As we know, entanglement is the most important resource in quantum information, quantum computation, et al. If we have got all the information of the system and can write down its state, we can use many criterions to judge whether the system is entangled or not. The most useful criterion is the “Peres criterion”. But in the laboratory it is very difficult to get all the information of the system. In this limited condition, how can we detect entanglement?
The physicist’s logic is very simple. In principle only Hermite physical quantity is measurable. For example susceptibility is measurable. If we can find some physical quantity that must be fulfilled some condition if the system is separable, we can use it to detect the entanglement. This quantity is called entanglement witness.
To get the condition of entanglement, uncertainty relation and Schwarz inequality should be used together with the definition of the separable state. Most of the witness are the variance of some quantity which could be detected directly. But no witness is good enough to detect all entanglement. For example, it is very difficult to detect the entanglement of the Bell state through entanglement witness. Many new witness will be introduced to improve detection efficient and detect more entangled states.