Combinatorics and Computing Weekly Seminar
Optimal Convergence Rates in Trace Distance and Relative Entropy for the Quantum Central Limit Theorem
Optimal Convergence Rates in Trace Distance and Relative Entropy for the Quantum Central Limit Theorem
Salman Beigi, IPM
16 OCT 2024
14:00 - 15:00
As a landmark in probability theory, the Central Limit Theorem (CLT) asserts that the normalized sum of independent and identically distributed random variables with zero mean converges to a Gaussian random variable. Depending on the structure of the underlying distribution, this convergence can be formulated with respect to different topologies or metrics. To name two famous such CLTs, the Lindberg--Levy CLT states convergence in distribution assuming the finiteness of the second moment, and the Berry--Esseen CLT asserts the uniform convergence of cumulative distribution functions at the rate of $O(1/sqrt n)$ assuming the finiteness of the third absolute moment. In this talk, we focus on the CLT in terms of stronger metrics, namely the total variation distance and the relative entropy. In fact, we are interested in the quantum version of CLT in terms of such strong metrics. We prove that the convergence rate of the quantum CLT in terms of the trace distance is $O(1/sqrt{n})$, and in terms of the relative entropy is $O(1/n)$, assuming some conditions on the moments. In the talk we mostly explain the results and techniques in the classical case, to make it accessible for the wide range of audiences of the seminars, and at the end explain the techniques in the quantum case.
Zoom room information:
https://us06web.zoom.us/j/85237260136?pwd=MFSZoKdmRXAjfaSaBzbf19lTaaKglf.1
Meeting ID: 852 3726 0136
Passcode: 362880
Venue: Niavaran, Lecture Hall 1