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URL:/institute-for-quantum-computing/events/tc-fraser-p
 hd-thesis-defence
SUMMARY:TC Fraser PhD Thesis Defence
CLASS:PUBLIC
DESCRIPTION:Summary \n\nAN ESTIMATION THEORETIC APPROACH TO QUANTUM REALIZA
 BILITY PROBLEMS\n\nThis thesis seeks to develop a general method for solvi
 ng so-called\nquantum realizability problems\, which are questions of the 
 following\nform under which conditions does there exists a quantum state\n
 exhibiting a given collection of properties? The approach adopted by\nthis
  thesis is to utilize mathematical techniques previously developed\nfor th
 e related problem of property estimation which is concerned with\nlearning
  or estimating the properties of an unknown quantum state. Our\nprimary re
 sult is to recognize a correspondence between (i) property\nvalues which a
 re realized by some quantum state\, and (ii) property\nvalues which are oc
 casionally produced as estimates of a generic\nquantum state. In Chapter 3
 \, we review the concepts of stability and\nnorm minimization from geometr
 ic invariant theory and non-commutative\noptimization theory for the purpo
 ses of characterizing the flow of a\nquantum state under the action of a r
 eductive group.\n\nIn particular\, we discover that most properties of qua
 ntum states are\nrelated to the gradient of this flow\, also known as the 
 moment map.\nAfterwards\, Chapter 4 demonstrates how to estimate the value
  of the\nmoment map of a quantum state by performing a covariant quantum\n
 measurement on a large number of identical copies of the quantum\nstate. T
 hese measurement schemes for estimating the moment map of a\nquantum state
  arise naturally from the decomposition of a large\ntensor-power represent
 ation into its irreducible sub-representations.\n\nThen\, in Chapter 5\, w
 e prove an exact correspondence between the\nrealizability of a moment map
  value on one hand and the asymptotic\nlikelihood it is produced as an est
 imate on the other hand. In\nparticular\, by composing these estimation sc
 hemes\, we derive necessary\nand sufficient conditions for the existence o
 f a quantum state jointly\nrealizing any finite collection of moment maps.
  Finally\, in Chapter 6\nwe apply these techniques to the quantum marginal
 s problem which aims\nto characterize precisely the relationships between 
 the marginal\ndensity operators describing the various subsystems of compo
 site\nquantum state. We make progress toward an analytic solution to the\n
 quantum marginals problem by deriving a complete hierarchy of\nnecessary i
 nequality constraints.\n 
DTSTAMP:20250522T063310Z
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