No abstract

We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence-bond states proposed in the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest neighbor interactions, and the partially dimerized or AKLT/VBS (Valence Bond Solid) states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.

We investigate the possibility of chaining qubits by letting pairs of nearest neighbours qubits dissipating into common environments. We then study entanglement dynamics within the chain and show that steady state entanglement can be achieved.

We consider a one-parameter family of matrix product states of spin one particles on a periodic chain and study in detail the entanglement properties of such a state. In particular we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other and show that the derivative of both these properties diverge when the parameter g of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point, the character of the matrix product state which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that entanglement of two sites have scailing behavior near the critical point.

We study the LOCC-assisted quantum capacity of bosonic dephasing channel with energy constraint on input states. We start our analysis by focusing on the energy-constrained squashed entanglement of the channel, which is an upper bound for the energy-constrained LOCC-assisted quantum capacity. As computing energy-constrained squashed entanglement of the channel is challenging due to a double optimization (over the set of density matrices and over the isometric extensions of a squashing channel), we first derive an upper bound for it, and then we discuss how tight that bound is for energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel. We prove that the optimal input state is diagonal in the Fock basis. Furthermore, we prove that the optimal squashing channel in the set of weakly-degradable and anti-degradable channels is a weakly-self-complementary channel. Restricting the search for optimal squashing channel to the set of weakly-self-complementary one-mode Gaussian quantum channels, we derive explicit upper and lower bounds for the energy-constrained LOCC-assisted quantum capacity of the bosonic dephasing channel in terms of its quantum capacity with different noise parameters. Thanks to the tightness of these bounds we provide a very good estimation of the LOCC-assisted quantum capacity of bosonic dephasing channel. We also study weakly-self-complementary qubit squashing channels which gives tighter upper bound to the energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel.