A Guide to Higher-order quantum operations

Philip Taranto, Simon Milz, Mio Murao +2 more

An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. $\textit{Higher-order quantum operations}\unicode{x2014}$quantum operations that transform quantum operations$\unicode{x2014}$are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.

Stanley Gudder

Neil Dowling, Silvia Pappalardi

The local-operator entanglement (LOE) measures the classical simulability of a Heisenberg operator and is conjectured to witness many-body chaos in locally interacting systems. Using tools from free probability, we analytically compute its asymptotic value for Haar random dynamics for all Rényi indices. We find that to the first two orders in large dimension, for traceless operators it exactly reproduces the Page curve for random states. In contrast to higher-order out-of-time-ordered correlators (OTOCs), which depend on more complex operator correlations via free cumulants, the leading-order LOE is independent of the initial operator. \add{We detail exactly how these higher-order correlations are encoded in the coefficients of the finite-size corrections. Therefore, guided by our Haar result, we argue that the long-time value of the LOE entropies in chaotic systems is robust: they depend only on a simple sub-sector of the hierarchy of correlations encoded by the full Eigenstate Thermalization Hypothesis. Our results therefore provide a conceptual boundary between operator simulability and higher-order scrambling diagnostics.

Yuan Li, Xiu-Hong Sun, Yan-Ni Dou

Xiu-Hong Sun, Yuan Li

Michael Antesberger, Marco Túlio Quintino, Philip John Walther +1 more

The field of indefinite causal order (ICO) has seen a recent surge in interest. Much of this research has focused on the quantum switch, wherein multiple parties act in a superposition of different orders in a manner transcending the quantum circuit model. This results in a new resource for quantum protocols, and is exciting for its relation to issues in foundational physics. The quantum switch is also an example of a higher-order quantum operation, in that it transforms not only quantum states but also other quantum operations. To date, no quantum process without a definite causal order has been completely experimentally characterized. Indeed, past work on the quantum switch has confirmed its ICO by measuring causal witnesses or demonstrating resource advantages, but the complete process matrix has been described only theoretically. Here we report our performing higher-order quantum process tomography. However, doing so requires exponentially many measurements with a scaling worse than that of standard process tomography. We overcome this challenge by creating a new fiber-based quantum switch using active optical elements to deterministically generate and manipulate time-bin encoded qubits. Moreover, our new architecture for the quantum switch can be readily scaled to multiple parties. By reconstructing the process matrix, we estimate its fidelity and tailor different causal witnesses directly for our experiment. To achieve this, we measure a set of tomographically complete settings, which also spans the input operation space. Our tomography protocol allows the characterization and debugging of higher-order quantum operations with and without an ICO, while our experimental time-bin techniques could enable the creation of a new realm of higher-order quantum operations with an ICO. Published by the American Physical Society 2024