manipulation of discrete quantum states has long held interest , most recently for application to quantum information processing . in the simplest realizations one deals with a nondegenerate two - state system ( a two - level atom ) and through pulsed resonant coherent excitationproduces a specified superposition of the two states , starting from a single state . following the availability of laser light sources , much attention centered on producing complete excitation , i.e. complete population transfer from the ground state to an excited state .more recently interest has shifted to the production of specific superpositions of the two states , and to producing transitions between superposition states . in the context of quantum information a two - state system serves as a qubit ; more general tasks involve multiple quantum states ( a qunit ) , possibly driven by pulsed nonresonant light .the goal of quantum - state manipulation is to realign the statevector from some given hilbert - space direction at an initial time to some prescribed state at a final time , each of these statevectors being defined by the set of complex - valued probability amplitudes associated with a set of quantum states .expressed in matrix form , the goal is to obtain a propagator matrix that transforms the vector of components ,   as noted in the following section , there exists an extensive literature describing analytic solutions to the two - state system for a variety of pulsed - excitation scenarios .there exist also numerous analytic solutions to multistate systems .  herewe consider , as a particular generalization of the two - state system , a two - level system involving two sets of degenerate sublevels : a less energetic set of states with common energy ( the _ lower _ set ) , and a more energetic set of states that share energy ( the _ upper _ set ) .thus we consider a hilbert space of dimension . for definitenesswe here assume that .we assume that interactions produced by laser fields can induce direct transitions between lower sublevels and upper sublevels , but not directly within either manifold of states ( i.e. we allow electric - dipole transitions only ) .  because the statevector has unit length at all times , any such change can be regarded as a rotation , and any allowable motion in several dimensions can be decomposed into a succession of two - dimensional rotations , analogous to the three euler angles that define an arbitrary rotation in euclidean space .however , length - preserving changes in multi - dimensional space can also be produced by a succession of reflections . when expressed in matrix formthese are generalizations of the householder reflections used in matrix calculus .as has been shown previously , such unitary operations can be implemented very efficiently using coherent excitation techniques ; as subsequently noted , the use of quantum householder reflections ( qhr ) permits efficient quantum - state engineering of transitions between arbitrary superpositions .the qhr implementation proposed earlier requires a particular multistate linkage pattern , in which a set of low - lying degenerate states all link , via radiative interaction , with a _ single _ upper state a generalization of the tripod linkage termed an _ n - pod_.such a linkage pattern occurs with a lower level having angular momentum ( three sublevels ) excited to an upper level having ( a single sublevel ) but the pattern is difficult to realize for more than 3 lower states .it is therefore desirable to extend the qhr technique to more general linkage patterns .the present manuscript describes a procedure that allows implementation of the qhr when there are multiple states in the upper set .such an extension makes possible the application of qhr to arbitrary angular momentum states , as occur with free atoms and molecules . as we show , the propagator for such situations is not a single qhr , but a product of qhrs , moreover with orthogonal vectors .the expression for the resulting propagator has a clear geometric interpretation as the effect of a succession of reflections , i.e. _ coupled mirrors_.  the key to this extension of the qhr is a transformation of the underlying basis states , the so - called morris - shore ( ms ) transformation .this replaces the original system , with its multiple linkages between states , by a set of independent nondegenerate two - state systems , thereby allowing us to utilize the considerable literature of analytic solutions to two - state systems and produce analytic solutions for degenerate multistate systems .in this paper we present the solution of the degenerate two - level problem in a simple closed form involving sums of projectors of ms states .these expressions are useful for deriving analytical solutions , generalizations of known two - state solutions , for systems having degenerate levels . for certain conditions , wherein the transition probabilities between states from different sets vanish , the propagator for states of the lower set is given by a product of qhrs with orthogonal vectors , each vector being a bright state ; a similar property applies to the upper set .we discuss properties of this novel geometric object , the coupled mirrors . to illustrate the procedure we develop a useful explicit analytic formalism for two upper states and present some examples .we consider the controlled alteration of a multistate system , expressed as a redirection of the statevector , induced by a set of laser pulses .the simplest example of coherent excitation occurs when there are just two nondegenerate states , indexed 1 and 2 .the time - dependent schrdinger equation prescribes the changes in hilbert space as ( here and henceforth we set , thereby making no distinction between energy and frequency units ) the column vector has the complex - valued probability amplitudes as elements , ^t ] , .\label{v}\ ] ]   a significant property of the linkage pattern shown in fig .[ fig - link ] is that a transformation of hilbert - space coordinates the morris - shore ( ms ) transformation reduces the dynamics to that of a set of independent two - state systems , together with decoupled states . when , as is the case with fig .[ fig - link ] , the additional states are a part of the lower - level manifold .they have no connection with excited states , and hence they can not produce excitation followed by fluorescence ; they are termed _ dark states _ , by contrast to the states that can produce excitation ( and thence fluorescence ) , the _bright states_. the number of bright states is the lesser of and , in this case .figure [ fig - ms ] shows the new linkage pattern in the ms basis.   transformed by the morris - shore transformation into a set of independent nondegenerate two - state systems , with couplings , and a set of decoupled ( dark ) states ., width=302 ]  because , by assumption , all elements of the interaction matrix share a common time dependence , the required ms transformation is achieved by a constant unitary transformation .  \label{w}\]]here is a unitary -dimensional square matrix ( ) which transforms the lower set of states , and is a unitary -dimensional square matrix ( ) which transforms the upper set .this transformation casts the dynamics into the ms basis , with new ms bright lower states , dark lower states , and upper states .the transformed ms hamiltonian has the form ,  \label{h - ms}\]]where  because the matrices and must be diagonal ( possibly after removing null rows and rearanging the basis states ) , the matrices and are defined by the conditions that they diagonalize and , respectively . has generally nonzero eigenvalues ( ) .the -dimensional matrix has the same eigenvalues as and additional zero eigenvalues . from the vector form ( [ v ] ) of obtain   .\label{v+v}\end{aligned}\ ] ]   is the gram matrix for the set of vectors , ( ) . thus if all these vectors are linearly independent then and all eigenvalues of are nonzero ; however , this assumption of independence is unnecessary .the ms hamiltonian ( [ h - ms ] ) has the explicit form .\label{h - ms explicit}\ ] ] the structure of shows that in the ms basis the dynamics decomposes into sets of decoupled single states and independent two - state systems ( ) , each composed of a lower state and an upper state , and driven by the two - state rwa hamiltonians , \quad ( n=1,2,\ldots , \nb ) .\label{hn}\ ] ] each of these two - state hamiltonians has the same detuning , but they differ in the couplings .each of the new lower states is an eigenstate of that corresponds to a specific eigenvalue of .similarly the new upper state is the eigenstate of , corresponding to the same eigenvalue .the square root of this common eigenvalue , , represents the coupling ( half the rabi frequency ) in the respective independent ms two - state system .the zero eigenvalues of correspond to decoupled ( dark ) states in the lower set ( we assume thoughout that ; therefore , dark states , if any , are in the lower set ) .the dark states are decoupled from the dynamical evolution because they are driven by one - dimensional null hamiltonians .the eigenvectors of and form the transformation matrices and , , \quad \b=\left [  \begin{array}{c } \langle \msb_1| \\  \vdots \\\langle \msb_{\nb}|% \end{array}\right ] .  \label{a b}\ ] ] they obey the completeness relations  [ completeness ]   the propagator for the independent ms two - state system ( ) is unitary and therefore can be parameterized in terms of the complex - valued cayley - klein parameters and , where and . in what followswe express the desired control of statevector motion in terms of constraints on these two - state cayley - klein parameters .the unimportant phase factor originates from the chosen representation of the hamiltonian ( [ h ] ) , which facilitates the application of the ms transformation . in the interaction representation , where the diagonal elements are nullified and the detunings appear in phase factors multiplying the couplings , the factor disappears .( note : this phase factor has been omitted in eqs .( 17 ) , ( 18 ) and ( 20 ) of ref . , where it should have been associated with and ; however , it did not affect any result there . )  by taking into account the ms propagators ( [ un ] ) for the two - state ms systems , the ordering of the states ( [ a b ] ) , and the ms hamiltonian ( [ h - ms explicit ] ) , we write the propagator of the full system in the ms basis as .\label{u - ms}\ ] ]  it is straightforward to show that the propagator in the original basis reads  [ u full ] ,  \label{u}\ ] ] where   the propagator connects states within the lower set , connects states within the upper set , and and mix states from the lower and upper sets . by using the completeness relations ( [ completeness a ] ) and ( [ completeness b ] ) we find  .\end{aligned}\ ] ]hence the propagator does not depend on the decoupled states  .this has to be expected because , owing to their degeneracy , the choice of the decoupled states is not unique : any superposition of them is also a zero - eigenvalue eigenstate of . because the dynamics in the original basis must not depend on such a leeway of choice, the full propagator must not depend on the decoupled states at all , and this is indeed the case .   equation ( [ u ] ) expresses the dynamics of the degenerate two - level system in terms of the dynamics of the independent non - degenerate two - state systems , each with the same detuning and pulse shape of the couplings but with different coupling strengths .therefore , eq . ( [ u ] )allows one to generalize any analytical non - degenerate two - state solution to a pair of degenerate levels .recently , such a generalization of the landau - zener model to two degenerate levels has been presented .this generalization displays several interesting properties , for instance , not all transition probabilities between degenerate states are defined for infinite time duration .other analytical solutions involving two degenerate levels have been derived for five chainwise - coupled states in or linkage configurations .we present below another interesting aspect of the solution ( [ u full ] ) : its geometrical nature .of particular significance is the special case when the cayley - klein parameters are all equal to zero , all transition probabilities in the ms basis , as well as these in the original basis from the lower set to the upper set , vanish , i.e. . equation ( [ beta=0 ] ) implies that , or all ms two - state systems , where are arbitrary phases .  after substituting eqs .( [ beta=0 ] ) and ( [ alpha_n ] ) in eqs .( [ u full ] ) , the propagator in the original basis reads  [ u - qhr ] ,  \\ \u_\na & = & \i + \sum_{n=1}^{\nb}(e^{i\phi_{n}}-1)|\msa_n\rangle \langle \msa_n|,\\  \label{ua } \u_\nb & = & e^{-i\delta } \left [ \i + \sum_{n=1}^{\nb}(e^{-i\phi_{n}}-1)|\msb_n\rangle \langle \msb_n| \right ] ,  \label{ub}\end{aligned}\ ] ]  where the propagator operates in the lower set of states and acts in the upper set .taking into account that the coupled lower states are orthonormal basis vectors , i.e. , as are the upper ms states  , we rewrite and as the products    where individual matrices are , in fact , generalizations of the matrices used to produce householder reflection in matrix computations .we refer to them as _ generalized quantum householder reflection _ ( qhr ) operators .the generalized qhr operator ( [ householder ] ) is unitary , , and its determinant has unit magnitude , . for qhr operator reduces to the identity , , while for , the qhr operator reduces to the standard reflection ,   the orthogonality of the qhr vectors in eq .( [ ua - m ] ) automatically ensures the commutation of the qhrs , = 0.\]]therefore their ordering in the product in eq .( [ ua - m ] ) is unimportant .the same argument applies to eq .( [ ub - m ] ) .the importance of qhr derives from the fact that any -dimensional unitary matrix can be decomposed into a set of at most generalized qhrs .the resulting ease with which pulse sequences can be designed to realize the qhr therefore enables one to synthesize any desired unitary transformation of a qunit state , for example a quantum fourier transform or transition between any two pure or mixed qunit states .   in the special case when the interaction vectors are _ orthogonal _ , , the ms eigenvalues andthe ms states simplify greatly .then the matrix of eq .( [ v+v ] ) becomes diagonal , and hence its eigenvalues are ( ) .moreover , the eigenstates of  the ms states in the upper set coincide with the original states , the coupled ms states in the lower set the eigenstates of  are readily found from eq .( [ vv+ ] ) , the propagator in the lower set is a product of qhrs , with the normalized interaction vectors serving as qhr vectors , whereas the propagator in the upper set is a phase gate ,    the advantage of having orthogonal interaction vectors is that they serve as qhr vectors .the implication is that in order to construct a pre - selected coupled - qhr transformation ( [ ua ortho ] ) , the required couplings are directly obtained from eq .( [ ms states orthogonal ] ) . otherwise , in the general case of non - orthogonal interaction vectors , a set of pre - selected qhr vectors ( ) , defined as the eigenvectors of , demand numerical derivation of the required couplings from eq .( [ vv+ ] ) .the condition ( [ alpha_n ] ) can be realized with the rosen - zener model , which assumes constant detuning and hyperbolic - secant time dependence for the couplings , with pulse duration ,    the independent ms two - state systems share the same detuning and pulse shape , but have different ms couplings . for the rosen - zener modelthe cayley - klein parameters ( ) read     where is euler s -function . using the reflection formula , we find follows that the condition is satisfied when  .the phase of depends on the detuning , but not on the corresponding coupling , and for an arbitrary integer we find .\ ] ] hence the qhr phase can be produced by an appropriate choice of the detuning .this result shows that even though the couplings for the ms two - state systems are not the same , the phases of the cayley - klein parameters coincide , .this feature is unique for the sech pulse . for other non - resonant pulses ,e.g. gaussian , the phase would depend also on the coupling and therefore will be generally different for each ms two - state system .for exact resonance ( ) , the cayley - klein parameter reads , where the pulse area is . when the pulse area is ( ) , the phase is equal to ; hence we obtain a physical realization for the standard qhr ( [ stanqhr ] ) .this result is not resticted to the sech pulse ( [ sech ] ) but it is valid for any pulse shape with such an area .when the pulse areas are multiples of , the phases vanish , , and the corresponding qhrs reduce to the identity .resonant pulses therefore do not produce variable qhr phases , which can be used as free parameters .far - off - resonant pulses provide the opportunity for easy adjustment of the phase , albeit only approximately .then the condition ( [ alpha_n ] ) is fulfilled automatically because of the smallness of the transition probabilities in each ms system . specifically ,if the common detuning exceeds sufficiently much the largest ms coupling , then all transition probabilities will be negligibly small , . by adiabatic elimination of each upper ms state one finds for a sech pulse the integral is equal to , and for a gaussian to .because each ms coupling is generally different , each phase will also be different .any desired phase , or a set of such phases , can easily be produced by choosing the original couplings , and hence the ms couplings , appropriately .above , we described the dynamics of the degenerate two - level quantum system in the general case when the lower and upper levels had arbitrary degeneracies , and respectively . in this section , we will illustrate these results with a specific example : when the upper set consists of just two degenerate states , i.e. .this case is insteresting because of the possible implementations in different real physical systems , several examples of which will be presented below . moreover, this special case allows for an elegant analytical treatment .we retain the notation for the lower states  , and we denote the two upper states and .the interaction matrix ( [ v ] ) reads  \equiv \left [ \left\vert v^{\prime } \right\rangle , \left\vert v^{\prime \prime } \right\rangle \right ] ,  \label{v2}\ ] ] where and are -dimensional interaction vectors comprising the couplings between the lower states and the corresponding upper state .the product reads .\ ] ]  with the introduction of parameters and through the definitions    we write the eigenvalues and the associated eigenvectors ( ) within the upper set as   and within the lower set of states .they are the eigenstates of the -dimensional matrix ,    which correspond to the ( nonzero ) eigenvalues ( [ eigenvalues ] ) .we construct them as superpositions of the interaction vectors and , and find after simple algebra  [ psi_a ]    after we have found the explicit form of the ms states ( [ psi_b ] ) and ( [ psi_a ] ) , we obtain the exact form of the propagators and ,  ,\end{aligned}\ ] ]  with and being the phases of the cayley - klein parameters for the ms two - state propagators ( [ un ] ) .it is easy to verify that the bright states ( [ a- ] ) and ( [ a+ ] ) are eigenstates of the propagator with eigenvalues and , respectively .physically this means that if the qunit starts in one of these states it will end up in this same state , acquiring only a phase factor .this occurs because of the conditions ( [ beta=0 ] ) and the independence of the different ms two - state systems .if the qhr phases are equal , , then any superposition of and is also an eigenvector of .the other eigenstates of the propagator are all degenerate , with unit eigenvalue , and they are orthogonal to and . for a qutrit ( )there is only one such eigenvector ( up to an unimportant global phase factor ) and it is proportional to . for higher - dimensional qunits ,any vector in a hyperplane orthogonal to and is an eigenvector of .  in the special casewhen the vectors and are orthogonal , , the expressions simplify .then and , and hence    then the propagator in the lower set is a product of qhrs , in which the interaction vectors and serve as qhr vectors , while the propagator in the upper set is a phase gate ,     following are examples of linkage patterns amongst angular - momentum states , which allow application of the qhr theory .   , the lower set , shown coupled by arbitrary polarization of electric - dipole radiation to the two sublevels of , the upper set .the states are labeled by their magnetic quantum number ., width=264 ]  figure [ fig - j3h1h ] shows linkage patterns possible with arbitrary polarization between the four magnetic sublevels of , the lower set , and the two of , the upper set .the interaction matrix has the elements ( with the clebsch - gordan coefficients included ) , where the subscripts , and 0 refer to right circular ( ) , left circular ( ) and linear ( ) polarizations .level coherently coupled to a level by three polarized ( , and ) pulsed laser fields , with sech shape , and detuning .then , , , , , .the arrows on the right indicate the values derived by the qhr theory ., width=302 ]  figure [ fig - populations ] shows an example of time evolution of the populations of the magnetic sublevels in the level , starting with all population in state .the conditions are realized approximately , by using large detuning from the upper level . in the end of the interaction ,the numerically calculated populations are seen to approach the values predicted by the qhr theory ( the arrows on the right ) .   the linkage pattern for electric - dipole couplings between sublevels of , the lower set , and , the upper set will , for polarization expressed as a combination of left- and right - circular polarization , appear as two uncoupled systems : three states form a -linkage , while five form an m - linkage , as shown in fig .[ fig - j2j1 ] .the interaction matrix is then $ ] and .   configuration ., width=302 ]   the ms transformation can be applied not only to a pair of degenerate levels but also to a ladder of degenerate levels , as long as there is only a single detuning .figure [ fig - j0j1j0 ] illustrates an example in an angular momentum basis between the magnetic sublevels of three levels with angular momenta .the interaction matrix has the form and the formalism of this section applies . in this caseall fields can be changed independently .have the same detuning from the three intermediate sublevels of .the various linkages are invoked by adjusting the direction of the polarization with respect to the quantization axis . the sublevels with formthe lower set , while those with form the upper set.,width=207 ]we have here extended the earlier work on qhr in hilbert space to allow more general linkage patterns between the quantum states , with particular attention to degenerate sublevels that occur with angular momentum states .the extension relies on the use of the morris - shore transformation to reduce the original multi - linkage hamiltonian to a set of independent two - state systems , thereby allowing the utilization of various known two - state analytic solutions . such solutions , when expressed in terms of cayley - klein parameters , readily lead to conditions upon the pulse areas and the time - varying detunings of the excitation pulses .we have found that the propagator within the lower ( or upper ) set of degenerate states represents coupled quantum - state mirrors .the realization of these coupled qhrs within the lower ( or upper ) set of states requires certain conditions on the interaction parameters , specifically that all transition probabiities between the lower and upper sets must vanish .we have proposed three physical realizations of this condition : with resonant , near - resonant ( hyperbolic - secant ) , and far - off - resonant pulses .resonant and near - resonant pulses provide exactly zero transition probabilities , but offer less flexibility in the qhr phases ; moreover they require a carefully chosen pulse area in each ms two - state system , which leads to a number of conditions on the interaction parameters .far - off - resonant pulses fulfill the zero - probability conditions only approximately , but offer much more flexibility. then the only restriction is for sufficiently large detuning , without specific constraints on pulse areas , because the zero - probability conditions are fulfilled simultaneously in all ms two - state systems .  for angular - momentum states ,there are six independent interaction parameters : three polarization amplitudes , two relative phases between different polarizations , and the common detuning .therefore the constructed qhr has six free parameters ( with the far - off - resonance realization ) . for a more general linkage, the number of independent parameters can be much larger .in a subsequent paper , we shall describe the mathematical aspects of this largely unknown operator of coupled reflections , which however , as we have shown here , arises naturally in quantum systems .in particular , we shall show how one can factorize , and therefore synthesize , an arbitrary u( ) propagator by such objects .the procedure is more efficient than a set of rotations that would produce the same transformation .we conclude by pointing out that the confinement of the statevector evolution to the lower set of states , and the availability of simple and efficient tools for its engineering , such as coupled qhrs , can be an essential ingredient for decoherence - free quantum computing .quantum information processing , and the needed manipulation of statevectors in hilbert space , has long interested sir peter knight , in whose honor the present issue of journal of modern optics has been assembled .we are pleased to offer the present article in his honor .   99 m.a .nielsen and i.l .chuang , _ quantum computation and quantum information _( cambridge university press , cambridge , 2000 ) ; d. bouwmeester , a. ekert and a. zeilinger , _ the physics of quantum information : quantum cryptography , quantum teleportation , quantum computation _( springer - verlag , berlin , 2000 ) ; s. bose , p.l .knight , m. murao , m.b .plenio and v. vedral , phil .lond . a * 356 * 1823 ( 1998 ) .householder , j. acm * 5 * 339 ( 1958 ) ; j.h .wilkinson , comput .j. * 3 * 23 ( 1960 ) ; j.h .wilkinson , numer .* 4 * 354 ( 1962 ) ; j.m .ortega , numer . math . * 5 * 211 ( 1963 ) ; d.j .mueller , numer . math . * 8 * 72 ( 1966 ) .z. kis , a. karpati , b.w .shore and n.v .vitanov , phys .a * 70 * 053405 ( 2004 ) ; z. kis , n.v .vitanov , a. karpati , c. barthel and k. bergmann , phys .a * 72 * 033403 ( 2005 ) ; a.a .rangelov , n.v .vitanov and b.w .shore , phys .a * 74 * 053402 ( 2006 ) .diels and w. rudolph , _ ultrashort laser pulse phenomena : fundamentals , techniques , and applications on a femtosecond time scale _ ( san diego : academic , 1996 ) ; m. wollenhaupt , v. engel and t. baumert , annu .phys . chem . *56 * 25(2005 ) ; t. brixner , t. pfeifer , g. gerber , m. wollenhaupt and t. baumert , in _ femtosecond laser spectroscopy _ , edited by hannaford p ( new york : springer , 2005 ) ,. 9 .ackerhalt , j.h .eberly and b.w .shore , phys .a * 19 * 248 ( 1979 ) ; p.m. radmore and p.l .knight , j. phys .b * 15 * 561 ( 1982 ) ; c.e .carroll and f.t .hioe , phys . rev .a * 36 * 724 ( 1987 ) ; c.e .carroll and f.t .hioe , j . math. phys . * 29 * 487 ( 1988 ) .eberly , b.w .shore , z. bialynicka - birula and i. bialynicki - birula , phys .rev . a * 16 * 2038 ( 1977 ) ; z. bialynicka - birula , i. bialynicki - birula , j.h .eberly and b.w .shore , phys .rev . a * 16 * 2048 ( 1977 ) ; b.w .shore and j.h .eberly , opt .* 24 * 83 ( 1978 ) ; r.j .cook and b.w .shore , phys .a * 20 * 539 ( 1979 ) .

Abstract: manipulation of superpositions of discrete quantum states has a mathematical counterpart in the motion of a unit - length statevector in an -dimensional hilbert space . any such statevector motion can be regarded as a succession of two - dimensional rotations . but the desired statevector change can also be treated as a succession of reflections , the generalization of householder transformations . in multidimensional hilbert space such reflection sequences offer more efficient procedures for statevector manipulation than do sequences of rotations . we here show how such reflections can be designed for a system with two degenerate levels a generalization of the traditional two - state atom that allows the construction of propagators for angular momentum states . we use the morris - shore transformation to express the propagator in terms of morris - shore basis states and cayley - klein parameters , which allows us to connect properties of laser pulses to hilbert - space motion . under suitable conditions on the couplings and the common detuning , the propagators within each set of degenerate states represent products of generalized householder reflections , with orthogonal vectors . we propose physical realizations of this novel geometrical object with resonant , near - resonant and far - off - resonant laser pulses . we give several examples of implementations in real atoms or molecules .


let be a finite - dimensional linear vector space with the dual space as the space of all linear function on .we assume is a proper , -strongly convex ( for strongly convex case and for convex case ) , and lower semicontinuous function satisfying f(x)-f(y ) _ *l _ x - y^  x , y v , where denotes the gradient of at for  ] ) , or nonsmooth with bounded variation of subgradients ( ) .hence the objective is -strongly convex with .we assume that the first - order black - box oracle of the objective is available .   over the past few decades , due to the dramatic increase in the size of data for many applications , first - order methodshave been received much attention thanks to their simple structures and low memory requirements .the efficiency of first - order methods can be poor ( a large number of function values and subgradients is needed ) for solving the general convex problems if the structure of the problem is not available . as a result , to develop practically appealing schemes ,it is necessary to make an additional restriction on problem classes . in particular, developing efficient methods for solving large - scale convex optimization problems is possible if the underlying objective has a suitable structure and the domain is simple enough .convexity and level of smoothness are two important factors playing key roles in construction of efficient schemes for such structured optimization problems .let be an optimizer of ( [ e.gfun ] ) and be an approximate solution given by a first - order method .we call an -solution of ( [ e.gfun ] ) if , for a prescribed accuracy parameter . in 1983 , nemirovski & yudin in derived optimal worst - case complexities for first - order methods to achieve an -solution for several classes of convex problems ( see table [ t.complexity ] ) .if a first - order scheme attains the worst - case complexity of a class of problems , it is called optimal .a special feature of these methods is that the corresponding complexity does not depend explicitly on the problem dimension . from practical point of view , studying the effect of an uniform boundedness of the complexity is very attractive and such methods are highly recommended when the prescribed accuracy is not too small , whereas the dimension of problem is considerably large .[ t.complexity ]  .list of the best known complexities of first - order methods for several classes of problems with respect to levels of smoothness and convexity ( cf . ) [ cols="<,^,^ " , ]in this paper , we propose several novel ( sub)gradient methods for solving large - scale convex composite minimization .more precisely , we give two estimation sequences approximating the objective function with some local and global information of the objective . for each of the estimation sequences , we give two iterative schemes attaining the optimal complexities for smooth , nonsmooth , weakly smooth , and smooth strongly convex problems .these schemes are optimal up to a logarithmic factors for nonsmooth strongly convex problems , and for weakly smooth strongly convex problems they attain a much better complexity than the complexity for weakly smooth convex problems . for each estimation sequence ,the first scheme needs to know about the level of smoothness and the hlder constant , while the second one is parameter - free ( except for the strong convexity parameter which we set zero if it is not available ) at the price of applying a backtracking line search .we then consider solutions of the auxiliary problems appearing in these four schemes and study the important cases appearing in applications that can be solved efficiently either in a closed form or by a simple iterative scheme . considering some applicationsin the fields of sparse optimization and machine learning , we report numerical results showing the encouraging behavior of the proposed schemes . + * acknowledgement .* i would like to thank arnold neumaier for his useful comments on this paper .golub , t. , slonim , d. , tamayo , p. , huard , c. , gaasenbeek , m. , mesirov , j. , coller , h. , loh , m. , downing , j. , caligiuri , m. : molecular classification of cancer : class discovery and class prediction by gene expression monitoring , science 286 , 531536 ( 1999 )              nesterov , y. : a method of solving a convex programming problem with convergence rate , doklady an sssr ( in russian ) , 269 ( 1983 ) , 543547 .english translation : soviet math . dokl ., * 27 * , 372376 ( 1983 )

Abstract: this paper discusses several ( sub)gradient methods attaining the optimal complexity for smooth problems with lipschitz continuous gradients , nonsmooth problems with bounded variation of subgradients , weakly smooth problems with hlder continuous gradients . the proposed schemes are optimal for smooth strongly convex problems with lipschitz continuous gradients and optimal up to a logarithmic factor for nonsmooth problems with bounded variation of subgradients . more specifically , we propose two estimation sequences of the objective and give two iterative schemes for each of them . in both cases , the first scheme requires the smoothness parameter and the hlder constant , while the second scheme is parameter - free ( except for the strong convexity parameter which we set zero if it is not available ) at the price of applying a nonmonotone backtracking line search . a complexity analysis for all the proposed schemes is given . numerical results for some applications in sparse optimization and machine learning are reported , which confirm the theoretical foundations .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore


with the quickly development of nano technology , the scale of electrical devices becoming smaller and smaller , and the ability of single nano machine is very limited due to its size , therefore , sharing information among nano machines is good for the use of single nano machine through the connection of them , which organise the nano network . achieving the communication of nano machines has two main ways which are nano electrical communication through carbon nanotubes and molecular communication through information molecules . among them , molecular communication(mc ) is the most potential way to achieve , which relies on the bio - inspired method and fits for the biological environment .+ at the open space of biological environment , the three dimensional space is fit for molecular communication to model channel . in mahfuzet al s articles , they elaborated on the spatiotemporal distribution of information messages under the binary concentration - encoded modulate methods .mahfuz s researches have a great impact on the study of molecular communication , which gives us a new modulate methods .for the binary concentration - encoded modulate methods , it s much fit for the biological environment , and without considering the inner structures of the molecules .also in akan s paper , he studied the ligand and receptor process of molecular communication , in order to study the details of how molecules bind with the receptors , which is based on the binary concentration - encoded modulate methods .+ all of their works help us understand the molecular communication clearly .as we know in molecular communication different modulation methods or different transportation environment can make different channel .most studies are based on the three dimensional environmental , and fews are concentrate on the one dimensional environment .comparing with the three dimensional , one dimensional can make sure the receiver receive all the information molecules for a long time , and the one dimensional situation is simple enough , therefore , it is worth for us to study it and it has a wealth of applications .for instance , in nariman s molecular communication test - bed , which is studied based on the one dimensional environment . also in extremely tiny pipe , the transmitter and receiver nano machines on sides of the pipe can regard as in the same line . additionally ,when transmitter and receiver nano machines are in extremely small distance , we can consider they are in one dimensional environment .+ in this paper , we mainly considered in the tiny slim pipe , how the continuous information molecules are transmitted . and how the information transmission is affected by the transmission frequency and the distance of transmitter and receiver .the study can help us know how the frequency affect the communication between nano machines . also can help us to place the nano machines in the future .+ the paper is organised as follows : in section [ sec_mc_model ] , we produced a molecular comunication model ; in section [ sec_sample_threshold ] we introduced the sample method and the threshold of the receiver ; in section [ sec_analysis ] , we analyzed some parameters which affect the model ; in section [ sec_conclusion ] , we make a conclusion of our work .we model a communication system composed of a pair of devices which are connected by an extremely slim pipe , each called a nanonetworking - enabled node(nen , ie ., nano node or nano robot) .the communication system is described as fig.[fig_mc_model ] . in molecular communication via diffusion(mcvd ) , the nens communicate with each other through the propagation of certain molecules via diffusion .the tn transmits the encoded information molecules in the propagation medium , the communication phase being known as the sending phase .the transmitted molecules propageated by the channel , the bind with the rn , which is known as the ligand - receptor binding .since the tn and rn nano machine are connected by the extremely slim pipe , surely , we can consider the tn and rn in the one dimensional environment , then we can study its spatiotemporal distribution of signal strength in one dimensional environment .the emission of information can be an instantaneous or a continuous emission .as the instantaneous release of information molecules , in a single puff , would be ideal design for communications requiring rapid fade - out .on the other hand , the source emitting message molecules continuously at a constant rate , to some extent might be useful for status telemetry , navigational beacons or periodic sampling monitors . in this paperwe regard the emission is continuous .+ the spatiotemporal distribution of molecules transmitted by the tn that will be available at rn is calculated by the well - known roberts equation explained in .we assume that a point source type tn continuously emitting molecules at a rate molecules per second , x is taken as the distance down the pipe from the source , is the cross - sectional area of the long and slim pipe , and the concentration of molecules at distance and time instant , then in one dimensional situation , is described as follows eq.[equation_spatiotemporal_1d ] , where as the dummy variable of integration , d is the diffusion constant in .+ we use the on - off modulation method to transmit information . by transmit 1 bit , the tn will emitting a quantity of molecules , and transmit 0 bit , the tn will emitting nothing to the channel .we utilize stands fot the emitting frequency of tn , and is the emitting period . in the emitting period , we can emitting molecules continuously , or transmitting nothing as described in eq.[equation_on_off ] or fig.[fig_on_off_modulation ] .in this paper , we regard as a constant value which is the average molecules in the emitting period .as aforementioned we transmit `` 1 '' when emitting a quantity of molecules , and transmit `` 0 '' when emitting nothing .in this situation , the distance between tn and rn is x , as fig.[fig_single_duration_spray ] describes , we try to transmit bit sequences `` 10110 '' .when molecules were transmitted to the receiver(rn ) , the rn needs to decide the information is `` 1 '' or `` 0 .therefore , we need to set a threshold for the receiver , by comparing the available information molecules(aim ) to the threshold , we can know the information is ' ' 1 `` if the aim is great or equal than the threshold , and in vice versa , the information is ' ' 0 " . in the rn side , it s no need to sample the value of aim for each time instant , according to the nyquist shannon sampling theorem , we can get the sampling value at time instant of is , where is the length of the bit information , we only need to compare the to the threshold . in fig.[fig_single_duration_spray ] , the red arrow is the sampling instant , and the interference is the noise information molecules from the previous molecules informations , in this paper , for a simple description , we only need to consider one bit information of the previous information molecules .in this section , we study how the distance and emitting frequency affect the transmission under the environment of air medium . and at last, we do the simulation to testify the difference of information transmission in one dimensional(1-d ) situation and three dimensional(3-d ) situation . in this section ,we only transmit two bit information `` 10 '' , for it can help us analysis the aim in the first bit `` 1 '' , and interference in the second bit `` 0 '' .we assume that the is 10000 molecules / sec , is 30 , the diffusion coefficient in air medium is =0.43 , we utilize the model in eq.[equation_spatiotemporal_1d ] to do the simulation at different distances , such as are 0.05,1,10 respectively .  through the fig.[fig_distance_varies ] , we find that with the increase of distance between tn an rn , the aim in the rn side is become smaller and smaller , but the interference is becoming larger and larger .therefore , we need to set the distance at short distance , in order to transmit enough information molecules , and improve the probability of successful information transmission .   assuming that the is 10000 molecules / sec ,the distance between tn and rn is , and the diffusion coefficient in air medium is . using the model in eq.[equation_spatiotemporal_1d ], we do the simulation at different emitting frequencies , such as , and .  through the fig.[fig_distance_varies ], we find that with the increase of , the aim in the rn side is become smaller and smaller , also the interference is becoming smaller and smaller .since the aim and interference increase or decrease simultaneously , we need to make a compromise for the change of frequency , in order to transmit enough information molecules , and improve the probability of successful information transmission.   we need to do some researches to study the differences between one dimensional situation and three dimensional situation .we use the same basic values of the parameters described before , except for the distance , the frequency .the three dimensional model of molecules transmission is adapted from .the simulation result shows in fig.[fig_compare_1d_3d ] .  from fig.[fig_compare_1d_3d ] , we know that the 3-d line higher than 1-d , and the interference is almost the same .in one dimensional situation can make sure the receiver the information molecules completely in certain time , however , the available information in three dimensional situation is higher .we think the reason is that in three dimensional situation , there are a large quantities of molecules in certain time slot , but to the one dimensional situation the information molecules needs to transmit sequencely which result the through out is smaller .in this paper , we studied continuous molecules were transmitted in one dimensional molecular communication .we use binary concentration - encoded method to encode information molecules into bit `` 1 '' or `` 0 '' , also we introduce sample theorem and threshold for the receiver to decode information molecules .to study the molecular communication deeper , we studied that how the metrics of distance and frequency affect the molecules concentration , and we learned that the longer the distance the lower of the available information molecules in rn , but higher in interferences , and the frequency has two sides on the available molecular communication , either positive or negative , therefore , we should make a compromise between the available information molecules and the interferences from the former information bits .finally , we made a comparison between one dimensional situation and three dimensional situation , and we find that in three dimensional situation , the rn side has a higher available information molecules , but can not make sure the rn receives the molecular information for one hundred percent .+ still there are many areas need to study , for the future research work , we can promote our research from three aspects .for the first , we can study how the variation of different emit rates affect the molecules concentration .for the second , we should find a method to make a cut - off between the frequency and the interference , in order to make the communication more accurate and high efficiency . for the third, we should consider to combine the advantages of the three dimensional and one dimensional situation to improve the available information molecules .this work was supported by the national natural science foundation of china ( 61173190 ) and the fundamental research funds for the central universities(gk201501008 ) .h. b. yilmaz , a. c. heren , t. tugcu , and c .- b .chae , `` three - dimensional channel characteristics for molecular communications with an absorbing receiver , '' _ communications letters , ieee _ , vol . 18 , no . 6 , pp .929932 , 2014 .m. u. mahfuz , d. makrakis , and h. t. mouftah , `` on the characterization of binary concentration - encoded molecular communication in nanonetworks , '' _ nano communication networks _ , vol . 1 , no . 4 , pp . 289300 , 2010 .     m. mahfuz , d. makrakis , and h. mouftah , `` a comprehensive analysis of strength - based optimum signal detection in concentration - encoded molecular communication with spike transmission , '' _ nanobioscience , ieee transactions on _ , vol . 14 , no . 1 ,6783 , jan 2015 .

Abstract: molecular communication as the most potential methods to solve the communication in nano scale , for it s derived from nature , and it becomes more and more prevalent . though molecular communication happens in three dimensional situation , there are also some situation that are in the one dimensional situation , especially when considering the transmitters and the receivers are in extremely short distance or in long slim pipe . in this paper , we introduce the one dimensional situation , and studied how the continuous information molecules transmitted in this situation , also introduced how to encode and decode the information molecules , and based on the molecular communication model , we studied some metrics of it , such as the distance between transmitter and receiver , the emitting frequency of transmitter . through the research we know that the distance and frequency are important metrics to the successful communication , which can direct us how to place the nano transmitters and receivers in the future nano network environment .


in 1984 , bennett and brassard first introduced a protocol for quantum key distribution ( qkd ) . inqkd two remote legitimate users ( alice and bob ) can establish an unconditionally secured key through the transmission of qubits . since the pioneering work of bennett and brassard several protocols for different cryptographic tasks have been proposed .while most of the initial works on quantum cryptography were concentrated around qkd , eventually quantum - states were applied to other ` post - coldwar ' cryptographic tasks .for example , in 1999 , hillery proposed a protocol for quantum secret sharing ( qss ) . in the same year , shimizu and imoto proposed a protocol direct secured quantum communication using entangled photon pairs .protocols for deterministic secured quantum communication ( dsqc ) were later proposed , in which the receiver can read out the secret message only after the transmission of at least one bit of additional classical information for each qubit .a set of protocols exist which does not require exchange of classical information .such protocols are generally referred to as protocols for `` quantum secure direct communication '' ( qsdc ) .dsqc and qsdc protocols are reducible to secure qkd protocols in the sense that the former equipped with a source of quantum randomness , yield the latter .a conventional qkd protocol generates the unconditionally secured key by quantum means but then uses classical cryptographic resources to encode the message .no such classical means are required in dsqc and qsdc . in recent past ,these facts have encouraged several groups to study dsqc and qsdc protocols in detail , see for example ( and references therein ) .  since the works of shimizu and imoto , and hillery _ et al . _ , several protocols of dsqc and qss are proposed .the unconditional security of the protocols is achieved by using different quantum resources .for example , dsqc schemes are proposed , a ) with and without maximally entangled state ( and references therein ) , b ) using teleportation , c ) using entanglement swapping , d ) using rearrangement of order of particles , etc .we are specifically interested in the dsqc protocols based on rearrangement of orders of particles . to be precise here we aim to provide a protocol of dsqc for a specific task / problem , which may be visualized as follows : charlie , alice and bob are three employees of a company .charlie is the director of the company and alice is a _ custodian _ of all the files of that company .now alice wishes to securely share some of the information with bob , say a file , which may be considered as a string of classical bits . for the present work ,we have chosen it to be classical information , though it can be easily generalized to quantum information . but to do so , alice requires permission from charlie , who , though permitting alice to do so , has imposed a restriction that bob can read the file only when charlie permits him to do so .this is a problem of practical importance because alice may transmit her quantum - encoded information to bob in advance ( say , because the quantum channel is available only then ) , but charlie may wish to give his permission at a later time , determined by , say , certain administrative constraints .  herewe propose a protocol of dsqc to solve the above mentioned problem by using rearrangement of order of particles and dense coding .since a secret is shared , the protocol may be considered as a protocol of secret sharing .this can also be visualized as an application of _ controlled dense coding _ ,in which the controller is charlie , who determines how much classical information is delivered to bob after alice sends him all her dense coding qubits . since the problem discussed here is of much practical importance , we made the protocol more realistic and relevant by considering the channel to be noisy . by doingso we generalize the idea of secret sharing through noisy quantum channel proposed in .the remaining part of the paper is organized as follows : in the next section we have described a protocol that realizes a cryptographic switch by using a pure entangled state ( bell state ) in an ideal scenario . in this section we put some restriction on the channel between alice and bob . in section iii , the effect of noise on the protocol is discussed with specific attention to the squeezed generalized amplitude damping channel . in sectioniv we have modified the protocol for a more general case where no restriction is imposed on the channel between alice and bob . finally , section v is dedicated for conclusions and discussions .let us first propose a solution for the above mentioned problem in an ideal scenario : here alice is semi - honest and bob can receive qubits from alice but can not send a qubit to alice ( for example , because he lacks a requisite license ) .thus the channel between alice and bob is one - way ( a restriction that we will relax in section iv ) . to begin with, we also assume that the channel is noiseless . by semi - honest , we mean that alice strictly follows the protocol and does not disclose the contents of the file to bob before obtaining charlie s permission . in other wordssince alice is honest to the protocol , she does not create a quantum channel between herself and bob to send the information through that .but after obtaining her share of entanglement from charlie she may try to help bob to bypass the control of charlie and to obtain the information before charlie permits bob to do so .we do not require bob to be semi - honest as even a dishonest bob can not obtain any information as long as alice respects the protocol . hereit would be apt to note that in the context of secure multiparty communication , the semi - honest and the malicious models are generally used as secutiy models .here we are interested about the semi - honest model , in which semi - honest participants ( alice in our case ) are assumed to follow the protocol ( and are thus honest ) but want to learn any other s secret ( in our case , alice and bob want to cooperate to learn , which bell state is prepared by charlie ; thus they are curious ) . in a semi - honest model ,a protocol is considered secure against a collusion of participants ( alice and bob in our case ) if by accumulating their data , these participants can not gain more information than what they can from the input and output of the protocol alone .in contrast to this in a malicious model , participants can deviate from the orginial protocol ( say alice and bob can use lm05 protocol to share the information without any involvement of charlie ) .the protocols proposed here and similar protocols are not secured under a malicious model .the use of the semi - honest model is well - jusitified in classical and quantum communication . in particular ,semi - honest models are very often used in modern classical cryptography , specially in the context of data mining . in the recent past ,the notion of a semi - honest user is also adopted in the field of multi - party secure quantum communication and different secured quantum communication protocols have already been proposed , where one or more participants are considered as semi - honest . to begin with, we will first propose a protocol for designing of cryptographic switch in an ideal scienario where the channel between alice and bob is noiseless and one - sided .the one sidedness of the the channel will essentially exclude the possible act of collusion between alice and bob . in section iv, the restriction of one - sidedness of the channel will be removed and then the security against a collusion of alice and bob would be obtained by rearrangement of order of particles . in this ideal situationour protocol works as follows :  1 .  after receiving alice s request, charlie prepares bell states ( not all the same ) and sends first qubits of all the bell states to alice and the second qubits to bob .but charlie does not disclose which bell state he has prepared .2 .  after receiving the qubits from charlie, alice understands that she has been permitted to send the information to bob .alice uses dense coding to encode two bits of classical information on each qubit and transmits her qubits to bob .when charlie plans to allow bob to know the secret information communicated to him , he discloses the bell state which he had prepared .since the initial bell state is known , bob now measures his qubits in bell basis and obtains the information encoded by alice .bob can perform step 5 ( i.e. , measurement in bell basis ) before step 4 but he will not obtain any meaningful information without the knowledge of the initial state . a bell measurement can reveal the bell state prepared by charlie if both the qubits are in possession of alice or bob . nowsince we have assumed that the channel between alice and bob is one - way , therefore , bob can not send his qubits to alice for bell measurement .alice can not send her qubits to bob as she is assumed to be a semi - honest party who strictly follows the protocol .her semi - honesty is motivated by the fact that , while she may wish to potentially cheat charlie , she wants her communication to bob to be secure both in the sense of being protected from the rest of the world ( in the usual qkd sense ) as well as being undetected by charlie .here we are considering a three - party quantum secret sharing scenario , in which charlie prepares a bell state , of which he transmits one half to alice and the other half to bob . from this viewpoint ,the latter receives an ensemble of bell - states :  denotes the parity bit and the phase bit : and .alice encodes two bits on her qubit using the four pauli operators of the superdense coding protocol , and sends the qubit to bob via a possibly insecure channel .  in the noiseless case ,bob measures the two qubits in his possession to obtain the state that corresponds to alice s encoding .however , bob can decode the full information only if charlie shares the full classical _ key _ information that would make the initial entangled state pure .more generally ( as detailed below ) , bob recovers alice s transmitted bits depending on the key information obtained from charlie .thus charlie acts as a _cryptographic switch _ who can determine the level of information alice sends to bob _after _ the full transmission of her qubit . in ref . , a related protocol is considered , in which charlie prepares a ghz state , from which he sends a qubit to alice and one to bob .after alice s encoding and transmission to bob , charlie measures his qubit in the basis , thereby collapsing the alice - bob state to one of the two bell states or .only when charlie reveals his 1 bit of key information does alice s dense coding protocol succeed .note that alice s and charlie s operations commute , so that we might as well consider that charlie prepares one of these two bell states , transmitting them to alice and bob .thus , our protocol generalizes that of ref . , in which all four bell states , instead of the above two alone , are considered .the key information in our case is thus 2 bits .we can thus consider a family of protocols in which the key information varies continuously as .our protocol is characterized by and that of ref . by .suppose bob s measurement yields the state .if he comes to know that charlie had transmitted the state , then he knows alice must have encoded , since .more generally , this idea can be compactly represented as : where the l.h.s is state prepared by charlie , which is transformed under alice s action to the state received by bob , given in the r.h.s .the sign denotes mod 2 addition , and ( ) represents , sequentially , the pauli , , and operators .  the problem can be treated as a communication situation in which alice is signaling bob by means of bell states .then the maximum information bob can extract from the pair of qubits is the holevo quantity . in our protocol ,where , bob can extract alice s 2 bits of information .we wish to parametrize the amount of key information charlie reveals by means of a single variable . to this end , we assume that the ensemble alice and bob receive from charlie , given by eq .( [ eq : bell ] ) , is a werner state of the kind where is projector to the bell state , , and .the amount of information provided by charlie is bits , where is shannon entropy .if ( ) , then bob knows charlie had sent out a state , and can work out alice s encoded information via eq .( [ eq : belltrans ] ) . similarly other werner statesare possible .the maximum information bob can extract from this ensemble is the holevo quantity for the ensemble ( [ eq : werner ] ) .( [ fig : holevononoise ] ) shows how bob s information increases with key information in the noiseless case ., as a function of the key information communicated by charlie , in the noiseless case . ] , the authors studied the effect of phase damping noise on the bit protocol . herewe consider noise to be a squeezed generalized amplitude damping channel acting on alice s qubit transmission .more realistically , we expect noise also to affect charlie s transmissions to alice and bob , but as these added complications offer no new theoretical insight , we ignore them in this work .the action of a squeezed generalized amplitude damping channel is given by the set of kraus operators   the details of the various parameters appearing in the above equation can be obtained from . in particular , they depend upon the bath squeezing parameter and temperature .it is readily checked that eq .( [ eq : new ] ) satisfies the completeness condition provided   in fig .( [ fig : holevosq ] ) , variation of bob s recovered information , quantified by the holevo quantity , as a function of bath squeezing , and charlie s information , is depicted .the holevo quantity increases with , but not as much as in the noiseless case ( fig .( [ fig : holevononoise ] ) ) : due to the randomness introduced by the noise .also , for a given information level , decreases with increase in squeezing .the relationship between the holevo quantity and the parameterizing angle ( [ eq : werner ] ) is shown in fig .( [ fig : holevo3 ] ) . for all casesshown the holevo quantity increases with the angle , with corresponding to a pure bell state .increase in angle corresponds to increase in the purity of the ensemble ( [ eq : werner ] ) leading to an increase in .as expected , noise causes a depletion in the holevo quantity .the maximum difference in , between the noiseless and noisy case , is seen to occur at , that is , when the state is pure the depletion due to noise is maximum .( [ fig : holevochr ] ) shows the effect , on the holevo quantity , of squeezing as a function of time , in particular , demonstrating its favorable influence in certain regimes . as seen from the figure , for sufficiently early times , squeezing fights thermal effects to cause an increase in the recovered information .the significance of noise is that alice and bob may consume some of the bell pairs to determine the noise level , and decide whether it is too high to permit secure information transfer ., as a function of the squeezing parameter , coming from the squeezed generalized amplitude damping channel , and key information communicated by charlie .the time of evolution , while temperature . ]  , as a function of the parameterizing angle ( [ eq : werner ] ) .the thick bold line corresponds to the noiseless case , the thin bold line to the case of noise with zero bath squeezing , while the large - dashed and small - dashed lines correspond to noise with squeezing equal to -0.2 and 0.3 , respectively .the time of evolution , while the temperature . ]  , as a function of the squeezed generalized amplitude damping channel parameters ( squeezing ) and ( time of evolution ) , assuming charlie communicates one bit of information .we note that , for sufficiently early times , squeezing fights thermal effects ( here ) to cause an increase in the recovered information . ]let us withdraw the restriction imposed on the channel from bob to alice .now we may modify the proposed protocol as follows to remain secured :  1 .  after receiving alice s request, charlie prepares bell states ( not all the same ) .he uses the bell states to prepare an ordered sequence with all the first qubits as ,  ] with all the second qubits resulting in an ordered sequence where the subscript denotes the order of a particle pair which is in the bell state .charlie scrambles the second qubits : that is , he disturbs the order of the qubits in to create a new sequence $ ] and sends it to bob .the actual order is known to charlie only .this ordering of information is part of the _ extended key _ of the modified protocol .3 .  after receiving the qubits from charlie, alice understands that she has been permitted to send the information to bob. since the sequence with alice and bob are different , even if alice or bob obtain the access of both and , they will not be able to find out the bell states prepared by charlie .thus any kind of collusion between alice and bob would fail .alice uses dense coding to encode two bits of classical information on each qubit and transmits her qubits to bob .when charlie plans to allow bob to know the secret information communicated to bob by alice , then charlie discloses the bell states which he had prepared and the exact sequence .6 .  since the initial bell states and exact sequence is known , bob now measures his qubits in bell basis and obtains the information encoded by alicethe proposed protocol allows to secretly share classical information in presence of a noisy quantum channel . in the protocol aliceshares the secret classical information with bob but a control is kept with charlie .thus the role of charlie may be viewed as that of a switch , which controls the channel between alice and bob .the protocol may also be viewed as a protocol of controlled secret sharing or a protocol of controlled dense coding . in the sequences and charlie can also insert some decoy photons and use them as check bits to ensure that there is no eavesdropping during his communication with alice and bob .  in sectionii we assumed the channel as one - way and in section iii we considered the effect of noise . the one - way channel assumption is a reasonable assumption when the channel is noisythis is so because the rate of success of an effort in which they illegally try to ascertain the bell state sent by charlie , for example by alice sending her qubit to bob first and bob measuring the bell state prepared by charlie and re - sending the qubit to alice for dense - coding operation , will be very small .this is so because it would require particles of the bell state to travel thrice through the noisy channel .high noise level opens the danger of an eavesdropper that neither alice nor bob would desire .this potentially provides an instance where noisy channel communication is useful .the restriction on the channel is removed in section iv and we have seen that our modified protocol works in a very general scenario . since it is easy to experimentally generate bell states , the proposed cryptographic switch may be realized experimentally and can be used for many practical problems of similar kind , say , in which the director of an organization wishes to keep a control over the time and amount of information to be disclosed to an employee of the company .the custodian of the files ( alice ) must not be worse than semi - honest , as she could otherwise create her own classical / quantum channel and communicate directly with bob .but there is always a potential chance that charlie can detect such communication .further , it is reasonable to assume that as a subordinate in an organization , she lacks the resources to create or purchase the entanglement required .further , the protocol is also valid even when charlie is semi - honest , fully honest charlie being one who does not get any part of the information sent by alice to bob .we have visualized the problem of cryptographic switch as a practical problem in a particular situation .many analogous examples exist , where similar situations arise .for example , one may think that the owner of a company ( charlie ) has asked his semi - honest assistant ( alice ) to send details of all his shares to a stock exchange broker , bob , to sell it in the stock market .but charlie wishes to keep an eye on stock fluctuations and to permit bob to sell his shares at some suitable time .these are only some specific examples , the idea of cryptographic switch is expected to be useful in various analogous situations of practical relevance , the idea being that the main information is priorly sent using a secure quantum channel , and ` switched on ' by a small classical key , when convenient .   21 c. h. bennett and g. brassard , quantum cryptography : public key distribution and coin tossing , in proceedings of the ieee international conference on computers , systems , and signal processing , bangalore , p. 175

Abstract: we illustrate using a quantum system the principle of a cryptographic switch , in which a third party ( charlie ) can control to a continuously varying degree the amount of information the receiver ( bob ) receives , after the sender ( alice ) has sent her information . suppose charlie transmits a bell state to alice and bob . alice uses dense coding to transmit two bits to bob . only if the 2-bit information corresponding to choice of bell state is made available by charlie to bob can the latter recover alice s information . by varying the information he gives , charlie can continuously vary the information recovered by bob . the performance of the protocol subjected to the squeezed generalized amplitude damping channel is considered . we also present a number of practical situations where a cryptographic switch would be of use .


the study of regularity of solutions of stochastic differential equations ( sdes ) has been a topic of great interest within stochastic analysis , especially since malliavin calculus was founded .one of the main motivations of malliavin calculus is precisely to study the regularity properties of the law of wiener functionals , for instance , solutions to sdes , as well as , properties of their densities .a classical result on this subject is that if the coefficients of an sde are functions with bounded derivatives of any order and the so - called hrmander s condition ( see e.g. ) holds , then the solution of the equation is smooth in the malliavin sense .then p. malliavin shows in that the laws of the solutions at any time are absolutely continuous with respect to the lebesgue measure and the densities are smooth and bounded .another approach is attributed to n. bouleau and f. hirsch where they show in absolute continuity of the finite dimensional laws of solutions to sdes based on a stochastic calculus of variations in finite dimensions where they use a limit argument .also , as a motivation of , d. nualart and m. zakai found related results on the existence and smoothness of conditional densities of malliavin differentiable random variables .it appears to be quite difficult to derive regularity properties for the densities of solutions to sdes with singular coefficients , i.e. non - lipschitz coefficients , in particular in the drift. nevertheless , some findings in this direction have been attained .let us for instance remark here the work by m. hayashi , a. kohatsu - higa and g. yki in where the authors show that sdes with hlder continuous drift and smooth elliptic diffusion coefficients admit hlder continuous densities at any time .their techniques are mainly based on an integration by parts formula ( ipf ) in the malliavin setting and estimates on the characteristic function of the solution in connection with fourier s inversion theorem .another result in this direction is due to s. de marco in where the author proves smoothness of the density on an open domain under the usual condition of ellipticity and that the coefficients are smooth on such domain .a remarkable fact is that hrmander s condition is skipped in this proof .moreover , estimates for the tails are also given .the technique relies strongly on malliavin calculus and an ipf together with estimates on the fourier transform of the solution .one may already observe that integration by parts formulas in the malliavin context are a powerful tool for the investigation of densities of random variables as it is the case in the work by v. bally and l. caramellino in where an ipf is derived and the integrability of the weight obtained in the formula gives the desired regularity of the density . as a consequence of the aforesaid result and t. nilssen give in a criterion to obtain regularity of densities of solutions to sdes according to how regular the drift is .the technique is also based on malliavin calculus and a sharp estimate on the moments of the derivative of the flow associated to the solution .this result is a slight improvement of a very similar criterion obtained by s. kusuoka and d. stroock in when the diffusion coefficient is constant and the drift may be unbounded .another related result on upper and lower bounds for densities is due to v. bally and a. kohatsu - higa in where bounds for the density of a type of a two - dimensional degenerated sde are obtained . for this case, it is assumed that the coefficients are five times differentiable with bounded derivatives .finally , we also mention the results by a. kohatsu - higa and a. makhlouf in where the authors show smoothness of the density for smooth coefficients that may also depend on an external process whose drift coefficient is irregular .they also give upper and lower estimates for the density .it is worth alluding the exceptional result by a. debussche and n. fournier in on this topic where the authors show that the finite dimensional densities of a solution of an sde with jumps lies in a certain ( low regular ) besov space when the drift is hlder continuous .the novelty is that their method does not use malliavin calculus as in the aforementioned works .it is therefore important to highlight that in this paper we do _ not _ use malliavin calculus or any other type of variational calculus and we see this as an alternative perspective for studying similar problems .instead , we employ control theory techniques to , shortly speaking , reduce the overall problem to a critical case for which many results in the literature are available . in particular , our technique entitles us to find a _worst case _ sde whose solution has an explicit density that dominates all densities of solutions to sdes among those with measurable bounded drifts .we believe this method is robust since no well - behaviour on the drift is needed other than merely boundedness and no markovianity of the system is assumed .certainly , no regularity is obtained but we are confident that the method can be exploited to gain more regularity of the densities .this paper is organised as follows . in section [ main results ]we summarise our main results with some generalisations to non - trivial diffusion coefficients and to any arbitrary dimension .we also give some insight on concrete properties of the bounds as well as some examples with graphics .section [ s : reduction ] is devoted to thoroughly prove the assertions of the main results .more specifically , we will give an argument based on a control problem to reduce the problem to one critical case .we will also prove in detail the properties adduced in the previous section .we denote the strictly positive numbers by , the trace of a matrix by and simply denotes either or .the skorokhod space is the set of all cdlg functions from to equipped with the skorokhod metric , c.f .* chapter vi.1 ) .the canonical space is the triplet where is the -algebra generated by the point evaluations and is the right - continuous filtration generated by the canonical process .we denote the generalised signum function by for any .this is the orthogonal projection to the unit euclidean sphere . for a complex number denote its real resp .imaginary part by resp . .further notations are used as in .in this section we present our main result and some direct consequences . in particular, we will find sharp explicit bounds for sdes with additive noise in the one - dimensional case and give some extensions to the -dimensional case with more general diffusion coefficients .  throughout this sectionlet be a filtered probability space with the usual assumptions on the filtration , i.e.  contains all -null sets and is right - continuous , be a -dimensional standard brownian motion and we define the process classes   the next results constitutes one of the core results of this section and will be proven in detail in the next section .[ t : main theorem , multi dimensional ] let , be a -dimensional standard brownian motion and .then has lebesgue density where denotes the volume of the -dimensional euclidean ball with and denotes the gamma function .moreover , satisfies for any , where and and are the unique solutions to the sdes for any .see at the end of section [ s : reduction ] .  if , then the functions , as well as some of their properties can be derived explicitly , cf .theorem [ t : alphabeta ] . in the multidimensional casewe can give some of their properties .let us summarise the formulas .[ summarythm ] let , and , be given as in theorem [ t : main theorem , multi dimensional ] .then where resp . denotes the distribution resp .density function of the standard normal law .for we have where for any .moreover , we have where for any .this is part of the statements of theorems [ t : alphabeta ] and [ t : multibounds ] below .  in what follows, we will derive bounds for the densities of solutions of general sdes .the following is an immediate consequence of theorem [ t : main theorem , multi dimensional ] .[ k : bound for sde with additive noise ] let , , be predictable and bounded by .then any weak solution of the sde has density at time which is bounded from below by and from above by where and are given in theorem [ t : main theorem , multi dimensional ] and is a -dimensional brownian motion .moreover , the bounds are optimal in the sense that for any there are two functionals , resp . for which the density of the solution to the sde , resp . attains the upper bound in , resp .the lower bound in .define and for any .then the bounds follow from theorem [ t : main theorem , multi dimensional ] .shifts of the processes , resp . attain the upper , resp .lower bounds at the given points .now we focus on our second main result which is an application of corollary [ k : bound for sde with additive noise ] .this time is given as a solution of an sde with measurable drift and a diffusion coefficient which is continuously differentiable .[ t : generelle sde ] let be predictable , be continuously differentiable and assume the following conditions .  1 . is an invertible matrix for any , .2 .  there is a function such that for any , where denotes the frchet derivative of with respect to .3 .  the function is bounded by some constant where denotes the hessian matrix of , i.e.  for any , .  then any solution of the sde has , at each time , lebegsue density and for every we have where , are defined as in theorem [ t : main theorem , multi dimensional ] . moreover ,if additionally is invertible for any fixed , then   define and for any .then it s formula yields theorem [ t : main theorem , multi dimensional ] states that has lebesgue density which admits the bounds for any , .  from the definition of we directly get for any , .if we assume that is invertible for any , then for any and , hence , the additional claim follows .  the conditions ( 1 ) to ( 3 ) appearing in theorem [ t : generelle sde ] simplify considerably in dimension .moreover , due to it - tanaka s formula we can relax the conditions on .let be a solution of the sde where , is a standard brownian motion , predictable and bounded by some constant , is a lipschitz continuous function with lipschitz bound and for some constant .then has lebesgue density and for any where and are defined as in theorem [ t : main theorem , multi dimensional ] when , and moreover , where is a uniform bound for .define .since is lipschitz continuous there is a function which is bounded by and . then it - tanaka s formula ( * ? ? ?* theorem vi.1.5 ) yields let and define which is predictable and bounded by . then the result follows from corollary [ k : bound for sde with additive noise ] .  in the next sectionwe will give precise definitions and mathematical computations of the functions and in dimension 1 and why these are the optimal bounds ( in the sense of corollary [ k : bound for sde with additive noise ] ) for the densities of sdes with bounded measurable drifts .before we do that , let us give some intuitive insight on the shape and behaviour of these bounds for the one - dimensional case .consider any one - dimensional process of the form as in theorem [ t : main theorem , multi dimensional ] .in particular , can be the solution to the following sde , , , , with bounded and predictable as in corollary [ k : bound for sde with additive noise ] . furthermore , denote by the density of at a fixed time . then theorem [ t : main theorem , multi dimensional ] grants that for any . in the following figure we can observe the functions and for different values of and see how they behave .we can see the function in orange and in green .any density lies between these two curves and these bounds are optimal in the sense that , for given we can find drifts and such that the associated densities , resp . for these drift coefficients satisfy , respectively , . as an illustrationwe just take the drift to be in blue and in red .starting at ( in green and orange ) with the respective densities when the drift coefficients are and ( blue and red ) at different times . ]  as we can see , both densities are bounded by and and the bounds are attained in for density of the process with drift ( in blue ) and in when the drift is ( in red ) .in this section we will see how to derive the functions and explicitly for the case as well as some of their properties , cf .theorem [ t : alphabeta ] .then we will show that these are indeed the bounds for the densities of any solution to sdes with bounded measurable drift by solving a stochastic control problem , cf .theorem [ t : control problem ] and thereafter we give the proof for theorem [ t : main theorem , multi dimensional ] . in the sequel , consider the process   c.f . for existence and ( pathwise ) uniqueness .moreover , at some point we will also use the property that the solution to equation ( [ e : signum sde ] ) is strong markov , even for the multidimensional case .this can be for instance justified using ( * ? ? ?* theorem 6.4.5 ) in connection with ( * ? ? ?* corollary ix.1.14 ) .[ 0:densities ] for every , resp . has density , resp . given by for and any where , resp . , denote the density , resp . the distribution function , of the standard normal law .the computation of the densities and in the previous lemma are relatively easy given the fact that the local - time of the brownian motion starting from 0 is symmetric and the joint law of and the local time of , is explicitly known , see . nevertheless , one is able to find reasonably explicit expressions for the densities of and which yield representations for and if .[ l : density minus ] for every , the density of is given by where , and is the first hitting time of the process at 0 whose density function is explicitly given by   let be the first time the process hits 0 , i.e. then it is clear , that for any  ] and  ] . as a consequence \\ & = { \mathrm{e}}_q \left[1_{\{b(t)\geq x - y , m(t ) < x\}}\frac{1}{z(t)}\right]\end{aligned}\ ] ] where is the equivalent measure w.r.t . defined by ( * ? ? ?* theorem 8.6.4 ) yields that the process , is a standard -brownian motion and is therefore the running maximum of the standard brownian motion , hence where denotes the joint density of and which is explicitly given , see , by we have finally , the above probability converges to the derivative of ( [ pym ] ) w.r.t . , that is   now we continue to compute .define the random variable .it is readily checked that and is -measurable because the event is in .then the strong markov property of and yield & = { \mathrm{e}}[f_{\varepsilon } ( y_x^{-}(\tau))1_{\{t\geq\tau_0^x\}}|\mathcal{f}_{\tau_0^x } ] \\  & = 1_{\{t\geq\tau_0^x\}}{\mathrm{e}}[f_{\varepsilon } ( y_x^{-}(\tau))|\mathcal{f}_{\tau_0^x } ] \\  & = 1_{\{t\geq\tau_0^x\}}{\mathrm{e}}[f_{\varepsilon}(y_0^{-}(\xi))]|_{\xi = \tau-\tau_0^x}\end{aligned}\ ] ] -a.s . as a consequence & = { \mathrm{e}}\left[{\mathrm{e}}[f_{\varepsilon } ( y_x^{-}(t))1_{\{t\geq\tau_0^x\}}|\mathcal{f}_{\tau_0^x } ] \right]\\ & = { \mathrm{e}}\left[1_{\{t\geq\tau_0^x\ } } { \mathrm{e}}[f_{\varepsilon}(y_0^{-}(\xi))]|_{\xi = \tau-\tau_0^x } \right]\\ & = { \mathrm{e}}\left[1_{\{t\geq\tau_0^x\ } } { \mathrm{e}}[f_{\varepsilon}(y_0^{-}(\xi))]|_{\xi = t-\tau_0^x } \right].\end{aligned}\ ] ] now, the density of is explicitly known by lemma [ 0:densities ] . thus = \int_0^t \int_{\mathbb{r } } f_{\varepsilon}(z ) q_{t - s}(0,z)\rho_{\tau_0^x}(s)ds.\ ] ]   we have for any and hence is a weak solution of for and starting point .hence , has the same law as for any .consequently , we have the claimed formula follows .the proof of this lemma follows completely the same ideas as in lemma [l : density minus ] .one of the main differences is that in this case the distribution of the stopping time has an atom at infinity , namely , from ( * ? ? ?* , formula 2.0.2 ) we have   now we are in a position to define the functions and for the one - dimensional case and study some of their properties . before we do that, we will need a technical result to prove one of the properties of these functions .define where the equalities hold -a.s . and here for any , .let with and define .then hence .* theorem ix.3.5 i ) ) yields a.s .observe that and hence -a.s .[ t : alphabeta ] let be the transition density of the markov process which is given in lemma [ l : density minus ] and the transition density for the markov process given in .define the functions by and where , and . then and where recall that , respectively are given as in lemma [ l : density plus ] , respectively as in lemma [ l : density minus ] .in addition , for each and the functions and are analytic in , lipschitz continuous in , symmetric , decreasing on and by symmetry increasing on  ] and ,\mathbb r) ] where . here, is defined as the density of at 0 .hence , .thus it is enough to show that is decreasing on for every .let .yields define .* proposition 2.1.5 a ) ) yields that is a stopping time because it is the first contact time with the closed set of the continuous process .observe , that for any  ] and is the other summand .it can be seen that is negative since . for the term we use exactly the same markov - argument as for the term in by defining . then and is -measurable .thus , the strong markov property of and and ( * ? ? ?* corollary 2.6.18 ) yield &= { \mathrm{e}}\left[1_{\{|y_y^{-}(\tilde \tau)|<\varepsilon\}}1_{\{t\geq\tau\}}|\mathcal{f}_{\tau}\right]\\ & = 1_{\{t\geq\tau\}}{\mathrm{e}}\left[1_{\{|y_y^{-}(\tilde \tau)|<\varepsilon\}}|\mathcal{f}_{\tau}\right]\\ & = 1_{\{t\geq\tau\}}{\mathrm{e}}\left[1_{\{|y_y^{-}(\xi)|<\varepsilon\}}|_{\xi=\tilde\tau - \tau}\right]\end{aligned}\ ] ] p - a.s . on the other hand , observe that by the definition of .so = 1_{\{t\geq\tau\}}{\mathrm{e}}\left[1_{\{|-y_x^{-}(\xi)|<\varepsilon\}}|_{\xi=\tilde\tau - \tau}\right]\ ] ] which implies that . as a result \leq 0\ ] ] which implies for every with , we show that has exponential tails .observe that for  ] , which yields   from now on , let us consider the processes and given in equation for the multidimensional case , i.e. , and a -dimensional standard brownian motion .we denote , and .theorem [ t : main theorem , multi dimensional ] guarantees that the density of any adapted process , , has bounds and .we start with a proposition which gives a different view on the functions and .namely , we define with and denote the volume of the -dimensional euclidean ball of radius then we have and cf . theorem [ t : main theorem , multi dimensional ] , where . in view of this equality, we are interested in the behaviour of the transition density of near zero which will be exploited in theorem [ t : multibounds ] below .[ p : alternative sde ] let be a filtered probability space .let be a -dimensional brownian motion and be the solution to the sde for any .define , for any , .then is a solution to the sde for which pathwise uniqueness holds .let .then and for any where denotes the frchet differential of , the hessian matrix of and denotes the unit matrix in .it s formula yields for any . since is a brownian motion, is a weak solution as required .it remains to show that the sde has unique weak solutions .let be a measure , equivalent to , such that is a standard -brownian motion .then the sde can be rewritten as ( * ? ? ?* theorem ix.3.5 ii ) ) yields that pathwise uniqueness holds for sde under .     define the processes , .it s formula yields where defines a new standard brownian motion w.r.t .* theorem iv.3.6 ) and it isometry ensure that is a -dimensional standard brownian motion .let be the solution of the sde for any and be the measure , equivalent to , such that is a -brownian motion where .then , we have similar arguments as in the proof of ( * ? ? ? * theoremix.3.7 ) show that for any , -a.s .observe that pathwise uniqueness holds for equation by proposition [ p : alternative sde ] and hence ( * ? ? ?* theorem ix.1.7 ii ) ) states that is a strong solution to equation .consequently , is -measurable and hence are independent processes .now given one has .this implies where in the last step we use the inequalities for every , -a.s . andthe fact that are independent processes .  by proposition [ p : alternative sde ]the law of under is the same as the law of under where and the law of is given in lemma [ l : density minus ] .hence , we have for any .in order to prepare our main result of this section we will start with a series of lemmas which aims at showing the continuity condition of ( * ? ? ?* theorem ix.2.11 ) .the needed continuity condition is summarised in lemma [ l : continuity of b ] .[ l : pfadabschaetzung ] let be a filtered probability space .let  ] and constant on .define for any .let be an adapted process which is bounded by , and define then where for .define and . then girsanov s theorem ( * ? ? ?* theorem iii.3.24 ) yields that is a -brownian motion starting in .define . then andhence by gronwall s lemma , see e.g. ( * ? ? ?* appendix ) , .we have for any where we used the cauchy - schwartz inequality twice and the fact that .we have :x+w(s)=0\ } ) \\       & = 0  \end{aligned}\ ] ] for by lebesgue s dominated convergence theorem where denotes the lebesgue measure on .[ l : an abschaetzung ] let be a filtered probability space and be a -dimensional standard brownian motion .let and be as in lemma [ l : pfadabschaetzung ] .let be a sequence of processes that converges in probability to .for any let be an adapted process which is bounded by , and define also , assume that converges in distribution to some process .define , .then in probability for . hence , in distribution .since has continuous sample paths for any , has -a.s . continuous sample paths .let .since is continuous we have for some .hence , by lemma [ l : pfadabschaetzung ] we have thus , we have :x_\infty(s)=0\ } ) & \leftarrow { \mathrm{e}}a_k(t , x_\infty ) \\                        & \rightarrow 0   \end{aligned}\ ] ] for and .thus :x_\infty(s)=0\})=0 ] for any .  in the next lemmathe martingales converge to the brownian motion but they , and hence the drift in , are not adapted to the same brownian motion .we show that they converge in our specific set - up .[ l : convergence of the marginals ] let be a filtered probability space .let be a -dimensional brownian motion , and where for any .assume that for any .then converges in distribution to the solution of the sde   by an independent enlargement of -argument , we may assume that there is a sequence of random variables which are indepent of , -measurable and that is centered normal on with variance where denotes the identity matrix in .define for any . then .define the -adapted process and .then has the same law as and , consequently , has the same law as for any .moreover , in probability .define for any , , , , where is the centered normal law with covariance matrix .then is the semimartingale characteristics of in the sense of ( * ? ? ?* definition ii.2.6 ) relative to the truncation function , .observe that and fulfil the conditions  ] and $ ] in the sense of .thus ( * ? ? ?* theorem ix.3.9 ) states that is tight .let be a subsequence of which converges in law and denote the limiting law by .lemma [ l : continuity of b ] yields that is -a.s . continuous .let be the canonical process on the canonical space .then ( * ? ? ?* theorem ix.2.11 ) yields that is , under , a semimartingale with characteristics .the continuous martingale part , denote it , of is a standard brownian motion because its semimartingale characteristics is .moreover ,   thus is a weak solution to the sde .* corollary ix.1.12 ) ) yields that the law of coincides with the law of the solution of the sde .consequently , any convergent subsequence of converges in law to . since is additionally tight , it , andhence , converges to .we are now in readiness to prove the core result of this section which is to solve a control problem . for dimension one, this problem has been studied by v. e. bene in in the markovian setting whose optimal control is indeed the _ signum _ function in dimension one .such solutions are known as _ bang - bang _ solutions .nevertheless , here we stress the fact that our case deals with multivalued controls , thus not _ bang - bang _ , in addition to the non - markovian setting as in the above mentioned result . for this reasonwe include a short proof based on the previous limit results .[ t : control problem ] let and be as in the beginning of section [ main results ] .let , and define and .then where for .in other words , an optimal control for the control problem above is given by .similarly ,   the control problem given in ( [ opplus ] ) can be interpreted as follows : one wishes to find the stochastic process among those in that minimises the probability that the underlying process is near zero . in other words ,we want the process to escape from 0 as much as possible . intuitively , the process is doing that . whenever is near zero on the positive line , the drift is positive and pushes even further away up and if is near zero from below the drift is negative and sends further down . for the control problem in ( [ opminus ] )the idea is similar , but there one wishes to maximise the probability of being close to zero , which clearly does .lemma [ l : convergence of the marginals ] yields that converges in law to .since has no atoms , we have for .thus , we have for .thus is an optimal control .define , and the brownian motion .then for any .theorem [ t : control problem ] states that for any , and . by definition we have observe that for any orthonormal transformation we have where here is a standard brownian motion and hence is a weak solution of for .consequently , has the same law as which implies .hence , we have lebesgue differentiation theorem ( * ? ? ?* corollary 2.1.16 ) yields that is a version of the lebesgue density of .consequently , the density of given by satisfies analogue arguments show that         n. bouleau and f. hirsch , _ proprits dabsolue continuit dans les espaces de dirichlet et applications aux quations diffrentielles stochastiques _ , seminaire de probabilits xx , lecture notes in mathematics , vol .1204 , springer , berlin , 1986 , pp .131161 .s. de marco , _ smoothness and asymptotic estimates of densities for sdes with locally smooth coefficients and applications to square root - type diffusions _ , the annals of applied probability ( 2011 ) , vol .4 , 12821321    m. hayashi , a. kohatsu - higa , and g. yki , _ local hlder continuity property of the densities of solutions of sdes with singular coefficients _ , journal of theoretical probability * 26 * ( 1991 ) , no .2013 , 11171134 .

Abstract: we consider a process given as the solution of a stochastic differential equation with irregular , path dependent and time - inhomogeneous drift coefficient and additive noise . explicit and optimal bounds for the lebesgue density of that process at any given time are derived . the bounds and their optimality is shown by identifying the worst case stochastic differential equation . then we generalise our findings to a larger class of diffusion coefficients .