Any measurement is intended to provide information on a system, namely knowledge about its state. However, we learn from quantum theory that it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. In this paper we address the issue of the interplay between information and disturbance for a general operational probabilistic theory. The traditional notion of disturbance considers the fate of the system state after the measurement. However, the fact that the system state is left untouched ensures that also correlations are preserved only in the presence of local discriminability. Here we provide the definition of disturbance that is appropriate for a general theory. Moreover, since in a theory without causality information can be gathered also on the effect, we generalise the notion of no-information test. We then prove an equivalent condition for no-information without disturbance--- atomicity of the identity ---namely the impossibility of achieving the trivial evolution---the identity ---as the coarse-graining of a set of non trivial ones. We prove a general theorem showing that information that can be retrieved without disturbance corresponds to perfectly repeatable and discriminating tests. Based on this, we prove a structure theorem for operational probabilistic theories, showing that the set of states of any system decomposes as a direct sum of perfectly discriminable sets, and such decomposition is preserved under system composition. As a consequence, a theory is such that any information can be extracted without disturbance only if all its systems are classical. Finally, we show via concrete examples that no-information without disturbance is independent of both local discriminability and purification.
Any measurement is intended to provide information on a system, namely knowledge about its state. However, we learn from quantum theory that it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. In this paper we address the issue of the interplay between information and disturbance for a general operational probabilistic theory. The traditional notion of disturbance considers the fate of the system state after the measurement. However, the fact that the system state is left untouched ensures that also correlations are preserved only in the presence of local discriminability. Here we provide the definition of disturbance that is appropriate for a general theory. Moreover, since in a theory without causality information can be gathered also on the effect, we generalise the notion of no-information test. We then prove an equivalent condition for no-information without disturbance--- atomicity of the identity ---namely the impossibility of achieving the trivial evolution---the identity ---as the coarse-graining of a set of non trivial ones. We prove a general theorem showing that information that can be retrieved without disturbance corresponds to perfectly repeatable and discriminating tests. Based on this, we prove a structure theorem for operational probabilistic theories, showing that the set of states of any system decomposes as a direct sum of perfectly discriminable sets, and such decomposition is preserved under system composition. As a consequence, a theory is such that any information can be extracted without disturbance only if all its systems are classical. Finally, we show via concrete examples that no-information without disturbance is independent of both local discriminability and purification.
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.
Abstract Every theory of information, including classical and quantum, can be studied in the framework of operational probabilistic theories—where the notion of test generalizes that of quantum instrument, namely a collection of quantum operations summing to a channel, and simple rules are given for the composition of tests in parallel and in sequence. Here we study the notion of compatibility for tests of a causal operational probabilistic theory. Following the quantum literature, we first introduce the notion of strong compatibility, and then we illustrate its ultimate relaxation, that we deem weak compatibility. It is shown that the two notions coincide in the case of observation tests—which are the counterpart of quantum POVMs—while there exist weakly compatible channels that are not strongly compatible. We prove necessary and sufficient conditions for a theory to exhibit incompatible tests. We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.
We study the relation of causal influence between input systems of a reversible evolution and its output systems, in the context of operational probabilistic theories. We analyse two different definitions that are borrowed from the literature on quantum theory—where they are equivalent. One is the notion based on signalling, and the other one is the notion used to define the neighbourhood of a cell in a quantum cellular automaton. The latter definition, that we adopt in the general scenario, turns out to be strictly weaker than the former: it is possible for a system to have causal influence on another one without signalling to it. Remarkably, the counterexample comes from classical theory, where the proposed notion of causal influence determines a redefinition of the neighbourhood of a cell in cellular automata. We stress that, according to our definition, it is impossible anyway to have causal influence in the absence of an interaction, e.g. in a Bell-like scenario. We study various conditions for causal influence, and introduce the feature that we call no interaction without disturbance , under which we prove that signalling and causal influence coincide. The proposed definition has interesting consequences on the analysis of causal networks, and leads to a revision of the notion of neighbourhood for classical cellular automata, clarifying a puzzle regarding their quantisation that apparently makes the neighbourhood larger than the original one.
In the general theory of quantum measurement, one associates a positive semidefinite operator on a d-dimensional Hilbert space to each of the n possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when n>d, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the Luders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
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.
We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, that is, to the fact that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamio\l{}kowski isomorphism in quantum theory. Such an isomorphism allows one to prove most of the basic features of quantum theory, like, e.g., existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.