We demonstrate how the key notions of Tononi et al.'s Integrated Information Theory (IIT) can be studied within the simple graphical language of process theories, i.e. symmetric monoidal categories. This allows IIT to be generalised to a broad range of physical theories, including as a special case the Quantum IIT of Zanardi, Tomka and Venuti.
We demonstrate how the key notions of Tononi et al.'s Integrated Information Theory (IIT) can be studied within the simple graphical language of process theories, i.e. symmetric monoidal categories. This allows IIT to be generalised to a broad range of physical theories, including as a special case the Quantum IIT of Zanardi, Tomka and Venuti.
This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi. Using the definitions of Classical Mathematical Integrated Information Theory. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also we will be using the definition of conditional density matrix to define repertoire.
This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi [2]. Using the definitions of Classical Mathematical Integrated Information Theory [1]. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also, we will be using the definition of conditional density matrix to define repertoire. We will be also discussing the relationship between consciousness and entanglement.
Originally conceived as a theory of consciousness, integrated information theory (IIT) provides a theoretical framework intended to characterize the compositional causal information that a system, in its current state, specifies about itself. However, it remains to be determined whether IIT as a theory of consciousness is compatible with quantum mechanics as a theory of microphysics. Here, we present an extension of IIT's latest formalism to evaluate the mechanism integrated information (φ) of a system subset to discrete, finite-dimensional quantum systems (e.g., quantum logic gates). To that end, we translate a recently developed, unique measure of intrinsic information into a density matrix formulation and extend the notion of conditional independence to accommodate quantum entanglement. The compositional nature of the IIT analysis might shed some light on the internal structure of composite quantum states and operators that cannot be obtained using standard information-theoretical analysis. Finally, our results should inform theoretical arguments about the link between consciousness, causation, and physics from the classical to the quantum.
Introduction Our starting point is a particular “canvas” aimed to “draw” theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this chapter, we consider the conceptual foundations for this canvas and how these can then be converted into mathematical structure. With very little structural effort (i.e., in very abstract terms) and in a very short time span, the categorical quantum mechanics (CQM) research program, initiated by Abramsky and this author, has reproduced a surprisingly large fragment of quantum theory [3, 45, 48, 56, 60–62, 170, 179]. It also provides new insights both in quantum foundations and in quantum information —for example, in [49–51, 58, 59, 64, 79, 80]—and has even resulted in automated reasoning software called quantomatic [71–73], which exploits the deductive power of CQM, which is indeed a categorical quantum logic [77]. In this chapter, we complement the available material by not requiring prior knowledge of category theory and by pointing at connections to previous and current developments in the foundations of physics. This research program is also in close synergy with developments elsewhere—for example, in representation theory [74], quantum algebra [176], knot theory [187], topological quantum field theory (TQFT) [132], and several other areas.
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-calculus, ZW-calculus, and so on. Each of these languages forms a †-symmetric monoidal category (†-SMC) and comes with an interpretation functor to the †-SMC of finite-dimensional Hilbert spaces. In recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of how to extend these languages beyond pure quantum mechanics to reason about mixed states and general quantum operations, i.e., completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map that allows one to get rid of a quantum system, an operation that is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction , which transforms any †-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction, we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However, this construction fails for some fringe cases like Clifford+T quantum mechanics, as the category does not have enough isometries.