Study On Quantum Effects In Subspace Of Hilbert Space Induced By Postselection

That all physical observables,including Hamiltonians,should be Hermitian operators has long been seen as one of the axioms in conventional quantum mechanics,because it has been assumed that the measured values of observables should be real,and the eigenspectrums of Hermitian observables are exactly real.However,with the development of quantum mechanics,it has been found that the Hermitian observables and the real measured values may be neither sufficient nor necessary conditions for each other.This is because new kinds of measured values can be induced on the subspace of Hilbert space by postselection,i.e.,weak values or modular values,which are both complex in general.At the same time,it has been found that some non-Hermitian parity-time(PT)-reversal symmetric systems may also have a real eigenspectrum,and their dynamics can be induced by postselection.In this paper,the theories and phenomena such as weak measurement and PT-symmetric system are brought into the theoretical framework of quantum effects in subspace of Hilbert space induced by postselection.The contents are as follows:(1)Measurement of modular values and connection to the annihilation operator.Based on the above,we propose a new method different from tomography to measure modular values,and we find that our annihilation operator measurement(AOM)method may significantly reduce the number of measurements from O(4n)to 0(2n)compared with the original tomography method.Based that,we can directly measure the probability amplitude of wave function.(2)Efficient simulation of the dynamics of n-dimensional PT-symmetric system with a local-operations-and-classical-communication(LOCC)protocol based on an embedding scheme.In this work,by combing through the history of the development of non-Hermitian quantum mechanics,we naturally extend the original embedding scheme from pure states to arbitrary mixed states.Based on the above groundwork,we further propose a LOCC protocol scheme to simulate the dynamics of a PT-symmetry unbroken system,and give a concise bound of its success probability.Our LOCC protocol scheme has several advantages over the original embedding scheme:the success probability is increased to λmin(λmin>1)times in general;this LOCC protocol has less dependence on but more flexibility in the selection of the metric operator and more adaptability in practical applications;and it is more consistent with the physical intuition.Finally,the physical or philosophical meaning behind the embedding scheme and the LOCC protocol is discussed.(3)Dynamics simulation and numerical analysis of arbitrary time-dependent PTsymmetric system based on density operators and the influence of quantum noises.We further study the influence of quantum noises on the simulation results with the technique of vectorization of density operators and matrixization of superoperators(VDMS),and we show the depolarizing(Dep)noise is the most fatal and should be avoided as much as possible.By theoretical analysis and numerical calculation,we find the problem of the chronological product usually has to be solved not only in the numerical calculation,but also even in the experiment,because the dilated higher-dimensional Hamiltonian is usually time-dependent.And the solution of the problem of chronological product is actually one of the key factors to the accuracy of dynamics simulation of the time-dependent PT-symmetric system,and even the most key factor.We prove that the trusted duration of the numerical calculation is actually bounded by the critical time Tc of convergence of Magnus series,while the implemented duration of the experimental running is actually bounded by the critical time Tl of the legitimacy of the dilation method.