability of CFT tools to tackle complex entanglement structures beyond pure states. Practically, the results can be pivotal in quantum computing and quantum information processing domains, especially within systems that rely on mixed entangled states.

Comparison with Numerical Results

Validation of theoretical predictions through numerical simulations is a critical component of this paper. The calculations in the harmonic chain model, a discrete analogue of the $c=1$ free boson CFT, confirm the theoretical predictions. The adherence of numerical data to predicted theoretical curves underscores the validity of the proposed framework.

Future Directions

The implications of this research lay the groundwork for future investigations into more complex system configurations and further explorations into non-standard lattice models, such as spin chains and itinerant fermions. Challenges persist, notably the analytic continuation for the replica index, underscoring a rich field for continued paper. Additionally, extending these techniques to finite temperature scenarios or massive QFTs might yield further insight into the nature of entanglement in these complex systems.

In conclusion, this paper advances the understanding of entanglement in QFT, providing robust theoretical tools and carefully validated conclusions. It offers a significant contribution to the field of quantum information theory, enlightening both theoretical pursuits and practical applications.