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Exploring and Constructing Multipartite Entangled States

Raissi, Zahra | 2017

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 50758 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Karimipour, Vahid; Memarzadeh, Laleh
  7. Abstract:
  8. Entanglement is considered to be one of the characteristic traits of quantum mechanics. Besides it plays a key role in quantum information science, being a resource for most of its applications such as quantum communication and quantum computation. The characterization (of different forms) of entanglement and its quantification play a central role in developing entanglement theory. By considering this fact, we describe a method for finding polynomial invariants under LOCC for a system of delocalized fermions shared between different parties, with global particle-number conservation as the only constraint. These invariants can be used to construct entanglement measures for different types of entanglement in such a system.Ideally, entanglement measures should allow to operationally quantify how useful a state is for quantum information applications. The maximally entangled states of two qubits, the so-called EPR states, are pure states of 2-qubits having reduced density matrices on each half of the system that are maximally mixed. A very intriguing question is whether systems made out of more than two parties can exhibit this property that all reduced states of at most half of the system size are maximally mixed.Such states are called Absolutely Maximally Entangled (AME) states and are pure multi-partite generalizations of the bipartite maximally entangled states. AME states are known to play an important role in quantum information processing when dealing with many parties.At the same time it is still largely unknown for which n and q AME states exist and how they can be constructed. We introduce a general method of gluing multi-partite states and show that entanglement swapping is a special class of a wider range of gluing operations. The gluing operation of two m and n qudit states consists of an entangling operation on two given qudits of the the two states followed by operations of measurements of the two qudits in the computational basis. We prove that when we glue two states by the third method, the k-uniformity of the states is preserved.AME states have also deep connections with apparently unrelated areas of mathematics such as combinatorial designs and structures, classical error correcting codes, and quantum error correcting codes (QECC). We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME(n; q) states of n parties with local dimension q a power of a prime for all n q + 1. Further, from a single AME state, we show how to produce an orthonormal basis of AME states. Also, we introduce quantum orthogonal srrays (QOA), combinatorial quantum objects which generalize orthogonal arrays, and investigate their application in quantum computer science. The main idea is to construct QOA such that it devide to classical part and quantum part. Our main results are on applications to find non-minimal support k-uniform state for larger set of n particles and local dimension q
  9. Keywords:
  10. Error Correction Codes ; Multipartite Entangled States ; Local Operation Assisted by Classical Communications (LOCC) ; Classical Error Correcting Codes ; Quantum Orthogonal Arrays (QOA) ; Invariant Polynomials

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