Complementary relationships between entanglement and measurement

Abstract

Complementary relationships exist among interference properties of particles such as pattern visibility, predictability, and distinguishability. Additionally relationships between average information gain Ḡ and measurement disturbance F for entangled spin pairs are well established. This article examines whether a similar complementary relationship exists between entanglement and measurement. For qubit systems, both measurements on a single system and measurements on a bipartite system are considered in regard to entanglement. It is proven that Ē + D ≤ 1 holds, where Ē is the average entanglement after a measurement is made and D is a measure of the measurement disturbance of a single measurement. Assuming measurements on a bipartite system shared by Alice and Bob, it is shown that Ē + Ḡ ≤ 1, where Ḡ is the maximum average information gain that Bob can obtain regarding Alice’s result. These results are generalized to arbitrary initial mixed states and non-Hermitian operators. In the case of maximally entangled initial states, it is found that D ≤ E_L and Ḡ ≤ E_L, where E_L is the loss of entanglement due to measurement by Alice. We conclude that the amount of disturbance and average information gain one can achieve is strictly limited by entanglement.