Determining the state of a quantum system, known as quantum tomography, is an important subroutine in many quantum information processing tasks. A general quantum state of an n-qubit system is specified by 4n − 1 free parameters, which practically impossible to measure for large n. Nevertheless, in many quantum information processing protocols, the aim is to prepare and manipulate pure states which are specified by 2(2n − 1) free parameters. It has been previously shown that we can harness the information that the state is pure to reduce the number of measurement settings required to specify the state. In particular, it has been shown that any two-qubit pure state is uniquely distinguished from any other two-qubit state, pure or mixed, by the outcomes of the measurement of only seven Pauli observables (rather than nine required in general). Similar result was established for three-qubit pure states with only 19 Pauli observables (here 27 are required in general). In this project we propose to study the number of Pauli observables sufficient to measure in order to uniquely distinguishing a 4-qubit pure state from any other 4-qubit state. We also aim at generalizing these results for n-qubit pure states. The student should have a background in linear algebra.
Quantum theory predicts that the knowledge of the whole is not equivalent, in general, to the knowledge of its parts. In this project we propose to study the relation between pure states and their reduced density matrices. In particular, it has been shown that the set of three-qubit pure states that are uniquely determined from any other pure state by their two-qubit reduced density matrices, equals the set of three-qubit pure states that are uniquely determined from any other state by their two-qubit reduced density matrices. These sets were shown to be not equal if instead of three-qubit pure states we are considering four-qubit pure states. The aim of this project is to determine whether this equality can be restore for four-qubit pure states when we are considering their three-qubit reduced density matrices. The student should have a background in linear algebra.
Communication complexity studies the number of bits that the participants of a communication system need to exchange in order to achieve a task, which has been established as a central subject in computer science connecting to a variety of fields. To prove the computational difficulty in communication complexity, various generic lower bound methods have been developed since the birth of this subject. A large amount work has been devoted to the comparisons among these methods, aiming to design an optimal one. This project aims to compare three most recent lower bound methods, which are also the most powerful methods, relative discrepancy bound, relaxed partition bound, and partition bound. All these three bounds can be formalized in terms of linear programs and share a lot of resemblances. This project requires the basic knowledge of linear programs, especially the transformation between the primal and dual programs. No background on communication complexity is necessary.
Braverman-Rao protocol is an elegant message-compression protocol. It studies such a communication task. Alice and Bob are given descriptions of distributions p and q, respectively. The task is Alice and Bob jointly sample according to p. Braverman and Rao showed that Alice and Bob only need to exchange D(p||q) bits. This protocol implies several well-known compression schemes such as Shannon’s compression scheme, Slepian-Wolf scheme etc, and has significant applications in communication complexity. This project aims to design a quantum Braverman-Rao protocol. Namely, Alice and Bob are given the descriptions of two quantum states and the task is that Bob gets a quantum state close to the one with the description in Alice’s hands. I have made partial progress on this project and plan to get a fully quantum Braverman-Rao protocol. This project requires the basic knowledge of (discrete) probability theory, information theory, and linear algebra.