The Quantum Separability Problem

Quantum separability is one of the central topics in today`s quantum physics. It describes the entanglement properties of mixed quantum states and is an essential aspect in a variety of applications (teleportation, cryptography, quantum computing, information processing and optics, to name just a few). The term entanglement has always been surrounded by a mysterious aura. In fact, the possibility of entangled states led Einstein to declare in his famous "EPR" paper with Rosen and Podolsky that quantum mechanics involves "spooky actions at a distance. We want to get a complete mathematical characterization of quantum separability for mixed quantum sta tes. It is a difficult problem, classified as NP-hard in the theory of computational complexity. Mixed states are represented mathematically by "density operators". These are positive, self-adjoint operators with trace one in a Hilbert space. Some necessary conditions for the separation of a mixed state are known. There are however at time being no general and usable conditions outside of some elementary low-dimensional cases and the Gaussian case (even the latter is not yet fully understood). We will deal with these questions with the help of advanced methods from pure and applied mathematics, such as operator theory, Gabor frames theory and symplectic geometry. Our approach is completely new and has not been used in previous work. Another instrument with which the question of separability has not yet been tackled is the so-called "principle of the symplectic camel", which is a remarkable property of mechanical systems. It will help us get a geometric description of the separability. Ideally, this project will lead to great theoretical advances in mathematics and quantum mechanics. Maurice de Gosson will be the principal investigator. He will continue his close collaboration with several international teams of mathematicians and mathematical physicists and will also initiate new national and international collaborations.

Quantum separability and the concept of entanglement are central to con- temporary quantum physics, underpinning many applications. The most renowned application is quantum teleportation, pioneered by 2022 Nobel laureates Alain Aspect and Anton Zeilinger. Beyond teleportation, these principles are integral to quantum cryptography, computing, information processing, and optics. Entanglement, often described as a mysterious feature of quantum me- chanics, was famously referred to by Einstein as "spooky action at a dis- tance" after the publication of the foundational "EPR" paper. In our re- search project, we tackle the formidable challenge of achieving a rigorous mathematical characterization of quantum separability, a problem deemed NP-hard in computational complexity theory. Mixed quantum states, represented mathematically as density operators, describe complex quantum systems in probabilistic terms. While some nec- essary conditions for separability are known-such as the "PPT criterion" introduced by Horodecki and Peres or the Werner-Wolf symplectic condition on covariance matrices-tractable sufficient conditions are scarce. These ex- ist primarily for low-dimensional cases or Gaussian states. Our research employs advanced mathematical techniques, particularly harmonic analysis and symplectic geometry, to address these gaps. A key result of our work demonstrates that quantum states can be disentangled using only symplectic rotations. This insight could pave the way toward establishing general sufficient conditions for quantum separability. Our contributions include a dozen peer-reviewed papers and the book Quantum Harmonic Analysis: An Introduction (published by De Gruyter), which synthesizes advanced research findings with an accessible introduction to quantum separability. This book bridges the gap between cutting-edge research and educational needs, making the subject more approachable for students and researchers alike.