A generalization for the quantum information retrieval protocol recently illustrated by Verlinde and van der Heijden for evaporating black holes is provided to inclusions of type III von Neumann factors. The physical interest of such scenario arises in Quantum Field Theory, where local algebras are type III von Neumann algebras. The formula obtained can be easily interpreted in terms of the statistical dimension of superselection sectors in the case of black holes undergoing charge evaporation, thanks to the index-statistics theorem, leading to a thermodynamic interpretation. A constraint on the values of the index leads to a final remark about the quantization of the charge emitted by the black hole during the evaporation process.
A generalization for the quantum information retrieval protocol recently illustrated by Verlinde and van der Heijden for evaporating black holes is provided to inclusions of type III von Neumann factors. The physical interest of such scenario arises in Quantum Field Theory, where local algebras are type III von Neumann algebras. The formula obtained can be easily interpreted in terms of the statistical dimension of superselection sectors in the case of black holes undergoing charge evaporation, thanks to the index-statistics theorem, leading to a thermodynamic interpretation. A constraint on the values of the index leads to a final remark about the quantization of the charge emitted by the black hole during the evaporation process.
A bstract We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the G → 0 limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II 1 subfactors. The subsequent evaporation and recovery steps are represented by Jones’s basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level “translation = teleportation”.
One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.
This paper provides an explanation of the type classification of von Neumann algebras, which has made many appearances in the recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading this paper will provide you with: (i) an argument for why “factors” are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into “effective density matrices”; (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and type II factors in terms of their “standard forms”; and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III 1 factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources the author has read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.
This essay was completed as partial fulfillment of the requirements forthe MASt in Applied Mathematics degree (Part III of the Mathemat-ical Tripos) carried out at the DAMPT, Cambridge. After reviewingthe essentials of C*-algebras and von Neumann algebras, we present theMurray–von Neumann classification of factors into types I, II and III.We then introduce the algebraic approach to Quantum Field Theory(QFT) and discuss existing arguments for the association of type IIIalgebras to local regions of spacetime. We explain why, as a conse-quence, the entanglement entropy of a region is not well defined, andhow relative entropy offers an alternative, reviewing the fundamentalsof entanglement entropy in QFT. Finally, we see how including grav-ity turns the algebra of observables outside a black hole into one oftype II∞, where a well-defined notion of entropy can be recovered. This work should be regarded as introductory notes to operator al-gebras with a particular emphasis on their applications in theoreticalphysics. They are intended to synthesize the mathematical content fora physics-oriented audience, using worked-out examples throughout toset a foundation and gradually build up to some direct examples of theusage of the framework. They are aimed at advanced undergraduatestudents or early postgraduate students, and only some basic knowl-edge on quantum mechanics, QFT, group theory and linear algebra areassumed
The black-hole--qubit correspondence has been proven to be ``useful for obtaining additional insight into one of the string black hole theory and quantum information theory by exploiting approaches of the other"[Phys. Rev. D 82, 026003 (2010)]. Though different classes of stringy black holes can be related to the well-known stochastic local operations and classical communication (SLOCC) entanglement classes of pure states, the string theory requires a more detailed classification than the SLOCC classification of three qubits. In this paper, we derive the entanglement family of three qubits under local unitary operations (LU), and use the black-hole--qubit correspondence to classify \textquotedblleft large\textquotedblright\ black holes into seven inequivalent families. In particular, we show that two black holes with 4 non-vanishing charges (q0, p1, p2, and p3) are LU equivalent if their difference is only in the signs of charges. Thus, the classification of black holes is independent of the signs of charges and is only related to the ratio of the absolute values of charges. This observation simplifies the classification task, as one would only need to consider either the classification of non-BPS black holes or the classification of BPS black holes, but not both. Moreover, through the LU classification, the physical basis for this black-hole--qubit correspondence can be observed, and a relation between the black-hole entropy and the von Neumann entanglement entropy is revealed. Therefore, the LU classification offers a more straightforward physical connection than the SLOCC classification. Based on the LU classification, we further study the properties of von Neumann entanglement entropy for each of the seven families, and find the black holes with the maximal von Neumann entanglement entropy.
The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission.
Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does.
We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional C∗-algebras throughout, modeling finite quantum systems that may have superselection sectors.