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arXiv:1507.00957

Introduction to quantum information theory and outline of two applications to physics: the black hole information paradox and the renormalization group information flow

Fabio Grazioso

Published Jul 3, 2015• 1 citation
arXivHTMLPDF

Abstract

This review paper is intended for scholars with different backgrounds, possibly in only one of the subjects covered, and therefore little background knowledge is assumed. The first part is an introduction to classical and quantum information theory (CIT, QIT): basic definitions and tools of CIT are introduced, such as the information content of a random variable, the typical set, and some principles of data compression. Some concepts and results of QIT are then introduced, such as the qubit, the pure and mixed states, the Holevo theorem, the no-cloning theorem, and the quantum complementarity. In the second part, two applications of QIT to open problems in theoretical physics are discussed. The black hole (BH) information paradox is related to the phenomenon of the Hawking radiation (HR). Consid- ering a BH starting in a pure state, after its complete evaporation only the Hawking radiation will remain, which is shown to be in a mixed state. This either describes a non-unitary evolution of an isolated system, contradicting the evolution postulate of quantum mechanics and violating the no-cloning theorem, or it implies that the initial information content can escape the BH, therefore contradicting general relativity. The progress toward the solution of the paradox is discussed. The renormalization group (RG) aims at the extraction of the macroscopic description of a physical system from its microscopic description. This passage from microscopic to macroscopic can be described in terms of several steps from one scale to another, and is therefore formalized as the action of a group. The c-theorem proves the existence, under certain conditions, of a function which is monotonically decreasing along the group transformations. This result suggests an interpretation of this function as entropy, and its use to study the information flow along the RG transformations.

DOI
10.48550/arxiv.1507.00957

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This review paper is intended for scholars with different backgrounds, possibly in only one of the subjects covered, and therefore little background knowledge is assumed. The first part is an introduction to classical and quantum information theory (CIT, QIT): basic definitions and tools of CIT are introduced, such as the information content of a random variable, the typical set, and some principles of data compression. Some concepts and results of QIT are then introduced, such as the qubit, the pure and mixed states, the Holevo theorem, the no-cloning theorem, and the quantum complementarity. In the second part, two applications of QIT to open problems in theoretical physics are discussed. The black hole (BH) information paradox is related to the phenomenon of the Hawking radiation (HR). Consid- ering a BH starting in a pure state, after its complete evaporation only the Hawking radiation will remain, which is shown to be in a mixed state. This either describes a non-unitary evolution of an isolated system, contradicting the evolution postulate of quantum mechanics and violating the no-cloning theorem, or it implies that the initial information content can escape the BH, therefore contradicting general relativity. The progress toward the solution of the paradox is discussed. The renormalization group (RG) aims at the extraction of the macroscopic description of a physical system from its microscopic description. This passage from microscopic to macroscopic can be described in terms of several steps from one scale to another, and is therefore formalized as the action of a group. The c-theorem proves the existence, under certain conditions, of a function which is monotonically decreasing along the group transformations. This result suggests an interpretation of this function as entropy, and its use to study the information flow along the RG transformations.
Thirty years ago, John Preskill concluded "that the information loss paradox may well presage a revolution in fundamental physics" and mused that "Conceivably, the puzzle of black hole evaporation portends a scientific revolution as sweeping as that that led to the formulation of quantum theory in the early 20th century." Many still agree with this assessment. On the other hand, it seems to me the "paradox" has little to do with the physical world but rather, at best, simply points out the possible inconsistency of two, already disparate, theories (mathematical models) of nature, general relativity and quantum mechanics, with virtually no conceivable observational consequences. The information paradox hinges on the concepts of a pure quantum state, the unitarity of quantum mechanics, and Hawking's semi-classical calculation of black hole evaporation. I used the qualifier "at best" above because, for me, the concept of a quantum state is far more restrictive than required by the paradox while unitarity is not a property of nature but rather of a mathematical model and is already violated by the process of making a measurement. Furthermore, the semi-classical calculation of Hawking is surely of limited applicability.
The black hole information puzzle arises from a discrepancy between conclusions drawn from general relativity and quantum theory about the nature of the radiation emitted by a black hole. According to Hawking's original argument, the radiation is thermal and its entropy thus increases monotonically as the black hole evaporates. Conversely, due to the reversibility of time evolution according to quantum theory, the radiation entropy should start to decrease after a certain time, as predicted by the Page curve. This decrease has been confirmed by new calculations based on the replica trick, which also exhibit its geometrical origin: spacetime wormholes that form between the replicas. Here we analyse the discrepancy between these and Hawking's original conclusions from a quantum information theory viewpoint, using in particular the quantum de Finetti theorem. The theorem implies the existence of extra information, WWW, which is neither part of the black hole nor the radiation, but plays the role of a reference. The entropy obtained via the replica trick can then be identified to be the entropy S(R∣W)S(R|W)S(R∣W) of the radiation conditioned on the reference WWW, whereas Hawking's original result corresponds to the non-conditional entropy S(R)S(R)S(R). The entropy S(R∣W)S(R|W)S(R∣W), which mathematically is an ensemble average, gains an operational meaning in an experiment with NNN independently prepared black holes: For large NNN, it equals the normalised entropy of their joint radiation, S(R1⋯RN)/NS(R_1 \cdots R_N)/NS(R1​⋯RN​)/N. The discrepancy between this entropy and S(R)S(R)S(R) implies that the black holes are correlated. The replica wormholes may thus be interpreted as the geometrical representation of this correlation. Our results also suggest a many-black-hole extension of the widely used random unitary model, which we support with non-trivial checks.
The black hole information paradox arises from an apparent conflict between the Hawk-ing black hole radiation and the fact that time evolution in quantum mechanics is unitary. The trouble is that while the former suggests that information of a system falling into a black hole disappears, the latter implies that information must be con-served. In this work we discuss the current divergence in views regarding the paradox, we evaluate the role that objective collapse theories could play in its resolution and we propose a link between spontaneous collapse events and microscopic virtual black holes. 1
In this article, we discuss through a principle of action the quantum information in relation to the information paradox.The information paradox, which arises in black hole physics, suggests that information can be lost when matter falls into a black hole, which contradicts the principle of information conservation in physics.We propose that the principle of action, which asserts that the action of a physical system is conserved, can be applied to information, showing that information is conserved in a similar way to energy and momentum.Through mathematical models and theoretical analysis, we show that the conservation of information can be understood as a result of the evolution of a system and probability amplitudes in quantum mechanics.
The renormalization group is a tool that allows one to obtain a reduced description of systems with many degrees of freedom while preserving the relevant features. In the case of quantum systems, in particular, one-dimensional systems defined on a chain, an optimal formulation is given by White's ‘density matrix renormalization group’. This formulation can be shown to rely on concepts of the developing theory of quantum information. Furthermore, White's algorithm can be connected with a peculiar type of quantization, namely, angular quantization . This type of quantization arose in connection with quantum gravity problems, in particular, the Unruh effect in the problem of black-hole entropy and Hawking radiation. This connection highlights the importance of quantum system boundaries, regarding the concentration of quantum states on them, and helps us to understand the optimal nature of White's algorithm.
The black hole information paradox remains one of modern physics’ most persistent conceptual fractures. Hawking’s original formulation suggested that black hole evaporation yields thermal radiation, apparently severing the relation between the initial state and the final emitted state. Quantum theory, however, requires unitary evolution and therefore forbids true information destruction. This paper proposes that the paradox may be more fruitfully interpreted not as a flat binary contradiction between preservation and loss, but as an unresolved boundary-state problem. To clarify that claim, the paper introduces the 101 Phase State as a symbolic formalism for intermediate informational resolution. In this model, +1 denotes preserved or recoverable informational order, -1 denotes apparent loss or thermal unreadability, and 0 denotes the boundary interface at which observational contradiction becomes structurally legible but not yet fully resolved.