### Quantum Information and Analysis Seminars at the University of South Carolina

Informal seminars that encourage learning, collaboration, and social activities between the participants,
are held weakly. Regular participants include graduate students
and faculty from the Departments of Computer Science and Engineering, Mathematics and Physics.

**Day and Time:** Fridays from 3:30 to 4:30 (with 2:15 to 3:15 being an alternative) Eastern Time.

**Place:** LeConte 348.

** Format:** The standard format of the seminar is in-person.

** Alternative Format:**
In the case that the
speaker cannot be with us in person, then the seminar is delivered and attended online via Zoom. In this case, the Zoom link is the following:

Join Zoom Meeting

**Meeting ID:** 813 7438 2767

**Passcode:** 314159

** Date:** 9/27/2024.

** Time:** 3:30-4:30pm.

** Speaker: ** Tiju Cherian John, University of Arizona.

** Title: ** What is Gaussianity in Non-Commutative Probability? And, Why Do We Care?

** Abstract: **
We examine non-commutative Gaussian measures, also known as quantum Gaussian states, within the framework of non-commutative probability theory. This concept, originally developed in quantum mechanics, has garnered renewed interest over the past three decades, particularly in continuous variable quantum information theory. Mathematically, the theory of Gaussian states remains largely unexplored. In this talk, we focus on quantum Gaussian states through the lens of classical probability, operator theory and quantum mechanics; and highlight key applications of this theory in modern physics.

** Date:** 4/18/2024.

** Time:** 3:30-4:30pm.

** Speaker: ** Angela Kolinko, (Computer Science and Engineering, USC).

** Title: ** Stabilizer formalism.

** Abstract: **
Stabilizer formalism is a powerful method for understanding a wide class of quantum operations. This theory presents us with a case of beautiful use of results from group theory in the field of quantum computation theory
In this talk, we plan to
- describe stabilizer formalism and corresponding measurement operation,
- state and sketch the proof of the Gottesman–Knill theorem,
- if time permits, present one of the applications of stabilizer formalism – stabilizer codes and their usage in quantum error correction.

** Date:** 3/29/2024.

** Time:** 2:30-3:30pm.

** Speaker: ** Zahra Khanian, Technical University of Munich.

** Title: ** Rate distortion theory for mixed states.

** Abstract: **
We consider the compression of asymptotically many copies of ensembles
of mixed quantum states where the encoder has access to a side information
system. The figure of merit is per-copy or local error criterion. Rate-distortion
theory studies the trade-off between the compression rate and the per-copy
error. The optimal trade-off can be characterized by the rate-distortion function,
which is the best rate given a certain distortion. We derive the rate-distortion
function of mixed-state compression. The rate-distortion functions in the
entanglement-assisted and unassisted scenarios are in terms of a single-letter
mutual information quantity and the regularized entanglement of purification,
respectively. Our compression scheme covers both blind and visible compression
models (and other models in between) depending on the structure of the side
information system.

The **recording** of the talk can be viewed here.

** Date:** 1/26/2024.

** Time:** 3:30pm.

** Speaker: ** Steve Dilworth

** Title: ** Transportation cost spaces and invariant projections

** Abstract: **
I will define the transportation cost space associated to a finite metric space M. It is a finite-dimensional normed space whose dual is the space of Lipschitz functions on M. I will present some examples which motivate an open question on the structure of the transportation cost space. Analysis of the `invariant' projections (from the edge space onto the space of Lipschitz functions) which commute with a group of isometries of the edge space has yielded useful information for the families of diamond and Laakso graphs. Recent results (with Kutzarova and Ostrovskii) for discrete tori and Hamming graphs which use this method will appear in a volume in honor of Professor Per Enflo. I will also discuss the limitations of the method, in particular the disappointing fact that it cannot answer the open question.

** Date:** 12/8/2023.

** Time:** 2:15pm.

** Speaker: ** Ralph Howard

** Title: ** Estimating the inradius of domains in Euclidean space.

** Abstract: **
We review and give proofs of theorems of
Blaschke, Lagunov, and Pestov and Ionin which give
lower bounds on the inradius of a domain in terms
of the boundary curvature. An example result,
due to Pestov and Ionin, is that a simple closed
curve in the plane which has curvature with respect
to the inner normal at most 1 surrounds a
disk of radius 1.

** Date:** 12/1/2023.

** Time:** 2:15pm.

** Speaker: ** Haonan Zhang

** Title: ** An inequality in Fourier analysis and its application to learning theory: classical and quantum

** Abstract: **
A fundamental problem from computational learning theory is to well-reconstruct an unknown function on the discrete hypercubes. One classical result of this problem for the random query model is the low-degree algorithm of Linial, Mansour, and Nisan in 1993. This was improved exponentially by Eskenazis and Ivanisvili in 2022 using a family of polynomial inequalities going back to Littlewood in 1930. Recently, quantum analogs of such polynomial inequalities were conjectured by Rouzé, Wirth, and Zhang (2022). This conjecture was resolved by Huang, Chen, and Preskill (2022) without knowing it when studying learning problems of quantum dynamics. In this talk, I will discuss another proof of this conjecture that is simpler and gives better estimates. As an application, it allows us to recover the low-degree algorithm of Eskenazis and Ivanisvili in the quantum setting. This is based on arXiv:2210.14468, joint work with Alexander Volberg (MSU).

** Date:** 11/17/2023.

** Time:** 2:15pm.

** Speaker: ** Frank (Peng) Fu

** Affiliation: ** Department of Computer Science and Engineering

** Title: ** Introduction to Compact Closed Categories (and Dagger Categories)

** Abstract: **
In this talk, I will first talk about how to model
the teleportation protocol with concepts from compact closed categories.
If time permit, I will give another brief introduction to dagger categories
and sketch how they can be used to model common concepts in quantum computing.

** Reading materials:** A categorical semantics of quantum protocols by Samson Abramsky and Bob Coecke,
and Dagger compact closed categories and completely positive maps
by Peter Selinger.

** Date:** 11/10/2023.

** Speaker: ** Frank (Peng) Fu

** Affiliation: ** Department of Computer Science and Engineering

** Title: ** An introduction to compact closed categories

** Abstract: **
Compact closed categories and their extensions can be used
as a framework to model many concepts in quantum computing
(e.g. scalars, inner products, even completely positive maps).
In this talk, I will give a brief introduction to compact closed categories and
their graphical language. As an example, I will sketch how they can be used to
understand the teleportation protocol.

** Reading materials:** A categorical semantics of quantum protocols by Samson Abramsky and Bob Coecke,
and Dagger compact closed categories and completely positive maps
by Peter Selinger.

** Date:** 11/3/2023.

** Speaker: ** Rabins Wosti

** Affiliation: ** Department of Computer Science and Engineering

** Title: ** Optimal lower bound of the average indeterminate length lossless quantum block encoding

** Abstract: **
Consider a general quantum source that emits at discrete
time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution
which may depend on the whole history. Also, fix two
positive integers $m$ and $l$.
We encode any
tensor product of $ml$ many states emitted by the quantum source by
breaking it into $m$ many blocks where each block has
length $l$, and considering
sequences of $m$ many isometries so that each isometry
encodes one of these blocks into the Fock space, followed by
the concatenation of their images. We only consider certain
sequences of such isometries that we call
``special block codes"
in order to ensure that the the string of encoded states is uniquely decodable. We compute the minimum average codeword length
of these encodings which depends on the quantum source
and the integers $m$, $l$, among all possible special block codes.
Our result extends the result of
[Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block,
i.e.\ for $m=1$. This is a joint work with G. Androulakis.

** Seminar notes:**
beamer file

** Date:** 10/27/2023.

** Speaker: ** George Androulakis

** Title: ** The Nussbaum-Szkola distributions and their use.

** Abstract: **
Abstract: We will review the use of Nussbaum-Szkola distributions in quantum information and in particular in computing quantum divergences. The talk will be based on joint works with T.C. John
arXiv:2308.02929,
arXiv:2203.01964,
arXiv:2303.03380.

** Seminar notes:**
beamer file

** Date:** 10/13/2023.

** Speaker: ** Stephen A. Fenner

** Affiliation: ** Department of Computer Science and Engineering

** Title: ** What is a quantum channel? Part III.

** Seminar notes:**
pdf file, and
pdf file for all three parts.

** Date:** 10/06/2023.

** Speaker: ** Stephen A. Fenner

** Affiliation: ** Department of Computer Science and Engineering

** Title: ** What is a quantum channel? Part II.

** Seminar notes:**
pdf file

** Date:** 9/29/2023.

** Speaker: ** Stephen A. Fenner

** Affiliation: ** Department of Computer Science and Engineering

** Title: ** What is a quantum channel? Part I.

** Seminar notes:**
pdf file

** Date:** 9/8/2023.

** Speaker: ** George Androulakis

** Title: ** The hyperfinite II_1 factor and its use in physics.

** Abstract: **
The hyperfinite II_1 factor is an infinite von Neumann algebra which is very similar to the nxn matrix algebra since it has a finite faithful tracial state. We will describe its construction and indicate some of its uses in physics.

** Seminar notes:**
Blog post

** Date:** 3/31/2023.

** Format: ** Online, via Zoom.

** Speaker: ** Sugata Gangopadhyay

** Affiliation: ** Department of Computer Science and Engineering, Indian Institute of Technology Roorkee.

** Title: ** A quantum algorithm to estimate the Gowers U2 norm and linearity testing of Boolean functions

** Abstract: ** We propose a quantum algorithm to estimate the Gowers U2 norm of a Boolean function, and extend it
into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are
epsilon-far from the set of linear Boolean functions, which seems to perform better than the classical
BLR algorithm.

** Article:** Jothishwaran, C.A., Tkachenko, A., Gangopadhyay, S. et al. A quantum algorithm to estimate the Gowers U2
norm and linearity testing of Boolean functions. Quantum Inf Process 19, 311 (2020).

** Date:** 3/3/2023.

** Speaker: ** George Androulakis

** Title: ** Petz-Renyi divergence between Gaussian states.

** Abstract: **
In the classical information theory, the Renyi divergence is an extension of the Kullback-Leibler divergence. Indeed, when the parameter of the Renyi divergence approaches 1, then the Renyi divergence approaches the Kullback-Leibler divergence.
Similarly, in quantum information theory, the Petz-Renyi divergence is an extension of the Umegaki relative entropy, i.e. when the parameter of the Petz-Renyi divergence approaches 1, then the Petz-Renyi divergence approaches the Umegaki relative entropy. Given two quantum states, it is an interesting problem to determine the values of the parameter of the Petz-Renyi divergence for which the Petz-Renyi divergence is finite. I will present a recent work with Tiju Cherian John where we solve this problem for a large class of Gaussian states. In particular, we verify a conjecture of Seshadreeshan, Lami and Wilde in some cases.

** Seminar notes:**
Blog post

** Date:** 2/17/2023.

** Speaker: ** George Androulakis

** Title: ** Introduction to Gaussian states.

** Abstract: **
The Gaussian states form a non-commutative analogue of the normal distribution. They are used in continuous variables quantum information in order to model quantum interactions of photons. In this introductory talk I will present their main definitions and examples.

** Seminar notes:**
Blog post

** Date:** 2/3/2023.

** Speaker: ** George Androulakis

** Title: ** Relative entropy via distribution of observables

** Abstract: **
The Umegaki relative entropy is an important tool in quantum information. The distribution of an observable with respect to a state is an important tool in quantum probability. We show how the first can be described via the second. The talk is based on joint work with Tiju Cherian John and can be found in https://arxiv.org/abs/2203.01964

** Seminar notes:**
Blog post.

** Date:** 1/27/2023.

** Speaker: ** George Androulakis

** Title: ** Classical and Quantum Divergences

** Abstract: **
We will overview classical and quantum divergences, and we will give a method for computing quantum divergences based on their classical counterparts. The talk will be based on joint work with Tiju Cherian John and can be found in https://arxiv.org/abs/2203.01964

** Seminar notes:**
Blog post.

** Dates:** 11/18 and 12/2 in 2022.

** Speaker: ** Tiju Cherian John

** Title: ** Violation of Bell's Inequality: A Quantum Probabilistic Take

** Abstract: **
We review the probabilistic underpinning of quantum mechanics as envisioned by John von Neumann and discuss a quantum mechanical example violating Bell's inequality.

** Seminar notes:**
Miro board.

** Dates: ** 10/28, 11/4 and 11/11 in 2022.

** Speaker: ** George Androulakis.

** Title: ** Bell's inequality

** Abstract: **
We will discuss the work of
John Stewart Bell.
Especially we are interested in Bell's 1964 article
On the Einstein Podolsky Rosen paradox.
Aspect, Clauser and Zeilinger played pioneering role in demonstrating a variant of Bell's result
Because of these works, these three researchers were awarded the
Nobel Prize in Physics on October 4th 2022

** Seminar notes:**
Miro board,
Blog post.

**Back to my homepage.****
**