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understanding and characterizing general features of the dynamics of open quantum systems is of great importance to physics , chemistry , and biology  @xcite . the non - markovian character is one of the most central aspects of an open quantum process , and attracts increasing attentions  @xcite . markovian dynamics of quantum systems is described by a quantum dynamical semigroup  @xcite , and often taken as an approximation of realistic circumstances with some very strict assumptions . meanwhile , exact master equations , which describe the non - markovian dynamics , are complicated  @xcite . based on the infinitesimal divisibility in terms of quantum dynamical semigroup , wolf _ et al . _ provided a model - independent way to study the non - markovian features  @xcite . later , in the intuitive picture of the backward information flow leading to the increasing of distinguishability in intermediate dynamical maps , breuer , laine , and piilo ( blp ) proposed a measure on the degree of non - markovian behavior based on the monotonicity of the trace distance under quantum channels  @xcite , as shown in fig . [ fig : sketch ] . the blp non - markovianity has been widely studied , and applied in various models  @xcite . ( color online )  sketch of the information flow picture for non - markovianity  @xcite . according to this scenario , the loss of distinguishability of the system s states indicates the information flow from the system to the reservoir . if the dynamics is markovian , the information flow is always outward , represented by the green thick arrow . non - markovian behaviors occurs when there is inward information flow , represented by the orange thin arrow , bringing some distinguishability back to the system.,width=226 ]    unlike for classical stochastic processes , the non - markovian criteria for quantum processes is non - unique , and even controversial . first , the non - markovian criteria from the infinitesimal divisibility and the backward information flow are not equivalent  @xcite . second , several other non - markovianity measures , based on different mechanism like the monotonicity of correlations under local quantum channels , have been introduced  @xcite . third , even in the framework of backward information flow , trace distance is not the unique monotone distance for the distinguishability between quantum states . other monotone distances on the space of density operators can be found in ref . @xcite , and the statistical distance  @xcite is another widely - used one . different distance should not be expected to give the same non - markovian criteria . the inconsistency among various non - markovianity reflects different dynamical properties .    in this paper , we show that the blp non - markovianity can not reveal the infinitesimal non - divisibility of quantum processes caused by the non - unital part of the dynamics . besides non - markovianity , `` non - unitality '' is another important dynamical property , which is the necessity for the increasing of the purity @xmath0 under quantum channels  @xcite and for the creating of quantum discord in two - qubit systems under local quantum channels  @xcite . in the same spirit as blp non - markovianity , we define a measure on the non - unitality . as blp non - markovianity is the most widely used measure on non - markovianity , we also provide a measure on the non - unital non - markovianity , which can be conveniently used as a supplement to the blp measure , when the quantum process is non - unital . we also give an example to demonstrate an extreme case , where the blp non - markovianity vanishes while the quantum process is not infinitesimal divisible . this paper is organized as follows . in sec . [ review ] , we give a brief review on the representation of density operators and quantum channels with hermitian orthonormal operator basis , and various measures on non - markovianity . in sec . [ sec : non - unital - nm ] , we investigate the non - unitality and the non - unital non - markovianity and give the corresponding quantitative measures respectively . in sec . [ sec : example ] , we apply the non - unital non - markovianity measure on a family of quantum processes , which are constructed from the generalized amplitude damping channels . section [ sec : conclusion ] is the conclusion . the states of a quantum system can be described by the density operator @xmath1 , which is positive semidefinite and of trace one . quantum channels , or quantum operations , are completely positive and trace - preserving ( cpt ) maps from density operators to density operators , and can be represented by kraus operators , choi - jamiokowski matrices , or transfer matrices  @xcite .    in this work , we use the hermitian operator basis to express operators and represent quantum channels . let @xmath2 be a complete set of hermitian and orthonormal operators on complex space @xmath3 , i.e. , @xmath4 satisfies @xmath5 and @xmath6 . any operator @xmath7 on @xmath3 can be express by a column vector @xmath8 through @xmath9 with @xmath10 . every @xmath11 is real if @xmath7 is hermitian .    in the meantime , any quantum channel @xmath12 can be represented by @xmath13 $ ] via @xmath14=t(\mathcal{e})r(\rho ) \label{eq:1},\ ] ] where @xmath15 is a @xmath16 real matrix with the elements @xmath17 furthermore , one can easily check that @xmath18 for the composition of quantum channels . here @xmath19 denotes the composite maps @xmath20 .    taking into the normalization of the quantum states , i.e. , @xmath21 , @xmath22 can be fixed as @xmath23 for any density operator @xmath1 by choosing @xmath24 with @xmath25 the identity operator . in such a case , @xmath4 for @xmath26 are traceless and generate the algebra @xmath27 . this real parametrization @xmath28 for density operators is also called as coherent vector , or generalized bloch vector  @xcite . in order to eliminate the degree of freedom for the fixed @xmath22 , we use the decomposition @xmath29 . therefore , any density operator @xmath1 can be expressed as @xmath30 with @xmath31 the generalized bloch vector and @xmath32 represents @xmath33 . under this frame , quantum channels can be represented by the affine map  @xcite @xmath34 where @xmath35 is a real matrix with the dimension @xmath36 and the elements of the vector @xmath37 reads @xmath38_{\mu } = \left\langle\lambda_{\mu},\mathcal{e}(\openone)\right\rangle / d,\ ] ] for @xmath26 . comparing eq . ( [ eq:1 ] ) with eq . ( [ eq:2 ] ) , one could find that @xmath39_{\mu\nu},\ ] ] for @xmath40 . thus , @xmath15 can be decomposed into the following sub - blocks : @xmath41.\ ] ]    reminding that a quantum channel @xmath42 is said to be unital if and only if @xmath43 @xcite , one could find that the necessary and sufficient condition for a unital map is that @xmath44 , namely , @xmath45 thus , @xmath37 describes the non - unital property of the quantum channel @xmath42 . the necessary and sufficient condition above could be easily proved by realizing that the bloch vector of @xmath46 is zero vector , i.e. , @xmath47 . based on the sub - block form of @xmath15 , @xmath44 is equivalent to that @xmath15 is block diagonal , i.e. , @xmath48 .    whether a quantum channel @xmath42 is completely positive ( cp ) can be reflected by the choi - jamiokowski matrix  @xcite @xmath49 where @xmath50 is the maximally entangled state . here @xmath51 is a basis in hilbert space . @xmath42 is cp if and only if the choi - jamiokowski matrix is positive . with the hermitian operator basis , @xmath52 is a @xmath16 matrix and can be written in the form  @xcite @xmath53 substituting this formula into eq . ( [ eq : c ] ) and utilizing eq . ( [ eq : t ] ) , one could express the choi - jamiokowski matrix as @xmath54 if @xmath42 is unital , it can be reduced into @xmath55      without the presence of correlation between the open system and its environment in the initial states , the reduced dynamics for the open system from @xmath56 to any @xmath57 can be expressed as @xmath58,\ ] ] which is a quantum channel . this indicates that @xmath59 is cpt . the unitary operator @xmath60 describes the time evolution of the closed entirety , and @xmath61 is the initial state of the environment . a quantum process @xmath62 is said to be infinitesimal divisible , also called as time - inhomogeneous or time - dependent markovian , if it satisfies the following composition law @xcite @xmath63 for any @xmath64 , where @xmath65 is also completely positive and trace preserving . various measures on the degree of the non - markovian behavior of quantum processes have been proposed and investigated  @xcite . almost all of the measures on the non - markovianity can be classified into three kinds , base on the degree of the violation of the following properties owned by the infinitesimal divisible quantum process :    \(i ) monotonicity of distance @xmath66 under cpt maps . that is @xmath67 for any quantum channel @xmath42 , where @xmath68 is an appropriate monotone distance under cpt maps on the space of density operators  @xcite , including trace distance , bures distance , statistical distance , relative entropy , and fidelity ( although fidelity itself is not a distance , it can be used to construct monotone distances ) and so on . some measures on non - markovianity by increasing of the monotone distance during the mediate dynamical maps @xmath65 have been given and discussed in refs . @xcite . the typical measure of this type , which would be used later in this paper , was first proposed by breuer , laine , and piilo in ref . @xcite , based on the monotonicity of trace distance @xcite @xmath69 where @xmath70 . interpreting the increase of the trace distance during the time evolution as the information flows from the environment back to the system , the definition of the blp non - markovianity is defined by @xmath71 where @xmath72 and @xmath73 for @xmath74 are two evolving states . \(ii ) positivity of the choi - jamiokowski matrix for cpt maps . the choi - jamiokowski matrix @xmath75 if and only if @xmath42 is a quantum channel , namely , @xmath42 is a cpt map . some measures on non - markovianity by the negativity of the choi - jamiokowski matrix for mediate dynamical maps @xmath65have been given and discussed in refs . @xcite .    in this work we would use one of these measures , which was proposed by rivas , huelga and plenio ( rhp ) in ref . they utilize the negativity of the choi - jamiokowski matrix @xmath76 for the mediate dynamical maps with the definition @xmath77 where @xmath78    \(iii ) monotonicity of correlations @xmath79 under local quantum channels . that is @xmath80\leq e(\rho^{ab})$ ] for any local quantum channel @xmath42 , where @xmath79 is an appropriate measure for the correlations in the bipartite states @xmath81 , including entanglement entropy and the mutual information . the corresponding measures on non - markovianity are given and discussed in refs . the non - markovianity measure @xmath82 is available to capture the non - markovian behavior of the unital aspect of the dynamics . but for the non - unital aspect , it is not capable . to show this , we use the hermitian orthonormal operator basis to express states and quantum channels . utilizing eq . ( [ eq : rho ] ) , the trace distance between two states @xmath83 and @xmath84 is given by @xmath85\cdot\bm{\lambda}\big|.\end{aligned}\ ] ] therefore , for the two evolving states , we get @xmath86 \cdot\bm{\lambda}\big| , \label{eq : dis}\ ] ] where @xmath83 , @xmath84 are initial states of the system .    from this equation one can see that the trace distance between any two evolved states is irrelevant to the non - unital part @xmath87 of the time evolution . then if there are two quantum channels , whose affine maps are @xmath88 and @xmath89 , respectively , the characteristic of trace distance between the evolving states from any two initial states can not distinguish these two channels . more importantly , @xmath87 may cause the non - divisibility of the quantum process @xmath90 , and this can not be revealed by @xmath82 .    on the other hand , the non - unital part @xmath87 has its own physical meaning : @xmath91 is necessary for the increasing of the purity @xmath92  @xcite . in other words , @xmath93 besides the non - markovian feature , the non - unitality is another kind of general feature of quantum processes . in analogy to the definition of blp non - markovianity , we defined the following measure on the degree of the non - unitality of a quantum process : @xmath94 } { \mathrm{d}t } \right| \mathrm{d}t,\ ] ] where @xmath95 is the initial state . obviously , @xmath96 vanishes if @xmath97 . since the non - unital aspect of the dynamics , which is not revealed by the trace distance , has its own speciality , we aim to measure the effect of non - unitality on non - markovian behavior . however , a perfect separation of the non - unital aspect from the total non - markovianity may be infeasible . therefore we require a weak version @xmath98 for measuring non - unital non - markovianity to satisfy the following three conditions : ( i ) @xmath99 vanishes if @xmath90 is infinitesimal divisible , ( ii ) @xmath99 vanishes if @xmath90 is unital , ( iii ) @xmath100 should be relevant to @xmath87 . based on these conditions , we introduce the following measure @xmath101 where @xmath102 with @xmath103 is the set of the trajectory states which evolve from the maximally mixed state , and @xmath104,\ ] ] with @xmath68 an appropriate distance which will be discussed below . the first condition is guaranteed if we require that @xmath66 is monotone under any cpt maps , i.e. , @xmath105\leq d(\rho_{1},\rho_{2})$ ] for any quantum channel @xmath42 . for the unital time evolution , the set @xmath106 only contains the maximally mixed state , so the above defined @xmath100 vanishes , and the second condition is satisfied . the third condition excludes the trace distance .    in this paper , we use the bures distance which is defined as @xmath107},\ ] ] where @xmath108 is the uhlmann fidelity  @xcite between @xmath83 and @xmath84 . here bures distance is an appropriate distance for @xmath100 because it obeys the monotonicity under cpt maps  @xcite and is relevant to @xmath87 . as here only the monotonicity of distance is relevant , for simplicity , we can also take the square of the bures distance or just the opposite value of unlmann fidelity as a simple version of monotone `` distance ''  @xcite . quantum relative entropy  @xcite @xmath110 $ ] , or its symmetric version @xmath111 , is another qualified candidate for the distance . noting that when the support of @xmath83 is not within the support of @xmath84 , namely , @xmath112 , @xmath113 will be infinite , so in such cases , quantum relative entropy will bring singularity to the measure of non - markovianity . also , hellinger distance  @xcite is qualified . although all of these distances are monotone under cpt maps , they may have different characteristics in the same dynamics , see ref . @xcite . the difference between non - unital non - markovian measure defined by eq . ( [ eq : definition_nun ] ) and the blp - type measures , including those which use other alternative distances , is the restriction on the pairs of initial states . comparing with the blp - type measures relying on any pair of initial states , the non - unital non - markovianity measure only relies on the pairs consisting of the maximally mixed state and its trajectory states . on one hand , this restriction makes the non - unital non - markovianity measure vanish when the quantum processes are unital , no matter they are markovian or non - markovian ; on the other hand , this restriction reflects that non - unital non - markovianity measure reveals only a part of information concerning the non - markovian behaviors . to illustrate the non - unital non - markovian behavior , we give an example in this section . we use the generalized amplitude damping channel ( gadc ) as a prototype to construct a quantum process . the gadc can be described by @xmath114 with the kraus operators @xmath115 given by  @xcite @xmath116 where @xmath117 and @xmath118 are real parameters . note that for any @xmath119 $ ] and any @xmath120 $ ] , the corresponding @xmath42 is a quantum channel . for a two - level system , the hermitian orthonormal operator basis can be chosen as @xmath121 , where @xmath122 is the vector of pauli matrices . with the decomposition in eq . ( [ eq : rho ] ) , the affine map for the bloch vector is given by @xmath123  @xcite , where @xmath124 the gadc is unital if and only if @xmath125 or @xmath126 . when @xmath126 , @xmath127 , the map is identity . a quantum process can be constructed by making the parameter @xmath117 and @xmath118 to be dependent on time @xmath128 . for simplicity , we take @xmath129 and @xmath130 , where @xmath131 is a constant real number . this is a legitimate quantum process , because @xmath90 is a quantum channel for every @xmath57 , and @xmath132 is the identity map . first , let us consider the @xmath82 for this quantum process . for any two initial states @xmath83 and @xmath84 , we have the trace distance @xmath133 & = \frac{1}{2}\mathrm{tr}\left|m(\mathcal{e}_{t})[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})]\cdot\frac{\bm{\sigma}}{\sqrt{2}}\right|\nonumber \\   & = \frac{1}{\sqrt{2}}\left|m(\mathcal{e}_{t})[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})]\right|,\end{aligned}\ ] ] where @xmath134 is the euclidean length of the vector @xmath31 , and we used the equality @xmath135 for pauli matrices . denoting @xmath136 by @xmath137 , we get @xmath138=\frac{e^{-t/2}}{\sqrt{2}}\sqrt{x^{2}+y^{2}+e^{-t}z^{2}},\label{eq : gadc_tr_distance}\ ] ] which implies @xmath139\leq0 $ ] for every time point @xmath57 and for any real numbers @xmath140 , @xmath141 , and @xmath142 . thus , the blp non - markovianity vanishes , i.e. , @xmath143 , although @xmath90 may be not infinitesimal divisible , which will become clear later . ( color online )  ( a )  evolution of trace distance and bures distance between two evolving states of a two - level system under the variant generalized amplitude damping channel , initially from the maximal mixed states @xmath144 and its trajectory state @xmath145 , respectively . ( b )  the evolution of @xmath146 defined by eq . ( [ eq : gt ] ) , whose integral with respect to time @xmath128 is rhp measure for non - markovianity . in these figures , the parameters are taken as @xmath147 and @xmath148.,width=188 ]    in order to investigate whether @xmath90 is infinitesimal divisible or not , we shall apply @xmath100 in the above model . the trajectory of the maximally mixed state under @xmath90 reads @xmath149 where @xmath150 taking these trajectory states as the initial states , we get the corresponding evolving states : @xmath151 then the fidelity reads @xmath152=\frac{1}{2}(h_{+}+h_{-}),\ ] ] where @xmath153 to compare with the behavior of trace distance , we also get @xmath154=\left|\eta_{t}w_{\tau}\right|/2 $ ] . with the expressions @xmath130 and @xmath129 , it is @xmath155=\frac{e^{-t}}{2}\left|\cos2\omega\tau\right|(1-e^{-\tau}).\label{eq : gadc_nonunital}\ ] ]    in fig . [ fig : db_dtr](a ) , we can see that while the trace distance between the evolving states @xmath156 and @xmath157 monotonously decreases with the time @xmath128 , the bures distance increases during some intermediate time intervals . from eq . ( [ eq : gadc_nonunital ] ) , one can see although @xmath158 $ ] depends on @xmath159 , it does not depend on @xmath160 . actually , from eq . ( [ eq : gadc_tr_distance ] ) one could find that for any two initial states , the trace distance between the evolving states is independent on @xmath160 . in this sense , the blp non - markovianity treats a family of quantum processes , which only differ with @xmath161 , as the same one . meanwhile , @xmath100 reveals the effects of @xmath161 on the infinitesimal non - divisibility and is capable of measuring it .    in order to compare with bhp measure , we also calculate the @xmath146 defined by eq . ( [ eq : gt ] ) . we get @xmath162\ ] ] with @xmath163 the mediate dynamical maps @xmath164 with infinitesimal @xmath165 are not completely positive when @xmath166 . from fig . [ fig : db_dtr](b ) , we can see that the increasing of the bures distance occurs in the regimes where @xmath166 , which coincides with the monotonicity of bures distance under cpt maps . in conclusion , we have shown that the measure for non - markovianity based on trace distance can not reveal the infinitesimal non - divisibility caused by the non - unital part of the dynamics . in order to reflect effects of the non - unitality , we have constructed a measure on the non - unital non - markovianity , and also defined a measure on the non - unitality , in the same spirit as blp non - markovianity measure .    like non - markovianity , the non - unitality is another interesting feature of the quantum dynamics . with the development of quantum technologies , we need novel theoretical approaches for open quantum systems . it is expected that some quantum information methods would help us to understand some generic features of quantum dynamics . we hope this work may draw attention to study more dynamical properties from the informational perspective . this work was supported by nfrpc through grant no . 2012cb921602 , the nsfc through grants no . 11025527 and no . 10935010 and national research foundation and ministry of education , singapore ( grant no . wbs : r-710 - 000 - 008 - 271 ) . m. m. wolf , quantum channels & operations guided tour , http://www-m5.ma.tum.de/ foswiki/ pub/ m5 /allgemeines/ michaelwolf/ qchannellecture.pdf[http://www-m5.ma.tum.de/ foswiki/ pub/ m5 /allgemeines/ michaelwolf/ qchannellecture.pdf ]
trace distance is available to capture the dynamical information of the unital aspect of a quantum process . 
 however , it can not reflect the non - unital part . 
 so , the non - divisibility originated from the non - unital aspect can not be revealed by the corresponding measure based on the trace distance . 
 we provide a measure of non - unital non - markovianity of quantum processes , which is a supplement to breuer - laine - piilo ( blp ) non - markovianity measure . a measure on the degree of the non - unitality 
 is also provided .