The Complementarity Relation And Joint Quatum Measurement

The principle of complementarity, which shows a quite distinct differ between the microcosmic and the macrocosmos, e.g., the wave-particle duality showed in many types of Young’s double-slit experiments, has been one of the most concerned fundamental problems from the early time of establishment of quantum mechanics. It has been presented fully and caused important applications in modern quantum information science. Meantime, the physical mechanism inducing the principle of complementarity is still under debate, whose focus is whether the principle of complementarity or Heisenberg uncertainty relation is more fundamental rule. On the other hand, Heisenberg uncertainty relation reveals that non-commuting observables can not been sharply measured simultaneously. However, some unsharp measurements of observables could be performed co-instantaneously; hence the conception of joint measurement is emerged. Aiming at the relation between the principle of complementarity and the joint measurement, the context of this thesis is as following:1. The quantificational complementarity relation, i.e., Jaeger-Shimony-Vaidman-Englert inequality in Mach-Zehnder interferometer, is the most typical and very important presentation of the complementarity principle. In this paper, we consider the joint measurement performed in the experimental framework, and obtain the sufficient and necessary condition of it, meaning the positive answer of a long-time open question. We find the relation between the jointly measurable condition and Jaeger-Shimony-Vaidman-Englert inequality, which exhibit that at least the wave-particle complementarity is enforced by a mathematical construction of quantum mechanics, i.e., the joint measurement, and derive a more stringent complementarity inequality. The result provides a new and effective method for quantum mechanical interpretation of the principle of complementarity, and shows the possibility of unifying the principle of complementarity and Heisenberg uncertainty relation.2. In fact, the joint measurement relative to Jaeger-Shimony-Vaidman-Englert inequality has a certain special form, and then the jointly measurable condition of arbitrary pair of observables (in2-dimensional Hilbert space) is still to be resolved. With the result of the jointly measurable condition in Mach-Zehnder interferometer as above, we make progress on this topic and get the sufficient and necessary condition of arbitrary two observables in2-dimensional system finally. Also, some different expressions of the condition are obtained by many research groups individually. Then we compare such results and find the equivalence among them (numerically). However, our result has an obvious advantage, which is that only one inequality is needed.3. Since the general joint measurement condition is obtained, it is possible to discover broad complementarity relations based on the method in Mach-Zehnder interferometer setup. So as an application of the general joint measurement condition, in this paper we consider the complementarity relation between the path distinguishablity and the interference pattern visibility in Mach-Zehnder interferometer with biased beam splitters, and find an interesting fact that, distinguishing four paths of quanta passing by is needed. The result concludes that the representation form of the principle of complementarity in this interferometer is unchanged.The chapter1introduces the principle of complementarity, the argument about the principle of complementarity and Heisenberg uncertainty relation, conception of joint measurement and the early works trying to connect the joint measurement with complementarity relation, or the principle of complementarity with Heisenberg uncertainty relation; the chapter2gives the joint measurements in Jaeger-Shimony-Vaidman-Englert experimental framework, i.e., Mach-Zehnder interferometer and its sufficient and necessary condition, and the relation between the condition and the complementarity inequality; the chapter3shows the joint measurement of two unsharp obsevables of a qubit and the chapter4gets the complementarity relation in a Mach-Zehnder interferometer with biased beam splitters with joint measurable condition. As the important conclusion of the research on the principle of complementarity, and to clearly display the joint measurement in the relative experiment, Jaeger-Shimony-Vaidman-Englert inequality is introduced detailedly in the chapter2, too.