Partial traces and entropy inequalities

Transcription

1 Linear Algebra and its Applications 370 (2003) Partial traces and entropy inequalities Rajendra Bhatia Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi , India Received 8 May 2002; accepted 5 January 2003 Subitted by C. Davis Abstract The partial trace operation and the strong subadditivity property of entropy in quantu echanics are explained in linear algebra ters Elsevier Inc. All rights reserved. Keywords: Quantu entropy; Density atrix; Partial trace; Operator convexity; Matrix inequalities 1. Introduction There is an interesting atrix operation called partial trace in the physics literature. The first goal of this short expository paper is to connect this operation to others ore failiar to linear algebraists. The second goal is to use this connection to present a slightly sipler proof of an iportant theore of Lieb and Ruskai [11,12] called the strong subadditivity (SSA) of quantu-echanical entropy. Closely related to SSA, and a crucial ingredient in one of its proofs, is another theore of Lieb [9] called Lieb s concavity theore (solution of the Wigner Yanase Dyson conjecture). An interesting alternate forulation and proof of this latter theore appeared in a paper of Ando [1] well-known to linear algebraists. Discussion of the Lieb Ruskai theore, however, sees to have reained confined to the physics literature. We believe there is uch of interest here for others as well, particularly for those interested in linear algebra. E-ail address: rbh@isid.ac.in (R. Bhatia) /03/$ - see front atter 2003 Elsevier Inc. All rights reserved. doi: /s (03)

2 126 R. Bhatia / Linear Algebra and its Applications 370 (2003) The partial trace Let H be a finite-diensional Hilbert space with inner product.,. and let L(H) be the space of linear operators on H. If A is positive sei-definite (A O) and has trace one (tr A = 1) we say A is a density atrix. Let H 1, H 2 be two Hilbert spaces of diensions n, respectively. Let H 1 H 2 be their tensor product. The partial trace tr H2 is a linear ap fro L(H 1 H 2 ) into L(H 1 ) defined as follows. Let e 1,...,e be an orthonoral basis for H 2. If A L(H 1 H 2 ), then tr H2 A is the linear operator A 1 on H 1 defined by the relation x,a 1 y = x e j, A(y e j ), (1) j=1 for all x,y H 1. It is easy to see that A 1 is well-defined (independent of the choice of the orthonoral basis {e j }). The partial trace A 2 = tr H1 A is defined analogously, and is an operator on H 2. If A is positive, then so are its partial traces A 1,A 2 ; and if A is a density atrix, then so are A 1,A 2. An operator A on H 1 H 2 is said to be decoposable if it can be factored as A = A 1 A 2 where A 1,A 2 are operators on H 1, H 2, respectively. If A 1,A 2 are density atrices, then so is their tensor product A = A 1 A 2 ; and in this case A 1 = tr H2 A, A 2 = tr H1 A. (The conditions tr A j = 1 are needed for this.) Let A be any operator or H 1 H 2 with partial traces A 1,A 2 and let B be a decoposable operator of the for B 1 I. Then one can see that tr AB = tr A 1 B 1. (2) For this choose orthonoral bases f 1,...,f n and e 1,...,e for H 1 and H 2 respectively and observe that tr AB = i,j f i e j,ab(f i e j ) = i,j f i e j, A(B 1 f i e j ) = i f i,a 1 B 1 f i = tr A 1 B 1. The relation (2) characterises the partial trace operation A A 1, and ay be taken as a definition instead of (1). (See the coprehensive review article by Wehrl [22, p. 242]. The definition given there restricts itself to A being a density atrix; the partial trace A 1 is then called the reduced density atrix.)

3 R. Bhatia / Linear Algebra and its Applications 370 (2003) The third definition of partial trace that we give now sees ore revealing. Let A be any operator on H 1 H 2 and write its atrix representation in a fixed orthonoral basis f i e j, 1 i n, 1 j. Partition this atrix into an n n block for A =[A ij ] 1 i, j n, (3) where each A ij is an atrix. Then the partial trace A 1 of A is the n n atrix A 1 =[tr A ij ] 1 i, j n, (4) i.e., the partial trace is obtained by replacing the atrix A ij in the decoposition (3) by the nuber tr A ij. The partial trace A 2 is defined analogously: split A into an block atrix with n n blocks and then replace each block by its trace. If B is a decoposable operator of the for B = B 1 I and if B 1 = [ b ij ] is the atrix of B 1 in the basis f 1,...,f n, then B can be written in n n block atrix for as B = [ b ij I ] 1 i, j n. Fro this one sees that if A and A 1 are as in (3) and (4), then tr AB = i,j b ij tr A ji = tr A 1 B 1. Thus this definition of the operation A A 1 leads to the sae object as before. Next we decopose this ap as a coposite of three aps. Let ω = e 2πi/ be the priitive th root of unity. Let U be the unitary diagonal atrix U = diag(1,ω,ω 2,...,ω 1 ). Let T be any atrix and let D(T ), the diagonal part of T, be the atrix obtained fro T by replacing all its off-diagonal entries by zeros. Then following the ideas in [4] we write D(T ) = 1 1 U k TU k. k=0 Next let W = U U U (ncopies). Let A be any operator on H 1 H 2 and let Φ 1 (A) = 1 1 W k AW k. (5) k=0

4 128 R. Bhatia / Linear Algebra and its Applications 370 (2003) If the atrix of A is partitioned as in (3) then Φ 1 (A) =[D(A ij )]. (6) Let V be the cyclic perutation atrix acting as Ve j = e j+1, 1 j, where e +1 = e 1. This is the atrix V = If D is an diagonal atrix, then 1 1 V k DV k = 1 diag (tr D,...,tr D), k=0 a diagonal atrix with all its diagonal entries equal. Let X = V V V (n copies), and for A L(H 1 H 2 ) let Φ 2 (A) = 1 1 X k AX k. (7) k=0 If A is partitioned as in (3), we have [ ] 1 Φ 2 Φ 1 (A) = (tr A ij )I, (8) where I is the identity atrix. Now let T (1,1) denote the (1,1) entry of a atrix T.If A is an n n partitioned atrix as in (3), let Φ 3 (A) be the n n atrix defined as Φ 3 (A) = [A (1,1) ij ]. (9) We have then tr H2 A = Φ 3 Φ 2 Φ 1 (A). (10) The expressions (5) and (7) for Φ 1, Φ 2 clearly display the as averaging operations. The ap Φ 3 is, upto a constant ultiple, picking out a principal subatrix. This too has an interpretation as an averaging operation [3,6]. All three are copletely positive aps [5], Φ 1, Φ 2 and Φ 3 Φ 2 Φ 1 are trace-preserving. The partial trace (third definition) and its generalisations have been studied in the atrix literature, though not under this nae. For exaple, de Pillis [7] has shown that the partial trace of a positive (sei-definite) atrix is positive. Generalisations, where the blocks are replaced not by their traces but by other functions ay be found in [7,15,16] and in references given therein.

5 3. Entropy inequalities R. Bhatia / Linear Algebra and its Applications 370 (2003) Let A be a density atrix. The (von Neuann) entropy of A is the non-negative nuber S(A) = tr(a log A). (11) One of its basic properties is concavity as a function of A: ( ) A + B S 2 S(A) + S(B). (12) 2 It is not difficult to prove this; see e.g., [2, Proble IX.8.14]. In fact, a uch stronger stateent follows fro Loewner s theory of operator onotone functions. The function f(t)= t log t is operator concave on (0, ) (see [2, Exercise V.2.13]). Soe deeper properties of entropy were proved by Lieb and Ruskai in Of these a few have found their way into the atrix theory literature [1,2]. We will touch upon these briefly on way to the Lieb Ruskai theore on SSA. Let A be a density atrix and K any self-adjoint operator. For 0 <t<1let S t (A, K) = 1 2 tr [At,K][A 1 t,k], (13) where [X, Y ] stands for the coutator XY YX. The quantity (13) is a easure of non-coutativity of A and K, and is called skew-entropy. This too is a concave function of A, a consequence of a ore general theore of Lieb [9]: Lieb s Concavity Theore. The function f(a,b)= tr X A t XB 1 t (14) of positive atrices A, B is jointly concave for each atrix X and for 0 t 1. Using the failiar identification of L(H) with H H, this stateent can be reforulated as: the function g(a, B) = A t B 1 t (15) of positive atrices A, B is jointly concave for 0 t 1. This forulation, and a proof of it, were given by Ando [1]. Other proofs of Lieb s theore that predate Ando s include ones by Epstein [8], Uhlann [21] and Sion [20]. Another quantity of interest is the relative entropy S(A B) = tr A(log A log B) (16) associated with a pair of density atrices A, B. If f is any convex function on the real line, then for Heritian atrices A, B tr [f (A) f(b)] tr [(A B)f (B)].

6 130 R. Bhatia / Linear Algebra and its Applications 370 (2003) This inequality (called Klein s inequality) applied to the function f(t)= t log t on the positive half-line shows that for positive atrices A, B, tr A(log A log B) tr (A B). If A, B are density atrices, then tr(a B) = 0, and hence S(A B) 0. (17) It is an easy corollary of Lieb s concavity theore that S(A B) is jointly convex in A, B. (18) (Choose X = I and differentiate the function (14) at t = 0.) Finally, we coe to the properties of S(A) and S(A B) related to partial traces. Let A 1,A 2 be density atrices. It is easy to see that S is additive over tensor products: S(A 1 A 2 ) = S(A 1 ) + S(A 2 ). (19) Now let A be a density atrix on H 1 H 2 and let A 1,A 2 be its partial traces. The subadditivity property of S says that S(A) S(A 1 ) + S(A 2 ). (20) This can be proved as follows. Fro (17) we have 0 S(A A 1 A 2 ) = tr A(log A log(a 1 A 2 )) = tr A(log A log(a 1 I) log(i A 2 )) = tr (A log A A 1 log A 1 A 2 log A 2 ) using (2) = S(A) + S(A 1 ) + S(A 2 ). Fro the definition (16) it is obvious that S(UAU UBU ) = S(A B) for every unitary atrix U. Hence, fro the representations (5) and (7) we see that S(Φ 2 Φ 1 (A) Φ 2 Φ 1 (B)) S(Φ 1 (A) Φ 1 (B)) S(A B). Note that Φ 2 Φ 1 (A) is a atrix of the special for (8). It is easy to see that S(Φ 3 Φ 2 Φ 1 (A) Φ 3 Φ 2 Φ 1 (B)) = S(Φ 2 Φ 1 (A) Φ 2 Φ 1 (B)). Thus we have S(A 1 B 1 ) S(A B). (21) A ore general result is known [13,14] and can be proved using our techniques: if Φ is any copletely positive, trace-preserving ap, then S(Φ(A) Φ(B)) S(A B).

7 R. Bhatia / Linear Algebra and its Applications 370 (2003) Now consider a tensor product H 1 H 2 H 3 of three Hilbert spaces. For siplicity of notation let us use the notation A 123 for an operator on this Hilbert space, and drop the index j when we take a partial trace tr Hj. Thus tr H3 A 123 = A 12, tr H1 A 12 = A 2, etc. SSA of entropy is the following stateent. Theore (Lieb Ruskai). Let A 123 be a density atrix on H 1 H 2 H 3. Then S(A 123 ) + S(A 2 ) S(A 12 ) + S(A 23 ). (22) Proof. Using the diinishing property (21) with respect to the partial trace tr H3 we see that S(A 12 A 1 A 2 ) S(A 123 A 1 A 23 ). We have seen while proving (20) that S(T T 1 T 2 ) = S(T ) + S(T 1 ) + S(T 2 ). So the above inequality can be rewritten as S(A 12 ) + S(A 1 ) + S(A 2 ) S(A 123 ) + S(A 1 ) + S(A 23 ), and on rearranging ters, as (22). The reader should note that the inequality (22) is siilar in for to others that look like S(E F)+ S(E F) S(E) + S(F). As we said in the introduction, our ephasis here has been on linear algebra and atrix inequalities. To understand the iportance of these results in physics the reader should turn to the original papers of Lieb and Ruskai, and to the review articles by Lieb [10] and by Wehrl [22,23] where references to other works (including alternate proofs and extensions) ay be found. See also the books [17,18] and the recent article [19]. Acknowledgeents The author thanks Centro de Estruturas Lineares e Cobinatórias, Universidade de Lisboa, for its hospitality in June 2001 when this work was done. He thanks J.A. Dias da Silva, R. Horn, E.H. Lieb and M.B. Ruskai for coents and references to the literature. References [1] T. Ando, Concavity of certain aps on positive definite atrices and applications to Hadaard products, Linear Algebra Appl. 26 (1979) [2] R. Bhatia, Matrix Analysis, Springer, 1997.

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