The quantum switch has been widely studied as a prototypical example of indefinite causal order in quantum information processing. However, the potential advantages of utilizing more general forms of indefinite causal orders remain largely unexplored. We study higher-order switches, which involve concatenated applications of the quantum switch, and we demonstrate that they provide a strict advantage over the conventional quantum switch in the task of quantum channel distillation. Specifically, we show that higher-order quantum switches enable the probabilistic distillation of any qubit Pauli channel into the identity channel with nonzero probability. This capability contrasts with the conventional quantum switch, which allows only a limited set of Pauli channels to be distilled with nonzero probability. We observe that, counterintuitively, the distillation rate generally increases the noisier the channel is. We fully characterize the asymptotic distillation rates of higher-order superswitches for qubit Pauli channels. Finally, we prove a no-go result for multiqubit generalizations.
The quantum switch has been widely studied as a prototypical example of indefinite causal order in quantum information processing. However, the potential advantages of utilizing more general forms of indefinite causal orders remain largely unexplored. We study higher-order switches, which involve concatenated applications of the quantum switch, and we demonstrate that they provide a strict advantage over the conventional quantum switch in the task of quantum channel distillation. Specifically, we show that higher-order quantum switches enable the probabilistic distillation of any qubit Pauli channel into the identity channel with nonzero probability. This capability contrasts with the conventional quantum switch, which allows only a limited set of Pauli channels to be distilled with nonzero probability. We observe that, counterintuitively, the distillation rate generally increases the noisier the channel is. We fully characterize the asymptotic distillation rates of higher-order superswitches for qubit Pauli channels. Finally, we prove a no-go result for multiqubit generalizations.
Indefinite causal order is an evolving field with potential involvement in quantum technologies. Here we propose and study one possible scenario of practical application in quantum communication: a compound entanglement distillation protocol that features two steps of a basic distillation protocol applied in a coherent superposition of two causal orders. This is achieved by using one faulty entangled pair to control-swap two others before a fourth pair is combined with the two swapped ones consecutively. As a result, the protocol distills the four faulty entangled states into one of a higher fidelity. Our protocol has a higher fidelity of distillation and probability of success for some input faulty pairs than conventional concatenations of the basic protocol that follow a definite distillation order. Our proposal shows the advantage of indefinite causal order in an application setting consistent with the requirements of quantum communication.
Abstract The standard quantum state discrimination problem can be understood as a communication scenario involving a sender and a receiver following these three steps: (i) the sender encodes information in pre-agreed quantum states, (ii) sends them over a noiseless channel, and (iii) the receiver decodes the information by performing appropriate measurements on the received states. In a practical setting, however, the channel is not only noisy but often also unknown, thus altering the states and making optimal decoding generally not possible. In this work, we study this noisy discrimination scenario using a protocol based on indefinite causal order. To this end, we consider the quantum switch and define its higher-order generalisations, which we call superswitches. We find that, for certain channels and ensembles, the guessing probability can be significantly improved compared to both single- and multiple-copy state discrimination.
Quantum compositions of channels can grant advantages over classical compositions in quantum information processing or thermodynamical tasks. One of such quantum composition is one with indefinite causal order, where multiple quantum channels are placed in a superposition of different causal or operation orders. The advantages of indefinite causal order were demonstrated in an experimentally realizable setup known as the quantum switch and were often attributed to this new quantum resource of indefinite causality, but some doubts and disagreements still remains as compositions without indefinite causal order can achieve the same advantages. In this work, we proposed that the quantum switch's uniqueness lies in its ability to exhibit coherent non-Markovianity, where memory effects acts across different paths of superposition. We showed that the quantum switch violates completely positive or CP-divisibility and thus has intrinsic non-Markovianity in its evolution, which allows for non-Markovian backflow of information from the environment. This backflow of information can then play a role in the many advantages seen in the quantum switch, and by extending the quantum switch operation such that the non-Markovianity is controllable, we showed that these advantages depend on the presence and amount of non-Markovianity, achieving the quantum switch case when the system is fully non-Markovian. Particularly, we demonstrated this for the quantum switch's advantages in communication capacities, work extraction via daemonic ergotropy, and refrigeration via heat extraction. Other quantum compositions that are often compared to indefinite causal order, such as the superposition of independent channels and superposition of trajectories, are also captured by this extended construction of the quantum switch, and we showed that their difference with the quantum switch lies in the presence or absence of non-Markovianity. Our work suggests that non-Markovianity is an important recipe for the quantum switch or indefinite causal order, and is a nonnegligible factor for its advantages.
Indefinite causal order is a key feature involved in the study of quantum higher-order transformations. Recently, intense research has been focused on possible advantages related to the lack of definite causal order of quantum processes. Quite often the quantum switch is claimed to provide advantages in information-theoretic and thermodynamic tasks. We address here the question of whether indefinite causal order is a resource for quantum thermodynamics. Inspired by previous results in the literature, we show that indefinite causal order is not necessary for the reported increase in free energy and ergotropy. More specifically, we show that a simple causally ordered process, which replaces the system's state with a new one before the final measurement, outperforms the quantum switch in all thermodynamic tasks considered so far. We further show that a similar advantage can also be achieved without completely discarding system if we allow for non-Markovian interactions between the system and an environment. We extend the analysis to more extreme examples of indefinite causal order, showing that they do not provide an advantage either. Finally, we discuss a possible way to study the advantages that may arise from indefinite causal order in a general scenario.
Recent results have shown that quantum theory is compatible with novel causal structures where events happen without a definite causal order. In particular, the quantum switch describes a process in which two quantum channels act in a coherent superposition of their two possible orders. Furthermore, the quantum switch can perform communication tasks that are impossible within the framework of the standard quantum Shannon theory. The present paper considers the scenario of one-shot heralded qubit communication using a higher-order quantum switch constructed from two quantum switches. Specifically, we show that two quantum switches put in a superposition of their alternative causal orders can transmit a qubit, without any error, with a probability strictly higher than that achievable with each quantum switch. We discuss three examples that demonstrate this communication advantage. Notably, a higher-order quantum switch not only can outperform useful quantum switches but also becomes useful as a resource even if the quantum switches making it up are useless.
Processes with indefinite causal orders (ICOs), such as the quantum switch, are higher-order quantum processes that superpose the order in which quantum operations are performed. Such coherent control yields computational advantages but is not faithfully captured by existing quantum programming languages: either they are restricted to the unitary case, and thus cannot combine ICOs with measurement, or they treat coherent control nonlinearly. In both cases, they do not realize the full computational power of ICOs. We introduce a higher-order quantum functional language that supports general quantum computation, not merely the permutation of channels, and whose linear type system allows quantum control to be well-defined beyond the unitary case, on arbitrary quantum channels. We equip this language with a small-step operational semantics that synchronizes measurement outcomes across superposed branches, using device references and a memory function. We also give a denotational semantics by means of completely positive maps. With linearity as the only constraint, some well-typed terms would denote unphysical maps. We therefore impose a typing discipline that goes beyond linearity, and interpret programs in the causal category Caus[CPM], under which every well-typed program is physically meaningful, a property that can be checked statically and efficiently. We prove soundness, and study the language's expressive power: it can express every quantum channel at first order, and at second order a large subclass of the so-called quantum circuits with quantum control (QC-QCs), containing the quantum switch. Last but not least, we show that this language is well-designed enough to be extended to the nonlinear setting with recursion.