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evaluating the amount of work extraction from reservoirs is an important issue in thermodynamics and nonequilibrium statistical mechanics . here , we focus on the work extraction by using information , where two distinct formalisms exist . one formalism concerns the measurement - feedback processes between two systems : a celebrated example is maxwell s demon . in this case , the generalizations of the second law and the fluctuation theorem have been obtained by including the _ mutual information _ @xcite between the system and the controller ( demon )  @xcite , and it has been applied to biochemical sensing  @xcite . we call this avenue of research as the measurement - feedback ( mf ) formalism . the other formalism concerns the work extraction from bit sequences called _ information reservoirs _ , where the shannon entropy of a long bit sequence is consumed to extract work ( see also fig . [ f : f : mjrevisit](a ) ) . a second - law - like inequality stronger than the conventional second law has been obtained in this formalism  @xcite . we call this as the information - reservoir ( ir ) formalism . although both of these two formalisms address the fundamental link between information and thermodynamics , the relation between the mf and the ir formalisms has not been fully understood . in previous researches , an important observation is that the ir formalism gives a stronger inequality than the mf formalism in some setups  @xcite . in contrast , in this paper we show that the mf formalism is also applicable to ir models , and leads to a stronger second - law - like inequality than the ir formalism . the central idea here is the additive decomposition of the total entropy production , which was first investigated in the context of a single measurement and feedback  @xcite , and then extended to general markov jump processes  @xcite . this decomposition assigns the _ partial entropy production _ to each transition path , for which a generalized second law is obtained . in addition , we show that the mandal - jarzynski ( mj ) model , a prominent ir model , can be regarded as a contracted model of an autonomous mf model even at the level of stochastic trajectories . in this special case , the newly obtained inequality is shown to be equivalent to the inequality with the mutual - information flow in terms of the mf formalism  @xcite . our result would lead to the comprehensive understanding of the link between information and thermodynamics . s : partial    we briefly review the notion of the partial entropy production introduced in ref . @xcite , which is an extension of the mf formalism . throughout this paper , we consider markov jump processes with discrete states and continuous time , and focus on the stationary states in the presence of a single heat bath . we normalize the inverse temperature @xmath0 as 1 for simplicity . the transition from a state @xmath1 to another state @xmath2 is written as @xmath3 , whose transition rate is denoted by @xmath4 . the stationary distribution @xmath5 satisfies the stationary condition @xmath6 we define the total entropy production @xmath7 with time interval @xmath8 as  @xcite @xmath9 where we denoted the number of jumps by @xmath10 , and the @xmath11-th jump occurs from @xmath12 to @xmath13 at time @xmath14 ( @xmath15 ) . the state at time @xmath16 is written as @xmath17 . @xmath18 is the heat absorption by the system from the heat bath with the transition @xmath3 , and @xmath19 represents the stochastic shannon entropy at the state @xmath2 . we note that the local detailed balance condition  @xcite is assumed . we now define the partial entropy production @xmath20 for a set of transitions , written as @xmath21 , which is a subset of all possible transitions . we first define the partial probability flux at the state @xmath2 as @xmath22 . we then define the partial entropy production @xmath23 as @xmath24 here , we defined @xmath25 , where @xmath26 takes @xmath27 if @xmath28 and takes @xmath29 otherwise . we also defined @xmath30 with @xmath31 . it is crucial that the partial entropy production @xmath20 is an additive decomposition of the total entropy production @xmath32 where @xmath33 is the complement of @xmath21 . the partial entropy production satisfies the fluctuation theorem : @xmath34 which leads to the generalized second law for the partial entropy production : @xmath35 it has been shown that the fluctuation theorem for autonomous measurement and feedback processes is obtained as a special case of eq . ( [ partial - ineq ] )  @xcite . we further discuss the significance of this inequality in sec . [ s : power ] . s : mj    we here explain the mj model which extracts work from a bit sequence  @xcite . in particular , we consider a simplified version of the mj model introduced in ref . the simplified mj model is a composite system of a bit sequence and a weight with the coupling reaction @xmath36 ( see fig . [ f : f : mjrevisit](a ) ) . here , 0 and 1 represent states of a bit , @xmath37 and @xmath38 represent ( relative ) positions of the weight on ladder - like energy levels with energy interval @xmath39 . the weight plays the role of a work storage . the bit sequence moves in one direction , from left to right , with the poisson process with the rate @xmath40 . the probability that the incoming bit is 1 is @xmath41 . when a new bit comes into the central site , the state of the weight is instantly labeled as @xmath38 ( @xmath37 ) corresponding to the state of the incoming bit 0 ( 1 ) , and the higher ( lower ) energy level is labeled as @xmath37 ( @xmath38 ) ( see fig . [ f : f : mjrevisit](b ) ) . under this labeling , the possible states of the composite system are only @xmath42 and @xmath43 . there are two possible transitions between @xmath37 and @xmath38 ( see fig . [ f : f : mjrevisit](c ) ) . one is the stochastic jump induced by the thermal bath with rates @xmath44 and @xmath45 , which satisfy @xmath46 . this is the coupling reaction of the up - down of the weight and the flip of a bit such that @xmath36 . we denote the set of these transitions by @xmath47 . the other is induced by the bit stream with rates @xmath48 and @xmath49 ; from the above - mentioned rule of the state labeling , the stream of the bit sequence also changes the states @xmath37 and @xmath38 . if @xmath41 is small ( i.e. , the entropy of the bit sequence is low ) , the weight can be raised on average by increasing the entropy of the bit sequence . the stationary probability of @xmath37 is given by @xmath50 , where @xmath51 and @xmath52 . since @xmath53 is the probability that the outgoing bit is 1 , the average energy gain per single incoming bit is given by @xmath54 , which we regard as work extraction . we note that the entropy change in the work storage is irrelevant to our argument , because the fluctuation of the state of the work storage does not affect the procedure of autonomous work extraction . this type of work storages for autonomous work extraction have been discussed in refs . s : power    we now apply the fluctuation theorem with the partial entropy production , discussed in sec .  [ s : partial ] , to the simplified mj model . we set @xmath21 in sec . [ s : partial ] to @xmath47 in sec . [ s : mj ] ( the red arrows in fig . [ f : f : mjrevisit](c ) ) . equality ( [ partial - ft ] ) and inequality ( [ partial - ineq ] ) are then written as @xmath55 and @xmath56 by using the stationary condition @xmath57 , inequality ( [ mj - ineq - mid ] ) reduces to a simple form : @xmath58 the left - hand side of ( [ restrict - mj ] ) represents the entropy change induced by the heat bath . we next extend our foregoing argument to general markov jump processes with states @xmath59 . in line with ref . @xcite , we divide pairs of transitions into two groups , which we refer to as _ thermal transitions _ and _ informational transitions_. we denote these sets of transitions by @xmath47 and @xmath60 , respectively ( see fig .  [ f : mj - gen ] ) . the work is extracted during the thermal transitions . by setting @xmath47 to @xmath21 , eq . ( [ partial - ineq ] ) reduces to @xmath61 where we defined @xmath62 . we now show the hierarchical structure of three upper bounds on @xmath63 with the simplified mj model : the conventional second law , an inequality derived from the ir formalism  @xcite , and the newly derived inequality ( [ restrict - mj ] ) . first , we compare the conventional second law and the inequality obtained from the ir formalism . to clarify the thermodynamic energy cost of the one - way stream of the bit sequence , we model the bit sequence by non - interacting particles with two components @xmath29 and @xmath27 ( fig . [ f : f : tape - bath ] ) , where the transition rates are the same as @xmath64 . particles in the bath stochastically come and go to the reaction site , which corresponds to the yellow cage of the mj model . the ratio of the particles @xmath29 and @xmath27 in the bath is @xmath65 , and therefore the gain ( loss ) of the chemical potential with the transition @xmath66 ( @xmath67 ) is @xmath68 . although the kinetics of the particle - bath model is equivalent to that of the bit sequence , the entropy production in the particle - bath and that in the bit sequence are different as shown below . the conventional second law for the total system with the particle - bath is given by @xmath69 on the other hand , the second law for the total system with the bit sequence has been derived from the ir formalism , which is given by  @xcite @xmath70 we refer to this inequality as the mj inequality . subtracting the left - hand side ( lhs ) of ( [ mj - ineq ] ) from the lhs of ( [ mj-2nd ] ) , we obtain @xmath71 where @xmath72 represents the kullback - leibler divergence that is non - negative  @xcite . inequality ( [ 2nd - vs - mj ] ) confirms that the mj inequality gives a stronger bound than the conventional second law . their difference @xmath73 characterizes an extra cost to move the bit sequence in one direction . in the particle - bath case , the outgoing particle from the reaction site is in state @xmath27 with probability @xmath74 , while a particle in the particle bath is in state @xmath27 with probability @xmath41 . thus , when a particle leaves the reaction site to the particle bath , the entropy production in the particle bath is calculated as @xmath75 . in contrast , in the bit sequence case , the ratio of bit 1 in the bit sequences on the left side of the cage ( i.e. , the particle bath releasing the incoming bit ) and that on the right side of the cage ( i.e. , the particle bath receiving the outgoing bit ) are different . therefore , a bit on the right side of the cage is in state 1 with probability @xmath74 , and when a bit leaves the reaction site to the bath , the entropy production in the bath is calculated as @xmath76 . the difference between the foregoing two entropy productions is equal to @xmath73 . this implies that the strength of the mj inequality comes not from the use of the bit sequence carrying information but from the one - directional operation , which inevitably needs the additional energy cost . we now compare two inequalities ( [ mj - ineq ] ) and ( [ restrict - mj ] ) ( i.e. , the mj inequality and the inequality derived from the partial entropy production method ) . subtracting the lhs of ( [ restrict - mj ] ) from the lhs of ( [ mj - ineq ] ) , we get @xmath77 here , @xmath78 characterizes the dissipation corresponding to the fact that the time evolution is not quasi - static  @xcite . the left - hand side of the mj inequality ( [ mj - ineq ] ) is just characterized by the entropy change between the initial state ( i.e. , the state of the incoming bit ) and the final state ( i.e. , the state of the outgoing bit ) . therefore , the mj inequality does not take into account the dissipation due to the free relaxation of a bit , while inequality ( [ restrict - mj ] ) can take it into account . such dissipation is unavoidable , as autonomous information engines inevitably involve relaxation processes , because of the very fact that they are autonomous . this is the physical reason why the partial entropy production method gives a stronger bound of the work extraction . more generally , we can show that inequality ( [ restrict - mj - gen ] ) is a stronger inequality than the generalized version of the mj inequality for the general ir formalism  @xcite . the setup for the generalized mj inequality is the same as that for inequality ( [ restrict - mj - gen ] ) . the general version of the mj inequality ( [ mj - ineq ] ) is written as  @xcite @xmath79 where we defined @xmath80 and @xmath81 . we then compare two inequalities ( [ mj - gen ] ) and ( [ restrict - mj - gen ] ) . subtracting the lhs of ( [ restrict - mj - gen ] ) from lhs of ( [ mj - gen ] ) , we obtain @xmath82 where we defined a positive constant @xmath83 as @xmath84 for each informational transition @xmath85 . therefore , inequality ( [ restrict - mj - gen ] ) is stronger than ( [ mj - gen ] ) . we next consider an autonomous measurement - feedback ( amf ) model  @xcite , and show the equivalence of the simplified mj model and the amf model . this equivalence allows us to directly compare the mj inequality and the second - law - like inequality with the mutual information flow . the amf model consists of a fluctuating particle on the tilted one - dimensional discrete lattice and the delta - function - like wall ( see fig .  [ f : info - rat](a ) ) . the position on the lattice is labeled as @xmath86 in mod 2 , and the delta - function - like walls are at the right side of 0 or 1 , which we label as @xmath87 or @xmath88 , respectively . the energy difference of the particle between a site and its right neighbor is @xmath39 . the transition rate of the position of the particle due to the thermal diffusion is given by @xmath89 with @xmath46 . the transition rate of the positions of the walls is given by @xmath90 which are equal to @xmath91 and @xmath92 , respectively . we note that @xmath93 is regarded as the chemical potential difference that drives the switching of the walls . owing to the chemical driving , the particle climes up against the potential bias when @xmath94 . we stress that the particle and the walls do not directly change the energy , but exchange only information . in this sense , the amf model is a typical example of an autonomous information - driven engine  @xcite . analogous to maxwell s demon , the walls measure the position of the particle , and change their own position depending on the measurement outcome , which is the feedback control . the amf model is a bipartite system with two variables @xmath95 , and thus we can define the entropy production associated only with the particle as @xmath96 , where @xmath97 is the heat absorbed by the subsystem @xmath98 , and @xmath99 is the change in the stochastic shannon entropy of @xmath98 . due to the presence of the autonomous measurement and feedback , @xmath100 can be negative on average and does not satisfy the fluctuation theorem in the conventional form . however , by taking into account the flow of stochastic mutual information @xmath101  @xcite , we obtain the modified fluctuation theorem in the stationary state  @xcite : @xmath102 here we defined the mutual information flow by @xmath103 , where @xmath104 with @xmath105 and kronecker delta @xmath106 , and @xmath107 . we stress that eq . ( [ infoft ] ) is a special case of eq . ( [ partial - ft ] ) : if we apply eq . ( [ partial - ft ] ) to the amf model by setting all transitions of @xmath98 to @xmath21 , we straightforwardly obtain eq . ( [ infoft ] ) . by applying jensen s inequality to eq . ( [ infoft ] ) , the generalized second law with the mutual information flow is obtained as  @xcite @xmath108 where @xmath109 represents the stationary entropy production rate of @xmath98 . this inequality is stronger than the conventional second law of the composite system . we now consider the contraction of the amf model by contracting two states @xmath110 and @xmath111 to a single state @xmath38 , and @xmath112 and @xmath113 to @xmath37 ( see fig . [ f : info - rat](b ) ) . this contraction means that the absolute positions of the particle and walls are replaced by the relative positions . since the amf model has the two types of transition paths corresponding to the thermal diffusion of the particle and the chemical switching of the walls , its contracted model also has two types of transition paths between @xmath37 and @xmath38 , which we refer to as the thermal transition and the informational transition . then , by denoting the transition rates of the thermal and informational transitions by @xmath114 and @xmath115 respectively , the contracted amf model is regarded equivalent to the simplified mj model introduced in sec . [ s : mj ] . correspondingly , each stochastic trajectory in the amf model is contracted to that in the simplified mj model , where the transition rates are kept unchanged : @xmath116 we denote as @xmath117 and @xmath118 the probabilities that trajectory @xmath119 from time 0 to @xmath120 and its contraction are realized in the amf model and the simplified mj model , respectively . we then obtain @xmath121 where the factor @xmath122 comes from the contraction of the initial state : if the initial state after the contraction is @xmath38 ( @xmath37 ) , the initial state before the contraction is either @xmath110 or @xmath111 ( @xmath112 or @xmath113 ) with probability @xmath122 . we next show that eq . ( [ infoft ] ) directly reduces to eq . ( [ mj - ft ] ) , and correspondingly , inequality ( [ infoineq ] ) reduces to inequality ( [ restrict - mj ] ) . therefore , for the case of the simplified mj model we can see the direct application of the mf formalism to an ir model without invoking the partial entropy production method . we first note that @xmath123 holds for @xmath124 or @xmath125 , where every line is just zero if @xmath126 , @xmath127 . we then obtain @xmath128 for any trajectory @xmath119 of the amf model . equality ( [ s - equiv ] ) implies that the two fluctuation theorems , eqs . ( [ infoft ] ) and ( [ mj - ft ] ) , are equivalent , and thus inequalities ( [ infoineq ] ) and ( [ restrict - mj ] ) are also equivalent . this clarifies the physical meaning of inequality ( [ restrict - mj ] ) from the viewpoint of the mf formalism : the bit sequence plays the equivalent role to the chemical fuel that drives the autonomous measurement and feedback . we have established the fundamental relation between the mf and the ir formalisms . inequality ( [ restrict - mj ] ) derived from the partial entropy production method provides a stronger inequality than that derived from the ir formalism ( [ mj - ineq ] ) . moreover , we have demonstrated that the contraction of the amf model , to which the fluctuation theorem with the mutual information flow is applicable , is equivalent to the simplified mj model . this allows us the direct comparison of the inequality with the mutual information flow and the mj inequality . we have also shown the hierarchy of the three second - law - like inequalities for the simplified mj model : the conventional second law ( [ mj-2nd ] ) , the mj inequality ( [ mj - ineq ] ) , and the inequality ( [ restrict - mj ] ) with the partial entropy production . an important insight is that the more source of dissipation is taken into account , the stronger inequality we obtain . from this perspective , the reason why inequality ( [ restrict - mj ] ) gives the strongest bound is the following : it specifies the entropy production only associated with the thermal transitions , and it takes into account the dissipation induced by the free relaxation of the bit . in general , the partial entropy production method would be useful to investigate stronger second - law - like inequalities for a variety of nonequilibrium dynamics . in addition , we discuss a possible extension of our result to quantum cases . recently , the refinement of quantum thermodynamics has been intensively studied from several perspectives  @xcite . these studies have revealed novel and fundamental aspects of the second law of thermodynamics . in addition , both of the mf formalism  @xcite and the ir formalism  @xcite for quantum systems has already been discussed . unifying these two formalisms in light of quantum thermodynamics would be an interesting future problem . the authors thank sosuke ito and kyogo kawaguchi for fruitful discussions and helpful comments . ns is supported by grant - in - aid for jsps fellows number 26 - 7602 . ts is supported by jsps kakenhi grant no . 25800217 and no . 22340114 , by kakenhi no . 25103003  fluctuation ` & ` structure " , and by platform for dynamic approaches to living system from mext , japan . 99      szilard l. szilard , z. phys . * 53 * , 840 ( 1929 ) . touchette h. touchette and s. lloyd , phys . rev . lett . * 84 * , 1156 ( 2000 ) . allahverdyan a. e. allahverdyan , d. janzing , and g. mahler , j. stat . p09011 ( 2009 ) . su2010 t. sagawa and m. ueda , phys . lett . * 104 * , 090602 ( 2010 ) . toyabe s. toyabe , t. sagawa , m. ueda , e. muneyuki , and m. sano , nat . phys . * 6 * , 988 ( 2010 ) . su2012 t. sagawa and m. ueda , phys . lett . * 109 * , 180602 ( 2012 ) . hsp j. horowitz , t. sagawa , and j. m. r. parrondo , phys . lett . * 111 * , 010602 ( 2013 ) . hartich d. hartich , a. c. barato , and u. seifert , j. stat . mech . p02016 ( 2014 ) . jordan2014 j. m. horowitz and m. esposito , phys . rev . x * 4 * , 031015 ( 2014 ) . hs j. m. horowitz and h. sandberg , new j. phys . * 16 * , 125007 ( 2014 ) . ss n. shiraishi and t. sagawa , phys . e * 91 * , 012130 ( 2015 ) . siks n. shiraishi , s. ito , k. kawaguchi , and t. sagawa , new j. phys . * 17 * , 045012 ( 2015 ) . bo s. bo , m. d. giudice , and a. celani , j. stat . p01014 ( 2015 ) . is s. ito and t. sagawa , nat . * 6 * , 7498 ( 2015 ) . hartich2 d. hartich , a. barato , and u. seifert , arxiv:1509.02111 ( 2015 ) . mj d. mandal and c. jarzynski , proc . ac . sci . * 109 * , 11641 ( 2012 ) . refri d. mandal , h. t. quan , and c. jarzynski , phys . lett . * 111 * , 030602 ( 2013 ) . deffner s. deffner and c. jarzynski , phys . x * 3 * , 041003 ( 2013 ) . strasberg p. strasberg , g. schaller , t. brandes , and c. jarzynski , phys . rev . e * 90 * , 062107 ( 2014 ) . merhav n. merhav , j. stat . p06037 ( 2015 ) . strasberg2 p. strasberg , j. cerrillo , g. schaller , and t. brandes , phys . e * 92 * , 042104 ( 2015 ) . boyd a. b. boyd , d. mandal , and j. p. crutchfield , arxiv:1507.01537 ( 2015 ) . campisi m. campisi , p. talkner , and p. hnggi , phys . lett . * 102 * , 210401 ( 2009 ) . goldstein s. goldstein , t. hara , and h. tasaki , arxiv:1303.6393 ( 2013 ) . brandao f. brando , m. horodecki , n. ng , j. oppenheim , and s. wehner , proc . sci . * 112 * , 3275 ( 2015 ) . esposito m. esposito , m. a. ochoa , and m. galpern , phys . * 114 * , 080602 ( 2015 ) .
there have been two distinct formalisms of thermodynamics of information : one is the measurement - feedback formalism , which concerns bipartite systems with measurement and feedback processes , and the other is the information reservoir formalism , which considers bit sequences as a thermodynamic fuel . in this paper , we derive a second - law - like inequality by applying the measurement - feedback formalism to information reservoirs , which provides a stronger bound of extractable work than any other known inequality in the same setup . in addition , we demonstrate that the mandal - jarzynski model , which is a prominent model of the information reservoir formalism , is equivalent to a model obtained by the contraction of a bipartite system with autonomous measurement and feedback . 
 our results provide a unified view on the measurement - feedback and the information - reservoir formalisms .    //