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Papers by Lukas Schneiderbauer
Journal of Physics A, May 31, 2016
We develop a systematic approach to determine and measure numerically the geometry of generic qua... more We develop a systematic approach to determine and measure numerically the geometry of generic quantum or "fuzzy" geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in R d including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace-or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.
BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterize... more BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterizes the semi-classical limit of a fuzzy brane described by a set of matrices.
Journal of High Energy Physics, Jul 1, 2020
We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional ... more We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional dilaton gravity and determine the rate of increase of holographic complexity for an evaporating black hole. The results are consistent with our previous numerical results for semi-classical black hole complexity using a volume prescription (CV) in the same model, but the CA calculation is fully analytic and provides a non-trivial positive test for the holographic representation of the black hole interior.
This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-... more This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-classical formula for the entanglement entropy of black hole radiation is examined. This entropy is an indicator of information loss and determines whether black hole evaporation is an information preserving process or destroys quantum information. Assuming information conservation, Page expressed the entanglement entropy as a function of time, which is referred to as the "Page curve." Using the improved formula for evaporating black hole solutions of a gravitational model introduced by Callan, Giddings, Harvey and Strominger (CGHS) and modified by Russo, Susskind and Thorlacius (RST), we find that the entanglement entropy follows the Page curve and thus is consistent with unitary evolution. Second, the notion of quantum complexity is explored in the context of black holes. The quantum complexity of a quantum state measures how many "simple operations" are required to create that state. Susskind conjectured that the quantum complexity of a black hole state corresponds to a certain volume inside the black hole. A modified conjecture equates the quantum complexity with the gravitational action evaluated for a certain region of spacetime which intersects the black hole interior. We test the complexity conjectures for semi-classical black hole solutions in the CGHS/RST model and find that both conjectures yield the expected behavior. v vi 3 Given any solution we can of course add terms whose variation is zero as we vary the conformal factor ρ and the result still solves equation (2.9). We will come back to this issue later on. Articles Article I JHEP03(2020)069
Description This work provides an introduction to quantum entanglement on a theoretical level. Am... more Description This work provides an introduction to quantum entanglement on a theoretical level. Among others it covers questions like: How can one define entanglement? How can one identify an entangled quantum system? How can entanglement be measured? Matura Graduation Project Title Student Administration Software Supervisor Dipl.-Ing. Peter Anzenberger Description Software project for a school in Austria to provide a platform helping teachers in their daily workflow, especially managing groups of students.
Quantum entanglement is a huge and active research field these days. Not only the philosophical a... more Quantum entanglement is a huge and active research field these days. Not only the philosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting, but also the possibilities to make use of it in our everyday life is thrilling. In the last few years many possible applications, mostly within the ’Quantum Information’ field, have been developed. Of course to make use of this feature one demands tools to control entanglement in a certain sense. How can one define entanglement? How can one identify an entangled quantum system? Can entanglement be measured? These are questions one desires an answer for and indeed many answers have been found. However today entanglement is not yet fully in control by mathematics; many problems are still not solved. This paper aims to provide a theoretical introduction to get a feeling for the mathematical problems concerning entanglement and presents approaches to handle entanglement identification or entanglement measures for s...
BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterize... more BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterizes the semi-classical limit of a fuzzy brane described by a set of matrices.
Journal of High Energy Physics, 2020
We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional ... more We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional dilaton gravity and determine the rate of increase of holographic complexity for an evaporating black hole. The results are consistent with our previous numerical results for semi-classical black hole complexity using a volume prescription (CV) in the same model, but the CA calculation is fully analytic and provides a non-trivial positive test for the holographic representation of the black hole interior.
Journal of High Energy Physics, 2020
A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapti... more A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapting the Quantum Ryu-Takayanagi (QRT) proposal to an analytically solvable semi-classical two-dimensional dilaton gravity theory. The Page time is found to be one third of the black hole lifetime, at leading order in semi-classical corrections. A Page curve is also obtained for a semi-classical eternal black hole, where energy loss due to Hawking evaporation is balanced by an incoming energy flux.
Journal of Physics A: Mathematical and Theoretical, 2016
We develop a systematic approach to determine and measure numerically the geometry of generic qua... more We develop a systematic approach to determine and measure numerically the geometry of generic quantum or "fuzzy" geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in R d including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace-or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.
University of Iceland, School of Engineering and Natural Sciences, Faculty of Physical Sciences, Sep 1, 2020
This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-... more This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-classical formula for the entanglement entropy of black hole radiation is examined. This entropy is an indicator of information loss and determines whether black hole evaporation is an information preserving process or destroys quantum information. Assuming information conservation, Page expressed the entanglement entropy as a function of time, which is referred to as the "Page curve." Using the improved formula for evaporating black hole solutions of a gravitational model introduced by Callan, Giddings, Harvey and Strominger (CGHS) and modified by Russo, Susskind and Thorlacius (RST), we find that the entanglement entropy follows the Page curve and thus is consistent with unitary evolution. Second, the notion of quantum complexity is explored in the context of black holes. The quantum complexity of a quantum state measures how many "simple operations" are required to create that state. Susskind conjectured that the quantum complexity of a black hole state corresponds to a certain volume inside the black hole. A modified conjecture equates the quantum complexity with the gravitational action evaluated for a certain region of spacetime which intersects the black hole interior. We test the complexity conjectures for semi-classical black hole solutions in the CGHS/RST model and find that both conjectures yield the expected behavior. v vi 3 Given any solution we can of course add terms whose variation is zero as we vary the conformal factor ρ and the result still solves equation (2.9). We will come back to this issue later on. Articles Article I JHEP03(2020)069
Journal of High Energy Physics, 2020
We obtain the holographic complexity of an evaporating black hole in the semi-classical RST model... more We obtain the holographic complexity of an evaporating black hole in the semi-classical RST model of two-dimensional dilaton gravity, using a volume prescription that takes into account the higher-dimensional origin of the model. For classical black holes, we recover the expected late time behaviour of the complexity, but new features arise at the semi-classical level. By considering the volume inside the stretched horizon of the evolving black hole, we obtain sensible results for the rate of growth of the complexity, with an early onset of order the black hole scrambling time followed by an extended period where the rate of growth tracks the shrinking area of the stretched horizon as the black hole evaporates.
Die vorliegende Arbeit beschreibt einen Algorithmus, um numerisch eine Naherung des semi-klassisc... more Die vorliegende Arbeit beschreibt einen Algorithmus, um numerisch eine Naherung des semi-klassischen Limes einer durch eine endliche Menge an endlich-dimensionalen Matrizen gegebenen NC-Brane Konfiguration zu finden. Diese Naherung ist numerisch beschrieben durch eine Sammlung von Punkten in R^m , die wiederum eine Mannigfaltigkeit eingebettet in R^m darstellen sollen. Zu diesem Zweck wird eine kurze Einfuhrung in die Theorie der sogenannten “nicht-kommutativen Geometrie” gegeben, die von wichtigen Beispielen begleitet wird. Einen Schwerpunkt dabei bilden die koharenten Zustande, die einen Grundstein fur den theoretischen Hintergrund dieses Algorithmus bilden. Nachdem der Ablauf des Algorithmus begrundet und beschrieben wurde, wird dieser unter anderem auf eine interessante Losung einer deformierten supersymmetrischen N = 4 Yang-Mills Theorie angewendet. Diese Losung wird im Detail diskutiert und diverse sowohl numerische als auch analytische Resultate werden prasentiert.
Journal of Physics A, May 31, 2016
We develop a systematic approach to determine and measure numerically the geometry of generic qua... more We develop a systematic approach to determine and measure numerically the geometry of generic quantum or "fuzzy" geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in R d including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace-or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.
BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterize... more BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterizes the semi-classical limit of a fuzzy brane described by a set of matrices.
Journal of High Energy Physics, Jul 1, 2020
We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional ... more We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional dilaton gravity and determine the rate of increase of holographic complexity for an evaporating black hole. The results are consistent with our previous numerical results for semi-classical black hole complexity using a volume prescription (CV) in the same model, but the CA calculation is fully analytic and provides a non-trivial positive test for the holographic representation of the black hole interior.
This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-... more This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-classical formula for the entanglement entropy of black hole radiation is examined. This entropy is an indicator of information loss and determines whether black hole evaporation is an information preserving process or destroys quantum information. Assuming information conservation, Page expressed the entanglement entropy as a function of time, which is referred to as the "Page curve." Using the improved formula for evaporating black hole solutions of a gravitational model introduced by Callan, Giddings, Harvey and Strominger (CGHS) and modified by Russo, Susskind and Thorlacius (RST), we find that the entanglement entropy follows the Page curve and thus is consistent with unitary evolution. Second, the notion of quantum complexity is explored in the context of black holes. The quantum complexity of a quantum state measures how many "simple operations" are required to create that state. Susskind conjectured that the quantum complexity of a black hole state corresponds to a certain volume inside the black hole. A modified conjecture equates the quantum complexity with the gravitational action evaluated for a certain region of spacetime which intersects the black hole interior. We test the complexity conjectures for semi-classical black hole solutions in the CGHS/RST model and find that both conjectures yield the expected behavior. v vi 3 Given any solution we can of course add terms whose variation is zero as we vary the conformal factor ρ and the result still solves equation (2.9). We will come back to this issue later on. Articles Article I JHEP03(2020)069
Description This work provides an introduction to quantum entanglement on a theoretical level. Am... more Description This work provides an introduction to quantum entanglement on a theoretical level. Among others it covers questions like: How can one define entanglement? How can one identify an entangled quantum system? How can entanglement be measured? Matura Graduation Project Title Student Administration Software Supervisor Dipl.-Ing. Peter Anzenberger Description Software project for a school in Austria to provide a platform helping teachers in their daily workflow, especially managing groups of students.
Quantum entanglement is a huge and active research field these days. Not only the philosophical a... more Quantum entanglement is a huge and active research field these days. Not only the philosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting, but also the possibilities to make use of it in our everyday life is thrilling. In the last few years many possible applications, mostly within the ’Quantum Information’ field, have been developed. Of course to make use of this feature one demands tools to control entanglement in a certain sense. How can one define entanglement? How can one identify an entangled quantum system? Can entanglement be measured? These are questions one desires an answer for and indeed many answers have been found. However today entanglement is not yet fully in control by mathematics; many problems are still not solved. This paper aims to provide a theoretical introduction to get a feeling for the mathematical problems concerning entanglement and presents approaches to handle entanglement identification or entanglement measures for s...
BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterize... more BProbe is a Wolfram Mathematica package. It is the implementation of an algorithm which rasterizes the semi-classical limit of a fuzzy brane described by a set of matrices.
Journal of High Energy Physics, 2020
We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional ... more We adapt the complexity as action prescription (CA) to a semi-classical model of two-dimensional dilaton gravity and determine the rate of increase of holographic complexity for an evaporating black hole. The results are consistent with our previous numerical results for semi-classical black hole complexity using a volume prescription (CV) in the same model, but the CA calculation is fully analytic and provides a non-trivial positive test for the holographic representation of the black hole interior.
Journal of High Energy Physics, 2020
A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapti... more A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapting the Quantum Ryu-Takayanagi (QRT) proposal to an analytically solvable semi-classical two-dimensional dilaton gravity theory. The Page time is found to be one third of the black hole lifetime, at leading order in semi-classical corrections. A Page curve is also obtained for a semi-classical eternal black hole, where energy loss due to Hawking evaporation is balanced by an incoming energy flux.
Journal of Physics A: Mathematical and Theoretical, 2016
We develop a systematic approach to determine and measure numerically the geometry of generic qua... more We develop a systematic approach to determine and measure numerically the geometry of generic quantum or "fuzzy" geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in R d including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace-or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.
University of Iceland, School of Engineering and Natural Sciences, Faculty of Physical Sciences, Sep 1, 2020
This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-... more This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-classical formula for the entanglement entropy of black hole radiation is examined. This entropy is an indicator of information loss and determines whether black hole evaporation is an information preserving process or destroys quantum information. Assuming information conservation, Page expressed the entanglement entropy as a function of time, which is referred to as the "Page curve." Using the improved formula for evaporating black hole solutions of a gravitational model introduced by Callan, Giddings, Harvey and Strominger (CGHS) and modified by Russo, Susskind and Thorlacius (RST), we find that the entanglement entropy follows the Page curve and thus is consistent with unitary evolution. Second, the notion of quantum complexity is explored in the context of black holes. The quantum complexity of a quantum state measures how many "simple operations" are required to create that state. Susskind conjectured that the quantum complexity of a black hole state corresponds to a certain volume inside the black hole. A modified conjecture equates the quantum complexity with the gravitational action evaluated for a certain region of spacetime which intersects the black hole interior. We test the complexity conjectures for semi-classical black hole solutions in the CGHS/RST model and find that both conjectures yield the expected behavior. v vi 3 Given any solution we can of course add terms whose variation is zero as we vary the conformal factor ρ and the result still solves equation (2.9). We will come back to this issue later on. Articles Article I JHEP03(2020)069
Journal of High Energy Physics, 2020
We obtain the holographic complexity of an evaporating black hole in the semi-classical RST model... more We obtain the holographic complexity of an evaporating black hole in the semi-classical RST model of two-dimensional dilaton gravity, using a volume prescription that takes into account the higher-dimensional origin of the model. For classical black holes, we recover the expected late time behaviour of the complexity, but new features arise at the semi-classical level. By considering the volume inside the stretched horizon of the evolving black hole, we obtain sensible results for the rate of growth of the complexity, with an early onset of order the black hole scrambling time followed by an extended period where the rate of growth tracks the shrinking area of the stretched horizon as the black hole evaporates.
Die vorliegende Arbeit beschreibt einen Algorithmus, um numerisch eine Naherung des semi-klassisc... more Die vorliegende Arbeit beschreibt einen Algorithmus, um numerisch eine Naherung des semi-klassischen Limes einer durch eine endliche Menge an endlich-dimensionalen Matrizen gegebenen NC-Brane Konfiguration zu finden. Diese Naherung ist numerisch beschrieben durch eine Sammlung von Punkten in R^m , die wiederum eine Mannigfaltigkeit eingebettet in R^m darstellen sollen. Zu diesem Zweck wird eine kurze Einfuhrung in die Theorie der sogenannten “nicht-kommutativen Geometrie” gegeben, die von wichtigen Beispielen begleitet wird. Einen Schwerpunkt dabei bilden die koharenten Zustande, die einen Grundstein fur den theoretischen Hintergrund dieses Algorithmus bilden. Nachdem der Ablauf des Algorithmus begrundet und beschrieben wurde, wird dieser unter anderem auf eine interessante Losung einer deformierten supersymmetrischen N = 4 Yang-Mills Theorie angewendet. Diese Losung wird im Detail diskutiert und diverse sowohl numerische als auch analytische Resultate werden prasentiert.