Weak Measurements And Quantum Measurement-related Issues

Quantum mechanics is the base of the modern physical theories, and the problem of quantum measurement is one of foundational problems of quantum mechanics. The re-searches on quantum measurement can improve our understanding of the objective word and provide advanced technologies for our experiments. This paper is focused on the problems of weak measurements, the relationship between the coupling strength and the information gain in the measurement, the classicality and quantumness of a quantum en-semble. The main points are summarized as follows:1. The theory of quantum measurement is briefly described, and the current the-ories and applications of weak measurements are introduced in this paper. Due to the approximations used in deriving the results of weak measurement, the existed measure-ment results in the previous literature are not valid when the pre-selected state|ψi> and the post-selected state|ψi> are approximatively orthogonal to each other. Thus, we need some more precise descriptions of weak measurements. In this paper, we calculate the shifts of the pointers without any approximation. Without using the concept of weak val-ue, the expressions we obtained are not only valid for the weak interaction cases, but also legitimate for the strong interaction cases. By changing the strength of the interaction, we find:when the coupling strength is weak, the shift of pointer with postselection can be can be much larger than the shift obtained without postselection, and the amplification effect disappears when the interaction strength is strong. We also give the maximum shifts of the measuring device for the two-dimensional systems by choosing appropriate PPS.2. The improvement of the signal-to-noise ratio (SNR) by weak measurements is studied. Based on the researches of Stern-Gerlach experiment, we find:without consider-ing the probability of obtaining the outcomes decrease due to postselection, the SNR can be significantly improved by weak measurements; neither SNR cannot be effectively im-proved when the probability decrease is considered. We also study the the enhancement of the measure sensitivity (MS) by weak measurements, similarly with the SNR, the MS cannot be efficiently enhanced if the probability decrease due to postselection is taken into account; the MS can be efficiently enhanced if the probability should not be taken into account.3. We investigate the relationships between the information gain and the interaction strength between the quantum systems and measuring apparatus. Based on the theories of quantum information, a general method is proposed to calculate the information gain of the measuring apparatus. The explicit expressions are obtain for the state of the mea-suring apparatus is a Gaussian type. For qubit systems, we prove that the information gain of the measuring apparatus increases monotonically with the coupling strength. A complementarity of information gain is given for the qubit systems in this paper.4. In this paper, we propose a quantity J, the ensemble classicality based on projec-tive measurement strategy, to measure the classicality of a given ensemble. The quantity J can tell how classical an ensemble is. When J=1the ensemble behaves like a purely classical ensemble; and when J<1the ensemble cannot be considered as a classical en-semble anymore. We have revealed that the more classical an ensemble is, the better an unknown state from the ensemble can be cloned. The quantity of J provides us with a tool to evaluate how well classical tasks such as cloning, deleting, and distinguishing could be accomplished for quantum ensembles. We also define the quantumness Q of an ensemble and we surprisingly find that the quantumness Q of an ensemble used in quantum key distribution is exactly the attainable lower bound of the error rate.