Quantum correlations and non-classical properties of hypergraph states

Sarkar, Ramita (2023) Quantum correlations and non-classical properties of hypergraph states. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Ramita Sarkar (15IP017)) 15IP017.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

In this thesis, different quantum correlations and non-classicality criteria of hypergraph states are studied. We, first introduce permutation symmetric states generated by hypergraphs and describe their combinatorial structures. We demonstrate that a quantum hypergraph state is k-separable if and only if the hypergraph has k-connected components. The permutation symmetric states remain invariant under any permutation. This combinatorial perspective motivates us to investigate multi-partite entanglement of permutation symmetric hypergraph states. Using generalised concurrence we measure entanglement up to ten qubits. A number of examples of these states are discussed. We also study coherence, discord and other quantum correlations in this system. In the following chapter, we introduce a general n qubit hypergraph state followed by several studies on quantum correlations and non-classical properties. We here study squeezing of hypergraph in different quadrature and finally find that hypergraph states are phase squeezed. We explored coherence in both number and phase basis. We study Mandel Q parameter, higher order anti-bunching criteria, Agarwal- Tara criteria for these states. A detailed investigation of the multiparty entanglement present in the 4-qubit quantum hypergraph states is presented, following a measurement-based geometrical approach. Considering a classification of the 4-party quantum system represented by a mathematical hypergraph based on the connections between its vertices, the genuine 4-party entanglement present in each bi-partition of the states have been measured. A strong correlation between the connectivity of the vertices of the hypergraphs and the genuine 4-party entanglement has been found. The equivalence of the genuine 4-party entanglement present in each bi-partition is shown considering similar connectivity of the vertices. This explicates the cyclic permutation symmetry of the multiparty entanglement present in the 4-qubit hypergraph states. Physically, one may expect the quantum systems with superposition of many states to behave in this symmetric manner while mapped into a network-type picture, which we have quantified, as well as classified in the next chapter. After that, we study the reciprocal of the mean quantum Fisher information (RMQFI), Χ² for general three qubit states, having graph and hypergraph states as special cases, for identifying genuine multi party entanglement characterised by Χ² < 1, and with Χ < 1 for phase estimation. We demonstrate that the most symmetric graph state and GHZ state have the lowest RMQFI values leading to highest statistical speed showing that both these states attain the Heisenberg limit in phase sensitivity. Unlike GHZ state, graph states have the same RMQFI values for measurement through different parameters, a property shared by the hypergraph states. Three qubit graph and hypergraph states can violate Bell’s inequality as QFI, FQ > N. Both GHZ state and the most symmetric graph state have the highest concurrence as well as the maximum QFI values equalling three for both these measures. Next, we study wave propagation in realistic medium, keeping in mind that quantum states are also waves which get affected by their propagation in medium. These waves come in the form of wave packet which in non-linear medium can form soliton those are extremely stable for propagation. More specifically we study a relativistic wave equation modified by the medium effect, the Westervelt equation. We present exact sub and supersonic soliton solutions of the one dimensional Westervelt equation, describing the propagation of pressure waves with high amplitude or frequency in nonlinear acoustic media. The sub and super-sonic waves only exist on a pedestal, in the absence of which the excitations propagate at the same speed as the small amplitude waves. For nonzero background, the soliton velocity is dependent on the linear acoustic properties such as, ambient density and sound speed, as also the nonlinearity coefficient. The nonlinearity coefficient is known to vary significantly with pathological conditions of tissue, often modelled by Westervelt equation, and hence these non-linear characteristic excitations may find application in tissue tomography as well as opto-mechanical devices. The use of light as an entangled photons to study brain tissue is a new domain of study.

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