Generalizations of Pauli channels

Transcription

1 Acta Math. Hugar. 24(2009, Geeralzatos of Paul chaels Dées Petz ad Hromch Oho 2 Alfréd Réy Isttute of Mathematcs, H-364 Budapest, POB 27, Hugary 2 Graduate School of Mathematcs, Kyushu Uversty, Japa Abstract The Paul chael actg o 2 2 matrces s geeralzed to a -level quatum system. Whe the full matrx algebra M s decomposed to parwse complemetary subalgebras, the trace-preservg lear mappgs M M are costructed such that the restrcto to the subalgebras are depolarzg chaels. The result s the ecessary ad suffcet codto of complete postvty. The ma examples appear o bpartte systems Mathematcs Subject Classfcato. Prmary 5A30, 94A40; Secodary 47N50. Key words ad phrases: Completely postve mappgs, complemetary subalgebras, chael, codtoal expectato. I ths paper we cosder partcular subalgebras of a full matrx algebra M = M (C. (By a subalgebra we mea *-subalgebra wth ut. A F-subalgebra s a subalgebra somorphc to a full matrx algebra M k. ( F s the abbrevato of factor, the ceter of such a subalgebra s mmal, CI. A M-subalgebra s a maxmal Abela subalgebra, equvaletly, t s somorphc to C. ( M s a abbrevato of MASA, the ceter s maxmal, t s the whole subalgebra. If A ad A 2 are subalgebras, the they are called quas-orthogoal or complemetary f the subspaces A CI ad A 2 CI are orthogoal (wth respect to the Hlbert-Schmdt er product A, B = Tr A B. Cocerg complemetary subalgebras we refer to [8], see also [6, 7, 9]. Complemetary M-subalgebras ca be gve by mutually ubased bases. Assume that ξ, ξ 2,..., ξ ad η, η 2,..., η are orthoormal bases such that ξ, η j = (, j.

2 If A s the algebra of all operators wth dagoal matrx the frst bass ad A 2 s defed smlarly wth respect to the secod bass, the A ad A 2 are complemetary M-subalgebras. There are examples such that M s the lear spa of parwse complemetary subalgebras the case whe s a power of a prme umber. If M s decomposed to complemetary subalgebras, the we costruct trace-preservg mappgs M M whch are completely postve uder some codtos. Itroducto If the parwse complemetary subalgebras A, A 2,..., A r of M are gve ad they learly spa the whole algebra M, the ay operator s the sum of the compoets the subspaces A CI ( r ad CI: A = (r Tr A I + r E (A, where E : M A s the trace-preservg codtoal expectato (whch s othg else but the orthogoal projecto wth respect to the Hlbert-Schmdt er product, see [0] about detals. It s easer to formulate thgs for matrces of trace 0. If Tr B = 0, the t has orthogoal decomposto B = r E (B. As a geeralzato of the Paul chael o a qubt, we defe a lear mappg α : M M such that r α(b = λ E (B or for a arbtrary A α(a = ( r Tr A r λ I + λ E (A, ( where λ R, r. We wat to fd the codto for complete postvty. The motvato s the followg well-kow example whch the complemetary subalgebras are geerated by the Paul matrces [0]. Example Let σ 0 = I ad σ, σ 2, σ 3 be Paul matrces,.e., [ ] [ ] [ σ =, σ 0 2 =, σ 0 3 = 0 2 ]

3 ad let E : M 2 M 2 be defed as for ω C, where λ R ad E (w 0 σ 0 + (w, w 2, w 3 σ = w 0 σ 0 + (λ w, λ 2 w 2, λ 3 w 3 σ (2 (w, w 2, w 3 σ = w σ + w 2 σ 2 + w 3 σ 3. Desty matrces are set to desty matrces f ad oly f λ. It s ot dffcult to compute the represetg block matrx X :=,j E(E j E j, we have +λ 3 λ 0 0 +λ λ X = 0 3 λ λ λ 0 λ 2 λ λ +λ 2 +λ Accordg to Cho s theorem [2] the postvty of ths matrx s equvalet to the complete postvty of E. X s utarly equvalet to the matrx +λ 3 λ +λ λ +λ 2 +λ λ λ λ Ths matrx s obvously postve f ad oly f λ λ 2 2 λ 3 2 ± λ 3 λ ± λ 2. (3 Ths s ecessary ad suffcet codto of complete postvty. It s ot obvous that codto (3 s symmetrc the three varables λ, λ 2, λ 3. Codto (3 actually determes the tetrahedro whch s the covex hull of the pots (,,, (,,, (,, ad (,,. Now we show the dea leadg to the geeralzato. The mappg E (2 has the form 3 E( = µ σ ( σ. =0 From the expaso of E(σ j we ca get equatos ad the soluto s the followg: µ 0 = 4 ( + λ + λ 2 + λ 3, µ = 4 ( + λ λ 2 λ 3, µ 2 = 4 ( λ + λ 2 λ 3, µ 3 = 4 ( λ λ 2 + λ 3. 3

4 If µ 0 for every, the E s a completely postve mappg. Therefore, + λ 3 ±(λ + λ 2, λ 3 ±(λ λ 2 or together ths s (3. (Actually, ths argumet gves that (3 s a suffcet codto for the complete postvty. Paul chaels form a mportat ad popular subject quatum formato theory [, 3, 4]. The mappgs ( were studed the paper [5] the case whe the subalgebras are maxmal Abela ad parwse complemetary. Our method s dfferet ad we allow o-commutatve subalgebras as well. The mappg ( restrcted to A has the form D λ D + ( λ I o desty matrces D. If 0 λ, the we ca say that D does ot chage wth probablty λ ad wth probablty λ t s set to the tracal state. Such mappgs are usually called as depolarzg chaels [0]. A smple example cludg o-commutatve subalgebras s the followg. Example 2 Cosder M 4 = M 2 M 2 ad the complemetary F-subalgebras A,..., A 4 geerated by the followg trplets of utares: σ 0 σ σ 0 σ 2 σ 0 σ 3, σ σ 0 σ 2 σ σ 3 σ, σ 2 σ 0 σ 3 σ 2 σ σ 2, σ 3 σ 0 σ σ 3 σ 2 σ 3. We take also the M-subalgebra A 5 geerated by σ σ, σ 2 σ 2, σ 3 σ 3. The codtoal expectatos E j : M 4 A j are covex combatos of automorphsms E j (A = 4 4 UjAU j, (4 where U j = I ad U j s are orthogoal utares from A j. Sce A 5 s a M-subalgebra, A 5 = A 5. The subalgebras A,..., A 4 are F-subalgebras geerated by the followg utares: σ σ 0 σ 2 σ 0 σ 3 σ 0, σ 0 σ σ σ 2 σ σ 3, σ 2 σ σ 0 σ 2 σ 2 σ 3, σ 3 σ σ 3 σ 2 σ 0 σ 3. (The above trplets geeratg A j ad A j ( j 4 are Paul trplets, see [7] for detals. Moreover, (Tr AI = ( 5 4 A + U 4 jkau jk. j= k=2 4

5 The lear mappg ( has the cocrete form α(a = ( 5 Tr A 5 λ 4 I + λ E (A, where the codtoal expectatos E j s expressed by the commutat, see (4. (The codto for complete postvty of α s Theorem 4. Our ma result s the ecessary ad suffcet codto for the complete postvty of mappgs lke ( whch ca be called geeralzed Paul chael. 2 Geeralzed Paul chaels Let A be a (utal *- subalgebra of M. Our am s to descrbe the codtoal expectato oto A by meas of a orthogoal system the commutat. Up to utary equvalece, a subalgebra A of M ca be wrtte as A = k M I m. The commutat A M s A = k I M m. Let N = k 2 ad let P be a mmal cetral projecto of A, that s, P = I I m. Proposto Let {U } N be a orthoormal bass of A. The the completely postve map F from M oto A gve by F (X = N U XU (X M s equal to F (X = k m Tr (P XP, where Tr s a partal trace from M M m oto M m. I partcular, f all /m are equal, the expectato from M oto A. F s the trace-preservg codtoal dma 5

6 Proof: If all /m are equal, the ther rato s equal to dma. Therefore t s suffcet to prove the frst asserto. Let {e (l j } l,j= s wrtte by ad {f (l j }m l,j= be matrx uts of M l U = k l l= s,t= U (l,st e(l st ad M ml, respectvely. The U for some U (l,st C. The operator W M N gve terms of ts matrx etres by the formula W,(l,s,t = m l U (l,st for N, l k ad s, t l s utary. Ideed, W ca be cosdered as the matrx whch takes the orthoormal bass { ml e (l st } of A to the orthoormal bass {U }. Hece we have N W,(l,s,t W,(l,s,t = δ ll δ ss δ tt. ad therefore N U (l,st U (l,s t = δ ll δ ss δ tt. (5 m l Let T be a partal sometry wth T T = e (l s s f (l t t ad T T = e (l 2 s 2 s 2 f (l 2 t 2 t 2. The we obta F (T = N U T U = = δ l l 2 δ s s 2 δ pq l p= N l2 l U (l 2,s 2 p U (l,s q e(l 2 ps 2 T e (l s q p= q= e (l ps m T e (l s p l by (5 so that F maps the off-dagoal part to 0, that s, f l l 2 the F (T = 0. Now let T = e (l s 2 s f (l t 2 t. The we obta whch shows the frst asserto. l F (T = δ s s 2 e (l pp f (l t m 2 t = δ s s 2 I l f t2 t l m l p= = l m l Tr l (T The commutat of M-subalgebras ad F-subalgebras are aga M-subalgebras ad F-subalgebras, ad both types t s possble to choose a orthogoal bass cosstg of utares, oly. Thus by a applcato of the prevous proposto, for such a 6

7 subalgebra A, the trace-preservg codtoal expectato s the covex combato of automorphsms: X m U dma XU (X M, where {U } s a orthogoal bass of A cosstg of utares. Bases cosstg of utares are mportat also quatum state teleportato [3]. Theorem Let {U : m} be a orthoormal system M. The the lear mappg α(a = µ U AU s completely postve f ad oly f µ 0 for every m. Proof: If µ 0 for every m, t s clear that α s completely postve. To prove the coverse, we frst show that W E j W E j,j s a projecto f Tr W W =. Ths s obvously self-adjot ad we ca compute that t s dempotet: ( ( W E j W E j W E kl W E kl,j k,l =,j,l W E j W W E jl W E l = Tr W W (,l W E l W E l. It follows that P k :=,j U k E j U k E j s a projecto for every k m. To show that they are parwse orthogoal, we compute the trace of P k P l : Tr P k P l = Tr Uk E j U k Ul E uv U l E j E uv =,j,j,u,v Tr U k E j U k U l E j U l =,j Tr E j U k U l E j U l U k. Due to the Lemma below ths equals Tr U k Ul Tr U l Uk = 0 whe k l. The complete postvty mples that ( µ k Uk E j U k E j = ( µ k Uk E j U k E j =,j k k,j k s postve, therefore µ k 0. 7 µ k P k

8 Lemma Tr E j XE j Y = (Tr X(Tr Y.,j Proof: Sce both sdes are blear the varables X ad Y, t s eough to check the case X = E ab ad Y = E cd. Smple computato gves that left-had-sde s δ ab δ cd. A physcst mght make a dfferet proof of the lemma: Tr E j XE j Y = Tr e e j X e j e Y = e j X e j e Y e,j,j,j ad the rght-had-sde s (Tr X(Tr Y. We also eed the ext lemma; the proof ca be foud [3]. Lemma 2 Let V, V 2,..., V 2 be matrces M. The the followg codtos are equvalet:. Tr V V j = δ j (, j 2, 2. 2 V AV = (Tr AI for every A M. The ext result cludes mportat partcular cases whch are formulated afterwards. Theorem 2 Let A,..., A r be parwse complemetary subalgebras of M such that ther commutats A,..., A r are parwse complemetary as well. The the trace-preservg codtoal expectatos E j : M A j ca be expressed by the orthoormal bases U j, U j2,..., U j(j A j, where U j = I, va the formula E j (A = (j U dma jau j (6 j ad the geeralzed Paul chael ( s completely postve f ad oly f + 2 λ dma j λ j for every r ad j λ j ( 2 dma j. To prove ths theorem we prepare the followg proposto. Proposto 2 Let A ad A 2 be complemetary subalgebras of M. The A ad A 2 are complemetary f ad oly f A A 2 learly spas M. Moreover, ths case the trace-preservg codtoal expectato E : M A ca be expressed as E (X = U XU (X M, dma 8

9 where {U } s a orthoormal bass of A. Proof: Assume A ad A 2 are complemetary. Let {U } ad {V j } be orthoormal bases of A ad A 2, respectvely, whch cosst of scalar multple of ther matrx uts. The the trace-preservg codtoal expectatos oto A ad A 2 are gve by the lear combatos of U ( U ad V j ( V j, respectvely, thaks to Proposto. Sce A ad A 2 are complemetary subalgebras, {V j U },j s a orthogoal system. Moreover the trace s wrtte by the lear combato of U V j ( V j U, because A ad A 2 are complemetary subalgebras f ad oly f the composto of two codtoal expectatos equals to Tr. But ths shows that {V j U },j learly spas the whole M thaks to Lemma 2. Coversely assume A A 2 learly spas the whole space M. Sce A s a subalgebra of M, A ca be wrtte as k A = M l I ml. Let Q be a mmal cetral projecto A 2 ad let {U (s bases of A ad A 2, respectvely, wth the assumpto U (s A 2 are complemetary ad spa{a A 2 } = M, { U (s M. Therefore by Lemma 2 ad Proposto, we have ad s,,j l= U (s Vj QV j U (s = Tr Q I j V j QV j = cq for some c > 0. These equatos mply, for s k, U (s QU (s = Tr Q c P s, } ad {V j } are orthoormal M s I ms. Sce A ad V j } s a orthoormal bass of where P s s a cetral projecto I s I ms. Now we take the trace to the above equato. The we have 2 s 2 Tr U (s QU (s s ( = Tr U (s U (s Tr Q = 2 s Tr Q ad so that Tr Q c Tr P s = Tr Q c sm s s m s = c. 9

10 Hece s /m s s equal to /c = dma for all s k ad so E = U (s ( U (s dma s the trace-preservg codtoal expectato oto A by Proposto. Smlarly, E 2 = Vj ( V j. dma 2 s the trace-preservg codtoal expectato oto A 2. Sce U (s Vj ( V j U (s = Vj U (s ( U (s s,,j s,,j s the ormalzed trace o M by Lemma 2, we obta 2 dma dma 2 composto E E 2 equals to Tr. s, j V j = ad so the Proof of Theorem 2. The frst asserto s already prove the above proposto. Due to the Lemma 2, we have (Tr AI = A + UjkAU jk + j=0 k=2 l t= W t AW t, where orthoormal system W t exted the orthoormal system U jk to a complete system the lear space M. I formula ( we use ths expresso for (Tr AI ad the assumed decomposto of the codtoal expectatos. So the expaso of α(a the coeffcet of Ujk AU jk s ad the coeffcet of A = ( I ( ( A ( j I λ s λ j + j + λ j dma j λ j. dma j Theorem tells us that completely postvty holds f ad oly f both are postve. Corollary Assume that M cotas parwse complemetary M-subalgebras A,..., A r. The the geeralzed Paul chael s completely postve f ad oly f for every r. Ths result appeared also [5]. + λ j 0 λ j

11 3 Bpartte chaels I ths secto we cosder subalgebras of M M. A subalgebra somorphc to M wll be called F-subalgebra. A M-subalgebra s a maxmal Abela subalgebra. Both kds of subalgebras are subspaces of dmeso 2. Theorem 3 Assume that A ad A 2 are F- or M-subalgebras of M M. If they are complemetary, the the commutats A ad A 2 are complemetary as well. Proof: Sce both kds of subalgebras are subspaces of dmeso 2, the dmeso of A A 2 s 4 so that A A 2 = M M. Therefore the commutats A ad A 2 are complemetary by Proposto 2. Theorem 4 Assume that M M s decomposed to parwse complemetary F- ad M-subalgebras A ( 2 +. The trace-preservg codtoal expectato of M M oto A s deoted by E. The lear trace-preservg mappg actg as α(b = 2 + s completely postve f ad oly f λ E (B (B M M, Tr B = λ j λ j 2 for every 2 +. Proof: Theorem 3 allows to use Theorem 2 ad the result follows. The theorem ca be appled Example 2. Note that decompostos of M 2 M 2 to F- ad M-subalgebras are dscussed [9], whle decomposto of M M to F-subalgebras s costructed [6] f = p k wth a prme umber p > 2. Ackowledgemet. The authors thak to Professor Tsuyosh Ado for commucato ad to the project of JSPS ad Hugara Academy of Sceces for support. Refereces [] C. H. Beett, C. A. Fuchs ad J. A. Smol, Etaglemet-ehaced classcal commucato o a osy quatum chael, pp Quatum Commucato, Computg, ad Measuremet (eds. O. Hrota, A.S. Holevo, ad C.M. Caves Pleum, New York, 997.

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