We provide a unified and exact framework for the variance-based uncertainty relations. This unified framework not only recovers some well-known previous uncertainty relations, but also fixes the deficiencies of them. Utilizing the unified framework, we can construct the new uncertainty relations in both product and sum form for two and more incompatible observables with any tightness we require. Moreover, one can even construct uncertainty equalities to exactly express the uncertainty relation by the unified framework, and the framework is therefore exact in describing the uncertainty relation. Some applications have been provided to illustrate the importance of this unified and exact framework. Also, we show that the contradiction between uncertainty relation and non-Hermitian operator, i.e., most of uncertainty relations will be violated when applied to non-Hermitian operators, can be fixed by this unified and exact framework. Quantum uncertainty relations 1-3 , expressing the impossibility of the joint sharp preparation of the incompatible observables 4,5 , are the most fundamental differences between quantum and classical mechanics 6-9. The uncertainty relation has been widely used in the quantum information science 10,11 , such as quantum non-cloning theorem 12,13 , quantum cryptography 14-17 , entanglement detection 18-22 , quantum spins squeezing 23-26 , quantum metrology 27-29 , quantum synchronization 30,31 and mixedness detection 32,33. In general, the improvement in uncertainty relations will greatly promote the development of quantum information science 18,28,34-36. The variance-based uncertainty relations for two incompatible observables A and B can be divided into two forms: the product form ΔA 2 ΔB 2 ≥ LB p 2,3,5,37,38 and the sum form ΔA 2 + ΔB 2 ≥ LB s 39-42 , where LB p and LB s represent the lower bounds of the two forms uncertainty relations, and ΔQ 2 is the variance of Q (To make sure that the quantity measuring the uncertainty will be a real number, the variance is taken as 〈(Q − 〈Q〉) † (Q − 〈Q〉) for non-Hermitian operators. Here the 〈Q〉 represents the expected value of Q). The product form uncertainty relation cannot fully capture the concept of the incompatible observables, because it can be trivial; i.e., the lower bound LB p can be null even for incompatible observables 39,40,43,44. This deficiency is referred to as the triviality problem of the product form uncertainty relation. In order to fix the triviality problem, Maccone and Pati deduced a sum form uncertainty relation with a non-zero lower bound for incompatible observables 44 , firstly showing that the triviality problem can be addressed by the sum form uncertainty relation. Thus, the sum form uncertainty relations were considered to be stronger than the product form uncertainty relations, and since then, lots of effort has been made to investigate the uncertainty relation in the sum form 18,39,45-48. However, most of the sum form uncertainty relations depend on the orthogonal state to the state of the system, and thus are diffic...

Abstract:The effects of different quantum feedback types on the estimation precision of the detection efficiency are studied. It is found that the precision can be more effective enhanced by a certain feedback type through comparing these feedbacks and the precision has a positive relation with detection efficiency for the optimal feedback when the system reach the state of dynamic balance. In addition, the bigger the proportion of | ⟩ is the higher the precision is and we will not obtain any information about the parameter to be estimated if | ⟩ is chosen as initial state for the feedback type . PACS number(s):

By constructing a quantum-control-assisted multipartite uncertainty relation, a general and efficient multipartite entanglement structure resolution analyzer is provided for all pure states with arbitrary number of parties in arbitrary finite dimension. The lower bounds of this multipartite uncertainty relation present different spectra for different multipartite entanglement structures, and they thus can be considered as a form of resolution analyzer for the multipartite entanglement structure. The resolution analyzer obtained delineates the detailed structure of multipartite entanglement with a suitable number of incompatible measurements, without a complete prior knowledge of the quantum state.

Non-Markovian dynamics detection is one of the most popular subjects in the quantum information science.In this paper, we construct a linear-entropy-based non-Markovianity witness scheme. The positive definiteness of the Choi state will be broken in the non-Markovian evolution, which can be witnessed by its linear entropy.Thus, the linear entropy of the Choi state can be used to witness the non-Markovian dynamics. The effectiveness of the proposed method is verified by an example of the pure dephasing channel. Also, we show that this method can be extended to the one based on Rényi entropy.

We construct a multi-observable uncertainty equality as well as an inequality based on the sum of standard deviations in the qubit system. The obtained equality indicates that the uncertainty relation can be expressed more accurately, and also can be used to detect the mixedness of the system. Meanwhile, the new uncertainty inequality can provide a tighter lower bound, and the tightness can be maintained at a high level even in an open system. Furthermore, the deficiency of the uncertainty relation, that the lower bound of the product form uncertainty relations can be null even for two incompatible observables, can be completely fixed by the new uncertainty relation.PACS number(s): 03.65.Yz; 03.65.Ud