Monogamy relations for relativistically causal correlations

Non-signalling conditions encode minimal requirements that any (quantum) systems put into spatial arrangements must satisfy in order to be consistent with special relativity. Recent works have argued that in scenarios involving more that two parties, conditions compatible with relativistic causality do not have to satisfy all possible non-signalling conditions but only a subset of them. Here we show that correlations satisfying only this subset of constraints have to satisfy highly non-local monogamy relations between the effects of space-like separated random variables. These monogamy relations take the form of new entropic inequalities between the various systems and we give a general method to derive them. Using these monogamy relations we refute previous suggestions for physical mechanisms that could lead to relativistically causal correlations, demonstrating that such mechanisms would lead to superluminal signalling.Comment: 14 pages, 4 figure

Toward correlation self-testing of quantum theory in the adaptive Clauser-Horne-Shimony-Holt game

Correlation self-testing of a theory addresses the question of whether we can identify the set of correlations realisable in a theory from its performance in a particular information processing task. Applied to quantum theory it aims to identify an information processing task whose optimal performance is achieved only by theories realising the same correlations as quantum theory in any causal structure. In [Phys. Rev. Lett. 125 060406 (2020)] we introduced a candidate task for this, the adaptive CHSH game. Here, we analyse the maximum probability of winning this game in different generalised probabilistic theories. We show that theories with a joint state space given by the minimal or the maximal tensor product are inferior to quantum theory, before considering other tensor products in theories whose elementary systems have various two-dimensional state spaces. For these, we find no theories that outperform quantum theory in the adaptive CHSH game and prove that it is impossible to recover the quantum performance in various cases. This is the first step towards a general solution that, if successful, will have wide-ranging consequences, in particular, enabling an experiment that could rule out all theories in which the set of realisable correlations does not coincide with the quantum set.Comment: 12+2 pages, 2 figures; v2: typos correcte

Analysing causal structures in generalised probabilistic theories

Causal structures give us a way to understand the origin of observed correlations. These were developed for classical scenarios, but quantum mechanical experiments necessitate their generalisation. Here we study causal structures in a broad range of theories, which include both quantum and classical theory as special cases. We propose a method for analysing differences between such theories based on the so-called measurement entropy. We apply this method to several causal structures, deriving new relations that separate classical, quantum and more general theories within these causal structures. The constraints we derive for the most general theories are in a sense minimal requirements of any causal explanation in these scenarios. In addition, we make several technical contributions that give insight for the entropic analysis of quantum causal structures. In particular, we prove that for any causal structure and for any generalised probabilistic theory, the set of achievable entropy vectors form a convex cone.Comment: 11+13 pages, 5 figures, v2: new examples and additional discussion added, v3 (published version): presentation improve

Quantum Causal Structure and Quantum Thermodynamics

This thesis reports progress in two domains, namely causal structures and microscopic thermodynamics, both of which are highly pertinent in the development of quantum technologies. Causal structures fundamentally influence the development of protocols for quantum cryptography and microscopic thermodynamics is crucial for the design of quantum computers. The first part is dedicated to the analysis of causal structure, which encodes the relationship between observed variables, in general restricting the set of possible correlations between them. Our considerations rely on a recent entropy vector method, which we first review. We then develop new techniques for deriving entropic constraints to differentiate between causal structures. We provide sufficient conditions for entropy vectors to be realisable within a causal structure and derive new, improved necessary conditions in terms of so-called non-Shannon inequalities. We also report that for a family of causal structures, including the bipartite Bell scenario and the bilocal causal structure, entropy vectors are unable to distinguish between classical and quantum causes, in spite of the existence of quantum correlations that are not classically reproducible. Hence, further development is needed in order to understand cause from a quantum perspective. In the second part we explore an axiomatic framework for modelling error-tolerant processes in microscopic thermodynamics. Our axiomatisation allows for the accommodation of finite precision levels, which is crucial for describing experiments in the microscopic regime. Moreover, it is general enough to permit the consideration of different error types. The framework leads to the emergence of manageable quantities that give insights into the feasibility and expenditure of processes, which for adiabatic processes are shown to be smooth entropy measures. Our framework also leads to thermodynamic behaviour at the macroscopic scale, meaning that for thermodynamic equilibrium states a unique function provides necessary and sufficient conditions for state transformations, like in the traditional second law

Non-Shannon inequalities in the entropy vector approach to causal structures

A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.Comment: 23 pages + appendix; v2: minor changes to Section IV A; v3: paper has been significantly shortened, an expanded version of the removed review section can be found in arXiv:1709.08988; v4: version to be published, supplementary information available as ancillary file

Smooth entropy in axiomatic thermodynamics

Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including microscopic systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy.Comment: 18 pages, 1 figure; book chapter in "Thermodynamics in the Quantum Regime - Recent Progress and Outlook", eds. F. Binder, L. A. Correa, C. Gogolin, J. Anders and G. Adesso; the chapter relies on results reported in MW's PhD thesis, arXiv:1807.0634

Self-Testing of Physical Theories, or, Is Quantum Theory Optimal with Respect to Some Information-Processing Task?

Self-testing usually refers to the task of taking a given set of observed correlations that are assumed to arise via a process that is accurately described by quantum theory, and trying to infer the quantum state and measurements. In other words it is concerned with the question of whether we can tell what quantum black-box devices are doing by looking only at their input-output behaviour and is known to be possible in several cases. Here we introduce a more general question: is it possible to self-test a theory, and, in particular, quantum theory? More precisely, we ask whether within a particular causal structure there are tasks that can only be performed in theories that have the same correlations as quantum mechanics in any scenario. We present a candidate task for such a correlation self-test and analyse it in a range of generalised probabilistic theories (GPTs), showing that none of these perform better than quantum theory. A generalisation of our results showing that all non-quantum GPTs are strictly inferior to quantum mechanics for this task would point to a new way to axiomatise quantum theory, and enable an experimental test that simultaneously rules out such GPTs.Comment: 6 pages; v2: close to published version; v3: typos correcte

Quantum physics needs complex numbers

Complex numbers, i.e., numbers with a real and an imaginary part, are essential for mathematical analysis, while their role in other subjects, such as electromagnetism or special relativity, is far less fundamental. Quantum physics is the only physical theory where these numbers seem to play an indispensible role, as the theory is explicitly formulated in terms of operators acting on complex Hilbert spaces. The occurrence of complex numbers within the quantum formalism has nonetheless puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum physics, where states and observables are represented by real operators, seemed much more natural. In fact, previous works showed that such "real quantum physics" can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Thus, are complex numbers really needed for a quantum description of nature? Here, we show this to be case by proving that real and complex quantum physics make different predictions in network scenarios comprising independent quantum state sources. This allows us to devise a Bell-type quantum experiment whose input-output correlations cannot be approximated by any real quantum model. The successful realization of such an experiment would disprove real quantum physics, in the same way as standard Bell experiments disproved local physics.Comment: 17 pages. MATLAB codes available under reques

Inability of the entropy vector method to certify nonclassicality in linelike causal structures

Bell's theorem shows that our intuitive understanding of causation must be overturned in light of quantum correlations. Nevertheless, quantum mechanics does not permit signalling and hence a notion of cause remains. Understanding this notion is not only important at a fundamental level, but also for technological applications such as key distribution and randomness expansion. It has recently been shown that a useful way to decide which classical causal structures could give rise to a given set of correlations is to use entropy vectors. These are vectors whose components are the entropies of all subsets of the observed variables in the causal structure. The entropy vector method employs causal relationships among the variables to restrict the set of possible entropy vectors. Here, we consider whether the same approach can lead to useful certificates of non-classicality within a given causal structure. Surprisingly, we find that for a family of causal structures that include the usual bipartite Bell structure they do not. For all members of this family, no function of the entropies of the observed variables gives such a certificate, in spite of the existence of nonclassical correlations. It is therefore necessary to look beyond entropy vectors to understand cause from a quantum perspective.Comment: 5 pages + appendix, v2: added references, v3: new title, added journal referenc