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DTSTART:20230312T070000
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DTSTART;TZID=America/Toronto:20230914T130000
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URL:https://uwaterloo.ca/institute-for-quantum-computing/events/tc-fraser-p
 hd-thesis-defence
LOCATION:QNC - Quantum Nano Centre 2101 200 University Avenue West Waterloo
  ON N2L 3G1 Canada
SUMMARY:TC Fraser PhD Thesis Defence
CLASS:PUBLIC
DESCRIPTION:AN ESTIMATION THEORETIC APPROACH TO QUANTUM REALIZABILITY PROBL
 EMS\n\nThis thesis seeks to develop a general method for solving so-called
 \nquantum realizability problems\, which are questions of the following\nf
 orm under which conditions does there exists a quantum state\nexhibiting a
  given collection of properties? The approach adopted by\nthis thesis is t
 o utilize mathematical techniques previously developed\nfor the related pr
 oblem of property estimation which is concerned with\nlearning or estimati
 ng the properties of an unknown quantum state. Our\nprimary result is to r
 ecognize a correspondence between (i) property\nvalues which are realized 
 by some quantum state\, and (ii) property\nvalues which are occasionally p
 roduced as estimates of a generic\nquantum state. In Chapter 3\, we review
  the concepts of stability and\nnorm minimization from geometric invariant
  theory and non-commutative\noptimization theory for the purposes of chara
 cterizing the flow of a\nquantum state under the action of a reductive gro
 up.\n\nIn particular\, we discover that most properties of quantum 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\nmom
 ent map of a quantum state by performing a covariant quantum\nmeasurement 
 on a large number of identical copies of the quantum\nstate. These measure
 ment schemes for estimating the moment map of a\nquantum state arise natur
 ally from the decomposition of a large\ntensor-power representation into i
 ts irreducible sub-representations.\n\nThen\, in Chapter 5\, we prove an e
 xact correspondence between the\nrealizability of a moment map value on on
 e hand and the asymptotic\nlikelihood it is produced as an estimate on the
  other hand. In\nparticular\, by composing these estimation schemes\, we d
 erive necessary\nand sufficient conditions for the existence of a quantum 
 state jointly\nrealizing any finite collection of moment maps. Finally\, i
 n Chapter 6\nwe apply these techniques to the quantum marginals problem wh
 ich aims\nto characterize precisely the relationships between the marginal
 \ndensity operators describing the various subsystems of composite\nquantu
 m state. We make progress toward an analytic solution to the\nquantum marg
 inals problem by deriving a complete hierarchy of\nnecessary inequality co
 nstraints.
DTSTAMP:20260420T125307Z
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