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Random Matrices and Applications in Quantum Information
This course will give an introduction to random matrices, with a special emphasis on their applications in the context of quantum information.
The course is primarily designed for students of the new Masters Study Program on Quantum Information; but anybody else interested in random matrices and/or quantum information theory is also very welcome.
The course will be coordinated with the course ``Mathematical Foundations of Quantum Information Theory'', so that it is possible (and actually recommended) to take both courses in parallel.
Prerequisites: No prior knowledge of physics is required. The course is primarily designed for Bachelor’s and Master’s students in mathematics or computer science. Students from physics and related fields are equally welcome. Participants are expected to be familiar with one- and multidimensional analysis and linear algebra as covered in Mathematics for Computer Scientists 1–3 or in the courses Analysis 1&2 and Linear Algebra 1.
Content
The basic objects of quantum information theory include pure states, represented by unit vectors; mixed states, represented by density matrices; quantum channels, represented by linear maps between matrix algebras; and unitary channels, given by conjugation with unitary matrices.
In the finite-dimensional setting, these are ultimately vectors, matrices, and linear maps, often equipped with an additional tensor-product structure. In small dimensions, such objects are relatively easy to visualize and analyze: a single qubit, for example, is described by a two-dimensional complex vector space. The interesting difficulties arise when one considers systems consisting of many qubits. The dimension of the corresponding state space then grows exponentially and very quickly becomes enormous.
In principle, the mathematics is still linear algebra. In practice, however, our geometric intuition—as three-dimensional beings—is poorly adapted to the geometry of very high-dimensional spaces. This course will introduce mathematical tools that help us understand what vectors, matrices, and linear maps look like in high dimensions.
A central idea will be to study random versions of the objects of interest: random vectors, random matrices, random states, and random quantum channels. Randomness here is not necessarily meant as a physical feature of the final object. Rather, it is a mathematical tool for exploring the geometry of the entire space of possible objects.
This point of view allows us to ask questions such as: How common are states or quantum channels with a given property? Are such objects exceptional, or do they form a substantial part of the relevant space? In high dimensions, one often encounters a remarkable phenomenon: many quantities are sharply concentrated around a typical value. Consequently, a property found by averaging may hold not merely on average, but for an overwhelming majority of all objects. This is the realm of concentration of measure.
Averaging over random vectors, matrices, or channels can therefore reveal generic high-dimensional behavior, while often being much more tractable than the direct analysis of an individual object. Random models can also provide existence proofs and benchmarks for explicit deterministic constructions: they tell us what a well-designed state, channel, code, or measurement should be expected to achieve.
Among the applications we will discuss are the Page curve, which describes the typical entanglement of random pure states, and the use of random quantum channels to show that two channels can sometimes perform better together than separately. By using an entangled input, it is possible to obtain a joint output that is less noisy, or more pure, than the best outputs of the individual channels would suggest. Both examples show how random constructions can reveal unexpected features of concrete quantum systems.
The course will study probabilistic and analytic methods for answering questions about random vectors and random matrices, with particular emphasis on applications to quantum information theory. Important examples will include Gaussian random vectors and matrices, random density matrices, and the uniform—or Haar—probability measure on the unitary group.
Averaging over large unitary groups has become an especially important tool in quantum information theory. The corresponding computational framework, known as Weingarten calculus, will be one of the central topics of the course.