First-order methods to recover the measurement operators and states of bipartite quantum correlations

First-order methods to recover the measurement operators and states of bipartite quantum correlations

Goverse, Vincent (TU Delft Electrical Engineering, Mathematics and Computer Science; TU Delft Applied Sciences)

de Laat, D. (mentor)

Delft University of Technology

2021-06-30

In this thesis, we start with giving a mathematical description of bipartite quantum correlations and how they are built up in the Tensor model. This is needed because we want to recover the state and the operators when only the bipartite quantum correlation is known. In the literature, there are see-saw algorithms to recover the state, but they are limited to only lower dimensions. In this thesis, we explore an alternative approach, where we directly minimize the function
f(ψ,{E_s^a},{F_t^b}) = ∑_a,b,s,t(P(a,b|s,t)- ψ^*(E_s^a ⊗ F_t^b)ψ)². Here, P(a,b|s,t) is the bipartite correlation,  ψ is the state vector, and E_s^a and F_t^b are the POVMs. Furthermore, ⊗ is the Kronecker product and ^* indicates the conjugate transpose of a vector. These variables are subject to constraints and some of them can easily be transformed into penalty functions. The matrices E_s^a and F_t^b have to be Hermitian positive semidefinite, for which we parameterize them by their Cholesky decompositions. The gradient of this (now unconstrained) problem can be explicitly determined with the use of Wirtinger calculus. This offers an elegant way to determine the gradient of real-valued functions with complex variables. Also, a total description of Wirtinger calculus is also given, including a proof that the gradient indeed points towards the direction of the steepest incline. We use first-order methods like gradient descent with backtracking line search and momentum-based gradient descent to find a minimum solution of the equation. If the cost function converges towards zero, we assume that the variables converge to a correct state and measurement operators.  These methods can find large correlations of approximately 3000 separate variables in 1.5 hours and are able to find many different other correlations and states. The algorithm had some problems finding the operators and state of a family of correlations that had four inputs and two outputs. For some correlations, the algorithms found states and operators of lower dimension than the correlations were build with. 

Bipartite Correlations
Gradient
Gradient descent
Quantum
Bipartite Quantum Correlations
penalty functions
Cholesky decomposition
momentum based
Wirtinger derivatives
Wirtinger
Complex derivatives

http://resolver.tudelft.nl/uuid:1958c8c9-8b06-4dfd-ab64-3fb34c363eab

Student theses

bachelor thesis

© 2021 Vincent Goverse