Time evolution of entanglement negativity from black hole interiors
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Springer
Received: March 21, 2019 Accepted: May 18, 2019 Published: May 28, 2019
Vinay Malvimat,a Sayid Mondalb and Gautam Senguptab a
Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan, Pune 411 008, India b Department of Physics, Indian Institute of Technology, Kanpur 208 016, India
E-mail: [email protected], [email protected], [email protected] Abstract: We investigate the time evolution of entanglement negativity following a global quench for mixed state configurations of two disjoint and adjacent intervals in a (1 + 1)dimensional conformal field theory (CF T1+1 ) dual to the eternal black hole sliced in half by an end of the world brane, through the AdS3 /CF T2 correspondence. To this end we obtain the time evolution of the holographic entanglement negativity for such mixed states from a dual bulk eternal black hole geometry and elucidate the relevant geodesic structures. The holographic entanglement negativity for such mixed states, following a global quench is described by half of the results for the eternal black hole. Significantly our results exactly match with the corresponding CF T1+1 computations. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 1812.04424
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)183
JHEP05(2019)183
Time evolution of entanglement negativity from black hole interiors
Contents 1
2 Entanglement entropy
5
3 Entanglement negativity 3.1 Two disjoint intervals 3.2 Two adjacent intervals
6 7 8
4 Holographic entanglement entropy
9
5 Holographic entanglement negativity 5.1 Two disjoint intervals 5.2 Two adjacent intervals
13 14 17
6 Summary and discussion
20
1
Introduction
Over the last few years quantum entanglement has emerged as a central issue in the study of diverse physical phenomena ranging from quantum many-body systems in out-ofequilibrium to the process of black hole formation and the information loss paradox. The entanglement for a bipartite pure state is characterized by the entanglement entropy which is the von Neumann entropy of the reduced density matrix. In (1 + 1)-dimensional conformal field theories (CF T1+1 ) the entanglement entropy may be computed through a replica technique as described in [1, 2]. However entanglement entropy fails to characterize mixed state entanglement as it typically involves correlations irrelevant to the entanglement of the specific mixed state. This subtle issue was addressed in quantum information theory by Vidal and Werner in [3] where the authors proposed a computable measure termed as entanglement negativity which characterized the upper bound on the distillable entanglement for the mixed state.1 The entanglement negativity is defined as the logarithm of the trace norm of the partially transposed density matrix with respect to a subsystem. In a series of interesting communications the authors in [5–7] described the computation of the entanglement negativity for bipartite mixed states in a CF T1+1 through a suitable rep
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