Probabilistic coding of quantum states

We discuss properties of probabilistic coding of two qubits to one qutrit and generalize the scheme to higher dimensions. We show that the protocol preservers entanglement between qubits to be encoded and environment and can be also applied to mixed states. We present the protocol which enables encoding of n qudits to one qudit of dimension smaller than the Hilbert space of the original system and then probabilistically but error-free decode any subset of k qudits. We give a formula for the probability of successful decoding

Violation of Clauser-Horne-Shimony-Holt inequality for states resulting from entanglement swapping

We consider violation of CHSH inequality for states before and after entanglement swapping. We present a pair of initial states which do not violate CHSH inequality however the final state violates CHSH inequality for some results of Bell measurements performed in order to swap entanglement.Comment: 3 pages, 2 figure

Mutually Unbiased Bases and Complementary Spin 1 Observables

The two observables are complementary if they cannot be measured simultaneously, however they become maximally complementary if their eigenstates are mutually unbiased. Only then the measurement of one observable gives no information about the other observable. The spin projection operators onto three mutually orthogonal directions are maximally complementary only for the spin 1/2. For the higher spin numbers they are no longer unbiased. In this work we examine the properties of spin 1 Mutually Unbiased Bases (MUBs) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator onto some direction z, the states of the other MUBs have to be squeezed. Then, we introduce the analogs of momentum and position operators and interpret what information about the spin vector the observer gains while measuring them. Finally, we study the generation and the measurement of MUBs states by introducing the Fourier like transform through spin squeezing. The higher spin numbers are also considered.Comment: 7 pages, 3 figures, comments welcom

Distillation of entanglement by projection on permutationally invariant subspaces

We consider distillation of entanglement from two qubit states which are mixtures of three mutually orthogonal states: two pure entangled states and one pure product state. We distill entanglement from such states by projecting n copies of the state on permutationally invariant subspace and then applying one-way hashing protocol. We find analytical expressions for the rate of the protocol. We also generalize this method to higher dimensional systems. To get analytical expression for two qubit case, we faced a mathematical problem of diagonalizing a family of matrices enjoying some symmetries w.r.t. to symmetric group. We have solved this problem in two ways: (i) directly, by use of Schur-Weyl decomposition and Young symmetrizers (ii) showing that the problem is equivalent to a problem of diagonalizing adjacency matrices in a particular instance of a so called algebraic association scheme.Comment: 22 pages, comments welcom

Distillation of entanglement from mixed states whose density matrix does not have full rank

Wydział FizykiW pracy doktorskiej pod tytułem „Destylacja splątania ze stanów mieszanych o niepełnym rzędzie macierzy gęstości” przedstawiono nowy protokół destylacji splątania. Jest on oparty na metodzie bisekcji i w niektórych przypadkach wykorzystuje jednokierunkowy protokół haszujący. Protokół ten zastosowano do następujących dwuqubitowych stanów splątanych: a) stanu mieszanego składającego się z czystego stanu splątanego i ortogonalnego do niego czystego stanu produktowego; b) stanu mieszanego składającego się z dwóch czystych stanów splątanych różniących się fazą i ortogonalnego do nich czystego stanu produktowego; c) stanu mieszanego składającego się z czystego stanu splątanego i dwóch czystych stanów produktowych (wszystkie stany są wzajemnie ortogonalne). Pokazano, że w przypadku stanów z punktów a) i b) protokół ten zawsze pozwala wydestylować splątanie, a w przypadku stanów z punktu c) protokół ten pozwala wydestylować splątanie dla pewnego zakresu parametrów charakteryzujących stany. Protokół ten porównano z innymi szeroko stosowanymi protokołami i pokazano, że w zastosowaniu do wymienionych stanów jest on od nich na ogół efektywniejszy. Wykorzystując zaproponowany protokół, znaleziono dolne ograniczenie na asystowaną klasyczną komunikacją w dwie strony kwantową pojemność następujących kanałów kwantowych: a) kanału tłumiącego amplitudę; b) kanału tłumiącego amplitudę i zmieniającego fazę; c) uogólnionego kanału tłumiącego amplitudę.In the thesis „Distillation of entanglement from mixed states whose density matrix does not have full rank” we presented new entanglement distillation protocol. It is based on bisection method and in some cases it uses one-way hashing protocol. The protocol was applied to the following two-qubit entangled states: a) mixed state which consists of pure entangled state and orthogonal pure product state; b) mixed state which consists of two pure entangled states with different phases and orthogonal pure product state; c) mixed state which consists of pure entangled state and two pure product states (all states are mutually orthogonal). It was shown that in the case of states from points a) and b) the protocol always enables to distill entanglement and in the case of states from point c) it enables to distill entanglement for certain range of parameters characterising those states. The protocol was compared with other widely used protocols and it was shown that in the case of mentioned states it is usually more effective. Using the proposed protocol, we found lower bound on quantum capacity assisted by two-way classical communication of the following quantum channels: a) amplitude damping channel; b) amplitude damping and phase-flip channel; c) generalised amplitude damping channel