Konstantinos Vardakis, "Iterative compression algorithm of quantum entanglement", Diploma Work, School of Electrical and Computer Engineering, Technical University of Crete, Chania, Greece, 2019
https://doi.org/10.26233/heallink.tuc.82665
This thesis addresses a central question in the field of Quantum Information Science and Technology. The object of concern is the quantum entanglement (i.e. quantum correlations) between two parts of a bipartite quantum system, each of which is mathematically described by a set of state vectors, all lying in a real vector space. The question then is formulated as follows: "Is it possible to reduce the number of vectors describing each subsystem of the bipartite quantum system and still have the same amount of total quantum entanglement ?" Hence the question, abbreviated to the name entanglement compression, is that of dimensional reduction of local sub-spaces composing the bipartite system, under the constraint of preserving the initial entanglement. In descriptive terms, the problem may be cast in the form: "can I have the same with less?" Given that quantum entanglement is treated par excellence as the new type of resources required by quantum technology, and the fact that the sub-systems, aimed to be dimensionally reduced, are composed by multiple qubits, the question then amounts to one of optimal handling of resources. Building upon previous works and recent developments, the thesis then proceeds to exploit a heuristic idea. Given that the coupling of sub-systems under study is mathematically determined by a coefficient-matrix, specifying the multi-tensor state vector of the bi-partite, a choice is made: use for coefficient-matrix any matrix describing a grayscale digital image. Digital images of e.g. Schroedinger, Lena are employed to build state vectors of multi-qubit bi-partite quantum systems. This situation motivates the exploitation of low rank matrix approximation techniques from image compression within the context of entanglement compression. Employing as quantitative measure of entanglement the quantum Rényi entropy of the marginal (reduced) density matrix of the bipartite system, the aim becomes to achieve dimensional reduction of the total bipartite state vector, while preserving (or optimizing on) the Rényi entropy of a quantum subsystem. The thesis shows that the first task of dimensional reduction is achieved via low rank approximation in the Singular Value Decomposition (SVD) of the image-state-vector matrix. A quantum algorithmic implementation of the classical technique to the quantum context is provided. The second task of the compression, that of entanglement preservation, is achieved via an algorithm, akin to Bell state generation quantum circuit, generalized to the context of multi-qubit systems. This part of the entanglement compression procedure utilizes tools from matrix analysis, such as pair-wise Hadamard product of matrices and related inequalities, to show that there is a trading between the two tasks, namely dimensional reduction and entanglement (entropy) preservation. This leads to building an iterative procedure, which involves unitary gates combined with higher dimensional oracle-driven projections acting on multi-qubit vectors. A thorough numerical investigation, based on exemplary cases of image-states, confirms the feasibility and the efficiency of quantum entanglement compression iterative procedure.