THE UNREASONABLE EFFECTIVENESS OF EXPONENTIALLY SUPPRESSED CORRECTIONS IN PRESERVING INFORMATION

2013 ◽  
Vol 22 (12) ◽  
pp. 1342030 ◽  
Author(s):  
KYRIAKOS PAPADODIMAS ◽  
SUVRAT RAJU
Keyword(s):  
Hilbert Space ◽  
Black Hole ◽  
Quantum Mechanics ◽  
Nonperturbative Effects ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Information Theoretic ◽  
Black Hole Evaporation ◽  
Strong Subadditivity ◽  
Unreasonable Effectiveness

We point out that nonperturbative effects in quantum gravity are sufficient to reconcile the process of black hole evaporation with quantum mechanics. In ordinary processes, these corrections are unimportant because they are suppressed by e-S. However, they gain relevance in information-theoretic considerations because their small size is offset by the corresponding largeness of the Hilbert space. In particular, we show how such corrections can cause the von Neumann entropy of the emitted Hawking quanta to decrease after the Page time, without modifying the thermal nature of each emitted quantum. Second, we show that exponentially suppressed commutators between operators inside and outside the black hole are sufficient to resolve paradoxes associated with the strong subadditivity of entropy without any dramatic modifications of the geometry near the horizon.

scholarly journals Quantum maximin surfaces

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Chris Akers ◽  
Netta Engelhardt ◽  
Geoff Penington ◽  
Mykhaylo Usatyuk
Keyword(s):  
Black Hole ◽  
Von Neumann Entropy ◽  
Quantum Fields ◽  
Extremal Surface ◽  
Von Neumann ◽  
Black Hole Evaporation ◽  
Generalized Entropy ◽  
General Proof ◽  
Strong Subadditivity ◽  
New Models

Abstract We formulate a quantum generalization of maximin surfaces and show that a quantum maximin surface is identical to the minimal quantum extremal surface, introduced in the EW prescription. We discuss various subtleties and complications associated to a maximinimization of the bulk von Neumann entropy due to corners and unboundedness and present arguments that nonetheless a maximinimization of the UV-finite generalized entropy should be well-defined. We give the first general proof that the EW prescription satisfies entanglement wedge nesting and the strong subadditivity inequality. In addition, we apply the quantum maximin technology to prove that recently proposed generalizations of the EW prescription to nonholographic subsystems (including the so-called “quantum extremal islands”) also satisfy entanglement wedge nesting and strong subadditivity. Our results hold in the regime where backreaction of bulk quantum fields can be treated perturbatively in GNħ, but we emphasize that they are valid even when gradients of the bulk entropy are of the same order as variations in the area, a regime recently investigated in new models of black hole evaporation in AdS/CFT.

A simple proof of the strong subadditivity inequality

2005 ◽  
Vol 5 (6) ◽  
pp. 507-513
Author(s):  
M.A. Nielsen ◽  
D. Petz
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Linear Algebra ◽  
Simple Proof ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Math Phys

Arguably the deepest fact known about the von~Neumann entropy, the strong subadditivity inequality is a potent hammer in the quantum information theorist's toolkit. This short tutorial describes a simple proof of strong subadditivity due to Petz [Rep. on Math. Phys. \textbf{23} (1), 57--65 (1986)]. It assumes only knowledge of elementary linear algebra and quantum mechanics.

scholarly journals Islands in linear dilaton black holes

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Georgios K. Karananas ◽  
Alex Kehagias ◽  
John Taskas
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Entanglement Entropy ◽  
Von Neumann Entropy ◽  
Two Dimensional ◽  
Extremal Case ◽  
Von Neumann ◽  
Dimensional Black Hole ◽  
Linear Dilaton ◽  
Linear Dilaton Black Hole

Abstract We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page’s time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the σ-model renormalization group (RG) equations. For those, we observe that the entanglement entropy is “running” i.e. it is changing along the RG flow with respect to the two-dimensional worldsheet length scale. At any fixed moment before Page’s time the aforementioned entropy increases towards the infrared (IR) domain, whereas the presence of islands leads the running entropy to decrease towards the IR at later times. Finally, we present a four-dimensional charged black hole that asymptotes to the linear dilaton background as well. We compute the associated entanglement entropy for the extremal case and we find that an island is needed in order for it to follow the Page curve.

scholarly journals Ontology in Quantum Mechanics

2021 ◽  
Author(s):  
Gerard ’t Hooft
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Black Hole ◽  
Quantum Mechanics ◽  
Evolution Equations ◽  
Black Hole Physics ◽  
Special Kind ◽  
Quantum Evolution ◽  
The Standard Model ◽  
Low Energies

It is suspected that the quantum evolution equations describing the micro-world as we know it are of a special kind that allows transformations to a special set of basis states in Hilbert space, such that, in this basis, the evolution is given by elements of the permutation group. This would restore an ontological interpretation. It is shown how, at low energies per particle degree of freedom, almost any quantum system allows for such a transformation. This contradicts Bell’s theorem, and we emphasise why some of the assumptions made by Bell to prove his theorem cannot hold for the models studied here. We speculate how an approach of this kind may become helpful in isolating the most likely version of the Standard Model, combined with General Relativity. A link is suggested with black hole physics.

scholarly journals Measurement theory in classical mechanics

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
So Katagiri
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Measurement Theory ◽  
Classical Mechanics ◽  
The State ◽  
Measurement Device ◽  
Von Neumann ◽  
Relative Interpretation ◽  
Classical And Quantum Mechanics

Abstract We investigate measurement theory in classical mechanics in the formulation of classical mechanics by Koopman and von Neumann (KvN), which uses Hilbert space. We show a difference between classical and quantum mechanics in the “relative interpretation” of the state of the target of measurement and the state of the measurement device. We also derive the uncertainty relation in classical mechanics.

scholarly journals Does spacetime inherently has von Neumann entropy?

2020 ◽  
Author(s):  
William Icefield
Keyword(s):  
Black Hole ◽  
Quantum Theory ◽  
Black Hole Entropy ◽  
Black Hole Thermodynamics ◽  
Von Neumann Entropy ◽  
Considerable Difficulty ◽  
Von Neumann ◽  
Thermodynamic Entropy

There has been considerable difficulty in equating thermodynamic entropy, suggested in classical and black hole thermodynamics, with von Neumann entropy. Successful derivations of black hole entropy from purely classical origins and recent doubts as to whether we can really equate von Neumann entropy with thermodynamic entropy open up the possibility that spacetime inherently encodes entropy. In this understanding, any quantum theory defined on some spacetime or worldsheet inherently calls for another quantum theory that explains entropy encoded by spacetime.

Dirac, Von Neumann, and the Derivation of the Quantum Formalism

2020 ◽  
pp. 203-218
Author(s):  
Jim Baggott
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Critical Stage ◽  
Von Neumann ◽  
Quantum Formalism ◽  
Conceptual Problems ◽  
Underlying Reality

The evolution of quantum mechanics through the 1920s was profoundly messy. Some physicists believed that it was necessary to throw out much of the conceptual baggage that early quantum mechanics tended to carry around with it and re-establish the theory on much firmer ground. It was at this critical stage that the search for deeper insights into the underlying reality was set aside in favour of mathematical expediency. All the conceptual problems appeared to be coming from the wavefunctions. But whatever was to replace them needed to retain all the properties and relationships that had so far been discovered. Dirac and von Neumann chose to derive a new quantum formalism by replacing the wavefunctions with state vectors operating in an abstract Hilbert space, and formally embedding all the most important definitions and relations within a system of axioms.

scholarly journals Purity-Based Continuity Bounds for von Neumann Entropy

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Junaid ur Rehman ◽  
Hyundong Shin
Keyword(s):  
Quantum States ◽  
Von Neumann Entropy ◽  
Trace Distance ◽  
Von Neumann ◽  
Information Theoretic

Abstract We propose continuity bounds for the von Neumann entropy of qubits whose difference in purity is bounded. Considering the purity difference of two qubits to capture the notion of distance between them results into bounds which are demonstrably tighter than the trace distance-based existing continuity bounds of quantum states. Continuity bounds can be utilized in bounding the information-theoretic quantities which are generally difficult to compute.

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