We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e., simplicial theories whose pure states are jointly perfectly discriminable. The usual classical theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving noncausal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement and classify their system-composition rules. We conclude by proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of $n$-local discriminability---no mixed state admits a purification. Our results hold also in the general case where the sets of states fail to be convex.
We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e., simplicial theories whose pure states are jointly perfectly discriminable. The usual classical theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving noncausal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement and classify their system-composition rules. We conclude by proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of n-local discriminability---no mixed state admits a purification. Our results hold also in the general case where the sets of states fail to be convex.
An operational probabilistic theory where all systems are classical, and all pure states of composite systems are entangled, is constructed. The theory is endowed with a rule for composing an arbitrary number of systems, and with a nontrivial set of transformations. Hence, we demonstrate that the presence of entanglement is independent of the existence of incompatible measurements. We then study a variety of phenomena occurring in the theory, some of them contradicting both classical and quantum theories, including cloning, entanglement swapping, dense coding, additivity of classical capacities, nonmonogamous entanglement, hypersignaling. We also prove the existence, in the theory, of a universal processor. The theory is causal and satisfies the no-restriction hypothesis. At the same time, it violates a number of information-theoretic principles enjoyed by quantum theory, most notably, local discriminability, purity of parallel composition of states, and purification. Moreover, we introduce an exhaustive procedure to construct generic operational probabilistic theories, and a sufficient set of conditions to verify their consistency. In addition, we prove a characterization theorem for the parallel composition rules of arbitrary theories, and specialize it to the case of bilocal-tomographic theories. We conclude pointing out some open problems. In particular, on the basis of the fact that every separable state of the theory is a statistical mixture of entangled states, we formulate a no-go conjecture for the existence of a local-realistic ontological model.
Abstract We introduce a class of probabilistic theories, termed Minimal Strongly Causal Operational Probabilistic Theories, where system dynamics are constrained to the minimal set of operations consistent with the set of states and permitting conditional tests. Specifically, the allowed instruments are limited to those derived from compositions of preparations, measurements, swap transformations, and conditional operations. We demonstrate that minimal theories with conditioning and a spanning set of non-separable states satisfy two quantum no-go theorems: no-information without disturbance and no-broadcasting. As a key example, we construct Minimal Strongly Causal Bilocal Classical Theory, a classical toy-theory that lacks incompatible measurements, preparation uncertainty relations, and is noncontextual (both Kochen–Specker and generalised), yet exhibits irreversibility of measurement disturbance, no-information without disturbance, and no-broadcasting. Therefore, the latter three properties cannot be understood per se as signatures of non-classicality. We further explore distinctions between a theory and its minimal strongly causal counterpart, showing that while the minimal strongly causal version of quantum theory diverges from full quantum theory, the same does not hold for classical theory. Additionally, we establish the pairwise independence of the properties of simpliciality, strong causality, and local discriminability.
In this chapter we introduce the principle of local discriminability, which stipulates the possibility of discriminating states of composite systems via local measurements on the component systems. The phenomenon of entanglement has relevant consequences on the accessibility of information in composite systems and on the complexity of characterization of transformations: local discriminability brings dramatic simplifications to the structure of the sets of states and transformations, reconciling the holism of entangled states with the reductionist scientific approach. In order to illustrate the structural relevance of the principle, we will briefly review some OPTs that do not satisfy it, contrasting the features of such theories with those of quantum and classical probabilistic theories, where the principle holds. Entanglement and Holism Holism is a widely known feature of quantum theory, related to the existence of entangled states. In quantum theory one has pure entangled states, whose marginal states are completely mixed. This situation corresponds to having maximal knowledge of the whole with minimal knowledge of the parts. The main manifestation of the holism of entanglement is the impossibility of reconciling the statistics of joint measurements on entangled states with local realism . Local realism is the assumption that any measurement on a system is just the reading of a pre-existing quantity encoded on the system. In other words, we cannot explain the statistics of measurements on the parts of a composite system as a manifestation of our ignorance of locally pre-existing values, as in classical OPT. Does entanglement make information about composite systems locally unaccessible? This would pose severe practical issues on the possibility of acquiring information about physical systems. For example, suppose that two distant telescopes observe a source of entangled photons from a stellar emission. Holism would imply that some features of the emission process could never be detected, unless we are able to make the entangled photons interact in a joint measurement. On the other hand, if a reductionist approach is possible, local observations of the two photons at the sites of the two telescopes would be enough to characterize the emitting source. Thanks to the local discriminability principle, this is actually the case with entangled quantum sources. Another consequence of holism is the fact that transformations generally cannot be characterized by local experiments, namely by applying the unknown transformation only to its input system.
Inspired by its fundamental importance in quantum mechanics, we define and study the notion of entanglement for abstract physical theories, investigating its profound connection with the concept of superposition. We adopt the formalism of general probabilistic theories (GPTs), encompassing all physical models whose predictive power obeys minimal requirements. Examples include classical theories, which do not exhibit superposition and whose state space has the shape of a simplex, quantum mechanics, as well as more exotic models such as Popescu-Rohrlich boxes. We call two GPTs entangleable if their composite admits either entangled states or entangled measurements, and conjecture that any two non-classical theories are in fact entangleable. We present substantial evidence towards this conjecture by proving it (1) for the simplest case of 3-dimensional theories; (2) when the local state spaces are discrete, which covers foundationally relevant cases; (3) when one of the local theories is quantum mechanics. Furthermore, (4) we envision the existence of a quantitative relation between local non-classicality and global entangleability, explicitly describing it in the geometrically natural case where the local state spaces are centrally symmetric.
We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms---Causality, Purity Preservation, Pure Sharpness, and Purification---identify a class of theories where every mixed state can be modelled as the marginal of a pure entangled state and where every unsharp measurement can be modelled as a sharp measurement on a composite system. This class of theories, called sharp theories with purification, includes quantum theory both with complex and real amplitudes, as well as a suitable extension of classical probability theory where classical systems can be entangled with other, non-classical systems. Theories satisfying our axioms support well-behaved notions of majorization, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. We conjecture that every theory admitting a sensible thermodynamics must be extendable to a sharp theory with purification.
Entanglement is one of the most striking features of quantum mechanics, and yet it is not specifically quantum.More specific to quantum mechanics is the connection between entanglement and thermodynamics, which leads to an identification between entropies and measures of pure state entanglement.Here we search for the roots of this connection, investigating the relation between entanglement and thermodynamics in the framework of general probabilistic theories.We first address the question whether an entangled state can be transformed into another by means of local operations and classical communication.Under two operational requirements, we prove a general version of the Lo-Popescu theorem, which lies at the foundations of the theory of pure-state entanglement.We then consider a resource theory of purity where free operations are random reversible transformations, modelling the scenario where an agent has limited control over the dynamics of a closed system.Our key result is a duality between the resource theory of entanglement and the resource theory of purity, valid for every physical theory where all processes arise from pure states and reversible interactions at the fundamental level.As an application of the main result, we establish a one-to-one correspondence between entropies and measures of pure bipartite entanglement.The correspondence is then used to define entanglement measures in the general probabilistic framework.Finally, we show a duality between the task of information erasure and the task of entanglement generation, whereby the existence of entropy sinks (systems that can absorb arbitrary amounts of information) becomes equivalent to the existence of entanglement sources (correlated systems from which arbitrary amounts of entanglement can be extracted).