Emergent Geometry from Entanglement
Structure
Sudipto Singha Roy, Silvia N. Santalla, Javier Rodríguez-Laguna,
and Germán Sierra
Abstract We attempt to reveal the geometry, emerged from the entanglement
structure of any general N -party pure quantum many-body state by representing
entanglement entropies corresponding to all 2 N bipartitions of the state by means of
a generalized adjacency matrix. We show this representation is often exact and may
lead to a geometry very different than suggested by the Hamiltonian. Moreover, in
all the cases, it yields a natural entanglement contour, similar to previous proposals.
The formalism is extended for conformal invariant systems, and a more insightful
interpretation of entanglement is presented as a flow among different parts of the
system.
Keywords Quantum entanglement · Geometry · Entanglement entropy
1 Introduction
Study of the distribution of quantum entanglement in different parts of the low-lying
states of quantum many-body Hamiltonians often unveils many interesting features
related to the physical system [1–3]. For instance, bounded growth of quantum
entanglement between a region and its exterior can be attributed to the fact that
interactions in the quantum many-body systems are typically local [4–7]. In this
work, we aim to explore the connection between the area-law for entanglement [4–
S. S. Roy · G. Sierra ()
Instituto de Física Teórica, UAM-CSIC, Universidad Autónoma de Madrid Cantoblanco, Madrid,
Spain
e-mail: german.sierra@uam.es
S. N. Santalla
Dept. de Física and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Carlos III
de Madrid, Madrid, Spain
e-mail: silvia.santalla@uc3m.es
J. Rodríguez-Laguna
Dept. de Física Fundamental, UNED, Madrid, Spain
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_33
347
Structure
Sudipto Singha Roy, Silvia N. Santalla, Javier Rodríguez-Laguna,
and Germán Sierra
Abstract We attempt to reveal the geometry, emerged from the entanglement
structure of any general N -party pure quantum many-body state by representing
entanglement entropies corresponding to all 2 N bipartitions of the state by means of
a generalized adjacency matrix. We show this representation is often exact and may
lead to a geometry very different than suggested by the Hamiltonian. Moreover, in
all the cases, it yields a natural entanglement contour, similar to previous proposals.
The formalism is extended for conformal invariant systems, and a more insightful
interpretation of entanglement is presented as a flow among different parts of the
system.
Keywords Quantum entanglement · Geometry · Entanglement entropy
1 Introduction
Study of the distribution of quantum entanglement in different parts of the low-lying
states of quantum many-body Hamiltonians often unveils many interesting features
related to the physical system [1–3]. For instance, bounded growth of quantum
entanglement between a region and its exterior can be attributed to the fact that
interactions in the quantum many-body systems are typically local [4–7]. In this
work, we aim to explore the connection between the area-law for entanglement [4–
S. S. Roy · G. Sierra ()
Instituto de Física Teórica, UAM-CSIC, Universidad Autónoma de Madrid Cantoblanco, Madrid,
Spain
e-mail: german.sierra@uam.es
S. N. Santalla
Dept. de Física and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Carlos III
de Madrid, Madrid, Spain
e-mail: silvia.santalla@uc3m.es
J. Rodríguez-Laguna
Dept. de Física Fundamental, UNED, Madrid, Spain
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_33
347