Experimental Transmission of Quantum Information Using a Superposition of Causal Orders

Experimental Demonstration of Input-Output Indefiniteness in a Single Quantum Device

Quantum theory allows information to flow through a single device in a coherent superposition of two opposite directions, resulting into situations where the input-output direction is indefinite. Here we introduce a theoretical method to witness input-output indefiniteness in a single quantum device, and we experimentally demonstrate it by constructing a photonic setup that exhibits input-output indefiniteness with a statistical significance exceeding 69 standard deviations. Our results provide a way to characterize input-output indefiniteness as a resource for quantum information and photonic quantum technologies and enable tabletop simulations of hypothetical scenarios exhibiting quantum indefiniteness in the direction of time.

Quantum Switch as a Thermodynamic Resource in the Context of Passive States

In recent years, many works have explored possible advantages of indefinite causal order, with the main focus on its controlled implementation known as quantum switch. In this paper, we tackle advantages in quantum thermodynamics, studying whether quantum switch is capable of activating a passive state, either alone or with extra resources (active control state) and/or operations (measurement of the control system). By disproving the first possibility and confirming the second one, we show that quantum switch is not a thermodynamic resource in the discussed context, though it can facilitate work extraction given external resources. We discuss our findings by considering specific examples: a qubit system subject to rotations around the x and y axes in the Bloch sphere, as well as general unitaries from the U(2) group; and the system as a quantum harmonic oscillator with displacement operators, as well as with a combination of displacement and squeeze operators.

Charging Quantum Batteries via Indefinite Causal Order: Theory and Experiment

In the standard quantum theory, the causal order of occurrence between events is prescribed, and must be definite. This has been maintained in all conventional scenarios of operation for quantum batteries. In this study we take a step further to allow the charging of quantum batteries in an indefinite causal order (ICO). We propose a nonunitary dynamics-based charging protocol and experimentally investigate this using a photonic quantum switch. Our results demonstrate that both the amount of energy charged and the thermal efficiency can be boosted simultaneously. Moreover, we reveal a counterintuitive effect that a relatively less powerful charger guarantees a charged battery with more energy at a higher efficiency. Through investigation of different charger configurations, we find that ICO protocol can outperform the conventional protocols and gives rise to the anomalous inverse interaction effect. Our findings highlight a fundamental difference between the novelties arising from ICO and other coherently controlled processes, providing new insights into ICO and its potential applications.

Superposed Quantum Error Mitigation

Overcoming the influence of noise and imperfections is a major challenge in quantum computing. Here, we present an approach based on applying a desired unitary computation in superposition between the system of interest and some auxiliary states. We demonstrate, numerically and on the IBM Quantum Platform, that parallel applications of the same operation lead to significant noise mitigation when arbitrary noise processes are considered. We first design probabilistic implementations of our scheme that are plug and play, independent of the noise characteristic and require no postprocessing. We then enhance the success probability (up to deterministic) using adaptive corrections. We provide an analysis of our protocol performance and demonstrate that unit fidelity can be achieved asymptotically. Our approaches are suitable to both standard gate-based and measurement-based computational models.

Device-independent certification of indefinite causal order in the quantum switch

Quantum theory is compatible with scenarios in which the order of operations is indefinite. Experimental investigations of such scenarios, all of which have been based on a process known as the quantum switch, have provided demonstrations of indefinite causal order conditioned on assumptions on the devices used in the laboratory. But is a device-independent certification possible, similar to the certification of Bell nonlocality through the violation of Bell inequalities? Previous results have shown that the answer is negative if the switch is considered in isolation. Here, however, we present an inequality that can be used to device-independently certify indefinite causal order in the quantum switch in the presence of an additional spacelike-separated observer under an assumption asserting the impossibility of superluminal and retrocausal influences.

Demonstration of a Quantum Switch in a Sagnac Configuration

The quantum switch is an example of a process with an indefinite causal structure, and has attracted attention for its ability to outperform causally ordered computations within the quantum circuit model. To date, realizations of the quantum switch have made a trade-off between relying on optical interferometers susceptible to minute path length fluctuations and limitations on the range and fidelity of the implementable channels, thereby complicating their design, limiting their performance, and posing an obstacle to extending the quantum switch to multiple parties. In this Letter, we overcome these limitations by demonstrating an intrinsically stable quantum switch utilizing a common-path geometry facilitated by a novel reciprocal and universal SU(2) polarization gadget. We certify our design by successfully performing a channel discrimination task with near unity success probability.

Measuring Incompatibility and Clustering Quantum Observables with a Quantum Switch

The existence of incompatible observables is a cornerstone of quantum mechanics and a valuable resource in quantum technologies. Here we introduce a measure of incompatibility, called the mutual eigenspace disturbance (MED), which quantifies the amount of disturbance induced by the measurement of a sharp observable on the eigenspaces of another. The MED provides a metric on the space of von Neumann measurements, and can be efficiently estimated by letting the measurement processes act in an indefinite order, using a setup known as the quantum switch, which also allows one to quantify the noncommutativity of arbitrary quantum processes. Thanks to these features, the MED can be used in quantum machine learning tasks. We demonstrate this application by providing an unsupervised algorithm that clusters unknown von Neumann measurements. Our algorithm is robust to noise and can be used to identify groups of observers that share approximately the same measurement context.

Existence of processes violating causal inequalities on time-delocalised subsystems

It has been shown that it is theoretically possible for there to exist quantum and classical processes in which the operations performed by separate parties do not occur in a well-defined causal order. A central question is whether and how such processes can be realised in practice. In order to provide a rigorous framework for the notion that certain such processes have a realisation in standard quantum theory, the concept of time-delocalised quantum subsystem has been introduced. In this paper, we show that realisations on time-delocalised subsystems exist for all unitary extensions of tripartite processes. This class contains processes that violate causal inequalities, i.e., that can generate correlations that witness the incompatibility with definite causal order in a device-independent manner, and whose realisability has been a central open problem. We consider a known example of such a tripartite classical process that has a unitary extension, and study its realisation on time-delocalised subsystems. We then discuss this finding with regard to the assumptions that underlie causal inequalities, and argue that they are indeed a meaningful concept to show the absence of a definite causal order between the variables of interest.

Causal nonseparability and its implications for spatiotemporal relations

Quantum nonseparability is a central feature of quantum mechanics, and raises important philosophical questions. Interestingly, a particular theoretical development of quantum mechanics, called the process matrix formalism (PMF), features another kind of nonseparability, called causal nonseparability. The PMF appeals to the notion of quantum process, which is a generalisation of the concept of quantum state allowing to represent quantum-like correlations between quantum events over multiple parties without specifying a priori their spatiotemporal locations. Crucially, since the PMF makes no assumption about the global causal structure between quantum events, it allows for the existence of causally nonseparable quantum processes. Such processes are said to have an indefinite causal structure. This work aims at investigating the philosophical implications of causal nonseparability, especially for the notion of spatiotemporal relations. A preliminary discussion will first study the formal connection between quantum and causal nonseparability. It will be emphasised that, although quantum processes can be seen as a generalisation of density matrices, the conceptual distinction between the two notions yields significant differences between quantum and causal nonseparability. From there, it will be shown that, depending on the interpretative framework, causal nonseparability suggests some kind of indeterminacy of spatiotemporal relations. Namely, within a realist context, spatiotemporal relations can be epistemically or metaphysically indeterminate. Finally, it will be argued that, in spite of the disanalogies between standard and causal nonseparability, similar implications for spatial relations can already be defended in the context of standard quantum mechanics. This work highlights the potentially very fruitful explorations of the implications of quantum features on the conception of spacetime, keeping in mind that quantum and spacetime theories are expected to be unified in a future theory of quantum gravity.

Quantum communication using a quantum switch of quantum switches

The quantum switch describes a quantum operation in which two or more quantum channels act on a quantum system with the order of application determined by the state of an order quantum system. And by suitably choosing the state of the order system, one can create a quantum superposition of the different orders of application, which can perform communication tasks impossible within the framework of the standard quantum Shannon theory. In this paper, we consider the scenario of one-shot heralded qubit communication and ask whether there are protocols using a given quantum switch or switches that could outperform the given ones. We answer this question in the affirmative. We define a higher-order quantum switch composed of two quantum switches, with their order of application controlled by another order quantum system. We then show that the quantum switches placed in a quantum superposition of their alternative orders can transmit a qubit, without any error, with a probability higher than that achievable with the quantum switches individually. We demonstrate this communication advantage over quantum switches that are useful as a resource and those that are useless. We also show that there are situations where there is no communication advantage over the individual quantum switches.

Experimental Realization of a Quantum Refrigerator Driven by Indefinite Causal Orders

Indefinite causal order (ICO) is playing a key role in recent quantum technologies. Here, we experimentally study quantum thermodynamics driven by ICO on nuclear spins using the nuclear magnetic resonance system. We realize the ICO of two thermalizing channels to exhibit how the mechanism works, and show that the working substance can be cooled or heated albeit it undergoes thermal contacts with reservoirs of the same temperature. Moreover, we construct a single cycle of the ICO refrigerator based on the Maxwell's demon mechanism, and evaluate its performance by measuring the work consumption and the heat energy extracted from the low-temperature reservoir. Unlike classical refrigerators in which the coefficient of performance (COP) is perversely higher the closer the temperature of the high-temperature and low-temperature reservoirs are to each other, the ICO refrigerator's COP is always bounded to small values due to the nonunit success probability in projecting the ancillary qubit to the preferable subspace. To enhance the COP, we propose and experimentally demonstrate a general framework based on the density matrix exponentiation (DME) approach, as an extension to the ICO refrigeration. The COP is observed to be enhanced by more than 3 times with the DME approach. Our Letter demonstrates a new way for nonclassical heat exchange, and paves the way towards construction of quantum refrigerators on a quantum system.

Semi-Device-Independent Certification of Causal Nonseparability with Trusted Quantum Inputs

While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are incompatible with a definite causal order. We explore a form of certification of causal nonseparability in a semi-device-independent scenario where the involved parties receive trusted quantum inputs, but whose operations are otherwise uncharacterized. Defining the notion of causally nonseparable distributed measurements, we show that certain causally nonseparable processes that cannot violate any causal inequality, including the canonical example of the quantum switch, can generate noncausal correlations in such a scenario. Moreover, by imposing some further natural structure to the untrusted operations, we show that all bipartite causally nonseparable process matrices can be certified with trusted quantum inputs.

Computational Advantage from a Quantum Superposition of Qubit Gate Orders

In an ordinary quantum algorithm the gates are applied in a fixed order on the systems. The introduction of indefinite causal structures allows us to relax this constraint and control the order of the gates with an additional quantum state. It is known that this quantum-controlled ordering of gates can reduce the query complexity in deciding a property of black-box unitaries with respect to the best algorithm in which the gates are applied in a fixed order. However, all tasks explicitly found so far require unitaries that either act on unbounded dimensional quantum systems in the asymptotic limit (the limiting case of a large number of black-box gates) or act on qubits, but then involve only a few unitaries. Here we introduce tasks (i) for which there is a provable computational advantage of a quantum-controlled ordering of gates in the asymptotic case and (ii) that require only qubit gates and are therefore suitable to demonstrate this advantage experimentally. We study their solutions with the quantum n-switch and within the quantum circuit model and find that while the n-switch requires to call each gate only once, a causal algorithm has to call at least 2n-1 gates. Furthermore, the best known solution with a fixed gate ordering calls O[n log_{2}(n)] gates.

Causality in a Qubit-Based Implementation of a Quantum Switch

We introduce a qubit-based version of the quantum switch, consisting of a variation of the Fermi problem. Two qubits start in a superposition state in which one qubit is excited and the other is in the ground state. However, it is not defined which is the excited qubit. Then, after some time, if a photon is detected, we know that it must have experienced an emission by one atom and then an absorption and re-emission by the other one, but the ordering of the emission events by both qubits is undefined. While it is tempting to refer to this scenario as one with indefinite causality or a superposition of causal orders, we show that there is still a precise notion of causality: the probability of excitation of each atom is totally independent of the other one when the times are short enough to prevent photon exchange.

Learning Temporal Quantum Tomography

Quantifying and verifying the control level in preparing a quantum state are central challenges in building quantum devices. The quantum state is characterized from experimental measurements, using a procedure known as tomography, which requires a vast number of resources. However, tomography for a quantum device with temporal processing, which is fundamentally different from standard tomography, has not been formulated. We develop a practical and approximate tomography method using a recurrent machine learning framework for this intriguing situation. The method is based on repeated quantum interactions between a system called quantum reservoir with a stream of quantum states. Measurement data from the reservoir are connected to a linear readout to train a recurrent relation between quantum channels applied to the input stream. We demonstrate our algorithms for representative quantum learning tasks, followed by the proposal of a quantum memory capacity to evaluate the temporal processing ability of near-term quantum devices.

Quantum and Classical Data Transmission through Completely Depolarizing Channels in a Superposition of Cyclic Orders

Completely depolarizing channels are often regarded as the prototype of physical processes that are useless for communication: any message that passes through them along a well-defined trajectory is completely erased. When two such channels are used in a quantum superposition of two alternative orders, they become able to transmit some amount of classical information, but still no quantum information can pass through them. Here, we show that the ability to place N completely depolarizing channels in a superposition of N alternative causal orders enables a high-fidelity heralded transmission of quantum information with error vanishing as 1/N. This phenomenon highlights a fundamental difference with the N=2 case, where completely depolarizing channels are unable to transmit quantum data, even when placed in a superposition of causal orders. The ability to place quantum channels in a superposition of orders also leads to an increase of the classical communication capacity with N, which we rigorously prove by deriving an exact single-letter expression. Our results highlight the more complex patterns of correlations arising from multiple causal orders, which are similar to the more complex patterns of entanglement arising in multipartite quantum systems.

Cyclic quantum causal models

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 they admit a definite causal order, i.e. have an acyclic causal structure. This work addresses causally nonseparable processes and offers a causal perspective on them through extending quantum causal models to cyclic causal structures. Among other applications of the approach, it is shown that all unitarily extendible bipartite processes are causally separable and that for unitary processes, causal nonseparability and cyclicity of their causal structure are equivalent.

Quantum Refrigeration with Indefinite Causal Order

We propose a thermodynamic refrigeration cycle which uses indefinite causal orders to achieve nonclassical cooling. The cycle cools a cold reservoir while consuming purity in a control qubit. We first show that the application to an input state of two identical thermalizing channels of temperature T in an indefinite causal order can result in an output state with a temperature not equal to T. We investigate the properties of the refrigeration cycle and show that thermodynamically, the result is compatible with unitary quantum mechanics in the circuit model but could not be achieved classically. We believe that this cycle could be implemented experimentally using tabletop photonics. Our result suggests the development of a new class of thermodynamic resource theories in which operations are allowed to be performed in an indefinite causal order.

Quantum Metrology with Indefinite Causal Order

We address the study of quantum metrology enhanced by indefinite causal order, demonstrating a quadratic advantage in the estimation of the product of two average displacements in a continuous variable system. We prove that no setup where the displacements are used in a fixed order can have root-mean-square error vanishing faster than the Heisenberg limit 1/N, where N is the number of displacements contributing to the average. In stark contrast, we show that a setup that probes the displacements in a superposition of two alternative orders yields a root-mean-square error vanishing with super-Heisenberg scaling 1/N^{2}, which we prove to be optimal among all superpositions of setups with definite causal order. Our result opens up the study of new measurement setups where quantum processes are probed in an indefinite order, and suggests enhanced tests of the canonical commutation relations, with potential applications to quantum gravity.