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The quantum switch, the canonical example of a process with indefinite causal order, has been claimed to provide various advantages over processes with definite causal orders for some particular tasks in the field of quantum metrology. In this work, we argue that some of these advantages in fact do not hold if a fairer comparison is made. To this end, we consider a framework that allows for a proper comparison between the performance, quantified by the quantum Fisher information, of different classes of indefinite causal order processes and that of causal strategies on a given metrological task. More generally, by considering the recently proposed classes of circuits with classical or quantum control of the causal order, we come up with different examples where processes with indefinite causal order offer (or not) an advantage over processes with definite causal order, qualifying the interest of indefinite causal order regarding quantum metrology. As it turns out, for a range of examples, the class of quantum circuits with quantum control of causal order, which are known to be physically realizable, is shown to provide a strict advantage over causally ordered quantum circuits as well as over the class of quantum circuits with causal superposition. Thus, the consideration of this class provides new evidence that indefinite causal order strategies can strictly outperform definite causal order strategies in quantum metrology.

Causal nonseparability is the property underlying quantum processes incompatible with a definite causal order. So far it has remained a central open question as to whether any process with a clear physical realisation can violate a causal inequality, so that its causal nonseparability can be certified in a device-independent way, as originally conceived. Here we present a method solely based on the observed correlations, which certifies the causal nonseparability of all the processes that can induce a causally nonseparable distributed measurement in a scenario with trusted quantum input states, as defined in [Dourdent et al., Phys. Rev. Lett. 129, 090402 (2022)]. This notably includes the celebrated quantum switch. This device-independent certification is achieved by introducing a network of untrusted operations, allowing one to self-test the quantum inputs on which the effective distributed measurement induced by the process is performed., Comment: 7+15 pages, 7 figures

We present the experimental measurement, on a quantum processor, of a series of polynomial lower bounds that converge to the quantum Fisher information (QFI), a fundamental quantity for certifying multipartite entanglement that is useful for metrological applications. We combine advanced methods of the randomized measurement toolbox to obtain estimators that are robust against drifting errors caused uniquely during the randomized measurement protocol. We estimate the QFI for Greenberg-Horne-Zeilinger states, observing genuine multipartite entanglement. Then, we prepare the ground state of the transverse field Ising model at the critical point using a variational circuit. We estimate its QFI and investigate the interplay between state optimization and noise induced by increasing the circuit depth., Comment: 24 pages, 13 figures

We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state of interest. This method involves performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multi-copy observables and present performance guarantees in terms of variance bounds which depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through numerical experiments estimating polynomial approximations of the von Neumann entropy and quantum state fidelities.

The operator entanglement (OE) is a key quantifier of the complexity of a reduced density matrix. In out-of-equilibrium situations, e.g. after a quantum quench of a product state, it is expected to exhibit an entanglement barrier. The OE of a reduced density matrix initially grows linearly as entanglement builds up between the local degrees of freedom, it then reaches a maximum, and ultimately decays to a small finite value as the reduced density matrix converges to a simple stationary state through standard thermalization mechanisms. Here, by performing a new data analysis of the published experimental results of [Brydges et al., Science 364, 260 (2019)], we obtain the first experimental measurement of the OE of a subsystem reduced density matrix in a quantum many-body system. We employ the randomized measurements toolbox and we introduce and develop a new efficient method to post-process experimental data in order to extract higher-order density matrix functionals and access the OE. The OE thus obtained displays the expected barrier as long as the experimental system is large enough. For smaller systems, we observe a barrier with a double-peak structure, whose origin can be interpreted in terms of pairs of quasi-particles being reflected at the boundary of the qubit chain. As $U(1)$ symmetry plays a key role in our analysis, we introduce the notion of symmetry resolved operator entanglement (SROE), in addition to the total OE. To gain further insights into the SROE, we provide a thorough theoretical analysis of this new quantity in chains of non-interacting fermions, which, in spite of their simplicity, capture most of the main features of OE and SROE. In particular, we uncover three main physical effects: the presence of a barrier in any charge sector, a time delay for the onset of the growth of SROE, and an effective equipartition between charge sectors., Comment: 13 + 24 pages with 3 + 7 figures

Quantum mechanics allows processes to be superposed, leading to a genuinely quantum lack of causal structure. For example, the process known as the quantum switch applies two operations ${\cal A}$ and ${\cal B}$ in a superposition of the two possible orders, ${\cal A}$ before ${\cal B}$ and ${\cal B}$ before ${\cal A}$. Experimental implementations of the quantum switch have been challenged by some on the grounds that the operations ${\cal A}$ and ${\cal B}$ were implemented more than once, thereby simulating indefinite causal order rather than actually implementing it. Motivated by this debate, we consider a situation in which the quantum operations are physically described by a light-matter interaction model. While for our model the two processes are indistinguishable in the infinite energy regime, restricting the energy available for the implementation of the operations introduces imperfections, which allow one to distinguish processes using different number of operations. We consider such an energetically-constrained scenario and compare the quantum switch to one of its natural simulations, where each operation is implemented twice. Considering a commuting-vs-anticommuting unitary discrimination task, we find that within our model the quantum switch performs better, for some fixed amount of energy, than its simulation. In addition to the known computational or communication advantages of causal superpositions, our work raises new questions about their potential energetic advantages., Comment: Updated version with a slightly modified abstract and conclusion emphasizing more on the exact assumptions and outcomes of the work. The Fig1 has a rewritten legend. Minor details were also added on various parts

We use hyper-entanglement to experimentally realize deterministic entanglement swapping based on quantum Elegant Joint Measurements. These are joint projections of two qubits onto highly symmetric, iso-entangled, bases. We report measurement fidelities no smaller than $97.4\%$. We showcase the applications of these measurements by using the entanglement swapping procedure to demonstrate quantum correlations in the form of proof-of-principle violations of both bilocal Bell inequalities and more stringent correlation criteria corresponding to full network nonlocality. Our results are a foray into entangled measurements and nonlocality beyond the paradigmatic Bell state measurement and they show the relevance of more general measurements in entanglement swapping scenarios.

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 argument 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. Remarkably, 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. We consider a known striking example of such a tripartite classical process that has a unitary extension, and study its realisation on time-delocalised subsystems. We then discuss the question of what a violation of causal inequalities implies in this setting, and argue that it is indeed a meaningful concept to show the absence of a definite causal order between the variables of interest., Comment: V2: Various improvements, argument on causal inequalities assumptions corrected and significantly strengthened. V3: Close to published version