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Impossibilities and Approximations in Quantum Information TheoryAuthor: Dhrumil Patel Date: 20190110 Report no: IIIT/TH/2019/5 Advisor:Indranil Chakrabarty AbstractQuantum information theory has gained a lot of attention recently due to its nonintuitiveness and its promise to help in developing new technologies like quantum cryptography, "quantum computer" which are known theoretically to handle very complex and intractable problems. A bit is the fundamental unit of classical information theory. But in quantum world there is qubit, which can be in both the states simultaneously termed as superposition. Superposition is a fundamental brick in quantum information theory. Due to it many types of cool phenomenons like quantum correlations i.e. entanglement, coherence etc. come into play and act as quantum resources for certain information theoretic tasks. It has been proved that quantum resources for completion of such tasks are more optimized and efficient than their classical counterparts. There are many restrictions imposed on quantum world in terms of nogo theorems out of which one we are interested in is "Nocloning" theorem. It states that there does not exists any quantum operation which can clone an arbitrary quantum state perfectly. Quantum Cryptography is one of the use case of such restrictions as it makes it more secure. We have proposed even more stronger restriction which is termed as "Nocloning of quantum coherence/superposition". Why stronger you may ask? It is known that cloning of arbitrary quantum state implies signaling. So the reverse is that nosignaling implies nocloning of quantum state. Similarly, cloning of quantum state implies cloning of quantum coherence. So the reverse is that nocloning of quantum coherence implies nocloning of quantum state. This states that nocloning of quantum coherence is stronger that nocloning of quantum state. Also we have proved how they are different from each other as one might think at first that they are the same. The theorem holds true with or without machine states in the picture. We then show state characterization i.e. given a approximate coherence cloner how many states are there whose coherence can be cloned perfectly. Another important thing of this paper is that the theorems are proved geometrically with a nice representation on the Bloch sphere. The state of a quantum system is described by a wave function. But in cloning of a quantum state, we try to clone both wave and particle nature of the system. While in cloning of quantum coherence, we just try to clone the wave aspect of the system. From this we can also comment on how it is related to Heisenbergâ€™s uncertainty principle. Coming to the another part of this thesis work. Another fascinating thing of quantum information theory is that we can use quantum resources to outperform some classical tasks or to perform some tasks whose classical counterparts does not exist. Quantum Superposition and Quantum Entanglement are used as quantum resources in various information theoretic tasks like Quantum ix teleportation, Super dense coding, entanglement swapping, remote state preparation, quantum cryptographic protocols, digital signature etc. It begins with the age old uncertainty principle which arises from noncommutativity of mutually complementary operators, on one face forbids us from doing many things whereas on the other face becomes the seed of great advantage in the domain of security and privacy.The fidelity of quantum information processing tasks like quantum teleportation, dense coding and entanglement swapping are related to Fully Entangled Fraction (FEF) which in turn also serves as an index that characterize nonlocal correlations and gives us a measure how much far a state is from the maximally entangled state. Here we try to approximately broadcast the FEF of an arbitrary quantum state acting as a resource which in turn broadcasts the fidelity of various information processing tasks. This process is extremely helpful when we need more number of quantum states with good resource abilities. We also prove nobroadcasting of nonlocality using a very famous BH state dependent local cloner. So to conclude, this thesis is all about a few impossibilities and approximations in quantum information theory. Full thesis: pdf Centre for Security, Theory and Algorithms 

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