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

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## Abstract

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.