Complementary Relationships between Entanglement and Measurement

Complementary relationships exist regarding interference properties of particles such as pattern visibility, predictability and distinguishability. Additionally, relationships are known between information gain $G$ and measurement disturbance $F$ for entangled spin pairs. The question of whether a similar complementary relationship between entanglement and measurement occurs is examined herein. For qubit systems, both measurement on a single system and measurements on a bipartite system are considered in regards to the entanglement. It is proven that $\overline{E}+D\le 1$ holds where $\overline{E}$ is the average entanglement after a measurement is made and for which $D$ is a measure of the measurement disturbance of a single measurement. For measurements on a bipartite system shared by Alice and Bob ,it is shown that $\overline{E}+\overline{G}\le 1$ where $\overline{G}$ is the maximum average information gain regarding Alice's result that can be obtained by Bob. These results are generalized for arbitrary initial mixed states and as well to non-Hermitian operators. In the case of maximally entangled initial states, it is found that $D\le E_{L}$ and $\overline{G}\le E_{L}$ where $E_{L}$ is the entanglement loss due to measurement by Alice. We conclude that the amount of disturbance and information gain that one can gain are strictly limited by entanglement.