Geometry of unital quantum maps and locally maximally mixed bipartite states

Rodriguez Ramos, C., 2022. Geometry of unital quantum maps and locally maximally mixed bipartite states. PhD, Nottingham Trent University.

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Abstract

In this thesis, we consider the geometry of the set of unital quantum maps and the geometry of the set of bipartite states with maximally mixed marginals. By the map-state duality, these two sets are isomorphic and can be considered by using the same mathematical formalism. When considering the geometry of unital quantum maps we encounter one crucial difference between two-dimensional systems and systems of higher dimensions. Unital qubit maps can be decomposed in terms of unitary maps. However, non-unitary maps need to be considered to decompose other unital qudit maps. To consider the geometry of unital quantum maps in higher dimensions, we construct a novel family of maps that includes both unitary and non-unitary unital quantum maps. For this family, we derive a criterion determining whether a given map of the family corresponds to an extreme point of the set of unital quantum maps. By applying the Choi-Jamiolkowski isomorphism over the family of maps, we consider the geometry of the set of locally maximally mixed bipartite states. In particular, we consider the problem of entanglement classification for the elements of this family of bipartite states. To do this, we find a set of invariants determining local unitary classes for our family. We also consider this family of bipartite states for qutrit systems. Remarkably, in this scenario, the chosen set of invariants can be used for the entanglement classification of the states of the family. For qutrit states, we consider the solutions of the equations giving unital quantum maps and locally maximally mixed bipartite states for the families previously considered. To do this, we construct an algorithm based on numerical methods to solve these equations. We also provide a graphical representation of the solutions given by the algorithm. Finally, we consider a constraint in the parameters of the equations allowing us to obtain solutions with analytical methods.

Item Type:	Thesis
Creators:	Rodriguez Ramos, C.
Contributors:	Name Role NTU ID ORCID Wilmott, C. Thesis supervisor SST3WILMOC orcid.org/0000-0003-4738-4227 Peytchev, E. Thesis supervisor CMP3PEYTCET orcid.org/0000-0001-5256-4383 Hetherington, T.J. Thesis supervisor SST3HETHET UNSPECIFIED
Date:	June 2022
Rights:	The copyright in this work is held by the author. You may copy up to 5% of this work for private study, or personal, non-commercial research. Any re-use of the information contained within this document should be fully referenced, quoting the author, title, university, degree level and pagination. Queries or requests for any other use, or if a more substantial copy is required, should be directed to the author.
Divisions:	Schools > School of Science and Technology
Record created by:	Melissa Cornwell
Date Added:	13 May 2024 13:48
Last Modified:	13 May 2024 13:48
URI:	https://irep.ntu.ac.uk/id/eprint/51428

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