New Perspectives and some Celebrated Quantum Inequalities

NEW PERSPECTIVES AND SOME CELEBRA TED QUANTUM INEQ UALITIES EDW ARD G. EFFR OS De dic ate d to Gert Pe dersen, whose r o guish and irr epr essible wi t i s misse d by al l. Abstra ct. Some of the important inequalities associated with quan- tum entrop y are immediate algebraic consequen ces of the Hansen-Peder- sen-Jensen inequality . A general argumen t is giv en in terms of the ma- trix p ersp ective of an op erator conv ex function. A matrix analogue of Mar ´ ec hal’s extended p ersp ectives provides additional inequalities , i n- cluding a p + q ≤ 1 result of Lieb. 1. I ntroduction Sev eral elegan t p ro ofs of inequalities due to Lieb [7] and to Lieb and Rusk ai [8], ha v e recen tly app eared (see Nielsen and Pe tz [12], Rusk ai [13 ]). W e p ro v e that one can use the “fully quan tized” Jensen in equalit y of F r ank Hansen an d Gert Pedersen [5] to eliminate all vesti ges of an alysis from their biv ariable argument s. W e then sho w that a matrix v ersion of Mar ´ ec h al’s extended p ersp ectiv es can b e used to formula te more elab orate join t matrix inequalities. In the concludin g section w e sugge st so me n atural links betw een matrix con v exity theory and the found ations of quan tum information theory . Since the basic difficulties are already apparen t in finite dimensions, we ha v e restricted our atten tion to fi nite matrices, a nd we ha ve a voi ded a n y attempt at f u ll generalit y ev en in that con text. I am v ery m uc h indebted to Mary Beth Ru s k ai, who corrected a num b er of errors in m y fi rst man u script, and wh o made me aw are of Lieb’s r esult in the third section. 2. The classical and ma trix n o tions of pe rspect ives Giv en a conv ex function f d efi ned on a con v ex set K ⊆ R n , th e p ersp e ctive g is defined on the subs et L = { ( x, t ) : t > 0 and x/t ∈ K } b y g ( x, t ) = f ( x/t ) t Date : January 28, 2008. Supp orted by the National Science F ou n dation DMS-0100883. 1 2 EDW ARD G. EFFROS (see [6]). It is a simple exercise to v erify that g ( x, t ) is a join tly con v ex function in the sense that if 0 ≤ c ≤ 1, then g ( cx 1 + (1 − c ) x 2 , c t 1 + (1 − c ) t 2 ) ≤ cg ( x 1 , t 1 ) + (1 − c ) g ( x 2 , t 2 ) . An elemen tary but imp ortant example is provi ded b y the contin uous conv ex function f ( x ) = x log x, w ith f (0) = 0 defined on [0 , ∞ ) ⊆ R . It follo ws that the p er s p ectiv e fu nction g ( x, t ) = t x t log x t = x log x − x log t is join tly con v ex. Letting p = ( p i ) and q = ( q i ) b e finite probab ility measur es with p i > 0 and q i > 0 , the con vexit y of f implies that the classical en trop y H ( p ) = − X p i log p i is conca ve, and the conv exit y of g implies that the relativ e entrop y ( q , p ) 7→ H ( q || p ) = X p i log p i − p i log q i is join tly conv ex on pairs of p robabilit y measures. W e recall that if f : [ a, b ] → R is con tin uous, and T is an n × n self-adjoin t matrix with sp ectrum in [ a, b ], then we can define f n ( T ) by s p ectral theory (or by using a basis in wh ic h T is diagonal). f is said to b e matrix c onvex if for eac h n ∈ N , the corresp ond ing f unction f n is con v ex on the self-adjoin t n × n matrices with sp ectrum in [ a, b ]. W e usu ally omit the subscrip t n . Theorem 2.1 (Hansen and Pedersen [5]) . If f is matrix c onvex, and A and B ar e m × n matric es with A ∗ A + B ∗ B = I n , then (2.1) f n ( A ∗ T A + B ∗ T B ) ≤ A ∗ f m ( T ) A + B ∗ f m ( T ) B . W e note that th eir pro of do es not en tail any analysis, but rather is based on a shrewd sequ ence of matrix manip ulations. As p oint ed out by Winkler [14], the result ma y b e restated that a real function f on an in terv al in R is a matrix conv ex fu nction if and only if th e sup ergrap h s of the f n form a matrix con v ex system. W e b egin with some matrix conv en tions. Giv en matrices L and R , w e let [ L, R ] = LR − R L . Let us sup p ose that L > 0 and R > 0. I f [ L, R ] = 0, i.e., the matrices comm ute, then we ma y find a b asis in w h ic h b oth matrices are diagonalize d. It follo ws that LR > 0, [ L, R − 1 ] = 0, and we ma y unam- biguously wr ite L R for the quotien t. W e also recall that for an y con tin uous function f , f ( L ) comm utes w ith any op erator comm u ting w ith L (including L itself ). Usin g simultaneously diagonalize d matrices, it is evident that we ha v e relations suc h as log L R − 1 = log L − log R . Theorem 2.2. Supp ose that f is op er ator c onvex. When r estricte d to p ositve c ommuting matric es, the “p ersp e ctiv e function ” (2.2) ( L, R ) 7→ g ( L, R ) = f  L R  R MA TRIX CONVEXITY 3 is jointly c onvex in the sense that if [ L j , R j ] = 0 ( j = 1 , 2 ), L = cL 1 + (1 − c ) L 2 , R = cR 1 + (1 − c ) R 2 , and 0 ≤ c ≤ 1 , then (2.3) g ( L, R ) ≤ cg ( L 1 , R 1 ) + (1 − c ) g ( L 2 , R 2 ) . Pr o of. The m atrices A = ( cR 1 ) 1 / 2 R − 1 / 2 and B = ((1 − c ) R 2 ) 1 / 2 R − 1 / 2 satisfy A ∗ A + B ∗ B = I . F rom Theorem 2.1, g ( L, R ) = Rf  L R  = R 1 / 2 f ( R − 1 / 2 LR − 1 / 2 ) R 1 / 2 = R 1 / 2 f  A ∗  L 1 R 1  A + B ∗  L 2 R 2  B  R 1 / 2 ≤ R 1 / 2  A ∗ f  L 1 R 1  A + B ∗ f  L 2 R 2  B  R 1 / 2 = ( cR 1 ) 1 / 2 f  L 1 R 1  ( cR 1 ) 1 / 2 + ((1 − c ) R 2 ) 1 / 2 f  L 2 R 2  ((1 − c ) R 2 ) 1 / 2 = cg ( L 1 , R 1 ) + (1 − c ) g ( L 2 , R 2 ) .  The follo wing is due to Lieb and Ru sk ai [8] (a r elated early discus sion ma y b e foun d in Lind blad [9]). Corollary 2.3. The r elative entr opy function ( ρ, σ ) 7→ S ( ρ || σ ) = T race ρ log ρ − ρ log σ is jointly c onvex on the strictly p ositive n × n density matric es ρ, σ . Pr o of. W e let M n ha v e the us u al Hilb ert sp ace str u cture determined by h X, Y i = T race X Y ∗ . Giv en p ositive d ensit y matrices σ and ρ, w e define op erators R and L on M n b y L ( X ) = σ X and R ( X ) = X ρ. Th en we h a v e that L and R are commuting p ositiv e op erat ors on the Hilbert sp ace M n . On the other hand th e fu nction f ( x ) = x log x is op erat or con v ex (see [1], p. 123), and thus S ( ρ || σ ) = h L R L log R L ( I ) , I i = h g ( L, R )( I ) , I i is join tly conv ex.  The follo wing is due to Lieb [7]. It wa s subsequent ly used by Lieb and Rusk ai to pro v e strong subadditivit y for relativ e entrop y [8]. Corollary 2.4. If 0 < s < 1 , then the function F ( A, B ) = T r ace A s K ∗ B 1 − s K is jointly c onc ave on the strictly p ositive n × n matric es A, B . 4 EDW ARD G. EFFROS Pr o of. Since f ( t ) = − t s is op erator conv ex (see [1] Th.5.1.9), g ( L, R ) = − L s R 1 − s is j oin tly con vex for appr opriately commuting operators. Again using the Hilb ert space str ucture on M n , we let L ( X ) = AX and R ( X ) = X B . It follo ws that ( A, B ) 7→ − T race A s K ∗ B 1 − s K = h g ( L, R )( K ∗ ) , K ∗ i is join tly conv ex.  V arious generalized en tropies ma y b e handled in m u c h the same man n er. 3. M ar ´ echal ’s pe r spective s P . Mar ´ ec h al has recen tly in tro duced an in teresting generalization of p er- sp ectivit y for conv ex fun ctions [10], [11]. This also has a n atur al matrix v ersion. F or this p urp ose we use Hansen and P edersen’s earlier result [4]. Theorem 3.1. If f is matrix c onvex, and f (0) ≤ 0 , and that A and B ar e m × n matric es with A ∗ A + B ∗ B ≤ I n , then f n ( A ∗ T A + B ∗ T B ) ≤ A ∗ f m ( T ) A + B ∗ f m ( T ) B . Giv en cont in uous functions f and h , and comm uting p ositiv e matrices L and R, w e define ( f ∆ h )( L, R ) = f  L h ( R )  h ( R ) Theorem 3.2. Supp ose that f is matrix c onvex with f (0) ≤ 0 and that h is matrix c onc ave with h > 0 . Then ( L, R ) 7→ ( f ∆ h )( L, R ) is jointly c onvex on p ostive c ommuting matric e s L , R i n the sense of (2.3). Pr o of. Let us supp ose that L = cL 1 + (1 − c ) L 2 and R = cR 1 + (1 − c ) R 2 where [ L j , R j ] = 0. W e ha v e that ch ( R 1 ) + (1 − c ) h ( R 2 ) ≤ h ( R ) , h en ce A = c 1 / 2 h ( R 1 ) 1 / 2 h ( R ) − 1 / 2 B = (1 − c ) 1 / 2 h ( R 2 ) 1 / 2 h ( R ) − 1 / 2 satisfy A ∗ A + B ∗ B = h ( R ) − 1 / 2 ch ( R 1 ) h ( R ) 1 / 2 + h ( R ) − 1 / 2 (1 − c ) h ( R 2 ) h ( R ) − 1 / 2 ≤ h ( R ) − 1 / 2 h ( R ) h ( R ) − 1 / 2 I = I . MA TRIX CONVEXITY 5 It follo w s from Theorem 3.1 that ( f ∆ h )( L, R ) = h ( R ) 1 / 2 f ( h ( R ) − 1 / 2 Lh ( R ) − 1 / 2 ) h ( R ) 1 / 2 = h ( R ) 1 / 2 f  A ∗  L 1 h ( R 1 )  A + B ∗  L 2 h ( R 2 )  B  h ( R ) 1 / 2 ≤ h ( R ) 1 / 2 A ∗ f  L 1 h ( R 1 )  Ah ( R ) 1 / 2 + h ( R ) 1 / 2 B ∗ f  L 2 h ( R 2 )  B h ( R ) 1 / 2 = ch ( R 1 ) 1 / 2 f  L 1 h ( R 1 )  h ( R 1 ) 1 / 2 + (1 − c ) h ( R 2 ) 1 / 2 f  L 2 h ( R 2 )  h ( R 2 ) 1 / 2 = c ( f ∆ h )( L 1 , R 1 ) + (1 − c )( f ∆ h )( L 2 , R 2 ) .  T o illustrate this constru ction, we repro v e a result of Lieb [7]. Corollary 3.3. Supp ose that 0 < p, q and that p + q ≤ 1 . Then the function ( A, B ) 7→ T race A q X ∗ B p X is jointly c onc ave on the p ositive n × n matric es. Pr o of. Since p + q ≤ 1, p + q is a conv ex co m bination of q and 1, i.e., w e ma y c ho ose 0 ≤ t ≤ 1 with p + q = (1 − t ) q + t 1. If w e let q = s , then p = − tq + t = (1 − q ) t = (1 − s ) t. Th us it suffices to sh o w that if 0 ≤ s, t ≤ 1, then ( A, B ) 7→ − T race A s X ∗ B (1 − s ) t X is join tly con v ex. The functions f ( x ) = − x s and h ( y ) = y t are op erator con v ex and conca v e, resp ect iv ely , and ( f ∆ h )( L, R ) = h ( R ) f  L h ( R )  = − R t L s R st = − L s R (1 − s ) t . If w e let L ( X ) = AX and R ( X ) = X B f or X ∈ M n , then it follo ws f rom the ab ov e theorem that ( A, B ) 7→ − T race A s X ∗ B (1 − s ) t X = h ( f ∆ h )( L, R )( X ∗ ) , X ∗ i is join tly conv ex.  4. m a trix convexity P erhaps the most in teresting asp ec t of Mar ´ ec hal’s constr u ction is that it b eha ve s w ell und er the F enc hel-Legendre transform, and u nder iteration. Søren Winkler form ulated an analogue of the F enchel-Leg endre du alit y for matrix conv ex f unctions [14], bu t the transforms are generally set-v alued mappings. F urther p r ogress migh t r esult if one could reformulate his theory in terms of comm uting pairs. It should also b e noted that other constructions 6 EDW ARD G. EFFROS in classical con v exit y theory , such as the linear f ractional transformations of con v ex fu nctions (see [2 ]) migh t also ha v e matrix generalizatio ns. Un til recen tly the theory of matrix conv exit y h as suffered from a lac k of examples and app lications. With the adv en t of quantum information theory (QIT), this situation h as dramatically changed. QIT pro vides a wea lth of remark able, purely non-classical tec hniques that migh t clarify some of the conceptual p r oblems in matrix con v exity theory . On the other hand, it seems lik ely th at matrix con vexit y will pr ovide an appropr iate framework for many of the calculations in QIT. A striking illustratio n of this phenomenon can b e found in [3]. Referen ces [1] Bhatia, R. Matrix analysis. Graduate T exts in Mathematics, 169. Springer-V erlag, New Y ork, 1997. xii+347 p p. [2] Bo yd, S . ; V andenberghe, L. Conv ex Op timization, to app ear. [3] Deveta k, I. ; Jun ge, M. ; King, C.; R usk ai, M. Mu ltiplicativity of completely b ou n ded p -norms implies a new additivity result. Comm. Math. Phys. 266 (2006), no. 1, 37–63. [4] Hansen, F.; Pedersen, G. Jensen’s inequality for op erators and L¨ ow ner’s theorem. Math. Ann. 258 (1981/82), no. 3, 229–241. [5] Hansen, F. ; P edersen, G. Jensen’s op erator inequality . Bull. London Math. So c. 35 (2003), no. 4, 553–564. [6] Hiriart-Urruty , J.B., Lemarchal, C. (1993) Conv ex An alysis and Minimization A lgo- rithms, I and I I, Springer V erlag, Berlin, Germany [7] Lieb, E. Con vex trace functions and the Wigner-Y anase-Dyson conjecture. Adv ances in Math. 11 (1973), 267–288. [8] Lieb, E.; Ruska i, M. Proof of the strong subadditiv ity of quantum-mechanical entrop y . With an app endix by B. S imon. J. Mathematical Phys. 14 (1973), 1938–1941. [9] Lindblad, G. Entrop y , information and quantum measurements. Comm. Math. Phys. 33 (1973), 305–322. [10] Mar ´ echal, P . On a functional op eration generating conv ex functions. I. Dualit y . J. Optim. Theory A ppl. 126 (2005), no. 1, 175–189. [11] Mar ´ echal, P . On a functional op eration generating con vex functions. I I. Algebraic prop erties. J. O ptim. Theory Appl. 126 (2005), no. 2, 357–366. [12] Nielsen, M. ; P etz, D. A simple p roof of the strong subadditivity inequality . Quantum Inf. Comput. 5 (2005), no. 6, 507–513. [13] Rusk ai, M. An other short and elementary pro of of strong subadditivity of quantum entrop y . Rep. Math. Phys. 60 (2007), no. 1, 1–12. [14] Winkler, S. The non- commutative Legendre-F enchel transform. Math. Scand. 85 (1999), no. 1, 30–48. Dep ar tme nt of Ma thema tics, UCLA, Los A n geles, CA 90095-1555 E-mail addr ess , Edw ard G. Effros: ege@math.ucla. edu