Bounds on continuous entanglement gain

• Main Author: Koo, Edmund Sui Ho
• Other Authors: Tomasz Paterek
• Format: Final Year Project
• Language: English
• Published: 2018
• Subjects: DRNTU::Science::Physics::Atomic physics::Quantum theory
• Online Access: http://hdl.handle.net/10356/75304
• Institution: Nanyang Technological University

Entanglement is a physical resource that is important in quantum teleportation, quantum dense coding and quantum cryptography. In this thesis, we investigate entanglement distribution between particles A and B (possibly located in different laboratories) via continuous interaction with an ancilla, C. We assume that A and B do not interact directly with each other, but only via C, and therefore the total Hamiltonian is of the form HAC + HBC. Our first result is the simplification of the expressions for HAC and HBC for a class of commuting Hamiltonians, i.e. [HAC, HBC] = 0 in which HAC is neither a free Hamiltonian on A nor a free Hamiltonian on C (which implies that A and C interact), and likewise HBC is neither a free Hamiltonian on B nor a free Hamiltonian on C. Using these simplifications, we looked at the time evolution of pure product states |αβγi and bi-product states |χiAB |γiC . We were able to analytically prove for pure product states that entanglement A : BC (or B : AC) is bounded by entanglement AB : C, that is the amount of entanglement in C. For bi-product states, we found a promising bound stating that entanglement gain, i.e. entanglement at time t minus initial entanglement is bounded by the entanglement in C. This is confirmed by extensive numerical simulations. We also considered the case where HAC realizes the swap operator SA−C at a particular time and swaps the state of A with C while HBC is just the identity operator. This scenario falls outside the class considered above. For this case, we managed to prove analytically (for bi-product states |χiAB |γiC ) the same bound that we conjectured above