IEICE TRANS.??, VOL.Exx??, NO.xx XXXX 200x 1

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1 IEICE TRANS.??, VOL.Exx??, NO.xx XXXX 00x PAPER Specal Secto o Iformato Theory ad Its Applcatos A Fudametal Iequalty for Lower-boudg the Error Probablty for Classcal ad Quatum ultple Access Chaels ad Its Applcatos Takuya KUBO, Nomember ad Hrosh NAGAOKA, ember arxv: v [cs.it 4 ar 05 SUARY I the study of the capacty problem for multple access chaels (ACs), a lower boud o the error probablty obtaed by Ha plays a crucal role the coverse parts of several kds of chael codg theorems the formato-spectrum framework. Recetly, Yag ad Oohama showed a tghter boud tha the Ha boud by meas of Polyasky s coverse. I ths paper, we gve a ew boud whch geeralzes ad stregthes the Yag-Oohama boud, ad demostrate that the boud plays a fudametal role dervg extesos of several kow bouds. I partcular, the Yag-Oohama boud s geeralzed to two dfferet drectos;.e, to geeral put dstrbutos ad to geeral ecoders. I addto we exted these bouds to the quatum ACs ad apply them to the coverse problems for several formato-spectrum settgs. key words: quatum chael, multple access chael, error probablty, formato-spectrum. Itroducto The capacty problem for multple access chaels(acs) has bee a mportat topc sce Shao [9 studed t. Ths problem s studed for several kds of settgs. For stace, the classcal case, Ahlswede [ foud the sgleletterzed capacty rego for statoary ad memoryless chaels, Ha [[3 foud the capacty rego for the geeral chaels by meas of formato spectrum method, ad Wter [ foud that for statoary ad memoryless chaels the quatum case. However, there rema some fudametal problems to be solved, cludg the expoetal covergece of the error probablty the strogcoverse rego for statoary memoryless chaels ad the geeral formato-spectrum formula for the capacty rego the quatum case. So we stll eed to look for good lower bouds o the error probablty. I ths paper, we dscuss lower bouds o the error probablty for the followg three settgs, whch are smlar but slghtly dfferet from each other. Settg LetX,X adybe arbtrary dscrete sets o whch a put dstrbuto p(x, x ) ad a chael W(y x, x ) are gve. For a reversed chaelg(x, x y), whch meas the probablty of decodg (or estmatg) the put (x, x ) from the observed outputy, the error probablty s defed by auscrpt receved February 3, 05. auscrpt revsed February 3, 05. The authors are wth the Graduate School of Iformato Systems, The Uversty of Electro-Commucatos. DOI: 0.587/tras.E0.??. Pe(g) := x,x,y p(x, x )W(y x, x )g(x, x y). () Settg LetX,X adybe arbtrary dscrete sets o whch a chael W(y x, x ) s gve. Gve a par of message sets ad wth = ad = together wth ecoders f (x m ) ad f (x m ), whch meas the probabltes of ecodg the message m ad m to the puts x ad x respectvely, we defe the error probablty for a arbtrary decoder g(m, m y) by Pe(g) := m,m f (x m ) f (x m )W(y x, x )g(m, m y). () x,x,y Settg 3 LetX,X ady be arbtrary dscrete set s o whch a chael W(y x, x ) s gve. Gve a par of codebooks C X adc X wth C = ad C =, we defe the error probablty for a arbtrary decoder g(m, m y) by Pe(g) := x C,x C,y W(y x, x )g(x, x y). (3) Note that Settg 3 ca be regarded as specal cases of both Settg ad Settg. That s, Settg 3 s obtaed by restrctg p(x, x ) to the product of the uform dstrbutos o the codebooks Settg, ad s obtaed by restrctg ecoders f, f to be determstc ad jectve Settg. I the study of the capacty problem, Settg 3 have bee maly dealt wth so far, as metoed below for [[3 ad [, whle Poor ad Verdú [8 dscussed a lower boud of the error probablty Settg ad Polyasky [7 used Settg hs meta-coverse argumet. I Settg 3, Ha [[3 showed the followg lower boud, whch s kow as the Ha boud. For a arbtrary postve umber γ, t holds that Pe(g) Pr{(X, X, Y) L L L 3 } 3γ, (4) Copyrght c 00x The Isttute of Electrocs, Iformato ad Commucato Egeers

2 IEICE TRANS.??, VOL.Exx??, NO.xx XXXX 00x Pr deotes the probablty defed by the jot dstrbuto p(x, x,y)= p u, (x )p u, (x )W(y x, x ) for the uform dstrbutos p u, ad p u, o the codebooks, ad L :={(x, x,y) W(y x, x )γ p(y x )}, (5) L :={(x, x,y) W(y x, x )γ p(y x )}, (6) L 3 :={(x, x,y) W(y x, x )γ 3 p(y)}, (7) 3 :=. (8) Ths boud s a AC exteso of the Verdú-Ha boud [0 ad plays a crucal role the coverse parts of several codg theorems for geeral AC chaels. Recetly Yag ad Oohama [ showed a tghter boud as follows. For a arbtrary codtoal dstrbuto q(y x, x ), a arbtrary dstrbutoπo{,, 3}, ad a arbtrary postve umberγ, t holds that Pe(g) Pr{(X, X, Y) L} γ 3 = π, (9) L :={(x, x,y) W(y x, x )γ q(y x, x )}, (0) q(y x, x )=π q(y x )+π q(y x )+π 3 q(y), () ad q(y x ), q(y x ) ad q(y) are the codtoal ad margal dstrbutos defed from the jot dstrbuto q(x, x,y)= p u, (x )p u, (x )q(y x, x ). () If we setπ = j j, γ = γ j j ad q(y x, x ) = p(y x, x ), we ca rewrte (9) ad (0) as follows. Pe(g) Pr{(X, X, Y) L} 3γ, (3) L={(x, x,y) W(y x, x )γ( p(y x )+ p(y x )+ 3 p(y))}. (4) Sce L L L 3 L, (9) s tghter tha (4). I what follows, we frst show a exteso of the Yag- Oohama boud (9) as Theorem secto, the Yag-Oohama boud s exteded from Settg 3 to Settg ad, addto, s slghtly stregtheed as s see subsecto 3.. We also see subsecto 3. that the theorem yelds a AC verso of the Poor-Verdú boud. I secto 4, we use Theorem aga to obta a exteso of the Yag- Oohama boud to Settg. I secto 5, we show that these results are aturally exteded to the quatum case. Lastly secto 6, we apply them to obta some asymptotc results whch correspod to the coverse parts of the geeral capacty theorems obtaed by Ha the classcal case. Cocludg remarks are gve secto 7.. A fudametal equalty o the error probablty for the classcal ACs The followg equalty plays a fudametal role ths paper. Theorem. I Settg gve secto, for a arbtrary decoderg, arbtraryα,α,α 3 0, a arbtrary probablty dstrbuto q(y) o Y, ad arbtrary oegatve-valued fuctos q (x,y), q (x,y) satsfyg that q(y) q (x,y) ad q(y) q (x,y) ( x, x,y), we have Pe(g) α [p(x, x.y) q α (x, x,y) +, x,x,y (5) q α (x, x,y)=α q (x,y)+α q (x,y)+α 3 q(y), (6) [t + = max{0, t}. (t R) (7) Proof. As the proof of Neyma-Pearso s Lemma, t follows from 0g(x, x y) that [(p(x, x.y) q α (x, x,y) + x,x,y {(p(x, x.y) q α (x, x,y)}g(x, x y) x,x,y Pe(g) α x,x,y q(y)g(x, x y) = Pe(g) α, (8) the secod equalty follows from q(y) q (x,y) ad q(y) q (x,y). 3. Corollares of Theorem Settg 3. A Yag-Oohama-type boud The Yag-Oohama boud s exteded to the geeral put dstrbutos the followg form. Corollary. Pe(g) Pr{p(X, X, Y)q α (X, X, Y)} α (9) Proof. Eq. (9) mmedately follows from (5), sce [p(x, x.y) q α (x, x,y) + x,x,y = (p(x, x.y) q α (x, x,y)) x,x,y x,x,y {p(x, x.y)>q α (x, x,y)} p(x, x.y) {p(x, x.y)>q α (x, x,y)} = Pr{p(X, X, Y)q α (X, X, Y)}, (0) {}s the dcator fucto. I Settg 3, the orgal Yag-Oohama boud (3) s obtaed from (9) by settgα =γ π /.

3 KUBO ad NAGAOKA: A FUNDAENTAL INEQUALITY FOR LOWER-BOUNDING THE ERROR PROBABILITY FOR CLASSICAL AND QUANTU ULTIPLE ACCESS CHANNELS AND 3. A Poor-Verdú-type boud Whle Corollary ca be regarded as a AC exteso of the Verdú-Ha boud [0 (or the Hayash-Nagaoka boud [4 the sese that arbtrary output dstrbutos are allowed), the followg boud correspods to the Poor-Verdú boud [8. Corollary. Pe(g) ( ) α Pr{p(X, X, Y) p α (X, X, Y)}, () p α (x, x,y)=α p(x,y)+α p(x,y)+α 3 p(y), () ad p(x,y), p(x,y) ad p(y) are margal dstrbutos defed from the jot dstrbuto p(x, x,y). Proof. If q = p, the rght had sde of (5) s rewrtte as (p(x, x.y) p α (x, x,y)) x,x,y ( α ) {p(x, x.y)> p α (x, x,y)} p(x, x.y) x,x,y {p(x, x.y)> p α (x, x,y)}, (3) the equalty follows from p(x, x,y) p(x, x,y) p(x,y), ad p(x, x,y) p(x,y). 4. Corollares of Theorem Settg p(y), A exteso of the Yag-Oohama boud to Settg, geeral stochastc ecoders are allowed, s also derved from Theorem as follows. Corollary 3. I Settg, for a arbtrary decoderg, arbtraryγ,γ,γ 3 0, a arbtrary dstrbuto q oy, ad arbtrary codtoal dstrbutos q (y x ), q (y x ) satsfyg that q(y) q (m,y) := x f (x m )q (y x ), (4) q(y) q (m,y) := x f (x m )q (y x ), (5) we have Pe(g) x,x,y γ ( m, m,y) p (x )p (x ) [ W(y x, x ) q γ (y x, x ) +, (6) q γ (y x, x )=γ q(y x )+γ q(y x )+γ 3 q(y), (7) p (x )= f (x m ), (8) m p (x )= f (x m ), (9) m 3 =. (30) Proof. Let a chael V from toybe defed by V(y m, m )= f (x m ) f (x m )W(y x, x ). (3) x,x The, replacgx,x ad W wth,, ad V Theorem ad lettg the put dstrbuto be uform o, we have Pe(g) m,m,y [ γ V(y m, m ) q γ (m, m,y), (3) + q γ (m, m,y)= γ q (m,y)+ γ q (m,y)+ γ 3 3 q(y). From the covexty of t [t +, we have Pe(g) m,m,x,x,y γ f (x m ) f (x m ) (33) [W(y x, x ) q γ (y x, x ) +. (34) Ths equalty mmedately derves the followg boud, whch s the drect exteso of the Yag-Oohama boud to Settg. Corollary 4. I Settg, for a arbtrary decoder g, a arbtrary dstrbutoπo{,, 3}, a arbtrary umberγ 0, ad a arbtrary chael q(y x, x ), we have π Pe(g) Pr{W(Y X, X )γ q(y X, X )} γ, (35) the radom varables X, X, ad Y are defed by the jot dstrbuto p(x, x,y)= f (x m ) f (x m )W(y x, x ), m,m (36) ad q s defed by ().

4 4 IEICE TRANS.??, VOL.Exx??, NO.xx XXXX 00x 5. Lower bouds o the error probablty for the quatum ACs I ths secto we exted the argumets of prevous sectos to classcal-quatum ACs. I the sgle access case, Hayash ad Nagaoka [4 exteded the Verdú-Ha boud to the quatum case, ad the preset authors [5, [6 exteded the Poor-Verdú boud. Applyg a smlar argumet to the oes developed there, we exted Theorem as preseted Theorem, from whch the correspodg results to Corollares -3 mmedately follow. We beg wth rewrtg Settg ad Settg to the quatum stuato. Settg 3 s omtted sce t s cluded Settg ad Settg. Settg Q LetX,X be arbtrary dscrete sets o whch a put dstrbuto p(x, x ) s gve. LetH be a arbtrary Hlbert space ads(h) be the set of desty operators oh ad W : X X S(H) be a classcal-quatum chael (a quatum chael, for short). Whe a POV (Postve Operator-Valued easure) Y={Y x,x }, whch satsfes that x,x Y x,x = I ad Y x,x 0 ( x, x ), represets a decodg (or estmatg) process, the error probablty s defed by Pe(Y) := p(x, x )Tr[W x,x Y x,x. (37) x,x Settg Q LetX,X be arbtrary dscrete sets,h be a arbtrary Hlbert space ad a quatum chael W :X X S(H) s gve. As Settg, gve a par of message sets ad wth = ad = together wth ecoders f (x m ) ad f (x m ), whch meas the probabltes of ecodg the message m ad m to the puts x ad x respectvely, we defe the error probablty for a arbtrary POV Y whose dexes are by Pe(Y) := m,m f (x m ) f (x m )Tr[W x,x Y m,m. (38) x,x Theorem s exteded as follows. Theorem. I Settg Q, for a arbtrary POV Y, arbtraryα,α,α 3 0, a arbtrary desty operatorσoh, ad arbtrary postve semdefte operatorsσ x,σ x satsfyg thatσ σ x adσ σ x ( x, x ), we have Pe(Y) α Tr[(p(x, x )W x,x σ α,x,x ) +, x,x (39) σ α,x,x =α σ x +α σ x +α 3 σ, (40) A + := A{A 0}. (4) Here ad the sequel, we use the otato{a B}={B A} to mea a projector oh whch s defed as follows. Whe A - B s spectrum-decomposed as A B= λ E, (4) {AB} := E. (43) :λ 0 Proof. As the classcal case, t follows from 0Y x,x I that Tr[(p(x, x )W x,x σ α,x,x ) + x,x Tr[(p(x, x )W x,x σ α,x,x )Y x,x x,x Pe(Y) α Tr[σY x,x x,x = Pe(Y) α, (44) the secod equalty follows fromσ σ x adσ σ x. Obvously, as Theorem derves Corollares ad, Theorem derves the followg corollares. Corollary 5. Pe(Y) p(x, x )Tr[W x,x {p(x, x )W x,x σ α,x,x } x,x α (45) Corollary 6. Pe(Y) ( α ) p(x, x )Tr[W x,x {p(x, x )W x,x W α,x,x }, x,x (46) W α,x,x =α W p,x +α W x,p+α 3 W p, (47) W p := p(x, x )W x,x, (48) x,x W p,x := p(x, x )W x,x, (49) x W x,p := p(x, x )W x,x. (50) x Corollary 5 s a AC exteso of the Hayash- Nagaoka boud, ad Corollary 6 s a quatum AC exteso of the Poor-Verdú boud.

5 KUBO ad NAGAOKA: A FUNDAENTAL INEQUALITY FOR LOWER-BOUNDING THE ERROR PROBABILITY FOR CLASSICAL AND QUANTU ULTIPLE ACCESS CHANNELS AND Corollary 3 s also exteded to the followg, whch ca be proved almost parallel wth the classcal oe, otg that the covexty of t [t + should be replaced wth the covexty of A Tr[A +. Corollary 7. I Settg Q, for a arbtrary POV Y, arbtraryγ,γ,γ 3 0, a arbtrary desty operatorσ oh, ad arbtrary desty operatorsσ x,σ x satsfyg that σ σ m := f (x m )σ x, x (5) σ σ m := f (x m )σ x, x (5) we have Pe(Y) γ ( m, m ) p (x )p (x )Tr[(W x,x σ γ,x,x ) +, (53) x,x σ γ,x,x =γ σ x +γ σ x +γ 3 σ, (54) p (x )= f (x m ), (55) m p (x )= f (x m ). (56) m 6. Applcatos of Theorem to the quatum formato spectrum settg,m() I ths secto, we show applcatos of Theorem to the quatum AC codg problems; the coverse parts of the ε-capacty rego problem ad the strog coverse rego problem, whch Ha [ [3 showed the classcal case. Let us troduce the settg of the quatum AC codg problem. Let X = {X () } = ad X = {X () } = be sequeces of dscrete sets, adh = {H () } = be a sequece of Hlbert spaces, for whch a sequece of quatum ACs W ={ :X () X () S(H () )} = s gve. Suppose that, for each, a par of ecoders ad a decoder are gve terms of codtoal probablty dstrbutos f ( m() ), f ( m() ) ad a POV Y() ={Y () } respectvely, {,..., () }, m() {,..., () }. The error probablty s the defed as follows: Pe () (Y () )=,m() () () Tr[ f ( m() ) f ( m() ) Y (),m(). (57) Here, we call a trple of ecoders ad decoder ( f (), f (), Y() ) whose error probablty equalsε a (, (), (),ε )-code. Now, we troduce the ε-capacty rego C(ε W). Defto. The ε-capacty rego C(ε W) s defed as C(ε W) :={(R, R ) {(, (), (),ε )-code} = s.t. lm supε ε, lm f lm f log () R, log () R }. (58) We also troduce C ( W) whch represets the complemet of the strog coverse rego. Defto. C ( W) :={(R, R ) {(, (), (),ε )-code} = s.t. lm f ε <, lm f lm f log () R, log () R }. (59) Next, we troduce the followg quattes. Defto 3. K(R, R p, p, σ) := lm sup,x() (x() )p() (x() )Tr[W() { e R σ () + e R σ () + e (R +R ) σ () }, (60) K (R, R p, p, σ) := lm f {,x() e R σ () (x() )p() (x() )Tr[W() + e R σ () + e (R +R ) σ () }, (6) p ={ } = ad p ={ } = are sequeces of probablty dstrbutos o X ad X, ad σ s a sequece of a trple of desty operators(σ (),σ (),σ () ) whch satsfes that σ () = ad σ () = for each. (x() (x() )σ() )σ() Wth these otatos, we have Theorem 3. (6) (63)

6 6 C(ε W) Cl({(R, R ) K(R, R p, p, σ)ε}), p, p σ Cl( ) deotes the closure operato. (64) Proof. If (R, R ) C(ε W), the from the dfto of C(ε W) there exsts a sequece of (, (), (),ε )-codes satsfyg that () e (R γ), (65) () e (R γ), (66) for a arbtrary postve umber γ ad all suffcetly large, ad lm supε ε. (67) Usg these codes, settg the sequeces of the put dstrbutos as (x() )= f () () (x() m() ), (68) (x() )= () f () (x() m() ). (69) Now, from Corollary 7, for arbtrary σ satsfyg (6) ad (63), we have ε 3e γ (x() )p() (x() ) [( Tr ( e γ () σ() + () σ() )) + () () σ(). + (70) From (65), (66) ad from the fact that for arbtrary Hermta operators A, B, Tr[A + Tr[B + f A B, we have ε 3e γ [( (x() )p() (x() )Tr A () B () [( (x() )p() (x() )Tr [ (x() )p() (x() )Tr = e (R γ) σ () = e (R γ) σ () ) A () + ) B () + { > B () (7) + e (R γ) σ () + e (R +R 3γ) σ (), (7) + e (R γ) σ () + e (R +R 4γ) σ (). (73) }, Therefore, t follows that [ ε (x() )p() (x() )Tr Hece, from (67) ad (74) we have IEICE TRANS.??, VOL.Exx??, NO.xx XXXX 00x { } B () 3e γ. (74) K(R γ, R γ p, p, σ)lm supε ε. (75) Sceγs arbtrary, (75) mples that (R, R ) Cl({(R, R ) K(R, R p, p, σ)ε}). (76) We also have Theorem 4. C ( W) Cl({(R, R ) K (R, R p, p, σ)<}). p, p σ (77) Proof. Let ( ) c deote the complemet adr( p, p, σ) := Cl({(R, R ) K (R, R p, p, σ) < }). Suppose that for arbtrary p, p, there exsts σ satsfyg that (R, R ) R( p, p, σ) c, whch meas that (R, R ) belogs to the rght had sde of (77). The (R γ, R γ) s also R( p, p, σ) c for suffcetly small postve umberγ sce R( p, p, σ) c s ope. Ths mples that K (R γ, R γ p, p, σ)=. (78) O the other had, for (, (), (),ε )-codes whch satsfes (65) ad (66) forγwhch s used (78) ad for all suffcetly large, the sequeces of the put dstrbutos whch are set as (68) ad (69) are clearly depedet. Furthermore, for such codes we have (74). Hece, from (74) ad (78) we have lm ε =. (79) Ths meas that (R γ, R γ) C ( W) c. For arbtrary postve umbers S < S ad S < S, f (S, S ) C ( W) c, the clearly (S, S ) C ( W) c from the defto of C ( W). Therefore, (R, R ) C ( W) c. The rest of ths secto s devoted to show how our results lead to the coverse parts of classcal capacty theorems obtaed by Ha [ [3. Frst, let W p = {( p(),,p() :=,p() :=,p(), )} be defed as (x() )p() (x() (x() )W )W(), (80), (8)

7 KUBO ad NAGAOKA: A FUNDAENTAL INEQUALITY FOR LOWER-BOUNDING THE ERROR PROBABILITY FOR CLASSICAL AND QUANTU ULTIPLE ACCESS CHANNELS AND := (x() )W. (8) The from Theorems 3 ad 4 we have C(ε W) Cl({(R, R ) K(R, R p, p, W p )ε}) (83) p, p ad C ( W) Cl({(R, R ) K (R, R p, p, W p )<}). (84) p, p I the classcal case, recallg that the Yag-Oohama boud mples the Ha boud, we ca easly show that ad K(R, R p, p, W p ) J(R, R X, X ) K (R, R p, p, W p ) J (R, R X, X ), J ad J are dfed [ [3. Therefore we have {(R, R ) Kε} {(R, R ) Jε}, (85) {(R, R ) K < } {(R, R ) J < }. (86) Note that Ha also proved ther drect parts [ [3, whch establsh capacty formulas: C(ε W)= Cl({(R, R ) Jε}), (87) p, p C ( W)= Cl({(R, R ) J < }), (88) p, p although (88) s ot explctly preseted [ [3. As a cosequece, we have Cl({(R, R ) Kε})= Cl({(R, R ) Jε}) p, p p, p (89) Cl({(R, R ) K < })= Cl({(R, R ) J < }) p, p p, p (90) geeral quatum ACs as applcatos of the fudametal equalty. It however remas to obta a good upper boud o the error probablty order to determe these regos. Refereces [ Ahlswede., R., ult-way commucato chaels., d It. Symp. If. Theory, pp Hugara Academy of Sceces, Budapest, 97. [ Ha, T. S., A formato-spectrum approach to capacty theorems for the geeral multple-access chael, IEEE Tras. Iform. Theory, vol. 44, o. 7, pp , 998. [3 Ha, T. S., Iformato-Spectrum ethods Iformato Theory, Sprger, 003. [4 Hayash,. ad Nagaoka, H., Geeral formulas for capacty of classcal-quatum chaels, IEEE Tras. Iform. Theory, vol. 49, o. 7, pp , 003. [5 Kubo, T. ad Nagaoka, H., Lower bouds of the error probablty estmatg classcal ad quatum states, QIT5 ( Japaese), Osaka, Japa, November 0. [6 Kubo, T. ad Nagaoka, H., Lower Bouds o the Error Probablty Classcal ad Quatum State Dscrmato, ISITA0, Hoolulu, USA, October 0. [7 Polyasky, Y., Chael codg: o-asymptotc fudametal lmts, Ph. D. thess, Departmet of Electrcal Egeerg, Prceto Uversty, 00. [8 Poor, H. V. ad Verdú, S., A lower boud o the probablty of error multhypothess testg, IEEE Tras. Iform. Theory, vol. 4, o. 6, pp , 995. [9 Shao, C. E., Two-way commucato chaels, 4th Berkeley Symp. ath. Stat. Prob., vol., pp Uversty of Calfora Press, Berkeley, CA, 96. [0 Verdú, S. ad Ha, T. S., A Geeral formula for chael capacty, IEEE Tras. Iform. Theory, vol. 40, o. 4, pp , 994. [ Wter, A., The Capacty of the Quatum ultple-access Chael, IEEE Tras. Iform. Theory, vol. 47, o. 7, pp , 00. [ Yag, H. ad Oohama, Y., Fte Blocklegth Aalyss for ultple Access Chaels ad Composte Hypothess Testg, SITA0 ( Japaese), Ota, Japa, December 0. the classcal case. I the quatum case, o the other had, sce we have ot prove ther drect parts, t s ot clear whether Theorem 3 ad 4 are tght. 7. Cocludg Remarks We have dscussed lower bouds o the error probablty for ACs several settgs. We have obtaed a fudametal equalty the classcal case (Theorem ) ad the quatum case (Theorem ). Usg the equalty the Yag-Oohama boud has bee geeralzed ad stregtheed several drectos ad exteded to the quatum case. We have also show coverse results o theε-capacty rego problem ad the strog coverse rego problem for