We define and analyze measures of correlations for bipartite states based on trace distance. For Bell diagonal states of two qubits, in addition to the known expression for quantum correlations using this metric, we provide analytic expressions for the classical and total correlations. The ensuing hierarchy of correlations based on trace distance is compared to those based on relative entropy and Hilbert–Schmidt norm. Although some common features can be found, the trace distance measure is shown to differentiate from the others in that the closest uncorrelated state to a given bipartite quantum state is not given by the product of the marginals, and further, the total correlations are strictly smaller than the sum of the quantum and classical correlations. We compare the various correlation measures in two dynamical non-Markovian models, locally applied phase-flip channels and random external fields. It is shown that the freezing behavior, observed across all known valid measures of quantum correlations for Bell diagonal states under local phase-flip channels, occurs for a larger set of starting states for the trace distance than for the other metrics.
This thesis studies four topics in quantum information theory using tools from representation theory and (high-dimensional) probability theory.First, we study the nonadditivity of minimum output von Neumann and Rényi entropy of quantum channels.A sketch of the proof by Aubrun, Szarek and Werner for nonadditivity of minimum output entropy is presented, and a slight simplification is given.We show that asymptotically H(•) Shannon entropy, or von Neumann entropy D(ρ, σ) Trace distance between two quantum states ρ, σ F(ρ, σ) Fidelity between two quantum states ρ, σ
We found the quasi inverse of qubit channels as a unitary map,
Ei, by minimizing the average trace distance between the input state to the channel and the output of the quasi inverse channel for arbitrary qubit channel
E and for arbitrary input states. The channel
E was assumed completely positive and trace-preserving. To find the quasi inverse for mixed states, we proposed an alternative definition of the quasi inverse based on the mean square of the trace distance (MSTD) of a channel. The definition based on the trace distance allowed easy generalization of the quasi inverse to mixed input states. The quasi inverse of the Pauli, generalized amplitude damping, mixed unitary, and tetrahedron channels calculated based on the MSTD agreed with the one computed using average fidelity in the special case of input states being pure.