Classical and Quantum Entanglement Detection

In classical physics, one can prepare any physical system composed of many independent parts in any feasible configuration or state just by acting on each part locally. For instance, in order to set up a car race in a particular configuration, it suffices to place each car on the right place and with the right speed. Namely, one does not need to act on several cars together. On the contrary, a striking feature of quantum mechanics is the existence of particle states that can only be prepared by bringing two or more particles together. Such states are said to be entangled, and they are a necessary ingredient for any quantum communication protocol, such as quantum cryptography, quantum teleportation or the certified generation of random numbers. Unfortunately, the problem of deciding if the mathematical description of a quantum state represents an entangled state (the classical entanglement problem) is a hard mathematical challenge. Establishing whether a physical device can prepare entangled states (the quantum entanglement problem) is even harder: since any measurement of a quantum system modifies the latter irreversibly, one needs to choose each measurement carefully so as not to incur into unnecessary preparations. This project aims to solve these two entanglement problems in a variety of relevant experimental scenarios, by combining ideas from quantum computing, convex optimization theory and statistical physics. Some of the expected outcomes of the project are: A quantum algorithm, implementable with present-day quantum computers, that solves the classical entanglement problem. A family of universal entanglement detection protocols, capable of discerning if an arbitrary source of few-part quantum states is indeed generating entanglement. A mathematical theory to describe general protocols for entanglement detection in quantum systems composed of hundreds or thousands of parts.

In classical physics, one can prepare any physical system composed of many independent parts in any feasible configuration or state just by acting on each part locally. For instance, in order to set up a car race in a particular configuration, it suffices to place each car on the right place and with the right speed. Namely, one does not need to act on several cars together. On the contrary, a striking feature of quantum mechanics is the existence of many-particle states that can only be prepared by bringing two or more particles together. Such states are said to be entangled, and they are a necessary ingredient for any quantum communication protocol, such as quantum cryptography, quantum teleportation or the certified generation of random numbers. Unfortunately, the problem of deciding if the mathematical description of a quantum state represents an entangled state (the classical entanglement problem) is a hard mathematical challenge. Establishing whether a physical device can prepare entangled states (the quantum entanglement problem) is even harder: since any measurement of a quantum system modifies the latter irreversibly, one needs to choose each measurement carefully so as not to incur into unnecessary preparations. Project P35509 advanced these two entanglement problems in a variety of relevant experimental scenarios. With regards to the classical problem, we showed how to detect entanglement among the atoms or molecules which make a solid object. With regards to the quantum problem, we presented an algorithm that generates optimal entanglement detection experiments. Key to our progress in quantum entanglement detection was the mathematical conceptualization of time-ordered processes, which allowed us to model general experimental protocols for entanglement detection. An interesting side result of the project is a new method to tackle the quantum marginal problem, i.e., the problem of determining if partial descriptions of a large array of particles are compatible with an overall quantum state. We advanced this long-standing question by uncovering an unexpected connection with the renormalization group, a well-studied notion in statistical physics for which K. G. Wilson received the Nobel Prize in 1982.