We show that the set of projections in an operator system can be detected using only the abstract data of the operator system. Specifically, we show that if is a positive contraction in an operator system  which satisfies certain ordertheoretic conditions, then there exists a complete order embedding of  into ( ) mapping to a projection operator. Moreover, every abstract projection in an operator system  is an honest projection in the C*-envelope of . Using this characterization, we provide an abstract characterization for operator systems spanned by two commuting families of projection-valued measures and discuss applications in quantum information theory.

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Quantum complexity theory is concerned with the amount of elementary quantum resources needed to build a quantum system or a quantum operation. The fundamental question in quantum complexity is to define and quantify suitable complexity measures. This non-trivial question has attracted the attention of quantum information scientists, computer scientists, and high energy physicists alike. In this paper, we combine the approach in [LBK + 22] and well-established tools from noncommutative geometry [Con90, Rie99, CS03] to propose a unified framework for resource-dependent complexity measures of general quantum channels. This framework is suitable to study the complexity of both open and closed quantum systems. We explore the mathematical consequences of the proposed axioms. The central class of examples in this paper is the so-called Lipschitz complexity [LBK + 22, PMTL21]. We use geometric methods to provide upper and lower bounds on this class of complexity measures [Nie06,NDGD06a,NDGD06b]. Finally, we study the dynamics of Lipschitz complexity in open quantum systems. In particular, we show that generically the Lipschitz complexity grows linearly in time and then saturates at a maximum value. This is the same qualitative behavior conjecture by Brown and Susskind [BS18, BS19].

IntroductionIn the early years of the twentieth century, the foundations for quantum mechanics were laid out by Dirac, Heisenberg, Bohr, Schrödinger, and others. In his work on the foundations of quantum mechanics, John von Neumann postulated that physical phenomena should be modeled in terms of Hilbert spaces and operators, with observables corresponding to self-adjoint operators and states corresponding to unit vectors. Motivated by his interest in the theory of single operators, he would introduce the notion of what is now termed a von Neumann algebra. Von Neumann and Francis Murray subsequently published a series of fundamental papers, beginning with "On rings of operators" [13], that develop the basic properties of these algebras and establish operator algebras as an independent field of study.In the years after Murray and von Neumann's initial work, the field of operator algebras developed rapidly and split into subfields including * -algebras and von Neumann algebras. Moreover, operator algebraists began to examine generalizations of these objects, such as operator spaces and operator systems. The importance of operator algebras can be witnessed by its applications in Voiculescu's free probability theory, Popa's deformation/rigidity theory, and Jones' theory of subfactors. These areas give us insight into numerous fields, including random matrix theory, quantum field theory, ergodic theory, and knot theory.In a landmark paper unraveling the isomorphism classes of injective von Neumann algebras, Connes proves that it is possible to construct a sequence of approximate embeddings for a large class of von Neumann algebras into finite-dimensional matrix algebras; Connes somewhat casually remarks that this property should hold for all separable von Neumann algebras. Formally, Connes' embedding problem, as this assertion is now called, asks if every type II 1 factor acting on a separable Hilbert space is embeddable into an ultrapower of the hyperfinite II 1 factor via a nonprinciple ultrafilter.Our goal is to unravel the meaning behind Connes' embedding problem and to highlight its significance by providing equivalent formulations that have driven research in the field.

Inspired by a well-known characterization of the index of an inclusion of II1 factors due to Pimsner and Popa, we define an index-type invariant for inclusions of operator systems. We compute examples of this invariant, show that it is multiplicative under minimal tensor products, and explain how it generalizes the quantum Lovász theta invariant for a matricial system defined by Duan, Severini, and Winter.