Quantum theory has an outstanding property, namely each state has its well defined purification - a state extremal in the set of states in larger Hilbert space. It is known that the classical theory and the theory of non-signaling boxes does not have purification for all of their states. These theories are examples of the so called generalized probabilistic theories (GPTs). However in any non-signaling GPT each state has a number of extensions to a larger system. We single out the most relevant among them, called a complete extension, unique up to local reversible operations on the extending system. We prove that this special, finite dimensional extension bares an analogy to quantum purification in that (i) it allows for an access to all ensembles of the extended system (ii) from complete extension one can generate any other extension. It then follows, that an access to the complete extension represents the total power of the most general non-signaling adversary. A complete extension of a maximally mixed box in two-party binary input binary output scenario is up to relabeling the famous Popescu-Rohrlich box. The latter thus emerges naturally without reference to the Bell's non-locality. However the complete extension is not a purification (a vertex) in the generic case. Moreover, we show that all convex discrete theories does not provide purification for almost all of it states. In particular the theory of contextuality does not possess purification. The complete extensions are by nature high-dimensional systems. We were able however to provide explicit structure of complete extension for the noisy Popescu-Rohrlich-boxes and the 3-cycle contextual box.
Quantum theory has an outstanding property, namely each state has its well defined purification - a state extremal in the set of states in larger Hilbert space. It is known that the classical theory and the theory of non-signaling boxes does not have purification for all of their states. These theories are examples of the so called generalized probabilistic theories (GPTs). However in any non-signaling GPT each state has a number of extensions to a larger system. We single out the most relevant among them, called a complete extension, unique up to local reversible operations on the extending system. We prove that this special, finite dimensional extension bares an analogy to quantum purification in that (i) it allows for an access to all ensembles of the extended system (ii) from complete extension one can generate any other extension. It then follows, that an access to the complete extension represents the total power of the most general non-signaling adversary. A complete extension of a maximally mixed box in two-party binary input binary output scenario is up to relabeling the famous Popescu-Rohrlich box. The latter thus emerges naturally without reference to the Bell's non-locality. However the complete extension is not a purification (a vertex) in the generic case. Moreover, we show that all convex discrete theories does not provide purification for almost all of it states. In particular the theory of contextuality does not possess purification. The complete extensions are by nature high-dimensional systems. We were able however to provide explicit structure of complete extension for the noisy Popescu-Rohrlich-boxes and the 3-cycle contextual box.
Entanglement generation is a fundamental dynamical capability in quantum information science and underpins many quantum advantages. While quantum theory enables it through unitary dynamics, boxworld, a generalized probabilistic theory admitting Popescu--Rohrlich boxes with supraquantum correlations, has no reversible transformation capable of generating entanglement. We show that this no-go picture changes fundamentally once reversibility is relaxed to pure-state preservation. We construct a pure-state-preserving transformation that maps every uncorrelated pure state to a Popescu--Rohrlich box and completely classify all pure-state-preserving entangling transformations in the simplest bipartite boxworld. Our results provide the first explicit mechanism for generating beyond-quantum entanglement without introducing mixing and demonstrate a physical distinction between reversibility and pure-state preservation that is obscured by the structure of quantum theory.
Generalized probabilistic theories (GPTs) provide a framework in which a range of possible theories can be examined, including classical theory, quantum theory, and those beyond. In general, enlarging the state space of a GPT leads to fewer possible measurements because the additional states give stronger constraints on the set of effects, the constituents of measurements. This can have implications for information processing. In box world, for example, a GPT in which any no-signaling distribution can be realized, there is no analog of a measurement in the Bell basis and hence the analog of entanglement swapping is impossible. A comprehensive study of measurements on multiple systems in box world has been lacking. Here we consider such measurements in detail, distinguishing those that can be performed by interacting with individual systems sequentially (termed wirings), and the more interesting set of those that cannot. We compute all the possible box-world effects for cases with small numbers of inputs, outputs, and parties, identifying those that are wirings. The large state space of box world leads to a small effect space and hence the effects of box world are widely applicable in GPTs. We also show some possible uses of nonwirings for information processing by studying state discrimination, nonlocality distillation, and the box-world analog of nonlocality without entanglement. Finally, we connect our results to the study of logically consistent classical processes and to the composition of contextuality scenarios. By enhancing understanding of measurements in box world, our results could be useful in studies of possible underlying principles on which quantum theory can be based.
A central question in quantum information is whether a given set of states can provide an advantage in some information-processing task. We establish a universal connection between two such questions: how well a set of states can be discriminated, and whether it can power tasks whose advantage stems from generalized contextuality. Working in the framework of generalized probabilistic theories (GPTs), which includes quantum and classical systems as special cases, we show that any set of states able to provide a nonclassical advantage in a contextuality-powered task must obey nontrivial upper bounds on the success probability of every state discrimination task using the full set. Contextual advantage therefore implies limited distinguishability, exposing a trade-off between two basic operational resources. Importantly, this is more than a statement about the idealized limit: perfect distinguishability is not needed to preclude contextuality. Our thresholds lie strictly below unity, so any set whose discrimination performance exceeds them while still imperfectly distinguishable is already guaranteed to admit a noncontextual explanation. The bounds follow from a simple geometric property, linear dependence of the state set, take a closed analytical form, and depend only on the prior of the task and the convex geometry of the states. As an illustration, we apply them to generalized parity-oblivious multiplexing, where sufficiently high success in the task implies that at least one sub-ensemble of the codebook cannot power any contextual advantage.
Deriving quantum mechanics from information-theoretic postulates is a recent research direction taken, in part, with the view of finding a beyond-quantum theory; once the postulates are clear, we can consider modifications to them. A key postulate is the purification postulate, which we propose to replace by a more generally applicable postulate that we call the complete extension postulate (CEP), i.e., the existence of an extension of a physical system from which one can generate any other extension. This new concept leads to a plethora of open questions and research directions in the study of general theories satisfying the CEP (which may include a theory that hyper-decoheres to quantum theory). For example, we show that the CEP implies the impossibility of bit-commitment. This is exemplified by a case study of the theory of non-signalling behaviors which we show satisfies the CEP. We moreover show that in certain cases the complete extension will not be pure, highlighting the key divergence from the purification postulate.
We study the task of lifting arbitrary quantum states and channels to purifications and Stinespring dilations, respectively, in both the probabilistic exact and deterministic approximate settings. We formalize this task through a general framework of quantum purification machines that, given a finite number of copies or uses of a black-box input, aim to output a corresponding purification or Stinespring dilation. In the probabilistic exact setting, we show that universality is not necessary to rule out such transformations: the simple requirement that a machine purifies two inputs of different rank with non-zero probability already implies that it cannot be described by a linear positive map. This simple argument captures a fundamental obstruction of quantum theory and recovers the impossibility of universal probabilistic purification from finitely many copies. In the approximate setting, we allow for general machines that are not required, in general, to produce a pure output. Using the minimum average error as our figure of merit, we derive analytical expressions for the performance of several physically motivated strategies as well as a general upper bound on the achievable error, which is tight in a specific regime. Our analysis reveals a trade-off: strategies that produce a pure output - among which we prove the optimal to be a strategy that produces as a fixed output a maximally entangled purification of the fully depolarizing channel - perform optimally between those considered for large environment dimension, while append-environment strategies that generally produce non-pure outputs perform better at small environment dimension.
Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of C is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, k-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.