In theories of communication, it is usually presumed that the involved parties perform actions in a fixed causal order. However, practical and fundamental reasons can induce uncertainties in the causal order. Here we show that a maximal uncertainty in the causal order forbids asymptotic quantum communication, while still enabling the noisy transfer of classical information. Therefore causal order, like shared entanglement, is an additional resource for communication. The result is formulated within an asymptotic setting for processes with no fixed causal order, which sets a basis for a quantum information theory in general quantum causal structures.
In theories of communication, it is usually presumed that the involved parties perform actions in a fixed causal order. However, practical and fundamental reasons can induce uncertainties in the causal order. Here we show that a maximal uncertainty in the causal order forbids asymptotic quantum communication, while still enabling the noisy transfer of classical information. Therefore causal order, like shared entanglement, is an additional resource for communication. The result is formulated within an asymptotic setting for processes with no fixed causal order, which sets a basis for a quantum information theory in general quantum causal structures.
Quantum theory is in principle compatible with scenarios where physical processes occur in an indefinite order, potentially yielding advantages in a broad range of information processing tasks. However, advantages in communication, the most basic form of information processing, have so far remained controversial and hard to prove. Here we provide a framework for assessing the role of causal order in communication, by comparing different causal structures under the constraint that the allowed operations must not generate signaling from signaling-incapable devices. Using this framework, we establish a clear-cut advantage of indefinite causal order, and, at the same time, we identify a series of fundamental limits to the communication power of causal structures in quantum mechanics. Notably, we find that a special form of indefinite causal order, obtained by coherently controlling the order of two processes, enhances the transmission of classical messages in a one-shot scenario, but no quantum operation with indefinite order can offer advantages over shared entanglement when asymptotically many uses of the same communication device are employed. Overall, our results unveil non-trivial relations between communication, causal order, entanglement, and no-signaling quantum processes.
The capacity of distant parties to send signals to one another is a fundamental requirement in many information-processing tasks. Such ability is determined by the causal structure connecting the parties, and more generally, by the intermediate processes carrying signals from one laboratory to another. Here we build a fully fledged resource theory of causal connection for all multi-party communication scenarios, encompassing those where the parties operate in a definite causal order and also where the order is indefinite. We define and characterize the set of free processes and three different sets of free transformations thereof, resulting in three distinct resource theories of causal connection. In the causally ordered setting, we identify the most resourceful processes in the bipartite and tripartite scenarios. In the general setting, instead, our results suggest that there is no global most valuable resource. We establish the signalling robustness as a resource monotone of causal connection and provide tight bounds on it for many pertinent sets of processes. Finally, we introduce a resource theory of causal non-separability, and show that it is – in contrast to the case of causal connection – unique. Together our results offer a flexible and comprehensive framework to quantify and transform general quantum processes, as well as insights into their multi-layered causal connection structures.
Requiring that the causal structure between different parties is well-defined imposes constraints on the correlations they can establish, which define so-called causal correlations. Some of these are known to have a "dynamical" causal order in the sense that their causal structure is not fixed a priori but is instead established on the fly, with for instance the causal order between future parties depending on some choice of action of parties in the past. Here we identify a new way that the causal order between the parties can be dynamical: with at least four parties, there can be some dynamical order which can nevertheless not be influenced by the actions of past parties. This leads us to introduce an intermediate class of correlations with what we call non-influenceable causal order, in between the set of correlations with static (non-dynamical) causal order and the set of general causal correlations. We then define analogous classes of quantum processes, considering recently introduced classes of quantum circuits with classical or quantum control of causal order---the latter being the largest class within the process matrix formalism known to have a clear interpretation in terms of coherent superpositions of causal orders. This allows us to formalise precisely in which sense certain quantum processes can have both indefinite and dynamical causal order.
Quantum mechanics allows operations to be in indefinite causal order, leading to advantages in computational and communication tasks. However, basic notions such as communication capacity have not yet been formalized. In light of this, through the process matrix formalism, we formulate different classical capacities for a bipartite quantum process. For one-way communication, we find that general quantum processes satisfy a generalization of the Holevo bound, i.e., we can send, at most, one bit per qubit. The result also holds for processes with indefinite causal order and with shared entanglement between the sender and the receiver. We further extend known bounds to classical channel capacities, showing that general processes cannot outperform causally separable ones. Next, we study bidirectional communication through a causally separable process. Our result shows that a bidirectional protocol cannot exceed the limit of a one-way communication protocol. Finally, we generalize this result to a multiparty broadcast communication protocol through a definite ordered process.
In this work, we study the transmission of classical information through three completely depolarizing channels in superposition of different causal orders. We exploit the quantum 3-switch as a resource for classical communication. We perform a kind of quantum control that was not accessible to the previously treated two-channel case. The fine and full quantum control achieved using selected combinations of causal orders let us uncover several features: nonmonotonous behavior on the transmission of information with respect to the number of causal orders involved, and different values of the transmission of information depending on the specific combinations of causal orders considered. Our results are a stepping stone to assess the efficiency of coherent quantum control and optimize resources in the implementation of indefinite causal structures. Finally, we suggest an optical implementation using standard telecom technology to test our predictions.
Causal nonseparability refers to processes where events take place in a coherent superposition of different causal orders.These may be the key resource for experimental violations of causal inequalities and have been recently identified as resources for concrete information-theoretic tasks.Here, we take a step forward by deriving a complete operational framework for causal nonseparability as a resource.Our first contribution is a formal definition for the specific notion of quantum control of causal orders, a stronger form of causal nonseparability-with the celebrated quantum switch as best-known example-where the causal orders of events for a target system are coherently controlled by a control system.We then build a resource theory-for both generic causal nonseparability as well as quantum control of causal orders-with a physically motivated class of free operations, based on process-matrix concatenations.We present the framework explicitly in the mindset with a control register.However, our machinery is totally versatile, being directly applicable also to scenarios with a target register alone.Moreover, an important subclass of our operations is free not only with respect to causal nonseparability and quantum control of causal orders but it also preserves the very causal structure of causal processes.Hence, our treatment contains, as a built-in feature, the basis of a resource theory of quantum causal networks too.As applications, first, we establish a simple sufficient condition for pure-process free convertibility.This imposes a hierarchy of quantum control of causal orders with the quantum switch at the top.Second, we prove that causal-nonseparability distillation exists.More precisely, we show how to convert multiple copies of a process with arbitrarily little causal nonseparability into fewer copies of a quantum switch.Our findings reveal conceptually new, unexpected phenomena, with both fundamental and practical implications.