A Note on Quantum Coherence

Abstract

In this paper, we discuss the coherence of the reduced state in system H A ⊗H B under taking different quantum operations acting on subsystem H B . Firstly, we show that for a pure bipartite state, the coherence of the final subsystem H A under the sum of two orthonormal rank 1 projections acting on H B is less than or equal to the sum of the coherence of the state after two orthonormal projections acting on H B , respectively. Secondly, we obtain that the coherence of reduced state in subsystem H A under random unitary channel ${\Phi }(\rho )={\sum }_{s}\lambda _{s}U_{s}\rho U_{s}^{\ast }$ acting on H B , is equal to the coherence of the state after each operation ${\Phi }_{s}(\rho )=\lambda _{s}U_{s}\rho U_{s}^{\ast }$ acting on H B for every s. In addition, for general quantum operation ${\Phi }(\rho )={\sum }_{s}F_{s}\rho F_{s}^{\ast }$ on H B , we get the relation $ C\left (\left ((I\otimes {\Phi })\rho ^{AB}\right )^{A}\right )\leq \sum \limits _{s}C\left (\left ((I\otimes {\Phi }_{s})\rho ^{AB}\right )^{A}\right ). $