Quantum networks play a key role in many scenarios of quantum information theory. Here we consider the quantum causal networks in the manner of entropy. First we present a revised smooth max-relative entropy of quantum combs, then we present a lower and upper bound of a type II error of the hypothesis testing. Next we present a lower bound of the smooth max-relative entropy for the quantum combs with asymptotic equipartition. At last, we consider the score to quantify the performance of an operator. We present a quantity equaling to the smooth asymptotic version of the performance of a quantum positive operator.
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 t…
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 al…
We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality recently proposed by Pawlowski et. al. (arXiv:0905.2992). We consider two entropic quantities, which we term measurement and mixing entropy. In classical and quantum theory, they are equal, being given by the Shannon and von Neumann entropies respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entr…
A central question for causal inference is to decide whether a set of correlations fit a given causal structure. In general, this decision problem is computationally infeasible and hence several approaches have emerged that look for certificates of compatibility. Here we review several such approaches based on entropy. We bring together the key aspects of these entropic techniques with unified terminology, filling several gaps and establishing new connections regarding their relation, all illustrated with examples. We consider cases where unobserved causes are classical, quantum and post-quant…
In order to understand the detailed mechanism by which a fundamental discreteness can provide a finite entanglement entropy, we consider the entanglement entropy of two classes of free massless scalar fields on causal sets that are well approximated by causal diamonds in Minkowski spacetime of dimensions 2,3 and 4. The first class is defined from discretised versions of the continuum retarded Green functions, while the second uses the causal set's retarded nonlocal d'Alembertians parametrised by a length scale $l_k$. In both cases we provide numerical evidence that the area law is recovered wh…
This thesis reports progress in two domains, causal structures and microscopic thermodynamics, both of which are pertinent in the development of quantum technologies. 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. Based on this, we develop techniques for deriving entropic constraints to differentiate between causal structures. We provide sufficient conditions for entropy vectors to be realis…
We introduce the term smooth entanglement entropy transfer, a phenomenon that is a consequence of the causality-cancellation property of the quantum gravity environment. The causality-cancellation of the quantum gravity space removes the causal dependencies of the local systems. We study the physical effects of the causality-cancellation and show that it stimulates entropy transfer between the quantum gravity environment and the independent local systems of the quantum gravity space. The entropy transfer reduces the entropies of the contributing local systems and increases the entropy of the q…
Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realisable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional…
Analysing causal structures in generalised probabilistic theories
Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所维也纳）；Department of Mathematics, University of York（约克大学数学系）
Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory encompassing quantum systems and gravity is expected to allow causally nonseparable processes featuring operations in indefinite causal order, defying that events be causally ordered at all. The first challenge has been addressed through the recent development of intrinsically quantum causal models, allowing causal explanations of quantum processes -- provided th…
Quantum causality extends the conventional notion of fixed causal structure by allowing channels and operations to act in an indefinite causal order. The importance of such an indefinite causal order ranges from the foundational---e.g. towards a theory of quantum gravity---to the applied---e.g. for advantages in communication and computation. In this review, we will walk through the basic theory of indefinite causal order and focus on experiments that rely on a physically realisable indefinite causal ordered process---the quantum switch.
Causal modelling provides a powerful set of tools for identifying causal structure from observed correlations. It is well known that such techniques fail for quantum systems, unless one introduces `spooky' hidden mechanisms. Whether one can produce a genuinely quantum framework in order to discover causal structure remains an open question. Here we introduce a new framework for quantum causal modelling that allows for the discovery of causal structure. We define quantum analogues for many of the core features of classical causal modelling techniques, including the Causal Markov Condition and F…
We prove an equivalence transformation between the correlation measure functions of the causally-unbiased quantum gravity space and the causally-biased standard space. The theory of quantum gravity fuses the dynamic (nonfixed) causal structure of general relativity and the quantum uncertainty of quantum mechanics. In a quantum gravity space, the events are causally nonseparable and all time bias vanishes, which makes it no possible to use the standard causally-biased entropy and the correlation measure functions. Since a corrected causally-unbiased entropy function leads to an undefined, obscu…
We develop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of interactive tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. The optimization method is illustrated in a number of ap…
