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

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,{Esa},{Ftb})
= ∑a,b,s,t(P(a,b|s,t)- ψ*(Esa ⊗ Ftb)ψ)2.
Here, P(a,b|s,t) is the bipartite correlation,  ψ is the state
vector, and Esa and Ftb 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 Esa and
Ftb 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.