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arXiv:gr-qc/0204007

Quantum Computing Spacetime

Paola Zizzi

Published Apr 2, 2002• 5 citations
arXivHTMLPDF

Abstract

A causal set C can describe a discrete spacetime, but this discrete spacetime is not quantum, because C is endowed with Boolean logic, as it does not allow cycles. In a quasi-ordered set Q, cycles are allowed. In this paper, we consider a subset QC of a quasi-ordered set Q, whose elements are all the cycles. In QC, which is endowed with quantum logic, each cycle of maximal outdegree N in a node, is associated with N entangled qubits. Then QC describes a quantum computing spacetime. This structure, which is non-local and non-casual, can be understood as a proto-spacetime. Micro-causality and locality can be restored in the subset U of Q whose elements are unentangled qubits which we interpret as the states of quantum spacetime. The mapping of quantum spacetime into proto-spacetime is given by the action of the XOR gate. Moreover, a mapping is possible from the Boolean causal set into U by the action of the Hadamard gate. In particular, the causal order defined on the elements of U induces the causal evolution of spin networks.

DOI
10.48550/arxiv.gr-qc/0204007

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A causal set C can describe a discrete spacetime, but this discrete spacetime is not quantum, because C is endowed with Boolean logic, as it does not allow cycles. In a quasi-ordered set Q, cycles are allowed. In this paper, we consider a subset QC of a quasi-ordered set Q, whose elements are all the cycles. In QC, which is endowed with quantum logic, each cycle of maximal outdegree N in a node, is associated with N entangled qubits. Then QC describes a quantum computing spacetime. This structure, which is non-local and non-casual, can be understood as a proto-spacetime. Micro-causality and locality can be restored in the subset U of Q whose elements are unentangled qubits which we interpret as the states of quantum spacetime. The mapping of quantum spacetime into proto-spacetime is given by the action of the XOR gate. Moreover, a mapping is possible from the Boolean causal set into U by the action of the Hadamard gate. In particular, the causal order defined on the elements of U induces the causal evolution of spin networks.
In the following we undertake to describe how macroscopic spacetime (or rather, a microscopic protoform of it) is supposed to emerge as a superstructure of a web of lumps in a stochastic discrete network structure. As in preceding work (mentioned below), our analysis is based on the working philosophy that both physics and the corresponding mathematics have to be genuinely discrete on the primordial (Planck scale) level. This strategy is concretely implemented in the form of cellular networks and random graphs . One of our main themes is the development of the concept of physical (proto)points or lumps as densely entangled subcomplexes of the network and their respective web, establishing something like (proto)causality . It may perhaps be said that certain parts of our programme are realizations of some early ideas of Menger and more recent ones sketched by Smolin a couple of years ago. We briefly indicate how this two-storey concept of quantum spacetime can be used to encode the (at least in our view) existing non-local aspects of quantum theory without violating macroscopic spacetime causality!
The Causal Set hypothesis for Quantum Gravity asserts that the smooth Lorentzian spacetime manifold of General Relativity is only an approximation to a fundamental microscopic discrete structure: a locally-finite partially-ordered set. The points of this set are akin to the point-events of the spacetime manifold, and the order-relations among these points are akin to the causal-relations among these point-events. The local-finiteness condition implies a fundamental discreteness and provides an important [conformal] factor needed for the continuum approximations. In this dissertation, we address two aspects toward the formulation of a quantum dynamics for causal sets. The first aspect concerns the dynamics of a zero-mass classical scalar field on a background causal set, Poisson-sampled from an Alexandrov interval in (1 + 1)-dimensional Minkowski spacetime. Our numerical simulations suggest that actions can be defined for such a scalar field, expressed solely in terms of the causal set and the scalar field, which approximates the corresponding classical action on the continuum Minkowski spacetime. This result may help us describe matter and gauge fields in a local and Lorentz-invariant way on a causal set, a discrete structure now being considered as the arena for spacetime physics. More importantly, it may help us formulate an action for the causal set itself, which would be used in a sum-over-histories approach to describe the quantum dynamics of causal sets. The second aspect concerns Quantum Measure Theory, a new approach to quantum dynamics inspired by the sum-over-histories approach, which is better suited to formulating the dynamics of causal sets and of other alternative structures for spacetime. To better understand the mathematical structure underlying this new approach, we derive some algebraic identities involving the sum-rules of the Quantum Measure and its generalizations. We conclude with a discussion on possible applications and open problems concerning these aspects.
Abstract Formalisms for higher order quantum processes provide a theoretical formalisation of quantum processes where the order of agents’ operations need not be definite and acyclic, but may be subject to quantum superpositions. This has led to the concept of indefinite causal structures (ICS) which have garnered much interest. However, the interface between these information-theoretic approaches and spatiotemporal notions of causality is less understood, and questions relating to the physical realisability of ICS in a spatiotemporal context persist despite progress in their information-theoretic characterisation. Further, previous work suggests that composition of processes is not so straightforward in ICS frameworks, which raises the question of how this connects with the observed composability of physical experiments in spacetime. To address these points, we compare the formalism of quantum circuits with quantum control of causal order (QC-QC), which models an interesting class of ICS processes, with that of causal boxes, which models composable quantum information protocols in spacetime. We incorporate the set-up assumptions of the QC-QC framework into the spatiotemporal perspective and show that every QC-QC can be mapped to a causal box that satisfies these set up assumptions and acts on a Fock space while reproducing the QC-QC’s behaviour in a relevant subspace defined by the assumptions. Using a recently introduced concept of fine-graining, we show that the causal box corresponds to a fine-graining of the QC-QC, which unravels the original ICS of the QC-QC into a set of quantum operations with a well-defined and acyclic causal order, compatible with the spacetime structure. Our results also clarify how the composability of physical experiments is recovered, while highlighting the essential role of relativistic causality and the Fock space structure.
Process matrices are a framework to model causal relations in the absence of a well-defined acyclic causal order. The framework is very general and does not even assume the existence of a background spacetime. As a result, it is an open question how the framework should be interpreted physically and how and even if composition can be defined. On the other hand, so-called causal boxes define a framework that allows for arbitrary composition. In this work, we treat quantum circuits with quantum control of causal order (QC-QC), a subset of process matrices, which can be interpreted as generalized quantum circuits, and process box, a subset of causal boxes, which can be interpreted as processes. We analyze their state spaces and define a notion of operational equivalence between QC-QCs and process boxes based on this analysis. We then explicitly construct for each QC-QC an operationally equivalent process box. This allows us to define composition of QC-QCs in terms of composition of causal boxes which is well-defined. We further show that process boxes admit a unitary extension and conjecture that the background spacetime can be assumed to have a specific simple form. Based on this conjecture, we construct an operationally equivalent QC-QC for each process box. Our results indicate that the only class of processes that can be physically implemented in a fixed background spacetime are those that can be interpreted as quantum circuits with quantum controlled superpositions of orders. Further, they also reveal that the composability issue can be resolved by embedding processes in a spacetime structure. This in turn sheds light on the connection between physical realizability in a spacetime and composability.
We provide a precise definition and analysis of quantum causal histories (QCHs). A QCH consists of a discrete, locally finite, causal pre-spacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the Wheeler–De Witt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.
Abstract We introduce a framework to study the emergence of time and causal structure in quantum many-body systems. In doing so, we consider quantum states which encode spacetime dynamics, and develop information theoretic tools to extract the causal relationships between putative spacetime subsystems. Our analysis reveals a quantum generalization of the thermodynamic arrow of time and begins to explore the roles of entanglement, scrambling and quantum error correction in the emergence of spacetime. For instance, exotic causal relationships can arise due to dynamically induced quantum error correction in spacetime: there can exist a spatial region in the past which does not causally influence any small spatial regions in the future, but yet it causally influences the union of several small spatial regions in the future. We provide examples of quantum causal influence in Hamiltonian evolution, quantum error correction codes, quantum teleportation, holographic tensor networks, the final state projection model of black holes, and many other systems. We find that the quantum causal influence provides a unifying perspective on spacetime correlations in these seemingly distinct settings. In addition, we prove a variety of general structural results and discuss the relation of quantum causal influence to spacetime quantum entropies.