Abstract We introduce superdensity operators as a tool for analyzing quantum information in spacetime. Superdensity operators encode spacetime correlation functions in an operator framework, and support a natural generalization of Hilbert space techniques and Dirac’s transformation theory as traditionally applied to standard density operators. Superdensity operators can be measured experimentally, but accessing their full content requires novel procedures. We demonstrate these statements on several examples. The superdensity formalism suggests useful definitions of spacetime entropies and spacetime quantum channels. For example, we show that the von Neumann entropy of a super-density operator is related to a quantum generalization of the Kolmogorov-Sinai entropy, and compute this for a many-body system. We also suggest experimental protocols for measuring spacetime entropies.
Abstract We introduce superdensity operators as a tool for analyzing quantum information in spacetime. Superdensity operators encode spacetime correlation functions in an operator framework, and support a natural generalization of Hilbert space techniques and Dirac’s transformation theory as traditionally applied to standard density operators. Superdensity operators can be measured experimentally, but accessing their full content requires novel procedures. We demonstrate these statements on several examples. The superdensity formalism suggests useful definitions of spacetime entropies and spacetime quantum channels. For example, we show that the von Neumann entropy of a super-density operator is related to a quantum generalization of the Kolmogorov-Sinai entropy, and compute this for a many-body system. We also suggest experimental protocols for measuring spacetime entropies.
A bstract In this paper, we investigate the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix defined on a single Cauchy surface. We present the construction of the spacetime density matrix in general quantum systems and its representation via the Schwinger-Keldysh path integral. We further introduce a super-operator framework, within which the spacetime density matrix appears as a special case, and discuss possible generalizations from this perspective. We also show that the spacetime density matrix satisfies a Liouville-von Neumann type equation of motion. When considering subsystems, a reduced spacetime density matrix can be defined by tracing over complementary degrees of freedom. We study the general properties of its moments and, in particular, derive universal short-time behavior of the second moment. We find that coupling between subsystems plays a crucial role in obtaining nontrivial results. Assuming weak coupling, we develop a perturbative method to compute the moments systematically.
This paper introduces a generalized definition of quantum information entropy, aiming to provide a more comprehensive framework for quantifying information in complex quantum systems. The core of this work lies in extending the conventional concept of entropy, traditionally applied to classical information, to the realm of quantum mechanics. We explore various types of quantum entropy, including von Neumann entropy, relative entropy, and a newly proposed generalized entropy, denoted as *Eg*. The definition of *Eg* is based on a novel approach that considers the entanglement structure and correlations within quantum states. We establish relationships between these different entropy measures, demonstrating their interconnectedness within the broader context of quantum information theory. The paper further outlines a mathematical formalism for calculating *Eg*, providing a tool for analyzing and characterizing quantum information. Our generalized entropy provides a more flexible and potentially more accurate measure of quantum information, particularly in scenarios involving complex quantum correlations and multi-partite quantum systems. ---
Information plays an important role in our understanding of the physical world. Hence we propose an entropic measure of information for any physical theory that admits systems, states and measurements. In the quantum and classical worlds, our measure reduces to the von Neumann and Shannon entropies, respectively. It can even be used in a quantum or classical setting where we are only allowed to perform a limited set of operations. In a world that admits superstrong correlations in the form of non-local boxes, our measure can be used to analyze protocols such as superstrong random access encodings and the violation of 'information causality'. However, we also show that in such a world no entropic measure can exhibit all the properties we commonly accept in a quantum setting. For example, there exists no 'reasonable' measure of conditional entropy that is subadditive. Finally, we prove a coding theorem for some theories that is analogous to the quantum and classical settings, providing us with an appealing operational interpretation.
Recent efforts to formulate quantum mechanics in a way that treats space and time on a more equal footing have led to a large variety of spacetime-oriented approaches. In this work we present a detailed study of spacetime states, the objects that play the role of quantum states in the recently introduced framework of spacetime quantum mechanics, and show that the main proposals in the literature are different manifestations of the same underlying object. Path integrals, quantum states over time, pseudo-density matrices, the Page and Wootters mechanism, superdensity operators, and timelike-entanglement proposals all arise from spacetime states through particular evaluations, reduced information, linear maps, or quantum channels. This unification provides explicit mathematical representations of these formalisms, reveals relations among them, and clarifies the spacetime information each one captures. We also study the broader relevance of the spacetime-state point of view for Leggett-Garg inequalities, OTOCs, temporal tensor networks, fermionic systems, relativistic QFTs, quantum reference frames, and classical physics, together with additional insights and perspectives revealed by the common unifying framework.
In modern quantum information theory one deals with an idealized situation when the spacetime dependence of quantum phenomena is neglected. However the transmission and processing of (quantum) information is a physical process in spacetime. Therefore such basic notions in quantum information theory as qubit, channel, composite systems and entangled states should be formulated in space and time. In this paper some basic notions of quantum information theory are considered from the point of view of quantum field theory and general relativity. It is pointed out an important fact that in quantum field theory there is a statistical dependence between two regions in spacetime even if they are spacelike separated. A classical probabilistic representation for a family of correlation functions in quantum field theory is obtained. A noncommutative generalization of von Neumann`s spectral theorem is discussed. We suggest a new physical principle describing a relation between the mathematical formalism of Hilbert space and quantum physical phenomena which goes beyond the superselection rules. Entangled states and the change of state associated with the measurement process in space and time are discussed including the black hole and the cosmological spacetime. It is shown that any reasonable state in relativistic quantum field theory becomes disentangled at large spacelike distances if one makes local observations. As a result a violation of Bell`s inequalities can be observed without inconsistency with principles of relativistic quantum theory only if the distance between detectors is rather small. We suggest a further experimental study of entangled states in spacetime by studying the dependence of the correlation functions on the distance between detectors.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.