Discrimination of two channels by adaptive methods and its application to quantum system

1 Discriminati on of t wo channels by adapti v e methods and its applic ation to quantum system Masahito Hayash i Abstract The optimal exponen tial error rate for ada ptiv e discrimination of two chann els i s discussed. In this problem, adapti ve choice of input signal i s allo wed. This problem is discussed in v arious settings. I t is pro v ed that adapti v e choice does not impro ve t he expo nential error rate in these settings. These results are applied to quantum state discrimination. Index T erms Simple hypothesis testing, Channel, Discrimination, Quantum state, One-way LOCC, Acti ve learning , Experimental design, Stein’ s lemma, Chernoff bound, Hoef fding bound, Han-Ko bayashi bound I . I N T R O D U C T I O N D ISCRIMIN A TING two distributions is treated as a fun damental problem in the field of statistical inferen ce. This problem can b e regarded as simple hypothesis testing because both hypotheses con sist of a single distribution. Many researchers, Stein, Chernoff[3], Hoef fding[16], and Han-K obayashi[1 0 ] hav e studied the asymptotic behavior when the number n of identical and independen t observations is sufficiently large. They formulated a simple h ypothesis testing/d iscrimination o f two distrib utions as an optimization problem an d d erived the respecti ve optimu m value, e. g., the optimal exponen tial erro r rate. W e call th ese op timum values the Stein bou nd, the Cher noff b ound , the Hoeffding bound, and the Han-K obayashi bound, respectively . Han [8], [9] later extended these results to the discriminatio n of two general sequences of distributions, including the Markovian case. Nagaoka-Hayashi [21] simplified Han’ s discussion and gener alized Han’ s extension of the Han-K obayashi bound . In th e pre sent pap er , w e c onsider an other extension of the ab ove results. That is, we extend the above results to the discrimination of two (classical) channels, in which two p robabilistic transition matric es are g iv en. Such a prob lem has appea red in Blahut[2]. In this problem , th e numb er o f applications of this channel is fixed to a gi ven co nstant n , and we can choose approp riate inputs for this purpose. I n th is case, we assume that the gi ven chann el is memoryless. If we use the same in put to all applications of the gi ven chan nel, the n output d ata obeys an ide ntical and independ ent distribution. T his prope rty ho lds ev en if we choose the input ran domly based on the same distribution on in put signals. This strategy is called the non-ad aptive method. In p articular, when the same input is applied to all ch annels, it is called the de terministic non-ad aptive method. If the inpu t is determined stochastically , it is called the sto chastic non-ad aptive meth od, which was treated by Blah ut[2]. In the non-ad aptive method, our task is cho osing the optimal input for d istinguishing two channels most efficiently . In the presen t paper, we assume that we can choose the k -th input signal b ased on the prece ding k − 1 output data. This strategy is called the adaptive method , which is the m ain f ocus of the p resent paper . In th e param eter estimation, such an a daptive metho d impr oves estimation perf ormanc e. That is, in the one-param eter estimation , the asymptotic estimation err or is boun ded by the in verse of th e optimum Fisher information. Howe ver , if we d o not apply the adaptive method, it is generally impossible to realize the optimum Fisher informatio n in all p oints at the same time. It is kn own that the adap tiv e metho d rea lizes th e optimum Fisher information in all points[1 3], [7]. Therefore, one may expect that the adap tiv e method improves the perform ance of discriminating two chan nels. As o ur main resu lt, we succeeded in pr oving that the ad aptive metho d canno t improve the non-ad aptive method in the sense of all of the above men tioned boun ds, i.e., the Stein bound, the Ch ernoff bound, the Hoef fding bound, and th e Han- K obayashi bou nd. Th at is, there is no difference between the no n-adap tiv e method and the a daptive method in these asy mptotic formu lations. Ind eed, as is p roven herein, the deterministic non-adaptive m ethod gi ves the optimu m p erform ance with respect to th e Stein bo und, the Chernoff boun d, and the Ho effding bound . Howe ver , in order to attain the Han-K obayashi boun d, in general, we need th e stochastic non -adaptive m ethod. On the o ther han d, the researc h field in q uantum inform ation has treated the discrimination of two quantum states. H iai- Petz[15] and Ogawa-Nagaoka[18 ] proved the quantum version of Stein’ s le mma. Audenaert et al. [ 1] and Nussbaum-Szkoła [23], [24] obtain ed the quantu m version o f the Chern off boun d. Ogawa-Hayashi [ 17] de riv ed a lower bound o f the quantum version of the Hoeffding b ound . Later , Hayashi [12] a nd Nagaoka [20] o btained its tight bo und based on the results by Aud enaert et al. [1] an d Nussbaum- Szkoła [23], [24]. Hayashi [11] ( in p.90 ) obtain ed the q uantum version of th e Han- K obayashi bou nd based on Na gaoka[ 19 ]’ s discussion. These discussions M. Hayashi is with Graduate School of Information Sciences, T ohoku Univ ersity , Aoba-ku , Senda i, 980-8579, Japan (e-mail: hayashi@math.is.tohoku .ac.jp) 2 assume that any measurement o n the n -tensor product system is allowed fo r testing the g iv en state. Hence, the next go al is the deriv ation of these bou nds under some locality restrictio ns on an n -par tite system for possible measur ements. One easy setting is restricting the present m easurement to be identical to that in the respective system. In this case, our task is the choice of the o ptimal measurement on the sing le system. By considering the mea surement an d the quantum state as th e input and the ch annel, respectively , we c an treat this problem by the no n-adap tiv e method of the classical channel. Another setting is restricting o ur measure ment to one-way local oper ations and classical communicatio ns (o ne-way LOCC). In the above- mentioned c orrespon dence, the o ne-way LOCC setting can be regarded as the ad aptive metho d of the classical chann el. He nce, applying the above argu ment to d iscrimination of two qu antum states, we can conclude that one-way co mmunica tion do es no t improve discr imination of two quan tum states in the respective asymptotic formulations. Furthermo re, the same p roblem appears in adap tiv e experimental d esign and active learnin g. In learning theory , we id entify the given system by using th e obtained sequ ence of inp ut and o utput pairs. In particular , in acti ve learning, we can ch oose the inputs using the precedin g d ata. Hence, the present result indicates that acti ve learning does not improve th e perfor mance of learning when the candidates of the unknown system are gi ven by o nly two classical channels. I n experimental d esign, we choose su itable design of o ur experim ent for inferrin g th e unk nown para meter . Ad aptive improvement for the design is allowed in adap tiv e experimen tal design. When the candida tes o f the un known par ameter are only two values, the obtained result can be applied. Th at is, adaptive impr ovement for design d oes not work. The r emainder of th e pre sent paper is organized as follows. Section II r evie ws the Stein bo und, the Chernoff bo und, the Hoeffding bound, and the Han-K obayashi bound in discrimination of two p robab ility distrib utions. In Sec tion III, we present our fo rmulation an d notatio ns of the adap ti ve method in the discrimin ation of two (classical) chann els, and d iscuss the adap tiv e- method versions of the Stein bound, the Cher noff boun d, the Ho effding bo und, and the Han-K obayashi b ound , r espectively . In Section I V, we co nsider a simple examp le, in which the stochastic non-adap ti ve metho d is requir ed for attaining the Ha n- K obayashi bound. In Sec tion V, we apply the present result to discrimination o f two quantum states by o ne-way LOCC. In Sections VI, VI I, and VI II, we prove the adaptive-method version s of Stein bound, th e Chernoff b ound , the Hoeffding boun d, and the Han-Kobayashi bound, respectiv ely . I I . D I S C R I M I N A T I O N / S I M P L E H Y P OT H E S I S T E S T I N G B E T W E E N T W O P RO B A B I L I T Y D I S T R I B U T I O N S In prepar ation for th e main to pic, we review the simple hyp othesis testing p roblem fo r the n ull hy pothesis H 0 : P n versus the alternative hyp othesis H 1 : P n , wh ere P n and P n are the n -th identical and indepen dent distributions of P an d P , re spectiv ely on the pro bability spa ce Y . Th e p roblem is to decide wh ich h ypoth esis is true based on n o utputs y 1 , . . . , y n . In the following, random ized tests are allowed as our decision. Hence, our d ecision method is described by a [0 , 1] -valued functio n f on Y n . When we observe n outputs y 1 , . . . , y n , we ac cept the alternativ e hypo thesis P with the probability f ( y 1 , . . . , y n ) . W e have two types of erro rs. In the first ty pe, the null hypothesis P is rejected despite being cor rect. In th e seco nd type , the alternative P is rejected d espite bein g cor rect. Hence, the first ty pe of error prob ability is given by E P n f , and the second ty pe of error probab ility is by E P n (1 − f ) . Note that E P describes the expecta tion u nder the distribution P . In the fo llowing, we assume that Φ( s | P k P ) := Z Y ( ∂ P ∂ P ( y )) s P ( d y ) < ∞ φ ( s | P k P ) := log Φ( s | P k P ) and φ ( s | P k P ) is C 2 -continu ous. In th e p resent p aper, we choo se the base o f the logarithm to be e . In the discrim ination o f two distrib utions, we treat two types of proba bilities equally . Then, we simply minimize the equ al sum E P n f + E P n (1 − f ) . Its optimal rate of exponential d ecrease is charac terized by the Chernoff bound[3]: C ( P , P ) := lim n →∞ − 1 n log(min f n E P n f n + E P n (1 − f n )) = − min 0 ≤ s ≤ 1 φ ( s | P k P ) . In order to tr eat these two erro r pro babilities asymm etrically , we often restrict the first type of erro r pro bability E P n f to b elow a particu lar threshold ǫ , and min imize the second type of e rror proba bility E P n (1 − f ) : β ∗ n ( ǫ ) := min f  E P n (1 − f )   E P n f ≤ ǫ  . Then, the Stein’ s lemma holds. For 0 < ∀ ǫ < 1 , the eq uation lim n →∞ 1 n log β ∗ n ( ǫ ) = − D ( P k P ) (1) holds, where the re lativ e entropy D ( P k P ) is defined by D ( P k P ) = Z Y − log ∂ P ∂ P ( y ) P ( dy ) . 3 Indeed , this lemma has the following variant for m. Define B ( P k P ) := sup { f n }  lim n →∞ − log E P n (1 − f n ) n     lim n →∞ E P n f n = 0  B ∗ ( P k P ) := inf { f n }  lim n →∞ − log E P n (1 − f n ) n     lim n →∞ E P n f n < 1  . Then, these two qua ntities satisfy the following relations: B ( P k P ) = B ∗ ( P k P ) = D ( P k P ) . As a furth er analysis, we focus on the decreasin g e xponen t o f the error prob ability of the first type under an expon ential constraint for the err or proba bility of the seco nd type . When the decrea sing expon ent of for the error pr obability of the seco nd type is greater than the relative entro py D ( P k P ) , the e rror probability of the second type con verges to 1 . In this case, we focus on the decr easing exponen t of th e probab ility of correctly accepting the nu ll hypothesis P . For th is purp ose, we define B e ( r | P k P ) := sup { f n }  lim n →∞ − log E P n f n n     lim n →∞ − log E P n (1 − f n ) n ≥ r  B ∗ e ( r | P k P ) := inf { f n }  lim n →∞ − log E P n (1 − f n ) n     lim n →∞ − log E P n (1 − f n ) n ≥ r  . Then, the two qu antities are calculated as B e ( r | P k P ) = min Q : D ( Q k P ) ≤ r D ( Q k P ) = sup 0 ≤ s ≤ 1 − sr − φ ( s | P k P ) 1 − s (2) B ∗ e ( r | P k P ) = min Q : D ( Q k P ) ≤ r D ( Q k P ) + r − D ( Q k P ) = sup s ≤ 0 − sr − φ ( s | P k P ) 1 − s . (3) The first expressions of (2) a nd (3) are illustrated b y Figs. 1 a nd 2. P P Q ( ) D Q P r = ( ) D Q P Fig. 1. Figu re of B e ( r | P k P ) P P Q ( ) D Q P r = Fig. 2. Figu re of B ∗ e ( r | P k P ) when r 0 ≥ r ≥ D ( P k P ) 4 Now , we define the new function B ( r ) : B e ( r ) :=  B e ( r | P k P ) r ≤ D ( P k P ) − B ∗ e ( r | P k P ) r > D ( P k P ) . Then, its gr aph is shown in Fig. 3. ( ) D P P ( ) e B r ( ) D P P r Graph of ( ) e B r ( ) , C P P 0 r Fig. 3. Graph of B e ( r ) In order to g iv e other characterization s o f (2), we in troduce a one-p arameter family P s,P, P ( dy ) := 1 Φ( s | P k P ) ( ∂ P ∂ P ( y )) s P ( d y ) , which is abbr eviated as P s . Then, since φ ( s ) is C 1 continuo us, D ( P s k P 1 ) = ( s − 1) φ ′ ( s ) − φ ( s ) s ∈ ( −∞ , 1 ] (4) D ( P 0 k P s ) = φ ( s ) − sφ ′ (0) s ∈ [0 , ∞ ) . (5) Since d ( s − 1) φ ′ ( s ) − φ ( s ) ds = − φ ′′ ( s ) < 0 , D ( P s k P 1 ) is mo noton ically d ecreasing with respect to s . As is men tioned in Theore m 4 of Blahut [2], whe n r ≤ D ( P k P ) , there exists s r ∈ [0 , 1] such that min Q : D ( Q k P ) ≤ r D ( Q k P ) = D ( P s r k P 0 ) . Then, (4) and (5) imply that r = D ( P s r k P 1 ) = ( s r − 1) φ ( s r ) − φ ( s r ) . Thus, we ob tain another expression. min Q : D ( Q k P ) ≤ r D ( Q k P ) = min s ∈ [0 , 1]: D ( P s k P ) ≤ r D ( P s k P ) . (6) On the othe r hand, d ds − sr − φ ( s | P k P ) 1 − s = − r + ( s − 1) φ ′ ( s ) − φ ( s ) (1 − s ) 2 = D ( P s k P 1 ) (1 − s ) 2 . (7) Since D ( P s k P 1 ) is mono tonically decreasing with respect to s , d ds − sr − φ ( s | P k P ) 1 − s = 0 if an d only if s = s r . The equ ation min Q : D ( Q k P ) ≤ r D ( Q k P ) = sup 0 ≤ s ≤ 1 − sr − φ ( s | P k P ) 1 − s (8) can be ch ecked. 5 In the following, we present som e explanation s co ncernin g (3). As is m entioned by Han-K obayashi[ 10 ] and Og awa- Nagaoka[ 18 ], when r 0 := D ( P −∞ k P 1 ) ≥ r ≥ D ( P k P ) , the relation B ∗ e ( r | P k P ) = D ( P s r k P 0 ) holds, where s r ∈ ( −∞ , 0 ] is define d as r = D ( P s r k P 1 ) = ( s r − 1) φ ( s r ) − φ ( s r ) . Thus, similar to (6) and (8), the relation min Q : D ( Q k P ) ≤ r D ( Q k P ) + r − D ( Q k P ) = D ( P s r k P ) = sup s ≤ 0 − sr − φ ( s | P k P ) 1 − s (9) holds, where s r ≤ 0 is defined by D ( P s r k P ) = r [ 18]. As mention ed by Nakagawa-Kanaya[22 ], when r ≥ r 0 , the r elation min Q : D ( Q k P ) ≤ r D ( Q k P ) + r − D ( Q k P ) = D ( P −∞ k P ) + r − D ( P −∞ k P ) = min Q : D ( Q k P ) ≤ r 0 ( D ( Q k P ) + r 0 − D ( Q k P )) + r − r 0 holds. This bound is attained by the following ran domized test. The hypoth esis P is acce pted with the probability only when the loga rithmic likelihood ratio takes the maximum value r 0 . Since D ( P s k P 1 ) < r , (7) implies that sup s ≤ 0 − sr − φ ( s | P k P ) 1 − s = lim s ≤−∞ − sr − φ ( s | P k P ) 1 − s = lim s ≤−∞ − sr 0 − φ ( s | P k P ) 1 − s + r − r 0 = min Q : D ( Q k P ) ≤ r 0 ( D ( Q k P ) + r 0 − D ( Q k P )) + r − r 0 . (10) Remark 1: The classical Hoeffding bound in information theory is due to Blahut[2] and Csisz ´ ar-Longo[4]. The correspondin g ideas in statistics were first put f orward by Hoeffding[16], from whom the b ound recei ved its name. So me authors p refer to refer this bo und as the Hoeffding-Blahu t-Csisz ´ ar - Long o bound. On the other hand, Han-K obayashi[1 0 ] ga ve th e first equation of (3), and proved that this equa tion a mong non- rando mized tests when r 0 ≥ r ≥ D ( P k P ) . They po inted out that the minim um min Q : D ( Q k P ) ≤ r D ( Q k P ) + r − D ( Q k P ) can be attained by Q satisfyin g D ( Q k P ) = r . Ogawa-Nagaoka[18 ]sho wed the second equ ation of (3) for this case. Nakagawa-Kanaya[2 2 ] proved the first equation when r > r 0 . I ndeed, as pointed by Nakagawa-Kanaya[22 ], when r > r 0 , any non-rand omized test cannot attain the minimu m min Q : D ( Q k P ) ≤ r D ( Q k P ) + r − D ( Q k P ) . In this case, the m inimum min Q : D ( Q k P ) ≤ r D ( Q k P ) + r − D ( Q k P ) cannot be attained by Q satisfying D ( Q k P ) = r . I I I . M A I N R E S U LT : A DA P T I V E M E T H O D Let us focus on two spaces, the set of inp ut signals X and the set of outputs Y . I n this case, the channel f rom X an d Y is described by the map f rom th e set X to the set of pro bability distributions on Y . That is, gi ven a chann el W W x represents the outpu t d istribution when the input is x ∈ X . When X and Y have finite elements, the channel is giv en by transition matrix. The main topic is th e discriminatio n of two classical chann els W and W . In p articular, we treat its asy mptotic an alysis when we can use the unk nown channel only n times. That is, we discrim inate two hypo theses, the null hypo thesis H 0 : W n versus the alternative hypoth esis H 1 : W n , where W n and W n are the n u ses of th e ch annel W and W Then, o ur problem is to de cide which hypo thesis is tru e b ased on n inputs x 1 , . . . , x n and n outp uts y 1 , . . . , y n . I n th is setting, it is allo wed to choose th e k -th input based on the previous k − 1 outp ut adaptively . W e choose th e k -th inp ut x k subject to the d istribution P k ( x 1 ,y 1 ) ,..., ( x k − 1 ,y k − 1 ) ( x k ) o n X . That is, the k -th input x k depend s on k conditional distributions ~ P k = ( P 1 , P 2 , . . . , P k ) . Hence, our decision method is de scribed by n co nditiona l distributions ~ P n = ( P 1 , P 2 , . . . , P n ) and a [0 , 1] -valued fun ction f n on ( X × Y ) n . In th is case, wh en we ch oose n in puts x 1 , . . . , x n and ob serve n ou tputs y 1 , . . . , y n , we accept the a lternative hypoth esis W with the pro bability f n ( x 1 , y 1 , . . . , x n , y n ) . That is, our scheme is illustrated by Fig. 4. In orde r to treat this problem mathematically , we introdu ce th e following n otation. For a channel W fro m X to Y an d a distribution P on X , we define two no tations, the distribution W P on X × Y and th e distribution W · P o n Y as W P ( x, y ) := W x ( y ) P ( x ) W · P ( x, y ) := Z X W x ( y ) P ( dx ) . Using the distribution W P , we d efine two quantities: D ( W k W | P ) := D ( W P k W P ) φ ( s | W k W | P ) := φ ( s | W P k W P ) . 6 Channel discr imination with adaptive improv ement 1 x 1 y 2 y n y 2 x n x Adaptive improvement is allowed or W W or W W or W W Fig. 4. The adapti ve method Based on k cond itional distributions ~ P k = ( P 1 , P 2 , . . . , P k ) , we defin e the following distrib utions: Q W , ~ P n := W P n W P n − 1 · · · W P 1 P W , ~ P n := P n · Q W , ~ P n − 1 Q s,W | W , ~ P n := P s,Q W, ~ P n ,Q W , ~ P n P s,W | W , ~ P n := P n · Q s,W | W , ~ P n − 1 . Then, the first type of erro r probab ility is g iv en b y E Q W, ~ P n f n , and th e second ty pe of erro r probab ility is by E Q W , ~ P n (1 − f n ) . In order to tre at this pro blem, we introduc e the follo wing quantities: C ( W, W ) := lim n →∞ − 1 n log( min ~ P n ,f n E Q W, ~ P n f n + E Q W , ~ P n (1 − f n )) β ∗ n ( ǫ ) := min ~ P n ,f n  E Q W , ~ P n (1 − f n )   E Q W, ~ P n f n ≤ ǫ  , and B ( W k W ) := sup { ( ~ P n ,f n ) } ( lim n →∞ − log E Q W , ~ P n (1 − f n ) n      lim n →∞ E Q W, ~ P n f n = 0 ) B ∗ ( W k W ) := inf { ( ~ P n ,f n ) } ( lim n →∞ − log E Q W , ~ P n (1 − f n ) n      lim n →∞ E Q W, ~ P n f n < 1 ) B e ( r | W k W ) := sup { ( ~ P n ,f n ) } ( lim n →∞ − log E Q W, ~ P n f n n     lim n →∞ − log E Q W , ~ P n (1 − f n ) n ≥ r ) B ∗ e ( r | W k W ) := inf { ( ~ P n ,f n ) } ( lim n →∞ − log E Q W, ~ P n (1 − f n ) n      lim n →∞ − log E Q W , ~ P n (1 − f n ) n ≥ r ) . W e obtain the following channel version of Stein’ s lemm a. Theor em 1: Assume that φ ( s | W x k W x ) is C 1 continuo us, an d lim ǫ → +0 φ ( − ǫ | W k W ) ǫ = sup x ∈X D ( W x k W x ) , (11) where φ ( s | W k W ) := sup x ∈X φ ( s | W x | W x ) = sup P ∈P ( X ) φ ( s | W k W | P ) , and P ( X ) is the set of distributions on X . Then, B ( W k W ) = B ∗ ( W k W ) = D := sup x ∈X D ( W x k W x ) . (12) The following is another expression of Stein’ s lemma. 7 Cor o llary 1: Under the same assumptio n, lim n →∞ − 1 n log β ∗ n ( ǫ ) = sup x ∈X D ( W x k W x ) . Condition (11) can b e replaced by an other condition . Lemma 1: When any element x ∈ X satisfies φ ′ (0 | W x k W x ) = D ( W x k W x ) and there exists a real nu mber ǫ > 0 su ch that C 1 := sup x ∈X sup s ∈ [ − ǫ, 0] d 2 φ ( s | W x k W x ) ds 2 < ∞ , (13) then cond ition (11) holds. In addition , we obtain a chan nel version of the Hoeffding b ound . Theor em 2: When sup x ∈X sup s ∈ [0 , 1] d 2 φ ( s | W x k W x ) ds 2 < ∞ (14) and sup x ∈X D ( W x k W x ) < ∞ , then B e ( r | W k W ) = sup x ∈X sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s = s up x ∈X min Q : D ( Q k W x ) ≤ r D ( Q k W x ) . (15) Cor o llary 2: Under the same assumptio n, C ( W, W ) = sup x ∈X − min 0 ≤ s ≤ 1 φ ( s | W x k W x ) . (16) These arguments imply that adapti ve improvement does not imp rove the performance in the above senses. For example, when we app ly the be st input x M := arg max x D ( W x k W x ) to all of n channe ls, we can achieve the optimal perfo rmance in the sense of the Stein bound. The same fact is true co ncernin g th e Hoeffding bound and the Cherno ff bo und. Pr o of: The relation C ( W, W ) = sup { r | B e ( r | W k W ) ≥ r } holds. Since sup n r    sup x ∈X sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s ≥ r o = sup x ∈X sup n r    sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s ≥ r o = sup x ∈X − min 0 ≤ s ≤ 1 φ ( s | W x k W x ) , the relation (16) holds. The chann el version of the Han-K obayashi bound is given as follo ws. Theor em 3: When φ ( s | W x k W x ) is C 1 continuo us, th en B ∗ e ( r | W k W ) = sup s ≤ 0 − sr − φ ( s | W k W ) 1 − s = inf P ∈P ( X ) sup s ≤ 0 − sr − φ ( s | W k W | P ) 1 − s = inf P ∈P 2 ( X ) sup s ≤ 0 − sr − φ ( s | W k W | P ) 1 − s , (17) where P 2 ( X ) is the distribution on X tha t takes positive p robability only on at m ost two elements. As shown in Section IV, the eq uality sup s ≤ 0 − sr − φ ( s | W k W ) 1 − s = inf x ∈X sup s ≤ 0 − sr − φ ( s | W x k W x ) 1 − s (18) does not nec essarily hold in general. In or der to under stand th e m eaning of this fact, we assume that the eq uation (18) doe s no t hold. Whe n we app ly the same inpu t x to all ch annels, the be st perf ormance canno t be achieved. Ho we ver , the best perfo rmance can be achie ved b y the following method . Assume that th e best inp ut distribution argmax P ∈P 2 ( X ) sup s ≤ 0 − sr − φ ( s | W k W | P ) 1 − s has th e suppo rt { x, x ′ } , and the prob abilities λ an d 1 − λ . T hen, app lying x o r x ′ to all channels with th e p robability λ an d 1 − λ , we ca n achieve the best p erform ance in the sense of the Han-K obayashi b ound . That is, the stru cture of optimal strategy of the Han -K obayashi bound is more com plex than th ose of the ab ove cases. 8 I V . S I M P L E E X A M P L E In this section , we treat a simp le e xample that d oes n ot satisfy (1 8). For four g iv en pa rameters p, q , a > 1 , b > 1 , we defin e the chan nels W and W : W 0 (0) := aq , W 0 (1) := 1 − aq, W 0 (0) := q , W 0 (1) := 1 − q , W 1 (0) := bq , W 1 (1) := 1 − bq , W 1 (0) := q , W 1 (1) := 1 − q . Then, we obtain lim s →−∞ φ ( s | W 0 k W 0 ) s = a, lim s →−∞ φ ( s | W 1 k W 1 ) s = b. In this case, D ( W 0 k W 0 ) = ap log a + (1 − ap ) log 1 − a p 1 − p D ( W 1 k W 1 ) = bq log b + (1 − bq ) log 1 − b q 1 − q . When a > b and D ( W 0 k W 0 ) < D ( W 1 k W 1 ) , the magnitude relation between φ ( s | W 0 k W 0 ) and φ ( s | W 1 k W 1 ) on ( −∞ , 0) depend s on s ∈ ( − ∞ , 0) . F or example, the case of a = 100 , b = 1 . 5 , p = 0 . 00 0 1 , q = 0 . 65 is shown in Fig. 5. In this c ase, B ∗ e ( r | W 0 k W 0 ) , B ∗ e ( r | W 1 k W 1 ) , and B ∗ e ( r | W k W ) are calculated by Fig. 6. Then, the inequality (18) does n ot hold. -1 -0.8 -0.6 -0.4 -0.2 0 s 0 0.2 0.4 0.6 0.8 1 Φ H s L Fig. 5. Magnitu de relation between φ ( s | W 0 k W 0 ) and φ ( s | W 1 k W 1 ) on ( − 1 , 0) . The upper solid line indica tes φ ( s | W 0 k W 0 ) , the dotted line indica tes φ ( s | W 1 k W 1 ) . 0.4 0.6 0.8 1 1.2 1.4 r 0 0.1 0.2 0.3 0.4 0.5 B * Fig. 6. Magnitu de relation between B ∗ e ( r | W 0 k W 0 ) , B ∗ e ( r | W 1 k W 1 ) , and B ∗ e ( r | W k W ) on ( − 1 , 0) . The upper solid line indic ates B ∗ e ( r | W 0 k W 0 ) , the dotted line indicates B ∗ e ( r | W 1 k W 1 ) , and the lo wer solid line indicat es B ∗ e ( r | W k W ) . 9 V . A P P L I C A T I O N T O A D A P T I V E Q UA N T U M S TA T E D I S C R I M I N AT I O N Quantum state discrim ination b etween two states ρ an d σ on a d -d imensional system H with n copies b y one-way LOCC is formu lated as follows. W e ch oose the first PO VM M 1 and ob tain the data y 1 throug h the m easuremen t M 1 . In the k -th step, we choose th e k -th PO VM M k (( M 1 , y 1 ) , . . . , ( M k − 1 , y k − 1 )) d epend ing on ( M 1 , y 1 ) , . . . , ( M k − 1 , y k − 1 ) . Th en, we o btain th e k -th data y k throug h M k (( M 1 , y 1 ) , . . . , ( M k − 1 , y k − 1 )) . Ther efore, this pr oblem can be regarded as classical channel discrim ination with the corresponden ce W M ( y ) = T r M ( y ) ρ and W M ( y ) = T r M ( y ) σ . That is, in this case, the set of input signal correspond s to the set of extremal p oints of the set o f PO VM s on the g iv en system H . Th e propo sed sche me is illustrated in Fig. 7. One-way adapt ive improvement ρ σ or ρ σ or ρ σ or Measurement 1 y 1 M Measurement 2 M Measurement n M 2 y n y Adaptive improv ement is allowed Fig. 7. Adap ti ve quantum state discriminat ion Now , we assum e that ρ > 0 and σ > 0 . In this case, X is compact, an d the map ( s, M ) → d 2 φ ( s | W M k W M ) ds 2 is continuous. Then, the con dition (13) holds. Ther efore, one-way improvement d oes not improve the performan ce in the sense of the Stein bound , th e Cherno ff bou nd, the Hoeffding bound, or the Han-Kobayashi bo und. That is, we o btain B ( W k W ) = B ∗ ( W k W ) = max M :POVM D ( P M ρ k P M σ ) B e ( r | W k W ) = max M :POVM sup 0 ≤ s ≤ 1 − sr − φ ( s | P M ρ k P M σ ) 1 − s B ∗ e ( r | W k W ) = sup s ≤ 0 − sr − max M :POVM φ ( s | P M ρ k P M σ ) 1 − s . Therefo re, th ere exists a difference between one-way LOCC and collective measu rement. V I . P R O O F O F T H E S T E I N B O U N D : ( 1 2 ) Now , we prove the Stein b ound : (12). For a ny x ∈ X , by choosin g the input x in n time s, we ob tain B ( W k W ) ≥ D ( W x k W x ) . T aking the supremu m, w e have B ( W k W ) ≥ sup x ∈X D ( W x k W x ) . Furthermo re, from the definition, it is tr ivial tha t B ( W k W ) ≤ B ∗ ( W k W ) . Therefo re, it is sufficient to sh ow the strong conv erse p art: B ∗ ( W k W ) ≤ D . (19) Howe ver , in prepa ration for the proo f o f (15), we present a proof of the weak co n verse p art: B ( W k W ) ≤ D (20) 10 which is weaker argument than (1 9), an d is valid without assump tion (11). In the following proof , it is essential to evaluate the KL-d iv ergence concern ing the obtained data. In order to p rove (2 0), we prove th at lim n →∞ − 1 n log E Q W , ~ P n (1 − f n ) ≤ D (21) when E Q W, ~ P n f n → 0 . (22) It follows from the definition s o f Q W , ~ P n and Q W , ~ P n that D ( Q W , ~ P n k Q W , ~ P n ) = n X k =1 D ( W k W | P W , ~ P k ) . Since − E Q W, ~ P n f n log E Q W , ~ P n f n ≥ 0 , info rmation processing ineq uality concerning the KL divergence yields the fo llowing: − h (E Q W, ~ P n (1 − f n )) − (E Q W, ~ P n (1 − f n )) log E Q W , ~ P n (1 − f n ) ≤ E Q W, ~ P n (1 − f n )(log E Q W, ~ P n (1 − f n ) − log E Q W , ~ P n (1 − f n )) + E Q W, ~ P n f n (log E Q W, ~ P n f n − log E Q W , ~ P n f n ) ≤ D ( Q W , ~ P n k Q W , ~ P n ) = n X k =1 D ( W k W | P W , ~ P k ) ≤ n D . (23) That is, − 1 n log E Q W , ~ P n (1 − f n ) ≤ D + 1 n h (E Q W, ~ P n (1 − f n )) E Q W, ~ P n (1 − f n ) . (24) Therefo re, ( 22) yields (2 1). Next, we prove the strong con verse part, i.e. , we show that E Q W, ~ P n (1 − f n ) → 0 (25) when r := lim n →∞ − log E Q W , ~ P n (1 − f n ) n > D . (26) Since Φ( s | Q W , ~ P n k Q W , ~ P n ) =Φ( s | Q W , ~ P n − 1 k Q W , ~ P n − 1 )  Z X  Z Y ( ∂ W ′ x n ∂ W x n ( y n )) s W x n ( dy n )  P s,W | W , ~ P n ( dx n )  , we obtain φ ( s | Q W , ~ P n k Q W , ~ P n ) = φ ( s | Q W , ~ P n − 1 k Q W , ~ P n − 1 ) + φ ( s | W k W | P s,W | W , ~ P n ) . (27) Applying (27) indu ctiv ely , we obtain the relation φ ( s | Q W , ~ P n k Q W , ~ P n ) = n X k =1 φ ( s | W k W | P s,W | W , ~ P k ) ≤ nφ ( s | W k W ) . (28) Since the info rmation quantity φ ( s | P k P ) satisfies the inform ation processing inequality , we have (E Q W, ~ P n (1 − f n )) 1 − s (E Q W , ~ P n (1 − f n )) s ≤ (E Q W, ~ P n (1 − f n )) 1 − s (E Q W , ~ P n (1 − f n )) s + (E Q W, ~ P n f n ) 1 − s (E Q W , ~ P n f n ) s ≤ e φ ( s | Q W, ~ P n k Q W , ~ P n ) ≤ e nφ ( s | W k W ) , for s ≤ 0 . T ak ing the logarith m, we obtain (1 − s ) log E Q W, ~ P n (1 − f n ) ≤ − s log E Q W , ~ P n (1 − f n ) + nφ ( s | W k W ) . (29) 11 That is, − 1 n log E Q W, ~ P n (1 − f n ) ≥ − s − 1 n log E Q W , ~ P n (1 − f n ) − φ ( s | W k W ) 1 − s . When lim n →∞ − log E P n (1 − f n ) n ≥ r , the inequality B ∗ e ( r | W k W ) ≥ lim n →∞ − 1 n log E Q W, ~ P n (1 − f n ) ≥ − sr − φ ( s | W k W ) 1 − s holds. T aking the supr emum, we obtain B ∗ e ( r | W k W ) ≥ sup s ≤ 0 − sr − φ ( s | W k W ) 1 − s . From cond itions (11) and (26), there exists a small re al numb er ǫ > 0 such tha t r > φ ( − ǫ | W k W ) − ǫ . Thus, sup s ≤ 0 − sr − φ ( s | W k W ) 1 − s ≥ ǫr − φ ( − ǫ | W k W ) 1 + ǫ > 0 . Therefo re, w e obtain (2 5). Remark 2: The tech nique of the strong conv erse p art excep t for (2 8) was de veloped by Nagaoka [ 19]. Hen ce, der i ving ( 28) can be regard ed as the main contribution in this section of the pr esent paper . Proof of Lem ma 1: It is sufficient for a proo f of (11) to show that the un iformity of the conver gence φ ( − ǫ | W x k W x ) ǫ − D ( W x k W x ) → 0 concer ning x ∈ X . Now , we ch oose ǫ > 0 satisfying cond ition (1 3). Th en, there exists s ∈ [ − ǫ, 0 ] such that φ ( − ǫ | W x k W x ) ǫ − D ( W x k W x ) = 1 2 ǫφ ( s | W x k W x ) ≤ C 1 2 ǫ . Therefo re, the condition (11) hold s. V I I . P R O O F O F T H E H O E FF D I N G B O U N D : ( 1 5 ) In this section, we prove the Hoeffding bound: (15). Since the ine quality B e ( r | W k W ) ≥ sup x ∈X sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s = sup x ∈X min Q : D ( Q k W x ) ≤ r D ( Q k W x ) is tri vial, we pr ove the opp osite ineq uality . In th e follo wing pr oof, th e geometric c haracterizatio n Fig. 1 and the weak and th e strong converse parts are essential. Equation (6) guarantees that sup x ∈X min Q : D ( Q k W x ) ≤ r D ( Q k W x ) = sup x ∈X min s ∈ [0 , 1]: D ( P s,W x , W x k W x ) ≤ r D ( P s,W x , W x k W x ) . For this purpo se, for arbitrary ǫ > 0 , we ch oose a cha nnel V : V x = P s ( x ) ,W x , W x by s ( x ) := argmin s ∈ [0 , 1]: D ( P s,W x , W x k W x ) ≤ r D ( P s,W x , W x k W x ) . Assume that a sequ ence { ( ~ P n , f n ) } satisfies lim n →∞ − 1 n log E Q W , ~ P n (1 − f n ) = r. By substituting V into W , the strong converse p art of the Stein bound:(2 5 ) implies that lim E Q V , ~ P n (1 − f n ) = 0 . The conditio n (13) can be checked by the f ollowing relations: dφ ( t | P s ( x ) ,W x , W x k W x ) dt = (1 − s ( x )) φ ′ ( s ( x )(1 − t ) + t | W x k W x ) (30) d 2 φ ( t | P s ( x ) ,W x , W x k W x ) dt 2 = (1 − s ( x )) 2 φ ′′ ( s ( x )(1 − t ) + t | W x k W x ) . (31) Thus, by substitutin g V an d W into W and W , the relation (24) implies that lim n →∞ − 1 n log E Q W , ~ P n (1 − f n ) ≤ sup x ∈X D ( V x k W x ) . Similar to (30) and (31), we can check the co ndition (13). 12 From the constru ction of V , we obtain lim n →∞ − 1 n log E Q W , ~ P n (1 − f n ) ≤ max x min Q : D ( Q k W x ) ≤ r − ǫ D ( Q k W x ) . The unifo rm continuity guarantees that lim n →∞ − 1 n log E Q W , ~ P n (1 − f n ) ≤ max x min Q : D ( Q k W x ) ≤ r D ( Q k W x ) . Now , we sho w the uniformity of the function r 7→ sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s concern ing x . As mentioned in p. 82 of Hayashi[11], the relatio n d dr sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s = s r s r − 1 holds, where s r := arg max 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s . Since d dr − sr − φ ( s | W x k W x ) 1 − s     s = s r = 0 , we have r = ( s r − 1) φ ′ ( s r | W x k W x ) − φ ( s r | W x k W x ) . Since − φ ( s r | W x k W x ) ≥ 0 , ( s r − 1) ≤ 0 , and φ ′′ ( s | W x k W x ) ≥ 0 , r ≥ ( s r − 1) φ ′ ( s r | W x k W x ) ≥ ( s r − 1) φ ′ (1 | W x k W x ) = (1 − s r ) D ( W x k W x ) . Thus, r D ( W x k W x ) ≥ (1 − s r ) . Hence, | s r s r − 1 | ≤ 1 1 − s r ≤ D ( W x k W x ) r ≤ sup x D ( W x k W x ) r . Therefo re, th e fun ction r 7→ sup 0 ≤ s ≤ 1 − sr − φ ( s | W x k W x ) 1 − s is unifor m continuous with respect to x . V I I I . P R O O F O F T H E H A N - K O B A Y A S H I B O U N D : ( 1 7 ) The inequa lity B e ( r | W k W ) ≥ sup s ≤ 0 − sr − φ ( s | W k W ) 1 − s . (32) has been shown in Section VI, and the inequality B e ( r | W k W ) ≤ inf P ∈P 2 ( X ) sup s ≤ 0 − sr − φ ( s | W k W | P ) 1 − s can be easily chec k by consider ing the input P . Therefor e, it is suf ficient to show the inequality inf P ∈P 2 ( X ) sup s ≤ 0 − sr − φ ( s | W k W | P ) 1 − s ≤ sup s ≤ 0 − sr − φ ( s | W k W ) 1 − s = sup s ≤ 0 inf P ∈P 2 ( X ) − sr − φ ( s | W k W | P ) 1 − s . (33) This relation seems to be guaranteed b y the mini-max the orem (Chap. VI Prop. 2.3 of [5]). Howev er , the function − sr − φ ( s | W k W | P ) 1 − s is no t necessarily concave concerning s while it is conve x concerning P . Hen ce, this relation cann ot be guaranteed by the mini-max theore m. 13 Now , we pr ove this inequality wh en the maximu m max s ≤ 0 − sr − φ ( s | W k W ) 1 − s exists. Since φ ( s | W x k W x ) is co n vex concer ning s , φ ( s | W k W ) is also con vex concern ing s . Then, we can d efine ∂ + φ ( s | W k W ) := lim ǫ → +0 φ ( s + ǫ | W k W ) − φ ( s | W k W ) ǫ ∂ − φ ( s | W k W ) := lim ǫ → +0 φ ( s | W k W ) − φ ( s − ǫ | W k W ) ǫ . Hence, the real n umber s r := arg max s ≤ 0 − sr − φ ( s | W k W ) 1 − s satisfies that (1 − s r ) ∂ − φ ( s r | W k W ) + φ ( s r | W k W ) ≤ − r ≤ (1 − s r ) ∂ + φ ( s r | W k W ) + φ ( s r | W k W ) . That is, there exists λ ∈ [0 , 1] such that − r = (1 − s r )( λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r | W k W )) + φ ( s r | W k W ) . (34) For an arbitrary real nu mber 1 > ǫ > 0 , there exists 1 > δ > 0 such that φ ( s + δ | W k W ) − φ ( s | W k W ) δ ≤ ∂ + φ ( s | W k W ) + ǫ (35) φ ( s | W k W ) − φ ( s − δ | W k W ) δ ≥ ∂ − φ ( s | W k W ) − ǫ. (36) Then, we choo se x + , x − ∈ X such that φ ( s r + λδ | W k W ) − δ ǫ ≤ φ ( s r + λδ | W x + k W x + ) ≤ φ ( s r + λδ | W k W ) (37) φ ( s r − (1 − λ ) δ | W k W ) − δ ǫ ≤ φ ( s r − (1 − λ ) δ | W x − k W x − ) ≤ φ ( s r − (1 − λ ) δ | W k W ) . (38) Thus, (37) implies th at φ ( s r + λδ | W x + k W x + ) − φ ( s r − (1 − λ ) δ | W x + k W x + ) δ ≥ φ ( s r + λδ | W k W ) − δ ǫ − φ ( s r − (1 − λ ) δ | W k W ) δ ≥ φ ( s r + λδ | W k W ) − φ ( s r + | W k W ) + φ ( s r + | W k W ) − φ ( s r − (1 − λ ) δ | W k W ) − δ ǫ δ ≥ λδ ∂ + φ ( s r | W k W ) + (1 − λ ) δ ( ∂ − φ ( s r + | W k W ) − ǫ ) − δ ǫ δ = λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r + | W k W ) − ǫ. (39 ) Similarly , (38) imp lies that φ ( s r + λδ | W x − k W x − ) − φ ( s r − (1 − λ ) δ | W x − k W x − ) δ ≤ λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r + | W k W ) + ǫ. (40) Therefo re, th ere exists a real numb er λ ′ ∈ [0 , 1] such that     ϕ ( s r + λδ | λ ′ ) − ϕ ( s r − (1 − λ ) δ | λ ′ ) δ − ( λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r + | W k W ))     ≤ ǫ. (41) where ϕ ( s | λ ′ ) := λ ′ φ ( s | W x + k W x + ) + (1 − λ ′ ) φ ( s | W x − k W x − ) . Thus, there exists s r ∈ [ s r − (1 − λ ) δ, s r + λδ ] such that   ϕ ′ ( s r | λ ′ ) − ( λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r | W k W ))   ≤ ǫ. (42) The relation (4 1) also implies that 0 ≤ ϕ ( s r − (1 − λ ) δ | λ ′ ) − ϕ ( s r | λ ′ ) ≤ ϕ ( s r − (1 − λ ) δ | λ ′ ) − ϕ ( s r + λδ | λ ′ ) ≤ [ ǫ − (( λ ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r | W k W ))] δ ≤ ( ǫ − ∂ − φ ( s r | W k W )) δ. (43) 14 Since φ ( s r − (1 − λ ) δ | W x + k W x + ) ≥ φ ( s r + λδ | W x + k W x + ) , relations (36) and (3 7) guara ntee th at 0 ≤ φ ( s r − (1 − λ ) δ | W k W ) − φ ( s r − (1 − λ ) δ | W x + k W x + ) ≤ φ ( s r − (1 − λ ) δ | W k W ) − φ ( s r + λδ | W k W ) + φ ( s r + λδ | W k W ) − φ ( s r + λδ | W x + k W x + ) ≤ ( ǫ − ∂ − φ ( s r | W k W )) ( s r + λδ − s r ) + δǫ ≤ ( ǫ − ∂ − φ ( s r | W k W )) δ + δǫ = (2 ǫ − ∂ − φ ( s r | W k W )) δ. Therefo re, 0 ≤ φ ( s r − (1 − λ ) δ | W k W ) − ϕ ( s r − (1 − λ ) δ | λ ′ ) ≤ λ ′ ( φ ( s r − (1 − λ ) δ | W k W ) − φ ( s r − (1 − λ ) δ | W x + k W x + )) + (1 − λ ′ )( φ ( s r − (1 − λ ) δ | W k W ) − φ ( s r − (1 − λ ) δ | W x − k W x − )) ≤ λ ′ ( ǫ − ∂ − φ ( s r | W k W )) δ + (1 − λ ′ ) δ ǫ ≤ ( ǫ − ∂ − φ ( s r | W k W )) δ. (44) Since (36) imp lies that φ ( s r − (1 − λ ) δ | W k W ) − φ ( s r | W k W ) ≤ ( ǫ − ∂ − φ ( s r | W k W )) δ, relations (43) and (4 4) guara ntee th at | ϕ ( s r | λ ′ ) − φ ( s r | W k W ) | ≤| ϕ ( s r | λ ′ ) − ϕ ( s r − (1 − λ ) δ | λ ′ ) | + | ϕ ( s r − (1 − λ ) δ | λ ′ ) − φ ( s r − (1 − λ ) δ | W k W ) | + | φ ( s r − (1 − λ ) δ | W k W ) − φ ( s r | W k W ) | ≤ (4 ǫ − 3 ∂ − φ ( s r | W k W )) δ ≤ C 2 δ, (45) where C 2 := 4 − 3 ∂ − φ ( s r | W k W )) ≥ 4 ǫ − 3 ∂ − φ ( s r | W k W ) . Note that the constant C 2 does not depen d on ǫ or δ . W e choose a real n umber r := (1 − s r ) ϕ ( s r | λ ′ ) + ϕ ′ ( s r | λ ′ ) . Then, ( 45), (42), and the inequality | s r − s r | ≤ δ imply that | r − r | ≤| (1 − s r ) ϕ ( s r | λ ′ ) − (1 − s r ) φ ( s r | W k W )) | + | ϕ ′ ( s r | λ ′ ) − ( λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r + | W k W )) | ≤| (1 − s r )( ϕ ( s r | λ ′ ) − φ ( s r | W k W )) | + | φ ( s r | W k W )( s r − s r ) | + | ϕ ′ ( s r | λ ′ ) − ( λ∂ + φ ( s r | W k W ) + (1 − λ ) ∂ − φ ( s r + | W k W )) | ≤ (1 − s r ) C 2 δ + | φ ( s r | W k W ) | δ + ǫ ≤ C 3 δ + ǫ, (46) where C 3 :=(2 − s r ) C 2 + | φ ( s r | W k W ) | ≥ (1 − s r + (1 − λ ) δ ) C 2 + | φ ( s r | W k W ) | ≥ (1 − s r ) C 2 + | φ ( s r | W k W ) | . Note that th e constant C 3 does not depend on ǫ or δ . The function − s r − ϕ ( s | λ ′ ) 1 − s takes the maximu m at s = s r . Using (4 5) and (46), we can ch eck that this m aximum is appr oximated by the value − s r r − φ ( s r || W k W ) 1 − s r as | − s r r − ϕ ( s r | λ ′ ) 1 − s r − − s r r − φ ( s r | W k W ) 1 − s r | ≤| − s r r − ϕ ( s r | λ ′ ) 1 − s r − − s r r − φ ( s r | W k W ) 1 − s r | + | − s r r − φ ( s r | W k W ) 1 − s r − − s r r − φ ( s r | W k W ) 1 − s r | ≤| s r r − s r r 1 − s r | + | ϕ ( s r | λ ′ ) − φ ( s r | W k W ) 1 − s r | + | − s r r − φ ( s r | W k W )( s r − s r ) (1 − s r )(1 − s r ) ≤ | ( s r ( r − r ) | + | r ( s r − s r ) | 1 − s r + | ϕ ( s r | λ ′ ) − φ ( s r | W k W ) 1 − s r | + | − s r r − φ ( s r | W k W ) (1 − s r + 1)(1 − s r ) | δ ≤ ( − s r + δ )( C 3 δ + ǫ ) + rδ 2 − s r + | C 2 ǫ 2 − s r | + | − s r r − φ ( s r | W k W ) (2 − s r )(1 − s r ) | δ ≤ C 4 ǫ + C 5 δ, (47) 15 where we cho ose C 4 and C 5 as follows. C 4 := − s r + 1 2 − s r + | C 2 2 − s r | ≥ − s r + δ 2 − s r + | C 2 2 − s r | C 5 := ( − s r + 1) C 3 + rδ 2 − s r + | − s r r − φ ( s r | W k W ) (2 − s r )(1 − s r ) | ≥ ( − s r + δ ) C 3 + rδ 2 − s r + | − s r r − φ ( s r | W k W ) (2 − s r )(1 − s r ) | . Note that the constants C 4 and C 5 do not depe nd on δ or ǫ . Since | − sr − ϕ ( s | λ ′ ) 1 − s − − s r − ϕ ( s | λ ′ ) 1 − s | ≤ − s 1 − s | r − r | ≤ | r − r | , (46) implies that | max s ≤ 0 − sr − ϕ ( s | λ ′ ) 1 − s − max s ≤ 0 − s r − ϕ ( s | λ ′ ) 1 − s | ≤ | r − r | ≤ C 3 δ + ǫ. (48) Since ϕ ( s | λ ′ ) ≤ φ ( s | W k W ) , (48) and (47) guar antee that 0 ≤ max s ≤ 0 − sr − ϕ ( s | λ ′ ) 1 − s − − s r r − φ ( s r | W k W ) 1 − s r ≤ ( C 4 + 1) ǫ + ( C 3 + C 5 ) δ. (49) W e define the distribution P λ ′ ∈ P 2 ( X ) by P λ ′ ( x + ) = λ ′ , P λ ′ ( x − ) = 1 − λ ′ . Since the fun ction x → log x is concave, the inequality ϕ ( s | λ ′ ) ≤ φ ( s | W k W | P λ ′ ) (50) holds. Hence, (4 9) and (50) imply that 0 ≤ inf P ∈P 2 ( X ) max s ≤ 0 − sr − φ ( s | W k W | P ) 1 − s − − s r r − φ ( s r | W k W ) 1 − s r ≤ max s ≤ 0 − sr − φ ( s | W k W | P λ ′ ) 1 − s − − s r r − φ ( s r | W k W ) 1 − s r ≤ ( C 4 + 1) ǫ + ( C 3 + C 5 ) δ. W e take the limit δ → +0 . After this limit, we take the limit ǫ → +0 . Then, we obtain (33). Next, we prove the inequality (33) wh en the m aximum ma x s ≤ 0 − sr − φ ( s | W k W ) 1 − s does not exist. The real number R := lim s →−∞ φ ( s | W k W ) s satisfies r ≥ − R . Thus, sup s ≤ 0 − sr − φ ( s r | W k W ) 1 − s = r + R. For any ǫ > 0 , there exists s 0 < 0 such that any s < s 0 satisfies that R ≤ φ ( s 0 | W k W ) − φ ( s | W k W ) s 0 − s ≤ R + ǫ. W e choose x 0 such that φ ( s 0 − 1 | W k W ) − ǫ ≤ φ ( s 0 − 1 | W x 0 k W x 0 ) ≤ φ ( s 0 − 1 | W k W ) . Thus, φ ( s 0 | W x 0 k W x 0 ) − φ ( s 0 − 1 | W x 0 k W x 0 ) ≤ φ ( s 0 | W k W ) − φ ( s 0 − 1 | W k W ) + ǫ ≤ R + 2 ǫ. Hence, for any s < s 0 , φ ( s 0 | W x 0 k W x 0 ) − φ ( s | W x 0 k W x 0 ) s 0 − s ≤ φ ( s 0 | W x 0 k W x 0 ) − φ ( s 0 − 1 | W x 0 k W x 0 ) ≤ φ ( s 0 | W k W ) − φ ( s 0 − 1 | W k W ) + ǫ ≤ R + 2 ǫ. 16 Thus, − r ≤ R ≤ lim s →−∞ φ ( s | W x 0 k W x 0 ) s ≤ R + 2 ǫ. Therefo re, sup s ≤ 0 − sr − φ ( s r | W x 0 k W x 0 ) 1 − s ≤ r + R + 2 ǫ. T aking ǫ → 0 , we obtain (33). I X . C O N C L U D I N G R E M A R K S A N D F U T U R E S T U DY W e have obtain ed a general asymptotic form ula for the discrimination of two classical chann els with adaptiv e imp rovement concern ing the several asym ptotic formulatio ns. W e have pr oved that any adapti ve meth od does not improve the asympto tic perfor mance. That is, the no n-adap tiv e method attains the optimu m perf orman ce in these asy mptotic fo rmulation s. Applying the obtained result to the discrimination of two qu antum states by o ne-way LOCC, we have sho wn that on e-way communication does not improve the asymptotic perf ormance in these senses. On the o ther hand, as shown in Section 3 .5 of Hay ashi[11], we can not improve th e asymptotic perfo rmance of the Stein bound even if we exten d th e cla ss of our measur ement to the sep arable PO V M in the n - partite system. Hence, two-way LOCC does not improve th e Stein bound. Howe ver , othe r asympto tic pe rforma nces in tw o-way LOCC an d separab le PO V M have not been solved. Theref ore, it is an interesting problem to solve whether two-way LOCC impr oves the asy mptotic perfo rmance for other tha n the Stein’ s b ound . Furthermo re, the discrimination o f two quantum channels (TP-CP maps) is an interesting related top ic. An open problem remains a s to wheth er cho osing inp ut qu antum states adaptively improves the discr imination performance in an asy mptotic framework. 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