We focus here on the uncertainty of an observable $Y$ caused by a precise measurement of $X$. We illustrate the effect by analyzing the general scenario of two successive measurements of spin components $X$ and $Y$. We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of $Y$. We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.

Under the scenario of generalized measurements, it can be questioned how much of quantum uncertainty can be attributed to measuring device, independent of the uncertainty in the measured system. On the course to answer the question, we suggest a new class of entropic uncertainty relation that differentiates quantum uncertainty from device imperfection due to the unsharpness of measurement. In order to quantify the unsharpness, we suggest and analyze the quantity that characterizes the uncertainty in the measuring device, based on Shannon entropy. Using the quantity, we obtain a new lower bound of entropic uncertainty with unsharpness and it has been shown that the relation can also be obtained under the scenario that sharp observables are affected by the white noise and amplitude damping.

We propose a schematic setup of quantum key distribution (QKD) with an improved secret key rate based on high-dimensional quantum states. Two degrees-of-freedom of a single photon, orbital angular momentum modes, and multi-path modes, are used to encode secret key information. Its practical implementation consists of optical elements that are within the reach of current technologies such as a multiport interferometer. We show that the proposed feasible protocol has improved the secret key rate with much sophistication compared to the previous 2-dimensional protocol known as the detector-device-independent QKD.

A Gaussian distribution of a quantum state with continuous spectra is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $\widehat{x}$ and $\widehat{p}$, the quantum entropic uncertainty relation can take a suggestive form, where the standard deviations ${\sigma }_x$ and ${\sigma }_p$ are featured explicitly. From the construction of the entropic uncertainty relation, it follows in a transparent manner that (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modified form, ${\sigma }_x{\sigma }_p\ge \hslash e^{\mathcal{N}}/2$; (ii) the additional factor $\mathcal{N}$ quantifies the quantum non-Gaussianity of the probability distributions of two observables; and (iii) the lower bound of the entropic uncertainty relation for a non-Gaussian continuous-variable (CV) mixed state becomes stronger with purity. The optimality of specific non-Gaussian CV states for the refined uncertainty relation has been investigated and the existence of a new class of CV quantum state is identified.

We propose an enhanced discrimination measurement for tripartite 3-dimensional entangled states in order to improve the discernible number of orthogonal entangled states. The scheme suggests 3-dimensional Bell state measurement by exploiting composite two 3-dimensional state measurement setups. The setup relies on state-of-the-art techniques, a multi-port interferometer and nondestructive photon number measurements that are used for the post-selection of suitable ensembles. With this scheme, the sifted signal rate of measurement-device-independent quantum key distribution using 3-dimensional quantum states is improved by up to a factor of three compared with that of the best existing setup.