Bin Chen and Shao-Ming Fei

1 College of Mathematical Science,Tianjin Normal University,Tianjin 300387,China

2 School of Mathematical Sciences,Capital Normal University,Beijing 100048,China

3 Max-Planck-Institute for Mathematics in the Sciences,D-04103 Leipzig,Germany

Abstract We investigate the average coherence with respect to a complete set of complementary measurements.By using a Wigner–Yanase skew information-based coherence measure introduced in Luo and Sun(2017 Phys.Rev.A 96,022130),we evaluate the average coherence of a state with respect to any complete set of mutually unbiased measurements and general symmetric informationally complete measurements,respectively.We also establish analytically the relations among these average coherences.

Keywords:average coherence,mutually unbiased measurements,general symmetric informationally complete measurements

1.Introduction

Quantum coherence,as one of the most significant quantum resources,has become a hot spot in recent years ever since Baumgratz et al[1]introduced the mathematical framework of quantifying the quantum coherence.Based on this framework,a variety of coherence quantifiers have been proposed,such as the l1norm of coherence,the relative entropy of coherence,the distance-based coherence,the coherence formation and the robustness of coherence[1–7].All these measures are indubitably based on two important concepts in the framework—incoherent states and incoherent operations.In[8],Luo et al established a quantitative link between coherence and quantum uncertainty.By identifying the coherence of a state(with respect to a measurement)as the quantum uncertainty of a measurement(with respect to a state),they introduced a coherence quantifier from an alternative perspective based on quantum uncertainty described by the famous Wigner–Yanase skew information[9].This new measure can be mathematically expressed as

Recently,Luo et al[10]studied the average coherence over any complete set of mutually unbiased bases(MUBs)[11,12],as well as the average coherence over all orthonormal bases in terms of the measurement-based coherence measure.They proved that these two averages are equivalent by direct evaluation.More concretely,letbe a complete set of d+1 MUBs in a d-dimensional Hilbert spacethe average coherence of ρ with respect tois defined as

The average coherence over all orthonormal bases is defined as

Another important quantity is the maximal coherence[8]

where Π is taken over all von Neumann measurements.It is obvious thatandare approximately equal when d is large enough.That is to say,the coherence of a state is almost maximal with respect to all orthonormal bases for high dimensional quantum systems[10].

Besides MUBs,there have been other types of complementary measurements—mutually unbiased measurements(MUMs)[13]and general symmetric informationally complete measurements(general SIC measurements)[14].These special quantum measurements have also many useful applications in quantum information theory.In[15],MUBs have been used to detect the entanglement of twoqudit,multipartite and continuous-variable quantum systems.The efficiency of the separability criteria based on MUBs subjects to the maximum number of MUBs,and can be improved by using MUMs and general SIC measurements instead[16–18].In this paper,we study the average coherence with respect to any complete set of MUMs and general SIC measurements,respectively.We evaluate the average coherence of a state with respect to these special types of quantum measurements.We find that the resulted average coherence is a constant multiple(related to the given measurements)of the maximal coherence as well as the average coherence with respect to all orthonormal bases.

2.Average coherence with respect to MUMs

We first recall some basic notions of MUBs and MUMs.Two orthonormal basesandofare said to be mutually unbiased if

In[13],Kalev and Gour generalize the concept of MUBs to MUMs.Two POVM measurements onb=1,2,are said to be MUMs if

as long as t is properly chosen such that all

Moreover,any complete set of MUMs can be expressed in such form[13].

We now investigate the average coherence with respect to MUMs in terms of the measurement-based coherence measure.Letbe a complete set of MUMs with the parameter κ.Similar to(2),we need to evaluate the following quantity,

whereIα(ρ,X)is the generalized skew information,usually called the Wigner–Yanase–Dyson entropy(WYD entropy)[19],which is given by

where 0<α<1.It is obvious that the WYD entropy Iα(ρ,X)reduces to the skew information when α=1/2.Like skew information,WYD entropy has many applications in quantum information theory,especially in characterizing the quantum uncertainties[20–22].It can be seen that

Next,we calculate the quantityFrom the construction of d+1 MUMs given above,one gets

where in the last equality,we have used the fact that[23].

Nevertheless,it has been proved that[22]

Therefore,we obtain

Combining equations(15),(16)and(19),we have

Here it is interesting that this quantityis tightly related to a measure of quantum uncertainty based on averaging WYD information,which is defined by[22]

where{Hi}is any complete orthogonal set of observables.One can easily seen thatMoreover,these two quantities are equivalent when a complete set of MUBs is taken into account,since κ=1 at this point.

From(14),we have the following conclusion.

Theorem 1.The average coherence of a stateρwith respect to thewith parameterκis given by

Moreover,it can be seen that

which implies that

whend→∞.This means that for high dimensional quantum systems,the‘closeness’of the average coherence with respect to MUMs to the maximal coherence depends heavily on the parameterandgets closer to the maximum coherence of whenincreases.

3.Average coherence with respect to general SIC measurements

In this section,we consider the average coherence of a state with respect to general SIC measurements.A set of d2positive-semidefinite operatorsonis said to be a general SIC measurements,if

where a is the efficiency parameter satisfyingif and only if all Pkare rank one projectors,which gives rise to an SIC-POVM.Like MUBs,the existence of SIC-POVMs in arbitrary dimension d is also an open problem.It has been only proved that there exist SIC-POVMs for a number of low-dimensional cases(see[25]and the references therein).However,there always exist a general SIC measurements for arbitrary d,which can be constructed explicitly[14].Letbe a set of d2?1 Hermitian,traceless operators acting onHd,satisfying Tr(Fk Fl)=δk,l.DefineThen the d2operators

form a general SIC measurements.Here t should be chosen such that Pk≥0,and the parameter a is given by

from the construction.

We now define the average coherence of a state ρ with respect to a general SIC measurementswith the parameter a as follows,

where we have used the fact that[23].

On the other hand,taking into account thatwe have

where the last equality follows from(19).Combining equations(29),(30)and the relation between the parameters t and a(27),we have

Therefore we obtain the following theorem:

Theorem 2.The average coherence with respect to a general SIC measurements with the parameterais given by

When a=1/d2,PGSMreduces to SIC-POVM.Then we have the average coherence of a state ρ with respect to a SICPOVM,

It is interesting to find the relations amongandRemarkably one sees thatThus the average coherence of a state provides an operational link between MUBs and SICPOVMs.This is also the case between MUMs and general SIC measurements,i.e.

where the constant multiple depends on the parameters κ and a.Furthermore,it is obvious thatwhich implies that

whend→∞.That is to say,for high dimensional systems,is much less than the maximal coherence,which is quite different from the case of

As an example,let us consider an arbitrary pure state ρ.Simple calculation shows thatandHence,one can see thatis almost the maximal,whileapproaches to the minimum coherence as d increases,see figure 1.In this sense,andcan be viewed as dual quantities to some extent in high dimensional systems.It is noteworthy that the above discussion is based on the assumption that there exist complete sets of MUBs and SIC-POVMs for arbitrary d.However,these results also apply tosincedue to the range of the parameter a.

Figure 1.The blue solid line is the dashed line is and the dotted–dashed line is

4.Conclusion

In summary,we have studied the average coherence with respect to complementary measurements.By evaluating the average coherence associated with MUMs and general SIC measurements,respectively,we have also established the relations among these quantities and the maximal coherence of quantum states.It has been shown that,for high dimensional systems,the quantitygets closer to the maximal coherence as the parameter inincreases.However,this is not the case for.Even for a SICPOVM,the quantityapproaches to zero when d becomes large.The reasons behind these results are worthy of investigation.One may conjecture that it is related to the number of measurements constituting a POVM.Our results can offer insight into quantum coherence and complementary measurements.It would be also interesting to study the measurement-based coherence measure for other types of measurements,and their relations amongand

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos.11805143 and 11675113,and Beijing Municipal Commission of Education(KZ201810028042).