New construction of mutually unbiased bases for odd-dimensional state space

We study the construction of mutually unbiased bases in Hilbert space for composite dimensions d which are not prime powers.We explore the results for composite dimensions which are true for prime power dimensions.We then provide a method for selecting mutually unbiased vectors from the eigenvectors of generalized Pauli matrices to construct mutually unbiased bases.In particular,we present four mutually unbiased bases in C^(15).

Security of quantum key distribution with virtual mutually unbiased bases

In a perfect quantum key distribution(QKD)protocol,quantum states should be prepared and measured with mutually unbiased bases(MUBs).However,in a practical QKD system,quantum states are generally prepared and measured with imperfect MUBs using imperfect devices,possibly reducing the secret key rate and transmission distance.To analyze the security of a QKD system with imperfect MUBs,we propose virtual MUBs to characterize the quantum channel against collective attack,and analyze the corresponding secret key rate under imperfect state preparation and measurement conditions.More generally,we apply the advantage distillation method for analyzing the security of QKD with imperfect MUBs,where the error tolerance and transmission distance can be sharply improved.Our analysis method can be applied to benchmark and standardize a practical QKD system,elucidating the security analysis of different QKD protocols with imperfect devices.

Experimental investigation of measurement incompatibility of mutually unbiased bases

Incompatible measurements are of fundamental importance to re-vealing the peculiar features of quantum theory,and are also use-ful resources in various quantum information tasks.In this work,we investigate the quantum incompatibility of mutually unbiased bases(MUBs)within the operational framework of quantum resource the-ory,and report an experimental validation via the task of state dis-crimination.In particular,we construct an experimentally friendly witness to detect incompatible MUBs,based on the probability of cor-rectly discriminating quantum states.Furthermore,we prove that the noise robustness of MUBs can be retrieved from violating the above witness.Finally,we experimentally test the incompatibility of MUBs of dimensionality ranging from 2 to 4,and demonstrate that it is more robust to noise,as either the dimensionality of measurements or the number of MUBs increases.Our results may aid the exploration of the essential roles of incompatible measurements in both theoretical and practical applications in quantum information.

Uncertainty relations for quantum coherence with respect to mutually unbiased bases

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corre- sponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases （MUBs）. To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.

Detecting high-dimensional multipartite entanglement via some classes of measurements

Mutually unbiased bases, mutually unbiased measurements and general symmetric informationally complete measure- ments are three related concepts in quantum information theory. We investigate multipartite systems using these notions and present some criteria detecting entanglement of arbitrary high dimensional multi-qudit systems and multipartite sys- tems of subsystems with different dimensions. It is proved that these criteria can detect the k-nonseparability （k is even） of multipartite qudit systems and arbitrary high dimensional multipartite systems of m subsystems with different dimensions. We show that they are more efficient and wider of application range than the previous ones. They provide experimental implementation in detecting entanglement without full quantum state tomography.

Families of Schmidt-number witnesses for high dimensional quantum states

Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. The Schmidt number is a quantity of the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with a Schmidt number less than k.

Geometry of skew information-based quantum coherence

We study the skew information-based coherence of quantum states and derive explicit formulas for Werner states and isotropic states in a set of autotensors of mutually unbiased bases(MUBs).We also give surfaces of skew information-based coherence for Bell-diagonal states and a special class of X states in both computational basis and in MUBs.Moreover,we depict the surfaces of the skew information-based coherence for Bell-diagonal states under various types of local nondissipative quantum channels.The results show similar as well as different features compared with relative entropy of coherence and l1 norm of coherence.

Separability criteria via some classes of measurements

Mutually unbiased bases (MUBs) and symmetric informationally complete (SIC) positive operator-valued measurements (POVMs) are two related topics in quantum information theory. They are generalized to mutually unbiased measurements (MUMs) and general symmetric informationally complete (GSIC) measurements, respectively, that are both not necessarily rank 1. We study the quantum separability problem by using these measurements and present separability criteria for bipartite systems with arbitrary dimensions and multipartite systems of multi-level subsystems. These criteria are proved to be more effective than previous criteria especially when the dimensions of the subsystems are different. Furthermore, full quantum state tomography is not needed when these criteria are implemented in experiment.