Abstract: This research-interaction seminar focuses on mathematical aspects of quantum information. In previous semesters we examined various applications of algebra, analysis, and geometry to quantum foundations, quantum cryptography, quantum computing, and other topics in theoretical physics. In this organizational meeting we'll discuss logistics, and then give some short advertisements for papers that we'd like to discuss during the fall semester. Suggestions and contributions are welcome! No previous experience in quantum theory is required, however linear algebra and (discrete) probability is a must. Seminar information is available at http://users.umiacs.umd.edu/~bclackey/QI-RIT2018Fall.html .
Abstract: I will talk about the basics of operational quantum theory, which presents quantum theory as a general probability theory focussing on systems and transformations. The natural language for this formalism is category theory. In this first lecture I will rapidly cover the most basic features of categories, build the framework for describing operational quantum theories, and discuss how these elements are represented in traditional quantum information science.
Abstract: Building on our description of operational quantum theories as categories we will discuss the physical concepts of causality, locality, and purity in the language of symmetric monoidal categories. Finally I will mention a set of additional axioms that singles out traditional quantum mechanics as a generalized probability theory.
Abstract: While diagrams are commonly used as an aid in proofs in quantum information, it is less common to see proofs themselves represented as pictures. Yet, some formal arguments can be more simply represented as pictures rather than equations. Graphical languages for quantum information have been developed (e.g., in the book "Picturing Quantum Processes" by B. Coecke and A. Kissinger) in which every picture-element has a formal meaning. In our work we use formal visual arguments to prove a new result on quantum self-testing: we show that N copies of the GHZ state can be self-tested by three non-communicating parties. I will discuss this proof and give an introduction to diagrammatic quantum information along the way.
Ref: S. Breiner, A. Kalev, C. Miller. "Parallel Self-Testing of the GHZ State with a Proof by Diagrams," https://arxiv.org/abs/1806.04744 .
Abstract: A unitary matrix can be block-diagonalized using the singular value decompostion of its submatrix. This property is sometimes referred to as "qubitization", which has wide applications in quantum computing. In this talk, I state and prove a particular version of qubitization, where the submatrix of the unitary admits an eigendecomposition.
 Guang Hao Low and Isaac L. Chuang, Hamiltonian simulation by qubitization, arXiv:1610.06546.
 Andras Gilyen, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, arXiv:1806.01838.
Abstract: The distinction between quantum correlation and classical correlation is usually studied with Bell inequalities. However, if we consider the geometry of quantum set in the probability space, Bell inequalities do not tell too much about it. In this work, we study the geometry of the set of quantum correlations arising from the convexity of the set itself. We study the scenario of two inputs and two outputs for two parties and derive nontrivial properties of the set in this case. We also show that even more complex features appear in Bell scenarios with more inputs or more parties. Finally, we discuss the limitations that the geometry of the quantum set imposes on the task of self-testing.
K. Goh, J. Kaniewki, E. Wolfe, T. Vertesi, X. Wu, Y. Cai, Y.-C. Liang, V. Scarani. Physical Review A 97, 022104. (2018)
Abstract: Random number generation protocols have been constructed that are strictly "device-independent" -- that is, they make no assumptions about the accuracy of the devices used. Such protocols make use of Bell violation experiments, and their security depends on physical assumptions (such as timing or non-communication). I will discuss a major breakthrough result that removes the need for the finer physical assumptions behind device-independent randomness, and relies instead on a computational hardness assumption. This result shows that, in principle, it is possible for a classical user to generate randomness simply by remote interaction with an untrusted client. The proof relies on the hardness of the Learning With Errors problem (LWE), which is used in postquantum cryptography.
Reference: Z. Brakerski, P. Christiano, U. Mahadev, U. Vazirani, T. Vidick. Certifiable Randomness from a Single Quantum Device. arXiv:1804.00640. (2018)
Abstract: This summer, 18-year-old Ewin Tang published a classical algorithm for recommendation systems that is "exponentially faster" than existing classical algorithms. His work was inspired by a quantum algorithm for the same task which had been one of few quantum machine learning algorithms that didn't suffer from numerous caveats. I'll talk about the general structure of Tang's algorithm and why it can be said to be "inspired" by the quantum algorithm. If time allows, I'll also give some proofs of the main results that justify Tang's algorithm.
Abstract: To program a quantum annealer for solving constrained optimization
problems, one must construct objective functions whose minima encode
the hard constraints imposed by the underlying problem. For such
"penalty models" we desire the additional property that the gap in the
objective value between states that meet and fail the constraints is
maximized amongst the allowable objective functions. We discuss why
this is a desirable property and mathematical methods to find optimal
penalty models using convex geometry and group theory. Our central
example will be constraints on the Hamming weight of bitstrings.
Abstract: The question of whether there can be unclonable quantum money that anyone can verify is one that has remained open for the last 40 years. Moving away from money, one can ask a more generalized question: can quantum states be used as copy-protected programs, that is, one that lets the user evaluate a function f but not create more programs for f? This talk will aim to formally define quantum money and copy-protection schemes, investigate their basic properties, and provide preliminaries from cryptography and quantum information. The goal of the talk is to provide a framework that can be used to study specific schemes.
Abstract: If n researchers participate in a nuclear bomb project, they share the classified information. However, they think there is a spy in the group, so they decide that the nuclear bomb can be accessed only by working collaboratively with any k of them (i.e., (k-1) people cannot access the bomb). How can this be achieved? I will talk about how GHZ states can be used to share the quantum secret between two parties, so that they need to work together to reconstruct the original information, and how to prevent an eavesdropper. At the end, we will see how this technique can be generalized to higher dimensions. The talk is mainly based on the paper Hillery, M., Buzek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59(3), 1829â1834 (1999).
Abstract: The study of quantum correlations started with John Bell's discovery of nonlocality and has led to real-world applications in cryptography, randomness certification and other fields. However, the structure of the set of quantum correlations is not well understood. Even to determine if a given correlation is quantum can be difficult in general. In this talk, I am going to introduce tools from semidefinite programming which can be used to determine if a given correlation is in the quantum set or an extreme point of the quantum set. Related basic concepts of semidefinite programming and quantum correlation will also be covered.
Abstract: Space can emerge from quantum entanglement. Indeed, the idea that non-trivial topologies/geometries might be related to entangled states is not new. Cohomology is a great way to detect global/topologically non-trivial quantities. We'll introduce chain complexes associated to multipartite states, and describe how the cohomology of such complexes output obstructions to factorizability of the state in the form of operators that exhibit non-local correlations. We'll also outline how the Euler characteristics of these complexes are related to mutual information. Such homological techniques are computationally accessible probes of a non-commutative space associated to a multipartite state; other geometric probes might provide further measures of entanglement.
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