one of the most striking features of quantum mechanics is that there are fundamental limits to how well we can measure quantities .for instance , an atom can be prepared so that it has either a well localized position or has a low spread in momentum .we can not , however , prepare it so that it has both a well localized position and a low spread in momentum . in terms of measurementthis means that a system can not be prepared so that a measurement of either position or momentum would both have low uncertainty .this is a general feature of quantum measurements and applies to any non - commuting observables that do not share a common eigenvector .this fact is embodied mathematically by the uncertainty principle , the most common form of which is \rangle|.\ ] ] the lower bound on this uncertainty product involves an expectation value and is thus state dependent .this can make it difficult to bound the uncertainty ; in particular it is possible for the lower bound to vanish . for example , the observables and do not commute , however the lower bound on the uncertainty will be zero if our system is prepared in the maximally mixed state , i.e. .these difficulties have led some authors to consider entropic uncertainty principles .these have the advantage of having non - trivial state independent lower bounds . both the standard uncertainty principle and the entropic uncertainty principles are derived under the assumption that the observables are measured on separate but identically prepared systems .they do not apply to the case where both quantities are measured at the same time on a single system . in this paperwe will investigate entropic uncertainty relations for joint measurements of two spin components of a spin-1/2 particle .a joint measurement is a single observation that gives information about two different observables .if two observables do not commute , then we can not jointly measure them using projective measurements alone .instead we must use generalized measurements , which can add extra uncertainty to the measurements .it seems that the price one pays for jointly measuring two observables is that one necessarily increases the uncertainty in the measurement process .this would appear to fit with the general feature of quantum mechanics to limit the amount of information that can be obtained from a system .this leads naturally to the question : how much information can be extracted from a single system concerning two observables ? to address this we must first determine how well we can measure the observables .an important question to ask in relation to joint measurements is how much extra uncertainty we have compared to when we measure each observable separately on identically prepared systems .an uncertainty principle for joint measurements would help in answering this question as it would give bounds on the uncertainty inherent in the process of jointly measuring two observables .the study of how well we can jointly measure two observables is also of interest with regards to assessing the security of quantum cryptography protocols .variance based uncertainty relations , which use the product of the variance as a measure of uncertainty , have been obtained for joint measurements .we shall use , instead , the entropy of the measurement outcomes as a means of quantifying the uncertainty in a joint measurement .one reason for this is that using entropy will lead to state independent lower bounds .in addition to this entropy can be interpreted as a measure of information .two different entropic quantities will be investigated , the sum of the marginal entropies and the joint entropy .lower bounds will be obtained for both quantities .the paper is organised in the following manner .a brief review of joint measurements will be given in section [ jmeasurements ] . in section [ eups ]we will discuss entropic uncertainty principles .a lower bound will be found in section [ jents ] for the joint entropy of a joint measurement of two spin components of a spin 1/2 particle . in section [ margents ] we will find lower bounds on the sum of the marginal entropies for the joint measurement of two spin componets of a spin 1/2 particle .finally we will discuss the results in section [ conc ] .a joint measurement of two observables is a simultaneous measurement of both observables upon the same quantum system . when the observables of interest commute then a joint measurement can be accomplished with standard von neumann or projective quantum measurements .if the observables do not commute , however , then we must describe our measurements in terms of the probability operator measure ( pom ) formalism .a detailed description of this generalized description of measurements can be found in .an important point to note is that any measurement can be described using a pom .conversely , any pom can be realized in terms of a projective measurement on a larger system .a condition that is frequently used in connection with joint measurements is the joint unbiasedness condition .this condition requires that the expectation values for the jointly measured observables are proportional to the expectation values of the observables measured separately .we thus require that and , where and are the jointly measured expectation values and and are constants of proportionality , the values of which are independent of the state to be measured . for examples of the joint unbiasedness condition applied to spin-1/2 systemssee .it is possible to relax this condition , leading to a more general description of joint measurements . in the remainder of this work we shall however consider only joint measurements that satisfy the unbiasedness condition .we consider joint measurements of two spin components for a spin-1/2 particle .let the spin observables that we seek to measure be and .the eigenvalues of these observables are .we will choose to assign the numerical values of to the results spin up and down also for a joint measurement of and .this means that and will vary between one and zero .it can be shown that the values of and will be restricted by the inequality  if a joint measurement scheme allows us to saturate inequality ( [ ineq ] ) , then , for given directions * a * and * b * , this measurement gives the largest possible value of for a given and vice versa .any joint measurement of two components of spin for which inequality ( [ ineq ] ) is saturated is in this sense an optimal joint measurement .a simple scheme for performing an optimal joint measurement of two spin components of a spin-1/2 particle is given in .in this scheme we introduce two new spin axes and , which we measure along .the orientation of the and axes relative to and is shown in figure [ fig1 ] .we will either measure along , with a probability , or along with a probability .if we measure along , then the result ` spin up ' is interpreted as both and being spin up , and the result ` spin down ' along is taken as and both spin down .if instead we measured along then the result ` spin up ' is taken as being spin up and being spin down .if we get the result ` spin down ' along then we take to be spin down and to be spin up .this measurement scheme thus provides simultaneous results for the two observables and .  it can shown that this measurement scheme can be used to implement an optimal joint measurement when we choose , and as  for more details on how to implement an optimal joint measurements of spin see .an alternative method of performing an optimal joint measurement of two spin components is to use quantum cloning .we have taken a joint measurement of two observables to be a single measurement that gives us information about both observables .this approach to joint measurements has been discussed by uffink who has proposed an alternative definition of a joint measurement . in the current formalism , if we want to jointly measure and , then we seek a pom such that and , where and are unsharp measurements of the observables and respectively .it has been argued by uffink that if is not a function of either or , then it should not be interpreted as a true joint measurement of these observables .the justification , however , for how the term joint measurement of and " is commonly used , is that the unsharp marginal observables approximate the two sharp observables .performing the joint measurement thus gives us information about both observables .for instance , one can use joint measurements to estimate the expectation values of two sharp observables .one could still argue , as uffink does , that we are not actually jointly measuring and .if we , however , apply this criterion consistently , then it leads to problems not only with joint measurements but with standard measurements . to see this we note that in any real measurement there will be experimental uncertainty in addition to the intrinsic quantum uncertainty .a measurement of an observable is thus not really measuring the observable .instead we measure some pom that approximates the desired observable .it is still common in physics to call this a measurement of the observable .standard entropic uncertainty principles use the sum of the marginal entropies to quantify the uncertainty in a measurement .for example if we are measuring the observables and , then the total uncertainty would be where is the probability of obtaining the outcome for a measurement of and is the probability of obtaining the outcome for a measurement of .for the case when the observables have non - degenerate outcomes , a strong lower bound on the sum of the entropies was found by maassen and uffink .their result states that for two non - degenerate observables and , with eigenvectors and respectively , we have that in certain situations one can find an entropic uncertainty principle that is stronger than that given in equation ( [ mu ] ) .for example , a stronger uncertainty principle exist for measurements of two spin components of a spin 1/2 particle .other extensions to the work of maassen and uffink have been found .it is important to note that the entropic uncertainty principle of maassen and uffink does not apply to joint measurements of and .rather , it was derived under the assumption that the observables would be measured on separate but identically prepared systems .if we wish to quantify the uncertainty in a joint measurement of two observables and , then we could use the sum of the marginal entropies , or the joint entropy .it is common to use the sum of the marginal entropies as a measure of uncertainty . to facilitate a comparison between our results and the existing literaturewe shall investigate the sum of the marginal entropies .the joint entropy , however , is also of interest for two reasons .firstly it deals directly with the joint probability , .the importance of this is that the existence of the joint probability distribution is a defining characteristic of joint measurements .the second reason why the joint entropy should be considered is that any lower bound on the joint entropy will also act as a lower bound on the sum of the marginal entropies .this can be seen by recalling the fact that entropy is subadditive , which means that  .it is important to note that nontrivial state independent lower bounds can be obtained for both and .these lower bounds will thus give us information about the minimum uncertainty that is inherent in the quantum mechanical measurement .we shall now look at obtaining lower bounds on both the sum of the marginal entropies and the joint entropy for a joint measurement of two observables on a qubit .a qubit is a spin-1/2 particle and thus we will consider a joint measurement of spin along two different directions .let our spin observables be and , where and are both unit vectors .we consider the entropy of an optimal joint measurement of spin , i.e. a measurement that saturates the inequality ( [ ineq ] ) .the reason for this is that optimal joint measurements provide the largest possible values for the sharpnesses and and this thus leads to probability distributions with lower entropy .for example if is the sharpness of a measurement of , which is measured on a state , then the probability of obtaining the results spin up for a measurement of is .it can thus be seen that as becomes closer to 1 , then will decrease .this shows that taking the sharpnesses of the joint measurement to be as large as possible leads to the marginal entropy being a small as possible , for a given state .one can also try to find a similar result for the joint entropy , i.e. try to show that for a given state the joint entropy is minimised by performing an optimal joint measurement .the difficulty with this is that the joint unbiasedness condition only specifies the form of the marginal probability distributions . in principlemany different joint probability distributions could lead to the same marginal distributions .this means that it is difficult to find a general proof that optimal joint measurements minimise the joint entropy .it is possible , however , to prove that optimal joint measurements are best for a restricted class of joint measurements .our first task is to obtain a lower bound on the joint entropy , .let and be the sharpnesses for a joint measurement of and respectively .we use the measurement scheme outlined in and in section [ jmeasurements ] .this provides an optimal joint measurement in which the joint probability distribution is where is the probability of measuring along the axis and where and are the probabilities of obtaining the result for a measurement of spin along and respectively .it can be shown that the joint entropy will be where and are the entropies for measurements of the spin along and respectively . from equation ( [ p ] ), we see that is specified completely by the choice of and .this means that the only part of ( [ jentropy ] ) that depends on the initial state is the convex sum we shall thus focus our attention on finding a state independent lower bound on this quantity .   if we wished to measure each spin component equallysharply then we would take .for this situation a tight lower bound can be obtained for ( [ convex ] ) . from equations ( [ m ] ) and ( [ l ] )it is clear that when .the observables and are thus complementary .the maassen and uffink relation , ( [ mu ] ) , then becomes this lower bound will be saturated when the state is prepared as an eigenstate of either or .if , then it is clear that the , where this lower bound can be saturated .consider now the case when and consequently .we can re - express the convex sum of entropies as \nonumber\\ + ( 1 - 2p)h(l)&\ge & p+(1 - 2p)h(l).\end{aligned}\ ] ] to obtain , our state must be an eigenstate of either or .if we take the state to be an eigenstate of , then .we thus find that the minimum of is , when and .for the case when , we can re - express the sum of the entropies as \nonumber\\ & -&(1 - 2p)h(m).\end{aligned}\ ] ] it can be seen that the minimum of the convex sum of the entropies is , which is obtained when the state is an eigenstate of .collecting these results we find that the lower bound on the joint entropy , for , is where .when , then the equality in this lower bound is obtained when the state is an eigenstate of . when , then the equality in the lower bound is obtained when the state is an eigenstate of .the lower bound is expressed in terms of , which refers to the specific measurement scheme that is described in and in section [ jmeasurements ] . in equation ( [ p ] ) , both and are expressed in terms of , and .it is thus possible to express the lower bound ( [ exactlb ] ) in terms of , and as where is the angle between and .it should be noted that the lower bound is only a function of and .the reason for this is that we are considering optimal joint measurements , which means that the value of is fixed by ( [ ineq ] ) .the lower bound thus depends only on the choice of spin components that we wish to jointly measure and not on the state .the state that achieves the minimum of , and thus saturates the inequality ( [ exactlb ] ) is an eigenstate of for or an eigenstate of for .using equations ( [ m ] ) , ( [ l ] ) and ( [ p ] ) it can be shown that when  thus we have the following results .when , the state that minimises the joint entropy will be a pure state with a bloch vector that is confined to the plane defined by and , where the bloch vector will lie exactly midway between and .for we have that the minimum entropy state will again be a pure state with a bloch vector that is confined to the plane , however , now the bloch vector will be midway between and .when the system is prepared in the minimum entropy state , we find that .it is interesting that the marginal entropies are equal when we minimise the joint entropy , . from a physical perspectivethis observation seems to be a consequence of the fact that we want to measure each qubit observable equally well , i.e. .we will now consider the case of joint measurements with .when , then and the spin observables and are not complementary .to obtain the lower bound we will again separate the analysis into two cases , when and when . for the first case , , we can re - express the convex sum of entropies as +(1 - 2p)h(l) ] . using ( [ mu ] ) and the fact that is non - positive for , we obtain the following lower bound , .the lower bound on the joint entropy will thus be this lower bound is expressed in terms of and , which both refer to one particular implementation of a joint measurement. it would be better to express the lower bound in terms of , , and , which apply to any joint measurement of two spin components . in order for us to achieve this, we must note that . by using equations ( [ ineq ] ) , ( [ m ] ) and ( [ l ] ) it can be shown that the lower bound ( [ lb1 ] ) may thus be re - written as    while we have derived equation ( [ jmlb1 ] ) for the specific example of the measurement scheme outlined in , the result will apply to any optimal joint measurement , i.e. one that satisfies the unbiasedness condition and saturates the inequality ( [ ineq ] ) .when the bound ( [ jmlb1 ] ) can not be saturated .one reason for this is that when we derived ( [ jmlb1 ] ) we made use of the relation ( [ mu ] ) .it was shown that for the case of two spin 1/2 particles , that equation ( [ mu ] ) can only be saturated when the two spin components are orthogonal to each other .a plot of ( [ jmlb1 ] ) is given in figure [ entplot2 ] , where the lower bound is plotted against and .the value of is chosen to be as large as equation ( [ ineq ] ) allows , given the choice of and .an interesting feature of figure [ entplot2 ] is that for any fixed choice of , the maximum value for is obtained when .this result is physically reasonable because corresponds to the two spin components being complementary to each other .the entropic uncertainty principle thus shows that we are most uncertain of the outcome when the two spin components are complementary .the entropic uncertainty principle , given in equation ( [ jmlb1 ] ) , allows us to quantify the notion of complementarity for the case when we jointly measure the two qubit observables .entropic uncertainty principles are often formulated in terms of the sum of the marginal entropies . for this reason it is useful to find a lower bound on the sum of the marginal entropies for the case of joint measurements .the lower bound in equation ( [ jmlb1 ] ) will act as a lower bound on both the joint entropy and the sum of the marginal entropies . in order for this lower bound to be tight with respect to would require that , which will not be true in general .if we wish to find better bounds on the sum of the marginal entropies , then we should work directly with this sum . in this sectionwe shall present two different approaches to bounding the sum of the entropies .the first approach will be to modify an argument presented in for obtaining the minimum of for two spin observables that are measured on separate but identically prepared systems .the second approach will be to use the concavity of entropy to obtain a simple lower bound .it is known that the bound of maassen and uffink , ( [ mu ] ) , can be saturated for complementary observables .this raises the question of whether it can also be saturated for observables that are not complementary .numerical investigations for systems described in a two dimensional hilbert space indicate that this is not the case .this has led to investigations into whether it is possible to improve upon equation ( [ mu ] ) for systems in a two dimensional hilbert space .a simple analytical lower bound , which could be saturated , was found in . for the case of measurements of two spin components , with an angle of between them ,the lower bound of took the form ,\cr 2h_2\left(\frac{1}{2}+\frac{1}{2}\sin(\eta/2)\right ) & \text { if } \eta\in [ \pi-\eta ' , \pi ] , \end{cases}\ ] ] where radians , and where .it can be seen that this lower bound only applies for a limited rangle of angles between the two spin components .outwith this range the minimum was found numerically .it is important to note that ( [ gr ] ) was derived for the case of measurements of and on separate but identically prepared systems .a similar result will now be derived for the case of joint measurements that are equally sharp , i.e. .if we have a qubit system prepared in the state , then the marginal probabilities for obtaining the results spin up or down for and measured jointly are given by  the entropic uncertainty will thus equal , where . in order to minimise the entropy , we must make the terms and as different from each other as possible .it is clear that if and are to be minimised then we must maximise and .this suggests that we should take to be confined to the plane of the bloch sphere that is defined by the vectors and .furthermore we should take the magnitude of to be as large as possible , which implies that .to summarise , we know that the state that minimises must be a pure state , with a bloch vector that is confined to the plane defined by the vectors and .these observations greatly simplify the problem as we have reduced the number of parameters that we need to minimise over from three to just one .we will parameterise the state as , where .the variable is the angle between and .the marginal probabilities will now be and .we shall now look at the situation where we measure and equally well and thus .   when , then the largest value for allowed by equation ( [ ineq ] ) is .numerical investigations show that assumes a global minimum at , for a large range of values for . when approaches then the minimum changes to a maximum .it is straightforward to show that {\theta=\eta/2}=0.\ ] ] evaluating the second derivative of at yields {\theta=\eta/2}=\nonumber\\ & \;&\frac{-\alpha}{\ln(2)}\left[\cos(\eta/2)\ln\left(\frac{1-\alpha\cos(\eta/2)}{1+\alpha\cos(\eta/2)}\right)+\frac{2\alpha\sin^2(\eta/2)}{1-\alpha^2\cos^2(\eta/2)}\right].\end{aligned}\ ] ] it can be seen that the second derivative , ( [ secondder ] ) , is positive at for .the value of is found by setting equation ( [ secondder ] ) equal to zero and solving for . by solving the equationnumerically we find that .we have thus found that when the angle between and is in the range , where , then this lower bound is optimal , in the sense that it can be saturated .for in the range it can be shown that the entropy is minimal when the vector is midway between the vectors and , and thus .this last fact may be established using an argument that is similar to the one that we employed to show that was a minimum for .this leads to the result that when  ] , where radians .an important feature of the lower bounds on and , with , was that they could be saturated .the states that achieves these lower bounds could then be obtained .it was found that for \cup[\pi-\eta']$ ] , where radians , the minimum entropy state was the same for both the joint entropy and the sum of the marginal entropies .furthermore , for the state that minimised the entropy . when was greater than , but less than , the state that minimised was different from that which minimised the joint entropy .numerical investigations found that the minimum entropy state for split into two states , one of which had a bloch vector that was close to , while the other had a bloch vector that was close to .this resulted in , for the states that minimised the sum of the marginal entropies . from a physical perspectivethis behaviour seems to be caused by the fact that as increases , the two spin components become more incompatible .this causes the uncertainty to increase as increases .eventually we reach a point were we can not minimise the sum of the marginal entropies by simultaneously having both and small .instead we must make one of the marginal entropies smaller than the other .it is interesting that this behaviour does not occur for the joint entropy .in fact the state that minimises always results in .a lower bound ( [ jmlb1 ] ) was also obtained on the joint entropy for the case when we measured one observable more sharply than the other .this lower bound , however , can not be saturated .the fact that entropy is subadditive means that the lower bound on the joint entropy will also act as a lower bound on the sum of the marginal entropies .this lower bound was compared to that which was provided by the generalised entropic uncertainty principle of krishna and parthasarathy .it was found that our lower bound was always better than the lower bound obtained using the generalised entropic uncertainty principle .one approach that we have not discussed is using information bell inequalities as a means of bounding the uncertainty of a joint measurement .this approach is similar to that discussed in with regards to using information bell inequalities to bound the sharpness of a joint measurement .unfortunately , this approach does not lead to a tight bound on the uncertainty of a joint measurement .the difficulty lies in the fact that the information bell inequalities are only violated by entangled states .in particular , maximally entangled states give the maximum violation .if we have an entangled state , then the single particle reduced states will be mixed states .the concavity of the entropy , however , means that the entropy is minimised only for pure states .we have not addressed the problem of finding entropic uncertainty principles in higher dimensions .one difficulty with this is that optimal joint measurements are not know for systems with dimension greater than 2 .it is still possible to construct joint measurements for obsevables in higher dimensions , and hence to find uncertainty principles for these measurements .the difficulty with this , however , is that any lower bound on the uncertainty would apply only to one specific measurement .if this measurement is not optimal , then one could in principle perform a different joint measurement that has less uncertainty than the given lower bound .it is important to note is that the entropic uncertainty principles that we have obtained contain uncertainty from two different sources .firstly we have the inherent uncertainty in preparation that is quantified by the standard entropic uncertainty principles ( [ mu ] ) .this uncertainty reflects the fact that we can not prepare a state where we are certain of the outcome of separate measurements of both or .secondly , there is a contribution to the uncertainty in our entropic uncertainty principles caused by the joint measurement .one can thus view the difference between the lower bounds in our entropic uncertainty principles , and the minimum of for separate measurements of and , as giving a coarse measure of the uncertainty that must be added in order to 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Abstract: we investigate the uncertainty associated with a joint quantum measurement of two spin components of a spin-1/2 particle and quantify this in terms of entropy . we consider two entropic quantities : the joint entropy and the sum of the marginal entropies , and obtain lower bounds for each of these quantities . for the case of joint measurements where we measure each spin observable equally well , these lower bounds are tight .


predicting interactions between small molecules and proteins is a key element in the drug discovery process .in particular , several classes of proteins such as g - protein - coupled receptors ( gpcr ) , enzymes and ion channels represent a large fraction of current drug targets and important targets for new drug development  .understanding and predicting the interactions between small molecules and such proteins could therefore help in the discovery of new lead compounds .various approaches have already been developed and have proved very useful to address this _ in silico _ prediction issue  .the classical paradigm is to predict the modulators of a given target , considering each target as a different problem .usual methods are classified into _ ligand - based _ and _ structure - based _ or _ docking _ approaches .ligand - based approaches compare a candidate ligand to the known ligands of the target to make their prediction , typically using machine learning algorithms  whereas structure - based approaches use the 3d - structure of the target to determine how well each candidate binds the target  .ligand - based approaches necessitate to know enough ligands of a given target with respect to the complexity of the ligand / non - ligand separation to produce accurate predictors .if few or no ligands are known for a target , one is compelled to use docking approaches , which in turn necessitate to know the 3d structure of the target and are very time consuming . if for a given target with unavailable 3d structure no ligand is known , none of the classical approaches can apply .this is the case for many gpcr as very few structures have been crystallized so far  and many of these receptors , referred to as _gpcr , have no known ligand .  an interesting way to solve this problem is to cast it in the _ chemogenomics _ framework .chemogenomics aims at mining the _ chemical space _ , which roughly corresponds to the set of all small molecules , for interactions with the _ biological space _ ,i.e. , the set of all proteins , in particular drug targets .a salient feature of the chemogenomics approach is the realization that some classes of molecules can bind `` similar '' proteins , suggesting that the knowledge of some ligands for a target can be helpful to determine ligands for similar targets .besides , this type of method allows for a more rational approach to design drugs since controlling a whole ligand s selectivity profile is crucial to make sure that no side effect occurs and that the compound is compatible with therapeutical usage .recent reviews  list several chemogenomic approaches to predict interactions between compounds and targets  .many of these chemogenomics methods rely on some fixed choice of which targets should be used when learning a predictor for a given target , the most extreme example being the learning of a predictor for a whole family or subfamily of targets  .most of them also need some specific procedure to choose which ligands of the selected targets are used and how they are used .we propose a method that uses existing and well tested machine learning algorithms , casting the interaction prediction problem in a joint ligand - target space .this embeds the sharing level threshold problem in a simple representation choice for which we also propose a systematic approach based on combinations of features of the ligand and features of the target .for the three families of targets of interest , we show that our approach outperforms the state - of - the - art individual svm , and gives good performances even for targets with no known ligand .we formulate the typical _ in silico _ chemogenomics problem as the following learning problem : given a collection of target / molecule pairs known to interact or not , estimate a function that would predict whether any chemical binds to any target . in this section we propose a rigorous and general framework to solve this problems , building on recent developments of kernel methods in bio- and chemoinformatics .much effort in chemoinformatics has been devoted to the more restricted problem of mining the chemical space for interaction with a single target , using a training set of molecules known to interact or not with the target .machine learning approaches , such as artificial neural networks ( ann ) or support vector machines ( svm ) , often provide competitive models for such problems .the simplest linear models start by representing each molecule by a vector representation , before estimating a linear function whose sign ( positive or negative ) is used to predict whether or not the small molecule is a ligand of the target .the weight vector is typically estimated based on its ability to correctly predict the classes of molecules in the training set .the _ in silico _ chemogenomics problem is more general because data involving interactions with different targets are available to train a model which must be able to predict interactions between any molecule and any protein . in order to extend the previous machine learning approaches to this setting , we need to represent a _ pair _ of target and chemicals by a vector , then estimate a linear function whose sign is used to predict whether or not can bind to . as before the vector can be estimated from the training set of interacting and non - interacting pairs , using any linear machine learning algorithm .to summarize , we propose to cast the _ in silico _ chemogenomics problem as a learning problem in the ligand - target space thus making it suitable to any classical linear machine learning approach as soon as a vector representation is chosen for protein / ligand pairs .we propose in the next sections a systematic way to design such a representation .a large literature in chemoinformatics has been devoted to the problem of representing a molecule by a vector , e.g. , using various molecular descriptors  .these descriptors encode several features related to the physico - chemical and structural properties of the molecules , and are widely used to model interactions between the small molecules and a single target using linear models described in the previous section  .similarly , much work in computational biology has been devoted to the construction of descriptors for genes and proteins , in order to represent a given protein by a vector .the descriptors typically capture properties of the sequence or structure of the protein , and can be used to infer models to predict , e.g. , the structural or functional class of a protein .  for our _ in silico_ chemogenomics problem we need to represent each pair of small molecule and protein by a single vector . in order to capture interactions between features of the molecule and of the protein that may be useful predictors for the interaction between and , we propose to consider features for the pair obtained by multiplying a descriptor of with a descriptor of . intuitively , if for example the descriptors are binary indicators of specific structural features in each small molecule and proteins , then the product of two such features indicates that both the small molecule and the target carry specific features , which may be strongly correlated with the fact that they interact .more generally , if a molecule is represented by a vector of descriptors and a target protein by a vector of descriptors , this suggests to represent the pair by the set of all possible products of features of and , i.e. , by the tensor product : remember that the tensor product in is a vector whose -th entry is exactly the product of the -th entry of by the -th entry of . this representation can be used to combine in a principled way any vector representation of small molecules with any vector representation of proteins , for the purpose of _ in silico _ chemogenomics or any other task involving pairs of molecules / protein . a potential issue with this approach , however , is that the size of the vector representation for a pair may be prohibitively large for practical computation and storage .for example , using a vector of molecular descriptors of size for molecules and representing a protein by the vector of counts of all -mers of amino - acids in its sequence ( ) results in more than 400k dimensions for the representation of a pair . in order to circumvent this issuewe now show how kernel methods such as svm can efficiently work in such large spaces .svm is an algorithm to estimate linear binary classifiers from a training set of patterns with known class  .a salient feature of svm , often referred to as the _ kernel trick _ , is its ability to process large- or even infinite - dimensional patterns as soon as the inner product between any two patterns can be efficiently computed .this property is shared by a large number of popular linear algorithms , collectively referred to as _ kernel methods _ , including for example algorithms for regression , clustering or outlier detection  .  in order to apply kernel methods such as svm for _ in silico _chemogenomics , we therefore need to show how to efficiently compute the inner product between the vector representations of two molecule / protein pairs .interestingly , a classical and easy to check property of tensor products allows to write the inner product between two tensor product vectors as a product of inner products : this factorization dramatically reduces the burden of working with tensor products in large dimensions .for example , in our previous example where the dimensions of the small molecule and proteins are vectors of respective dimensions and , the inner product in dimensions between tensor products is simply obtained from ( [ eq : inptp ] ) by computing two inner products , respectively in dimensions and , before taking their product .even more interestingly , this reasoning extends to the case where inner products between vector representations of small molecules and proteins can themselves be efficiently computed with the help of _ positive definite _kernels  , as explained in the next sections .positive definite kernels are linked to inner products by a fundamental result  : the kernel between two points is equivalent to an inner product between the points mapped to a hilbert space uniquely defined by the kernel .now by denoting we obtain the inner product between tensor products by :   in summary , as soon as two kernels and corresponding to two implicit embeddings of the chemical and biological spaces in two hilbert spaces are chosen , we can solve the _ in silico _ chemogenomics problem with an svm ( or any other relevant kernel method ) using the product kernel ( [ eq : product ] ) between pairs .the particular kernels and should ideally encode properties related to the ability of similar molecules to bind similar targets or ligands respectively .we review in the next two sections possible choices for such kernels .recent years have witnessed impressive advances in the use of svm in chemoinformatics  . in particular much workhas focused on the development of kernels for small molecules for the purpose of single - target virtual screening and prediction of pharmacokinetics and toxicity .for example simple inner products between vectors of classical molecular descriptors have been widely investigated , including physicochemical properties of molecules or 2d and 3d fingerprints  .other kernels have been designed directly from the comparison of 2d and 3d structures of molecules , including kernels based on the detection of common substructures in the 2d structures molecules seen as graphs  or on the encoding of various properties of the 3d structure of a molecules  .while any of these kernels could be used to model the similarities of small molecules and be plugged into ( [ eq : product ] ) , we restrict ourselves in our experiment to a particular kernel proposed by  called the _ tanimoto kernel _ , a classical choice that usually gives state - of - the - art performances in molecule classification tasks .it is defined as : where is a binary vector whose bits indicate the presence or absence of all linear path of length or less as subgraph of the 2d structure of .we chose in our experiment , i.e. , characterize the molecules by the occurrences of linear subgraphs of length or less , a value previously observed to give good results in several virtual screening task .we used the freely and publicly available __ chemcpp _ _ software to compute this kernel in the experiments .svm and kernel methods are also widely used in bioinformatics  , and a variety of approaches have been proposed to design kernels between proteins , ranging from kernels based on the amino - acid sequence of a protein  to kernels based on the 3d structures of proteins  or the pattern of occurrences of proteins in multiple sequenced genomes  .these kernels have been used in conjunction with svm or other kernel methods for various tasks related to structural or functional classification of proteins .while any of these kernels can theoretically be used as a target kernel in , we investigate in this paper a restricted list of specific kernels described below , aimed at illustrating the flexibility of our framework and test various hypothesis .* the _ dirac _ kernel between two targets is : this basic kernel simply represents different targets as orthonormal vectors .from ( [ eq : product ] ) we see that orthogonality between two proteins and implies orthogonality between all pairs and for any two small molecules and .this means that a linear classifier for pairs with this kernel decomposes as a set of independent linear classifiers for interactions between molecules and each target protein , which are trained without sharing any information of known ligands between different targets . in other words , using dirac kernel for proteins amounts to performing classical learning independently for each target , which is our baseline approach .* the _ multitask _ kernel between two targets is defined as : this kernel , originally proposed in the context of multitask learning , removes the orthogonality of different proteins to allow sharing of information . as explained in , plugging in amounts to decomposing the linear function used to predict interactions as a sum of a linear function common to all targets and of a linear function specific to each target : a consequence is that only data related to the the target are used to estimate the specific vector , while all data are used to estimate the common vector . in our frameworkthis classifier is therefore the combination of a target - specific part accounting for target - specific properties of the ligands and a global part accounting for general properties of the ligands across the targets .the latter term allows to share information during the learning process , while the former ensures that specificities of the ligands for each target are not lost .* while the multitask kernel provides a basic framework to share information across proteins , it does not allow to weight differently how known interactions with a protein should contribute to predict interactions with a target .empirical observations underlying chemogenomics , on the other hand , suggest that molecules binding a ligand are only likely to bind ligand similar to in terms of structure or evolutionary history . in terms of kernelsthis suggest to plug into  a kernel for proteins that quantifies this notion of similarity between proteins , which can for example be detected by comparing the sequences of proteins . in order to test this approach ,we therefore tested two commonly - used kernels between protein sequences : the mismatch kernel  , which compares proteins in terms of common short sequences of amino acids up to some mismatches , and the local alignment kernel  which measures the similarity between proteins as an alignment score between their primary sequences . in our experiments involving the mismatch kernel, we use the classical choice of -mers with a maximum of mismatch , and for the datasets where some sequences were not available in the database , we added to the kernel ( and normalized at on the diagonal ) in order to keep it valid .* alternatively we propose a new kernel aimed at encoding the similarity of proteins with respect to the ligands they bind . indeed , for most major classes of drug targets such as the ones investigated in this study ( gpcr , enzymes and ion channels ), proteins have been organized into hierarchies that typically describe the precise functions of the proteins within each family .enzymes are labeled with _ enzyme commission numbers _ ( ec numbers ) defined in  , that classify the chemical reaction they catalyze , forming a -level hierarchy encoded into 4 numbers .for example includes oxydoreductases , includes oxidoreductases that act on the aldehyde or oxo group of donors , is a subclass of with or as acceptor and is a subgroup of enzymes catalyzing the oxidation of formate to bicarbonate .these number define a natural and very informative hierarchy on enzymes : one can expect that enzymes that are closer in the hierarchy will tend to have more similar ligands .similarly , gpcrs are grouped into classes based on sequence homology and functional similarity : the _ rhodopsin _ family ( class a ) , the _ secretin _ family ( class b ) , the _ metabotropic _ family ( class c ) and a last class regrouping more diverse receptors ( class d ) .the kegg database  subdivides the large rhodopsin family in three subgroups ( amine receptors , peptide receptors and other receptors ) and adds a second level of classification based on the type of ligands or known subdivisions . for example , the rhodopsin family with amine receptors is subdivided into cholinergic receptors , adrenergic receptors , _etc_. this also defines a natural hierarchy that we could use to compare gpcrs .finally , kegg also provides a classification of ion channels .classification of ion channels is a less simple task since some of them can be classified according to different criterions like voltage dependence or ligand - gating .the classification proposed by kegg includes _ cys - loop superfamily , glutamate - gated cation channels , epithelial and related na+ channels , voltage - gated cation channels , related to voltage - gated cation channels , related to inward rectifier k+ channels , chloride channels _ and _ related to atpase - linked transporters _ and each of these classes is further subdivided according for example to the type of ligands ( _ e.g. _ , glutamate receptor ) or to the type of ion passing through the channel ( _ e.g. _ , na+ channel ) . here again , this hierarchy can be used to define a meaningful similarity in terms of interaction behavior .+ for each of the three target families , we define the hierarchy kernel between two targets of the family as the number of common ancestors in the corresponding hierarchy plus one , that is , where contains as many features as there are nodes in the hierarchy , each being set to if the corresponding node is part of s hierarchy and otherwise , plus one feature constantly set to one that accounts for the `` plus one '' term of the kernel .we extracted compound interaction data from the kegg brite database  concerning enzyme , gpcr and ion channel , three target classes particularly relevant for novel drug development .  for each family ,the database provides a list of known compounds for each target . depending on the target families , various categories of compoundsare defined to indicate the type of interaction between each target and each compound .these are for example _ inhibitor , cofactor _ and _ effector _ for enzyme ligands , _ antagonist _ or _ ( full / partial ) agonist _ for gpcr and _ pore blocker , ( positive / negative ) allosteric modulator , agonist _ or _ antagonist _ for ion channels .the list is not exhaustive for the latter since numerous categories exist .although different types of interactions on a given target might correspond to different binding sites , it is theoretically possible for a non - linear classifier like svm with non - linear kernels to learn classes consisting of several disconnected sets .therefore , for the sake of clarity of our analysis , we do not differentiate between the categories of compounds .we eliminated all compounds for which no molecular descriptor was available ( principally peptide compounds ) , and all the targets for which no compound was known . for each target , we generated as many negative ligand - target pairs as we had known ligands forming positive pairs by combining the target with a ligand randomly chosen among the other target s ligands ( excluding those that were known to interact with the given target ) .this protocol generates false negative data since some ligands could actually interact with the target although they have not been experimentally tested , and our method could benefit from experimentally confirmed negative couples .this resulted in data points for enzymes ( known enzyme - ligand pairs and generated negative points ) representing interactions between enzymes and compounds , training data points for gpcrs representing interactions between receptors and compounds and ion channel data points representing interactions between channels and compounds .besides , figure [ fig : distrib ] shows the distribution of the number of known ligands per target for each dataset and illustrates the fact that for most of them , few compounds are known .        for each target in each family , we carried out two experiments .first , all data points corresponding to other targets in the family were used for training only and the points corresponding to were -folded with .that is , for each fold , an svm classifier was trained on all points involving other targets of the family plus a fraction of the points involving , then the performances of the classifier were tested on the remaining fraction of data points for .this protocol is intended to assess the incidence of using ligands from other targets on the accuracy of the learned classifier for a given target .second , for each target we learned an svm classifier using only interactions that did not involve and tested on the points that involved .this is intended to simulate the behavior of our framework when making predictions for orphan targets , _i.e. _ , for targets for which no ligand is known .for the first protocol , since learning an svm with only one training point does not really make sense and can lead to `` anti - learning '' less than performances , we set all results involving the dirac target kernel on targets with only known ligand to .this is to avoid any artefactual penalization of the dirac approach and make sure we measure the actual improvement brought by sharing information across targets .we first expose the results obtained on the three datasets for the first experiment , assessing how using training points from other targets of the family improves prediction accuracy with respect to individual ( dirac - based ) learning .table [ tab : exp1 ] shows the mean success rate across the family targets for an svm with a product kernel using the tanimoto kernel for ligands and various kernels for proteins .  ) for ion channels with multitask , hierarchy and local alignment kernels . ]  ) for ion channels with multitask , hierarchy and local alignment kernels . ]  ) for ion channels with multitask , hierarchy and local alignment kernels . ]  for the enzymes and ion channels datasets , we observe significant improvements when the multitask kernel is used in place of the dirac kernel , on the one hand , and when the hierarchy kernel replaces the multitask kernel , on the other hand .for example , the dirac kernel only performs at an average accuracy of for the ion channel dataset , while the multitask kernel increases the accuracy to and the hierarchy kernel brings it to . for the enzymes , a global improvement of observed between the dirac and the hierarchy approaches .this clearly demonstrates the benefits of sharing information among known ligands of different targets , on the one hand , and the relevance of incorporating prior information into the kernels , on the other hand .  on the gpcr datasetthough , the multitask kernel performs worse than the dirac kernel , probably because some targets in different subclasses show very different binding behavior which results in adding more noise than information when sharing naively with this kernel .however a more careful handling of the similarities between gpcrs through the hierarchy kernel again results in significant improvement over the dirac kernel ( from to ) , again demonstrating the relevance of the approach .sequence - based target kernels do not achieve the same performance as the hierarchy kernel , although they perform relatively well for the ion channel dataset . in the case of enzymes ,it can be explained by the diversity of the proteins in the family and for the gpcr , by the well known fact that the receptors do not share overall sequence homology  .figure [ fig : kernels ] shows 3 of the tested target kernels for the ion channel dataset .the hierarchy kernel adds some structure information with respect to the multitask kernel , which explains the success rate increase .the local alignment sequence - based kernels fail to precisely re - build this structure but retains some substructures . in the cases of gpcr and enzymes , almost no structure is found by the sequence kernels , which , as alluded to above , was expectable and suggests that more subtle comparison of the sequences would be required to exploit the information they contain .figure [ fig : ratios ] illustrates the influence of the number of training points for a target on the improvement brought by using information from similar targets . as one could expect , the improvement is very strong when few ligands are known and decreases when enough training points become available . after a certain point ( around training points ) , using similar targets can even deteriorates the performances .this suggests that the method could be globally improved by learning for each target independently how much information should be shared , for example through kernel learning approaches  .  *4c target & enzymes & gpcr & channels + dirac & & & + multitask & & & + hierarchy & & & + mismatch & & & + local alignment & & & +        the second experiment aims at pushing this remark to its limit by assessing how each strategy is able to predict ligands for proteins with no known ligand .table [ tab : exp2 ] shows the results in that case .as expected , the classifiers using dirac kernels show random behavior in this case since using a dirac kernel with no data for the target amounts to learning with no training data at all .on the other hand we note that it is still possible to obtain reasonable results using adequate target kernels .in particular , the hierarchy kernel loses only for the ion channel dataset , for the gpcr dataset and compared to the first experiment where known ligands were used , suggesting that if a target with no known compound is placed in the hierarchy through , _e.g. _ in the case of gpcr homology detection with known members of the family using specific gpcr alignment algorithms  or fingerprint analysis  , it is possible to predict some of its ligands almost as accurately as if some of them were already available .  *4c target & enzymes & gpcr & channels + dirac & & & + multitask & & & + hierarchy & & & + mismatch & & & + local alignment & & & +we propose a general method to combine the chemical and the biological space in a principled way and predict interaction between any small molecule and any target , which makes it a vary valuable tool for drug discovery .the method allows to represent systematically a ligand - target couple , including information on the interaction between the ligand and the target .prediction is then performed by any machine learning algorithm ( an svm in our case ) in the joint space , which makes targets with few known ligands benefit from the data points of similar targets , and which allows to make predictions for targets with no known ligand .our information sharing process therefore simply relies on a description choice for the ligands , another one for the targets and on classical machine learning methods : everything is done by casting the problem in a joint space and no explicit procedure to select which part of the information is shared is needed . since it subdivides the representation problem into two subproblems , our approach makes use of previous work on kernels for molecular graphs and kernels for biological targets . for the same reason, it will automatically benefit from future improvements in both fields .this leaves plenty of room to increase the performance .results on experimental ligand datasets show that using target kernels allowing to share information across the targets considerably improve the prediction , especially in the case of targets with few known ligands .the improvement is particularly strong when the target kernel uses prior information on the structure between the targets , _e.g. _ , a hierarchy defined on a target class .although sequence kernels did not give very good results in our experiments , we believe using the target sequence information could be an interesting alternative or complement to the hierarchy kernel . further improvement could come from the use of kernel for structures in the cases where 3d structure information is available ( _ e.g. _ for the enzymes , but not for the gpcr ) .our method also shows good performances even when no ligand at all is known for a given target , which is excellent news since classical ligand based approaches fail to predict ligand for these targets in the one hand , and docking approaches are computationally expensive and not feasible when the target 3d structure is unknown which is the case of gpcr in the other hand .  in future work , it could be interesting to apply this framework to quantitative prediction of binding affinity using regression methods in the joint space .it would also be important to confirm predicted ligands experimentally or at least by docking approaches when the target 3d structure is available .we thank pierre mah for his help with chemcpp and kernels for molecules , and vronique stoven for insightful discussions on the biological aspects of the problem .attwood , t. k. , bradley , p. , flower , d. r. , gaulton , a. , maudling , n. , mitchell , a. l. , moulton , g. , nordle , a. , paine , k. , taylor , p. , uddin , a. , and zygouri , c. 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Abstract: predicting interactions between small molecules and proteins is a crucial ingredient of the drug discovery process . in particular , accurate predictive models are increasingly used to preselect potential lead compounds from large molecule databases , or to screen for side - effects . while classical _ in silico _ approaches focus on predicting interactions with a given specific target , new chemogenomics approaches adopt cross - target views . building on recent developments in the use of kernel methods in bio- and chemoinformatics , we present a systematic framework to screen the chemical space of small molecules for interaction with the biological space of proteins . we show that this framework allows information sharing across the targets , resulting in a dramatic improvement of ligand prediction accuracy for three important classes of drug targets : enzymes , gpcr and ion channels .


recent research has focused on protection or restoration of a quantum state subject to the influence of intrinsic and extrinsic noises .quantum noise is ubiquitous and is typically registered as disentanglement and loss of quantum coherence .several theoretical schemes have been proposed to control decoherence , ranging from dynamic decoupling control and feedback control , to weak measurement and error correction codes , etc .while the dynamic decoupling and weak measurement are implemented to protect the quantum states by active external interventions , the error correction in quantum codes can be achieved through a passive realization of the encoded systems . however , in all realistic applications , the errors in quantum operations utilized in quantum information processing can not be completely eliminated .besides , the intricacy in manipulating coupled multiple - qubit systems has posed a serious challenge in scalable realizations in various quantum information carriers .all these fundamental issues of decoherence and control are still active research subjects .  quantum decoherence can be modeled as a non - unitary process , and the non - unitary dynamics of a quantum open system can be described by a master equation . clearly , a state evolving unitarily in an ideally _ closed _ system can be recovered by one unitary reversal operation .therefore it is natural to ask , _ is it possible to recover an initial quantum state for a non - unitary evolution via a unitary quantum reversal operation ? _  the state of an quantum open system , denoted by the reduced density operator at time , may be written in the form of kraus ( operator sum ) representation : with .if these kraus operators are proportional to a set of unitary operators as , then restoration of the initial quantum state is possible , and such a decomposition is called a _ random unitary _ ( ru ) decomposition with ru - type kraus operators . in a well - defined situation where the environment is measurable , m. gregoratti and r. f. wernerexplicitly provide an error correction scheme which needs only one reversal operation on the system based on the outcome of a measurement performed on the environment .ideally , the success probability is always , and the fidelity of the recovered state is also .this restoration scheme is of interest in many scenarios where a quantum measurement may be performed on the environment under consideration .however , generally it is not clear whether or not the required ru decomposition always exists .in fact , it has been shown that the ru decomposition does not exist in a generic dissipative model . even in casesfor which it _ does _ exist , finding the ru - type kraus operators is often difficult . for the dephasing channel , buscemi _ et al ._ prove that if the dimension of the hilbert space satisfies , then the ru decomposition _ does _ always exist .moreover , strunz and co - workers have provided an explicit example of the ru decomposition for the one - qubit dephasing channel .however , the explicit construction of an ru decomposition for an -qubit system coupled to a general environment has not yet been proposed .the purpose of this paper is to explicitly construct an ru decomposition for an -qubit model in both the individual baths case and the common bath case .we investigate the feasibility of performing environment - assisted error correction for dephasing chanel .since a major limitation of the original scheme is that ru decomposition rarely exists for certain models , we provide an example of how to use non - ru decomposition to fully recover some entangled states , and thus overcome the ru limitation .moreover , in our scheme , the measurement performed on the environment is in the fock basis , the restoration , in many interesting cases , only depends on the parity of total photon numbers ( even or odd ) in the environment .this may enhance the feasibility of experimental realizations of the proposed restoration scheme .this paper is organized as follows : in sec .[ sec : ii ] , we review the general error correction scheme proposed in and developed in . in sec .[ sec : iii ] , we investigate the case of an -qubit model interacting with a common bath .particularly , we explicitly construct the ru decomposition and non - ru decomposition for this -qubit system .when the ru decomposition exists , the initial state may be recovered using the standard restoration scheme .interestingly , our work goes beyond the standard restoration scheme based on the ru decomposition .we show that , for a non - ru decomposition , a quantum operation can entail a full recovery of some entangled states embedded in the phase noise . in sec .[ sec : iv ] , we investigate another type of interaction between system and baths , qubits interacting with individual baths .we first derive the ru - type kraus decomposition for a single - qubit system associated with an explicit measurement basis .then , we extend the restoration scheme to the -qubit model interacting with individual dephasing baths . thus , we show that perfect restoration can always be achieved for a multiple - qubit system for both the common bath case and the individual baths case .the framework of open quantum systems is established from quantum theory for the total system ( system plus environment ) , which is governed by the schrdinger equation .the evolution of the total density matrix is given by , where ( setting ) is the total evolution operator .the evolution for the reduced density matrix of the system can be obtained by tracing over the environmental degrees of freedom  ] .\3 . if the total photon number in is even but not , then where ^{m_{k}}}{m_{k}!} ] , , and .the corresponding measurement operators for this set of kraus operators are , , and , where is the total photon number summed over all modes .they satisfy , , and are all non - negative . performing a measurement on this set of measurement operators , the possible outcomes give the non - ru decomposition as=l_{c , i}\rho_{s}(0)l_{c , i}^{\dagger},\;(i=1,2,3 ) \label{eq:14}\ ] ] with probability  ] ) of the kraus operators is always close to zero . therefore , it is safe to drop the high excitation states and the coefficient matrix becomes finite - dimensional .for example , according to fig .[ fig:2 ] , we can safely choose when , so then the coefficients matrix is .therefore , the equations for form a set of first order linear equations .since all the are known , i.e. the coefficient matrix is fully determined , it can be solved by computer rather quickly .finally , the measurement basis is just the first four measurement basis states ( ) just correspond to ru - type kraus operators , and the probabilities of finding the other measurement outcomes ( ) are zero .although the measurement basis states for ru decomposition can be easily numerically determined by the procedure above , the physical meaning is still unclear since they are superpositions of fock states , and more importantly , it may be difficult to perform such measurements .  in the common bath case, the fock basis does not admit an ru decomposition . to overcome this, we have two options .the first option is to construct an ru decomposition with kraus operators , which will shift the difficulty to finding the measurement operators .unfortunately , such measurement operators are difficult to realize . however, if we can overcome this by cleverly designing an experimental measurement procedure , then the advantage of this first choice is that the restoration scheme is applicable to arbitrary initial states .  on the other hand ,the second option is that we can make a compromise about what we choose for the initial states .if we are only interested in recovering some particular unknown entangled states ( and in most quantum information processing schemes , the initial states are usually the entangled states we discussed ) , then we can go beyond the ru decomposition scheme .the non - ru decomposition may give us a much simpler way of recovering entangled states , and simplify the implementation of the measurements .based on our current knowledge of experimental techniques , we show the whole scheme of the second option , and theoretically prove the feasibility of the first option .then , we leave the problem of designing a clever experiment to perform a measurement on superposition fock states to future investigations .   using the technique of constructing ru kraus decomposition for the two - qubit case, we can try to solve the general -qubit case .therefore , to gain some insight , we will explicitly construct the ru decomposition of the three - qubit case as a special example .first , we will review some properties of the dephasing channel which have been studied in . according to , the -qubit dephasing channel can be expressed as where is the entry - wise product of matrices and ( also called a schur or hadamard product ) .although the dimension of matrix is , the rank is =n+1 , \label{eq:29}\ ] ] i.e. only rows or columns give us useful information .therefore we can always shift the linearly independent rows ( columns ) to the top - left of the whole matrix , in which case it takes the form , , \label{eq:30}\ ] ] where is the linearly independent part ( \!=\ !n+1 ] .if the total photon number in is even , then where ^{m_{k}}}{m_{k}!} ] .the measurement required is just a parity measurement .for example , if an odd photon number is detected , one immediately concludes that the system must have collapsed into the state .since and are each proportional to a unitary matrix , it is easy to see that an inverse operation can fully restore the initial state as shown in ( [ eq:3 ] ) with and .the restoration procedure is similar to the two - qubit case plotted in fig .[ fig:1 ] .it should be noted that also finds another ru decomposition of this model , and they have shown how to derive the measurement operators . in their paper , the measurement basis is time - dependent , while in our scheme , the measurement basis is time - independent .the purpose of using this measurement basis ( parity of fock states ) is to find an explicit experiment realization of the scheme .based on the discussion of the restoration scheme for the single - qubit model , we can extend the scheme to qubits in the case of individual baths .consider the hamiltonian . in the case of individual baths , it is easy to see that the evolution of the total system can be written as a tensor product of all the participating subsystems .the kraus operators for the total can be obtained by employing the tensor products of the kraus operators for each subsystem , i.e. , where ( ) are the kraus operators for the -qubit system in the case of individual baths , and are the kraus operators of the subsystem . as discussed in the last section ,there are two kraus operators for each subsystem , so the index can be or for each subsystem from to .the total number of kraus operators of the -qubit system is , so the index ranges from to .  for simplicity , but without losing generality, we give an explicit example for the special case of , i.e. the two - qubit model . in this case , the two - qubit system can be described by the four kraus operators can be explicitly written as tensor products of the kraus operators of the single qubit case ; , , , and .the measurement basis for each subsystem has already been analyzed in the last section . in the two - qubit case, we need to measure both environment `` 1 '' and `` 2 '' , and each measurement could return one possible outcome , `` odd '' or `` even . 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Abstract: in this paper , we propose an explicit scheme to fully recover a multiple - qubit state subject to a phase damping noise . we establish the theoretical framework and the operational procedure to restore an unknown initial quantum state for an -qubit model interacting with either individual baths or a common bath . we give an explicit construction of the _ random unitary _ ( ru ) kraus decomposition for an -qubit model interacting with a common bath . we also demonstrate how to use only one unitary reversal operation to restore an arbitrary state with phase damping noise . in principle , the initial state can always be recovered with a success probability of . interestingly , we found that non - ru decomposition can also be used to restore some particular entangled states . this may open a new path to restore a quantum state beyond the standard ru scheme .