Any theory with definite causal structure has a defined past and future, be it defined by light cones or an absolute time scale. Entropy is a concept that has traditionally been reliant on a definite notion of causality. However, without a definite notion of causality, the concept of entropy is not all lost. Indefinite causal structure results from combining probabilistic predictions and dynamical space-time. The causaloid framework lays the mathematical groundwork to be able to treat indefinite causal structure. In this paper, we build on the causaloid mathematics and define a causally-unbiased entropy for an indefinite causal structure. In defining a causally-unbiased entropy, there comes about an emergent idea of causality in the form of a measure of causal connectedness, termed the Q factor.
Any theory with definite causal structure has a defined past and future, be it defined by light cones or an absolute time scale. Entropy is a concept that has traditionally been reliant on a definite notion of causality. However, without a definite notion of causality, the concept of entropy is not all lost. Indefinite causal structure results from combining probabilistic predictions and dynamical space-time. The causaloid framework lays the mathematical groundwork to be able to treat indefinite causal structure. In this paper, we build on the causaloid mathematics and define a causally-unbiased entropy for an indefinite causal structure. In defining a causally-unbiased entropy, there comes about an emergent idea of causality in the form of a measure of causal connectedness, termed the Q factor.
The Causaloid framework is an operational approach aimed to house both the radical aspects of General Relativity -- dynamic causal structure, and Quantum Theory -- indefiniteness, to provide a scaffolding that might be suitable for Quantum Gravity by providing a landscape of theories that allow for indefinite causal structure. One may consider it as a generalisation of generalised probability theories (or GPTs) where a priori regions are not assumed to have any given causal relationship, to incorporate the possibility of indefinite causal structure. Since its conception, there have been many advances in the field of indefinite causal structure mostly stemming from the work of Chiribella et al. on the quantum switch and supermaps and from Oreshkov et al. on causal inequalities and process matrices. These approaches have systems moving along wires and use Hilbert space structure. They violate the standard causality constraints of Quantum Theory and, in this sense, can be regarded as post-quantum. The Causaloid approach does not necessarily have systems moving along wires or Hilbert spaces. This is the first paper in a trilogy of papers aiming to close the gap between the Causaloid (that allows for GPTs) and post-quantum studies that employ Hilbert spaces. To do so in the present paper, we provide a diagrammatic language for the Causaloid framework along with new terminology for the three levels of physical compression (called Tomographic, Compositional, and Meta compression).
Introduction Indefinite causal structure poses particular problems for theory formulation since many of the core ideas used in the usual approaches to theory construction depend on having definite causal structure. For example, the notion of a state across space evolving in time requires that we have some definite causal structure so we can define a state on a space-like hypersurface. We will see that many of these problems are mitigated if we are able to formulate the theory in a formalism-local (or F-local) fashion. A formulation of a physical theory is said to be F-local if, in making predictions for any given arbitrary space-time region, we need only refer to mathematical objects pertaining to that region. This is a desirable property both on the grounds of efficiency and since, if we have indefinite causal structure, it is not clear how to select some other space-time region on which our calculations may depend. The usual ways of formulating physical theories (the time-evolving state picture, the histories approach, and the local-equations approach) are not F-local. We set up a framework for probabilistic theories with indefinite causal structure. This, the causaloid framework, is F-local. We describe how quantum theory can be formulated in the causaloid framework (in an F-local fashion). This provides yet another formulation of quantum theory. This formulation, however, may be particularly relevant to the problem of finding a theory of quantum gravity. The problem of quantum gravity is to find a theory that reduces in appropriate limits to general relativity and quantum theory (including, at least, those situations in which those two theories have been experimentally confirmed).
Causality has never gained the status of a 'law' or 'principle' in physics. Some recent literature has even popularized the false idea that causality is a notion that should be banned from theory. Such misconception relies on an alleged universality of the reversibility of the laws of physics, based either on the determinism of classical theory, or on the multiverse interpretation of quantum theory, in both cases motivated by mere interpretational requirements for realism of the theory. Here, I will show that a properly defined unambiguous notion of causality is a theorem of quantum theory, which is also a falsifiable proposition of the theory. Such a notion of causality appeared in the literature within the framework of operational probabilistic theories. It is a genuinely theoretical notion, corresponding to establishing a definite partial order among events, in the same way as we do by using the future causal cone on Minkowski space. The notion of causality is logically completely independent of the misidentified concept of 'determinism', and, being a consequence of quantum theory, is ubiquitous in physics. In addition, as classical theory can be regarded as a restriction of quantum theory, causality holds also in the classical case, although the determinism of the theory trivializes it. I then conclude by arguing that causality naturally establishes an arrow of time. This implies that the scenario of the 'block Universe' and the connected 'past hypothesis' are incompatible with causality, and thus with quantum theory: they are both doomed to remain mere interpretations and, as such, are not falsifiable, similar to the hypothesis of 'super-determinism'.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.
We adopt an operational mode of thinking to study spacetime fluctuations and their impacts on questions of quantum gravity. Particular emphasis is put on studying causal structure fluctuations. The study is guided by the principle of causal neutrality, which says that fundamental concepts and laws of physics should be stated without assuming a definite spacetime causal structure. Most traditional concepts and theories of physics violate this principle. A major theme of this thesis is to upgrade the traditional concepts and theories in accordance with the principle of causal neutrality. \n \nAmong other things, the thesis include works on the following. (1) An axiomatic derivation of the complex Hilbert space structure of quantum theory without assuming definite causal structure. (2) A so-called causally neutral quantum field theory (CNQFT) framework for algebraic quantum physics allowing indefinite causal structure. (3) A proposal that causal fluctuation regularizes quantum field ultraviolet divergences (UV) and that the UV regularizing correlation functions with causal fluctuations characterize the UV structure of physical states. (4) A study on quantitative measures of causality. (5) A so-called ``correlation networks'' framework to study quantum theory with indefinite causal structure, which generalizes previous frameworks. (6) New definitions of entanglement and entanglement measures that conform to the causal neutrality principle. (7) Ideas on finding a causally neutral analogue of Einstein's equation for quantum spacetime. (8) A simple theorem showing that the communication capacities of general correlations (with definite or indefinite causal structure) defined with respect to state transmission can be reduced to those of channels.
This paper introduces Quantum Causal Entropy (QCE), a novel framework that extends the concept of entropy to incorporate causal relationships within quantum systems. Traditional entropy measures uncertainty in a system, quantified by the Shannon entropy formula: [S] = - Σ p(s) * log(p(s)), where p(s) represents the probability of a state 's'. QCE posits that causal relationships also possess an entropy component, denoted as [Sc], which describes the degree of uncertainty regarding the *consequences* of a quantum event, reflecting the inherent unpredictability of their cause-effect dynamics. The methodology involves conducting quantum causal experiments, primarily through carefully designed quantum interference experiments, to empirically determine the causal entropy of specific quantum processes. This work lays the groundwork for establishing a fundamental theory of quantum causal physics, potentially unlocking new insights into the nature of reality and possessing significant implications for future technological advancements. The core contribution of this research lies in redefining entropy to encompass not just uncertainty but also the inherent causal structure of quantum phenomena.
General relativity is a deterministic theory with non-fixed causal structure. Quantum theory is a probabilistic theory with fixed causal structure. In this paper we build a framework for probabilistic theories with non-fixed causal structure. This combines the radical elements of general relativity and quantum theory. The key idea in the construction is physical compression. A physical theory relates quantities. Thus, if we specify a sufficiently large set of quantities (this is the compressed set), we can calculate all the others. We apply three levels of physical compression. First, we apply it locally to quantities (actually probabilities) that might be measured in a particular region of spacetime. Then we consider composite regions. We find that there is a second level of physical compression for the composite region over and above the first level physical compression for the component regions. Each application of first and second level physical compression is quantified by a matrix. We find that these matrices themselves are related by the physical theory and can therefore be subject to compression. This is the third level of physical compression. This third level of physical compression gives rise to a new mathematical object which we call the causaloid. From the causaloid for a particular physical theory we can calculate verything the physical theory can calculate. This approach allows us to set up a framework for calculating probabilistic correlations in data without imposing a fixed causal structure (such as a background time). We show how to put quantum theory in this framework (thus providing a new formulation of this theory). We indicate how general relativity might be put into this framework and how the framework might be used to construct a theory of quantum gravity.