North-Holland. Minimum relative entropies of low- ... Community and Organization Research Institute, University of California, Santa Barbara,. CA 93106, USA.
Physica A 182 (1992) 145-154 North-Holland
Minimum relative entropies of low-dimensional spin systems P a u l B. S l a t e r
Community and Organization Research Institute, University of California, Santa Barbara, CA 93106, USA Received 9 May 1991
Work of Band and Park in the mid-1970's in which they proposed-on informationtheoretic grounds- an alternative to the von Neumann entropy measure, S(p) = - T r p In p, of a density matrix (p) has apparently not been further applied. In this paper, however, specific measures are generated of the information-theoretic entropy of spin-i, spin-1 and two-photon mixed states. For this purpose, the minimum relative entropies of arbitrary mixed states with respect to uniform prior distributions over the pure states are determined. Though Band and Park did not specifically discuss this approach, it is contended that it is harmonious with their position. The duality theory of convex programming is employed to interrelate the von Neumann entropy and the minimum relative entropy measure adopted. Some concluding remarks are made on the possible use of such relative entropy indices in modeling the nonunitary (irreversible) evolution of quantum systems.
I. Introduction T h e e n t r o p y associated with a d e n s i t y m a t r i x ( p ) is usually t a k e n to be the v o n N e u m a n n m e a s u r e [1]
S(p) = - T r p In p ,
(1)
w h e r e b y In p is m e a n t [2] the m a t r i x , the e i g e n v a l u e s of which are the l o g a r i t h m s of those of p, a n d the associated e i g e n v e c t o r s are the s a m e as those of p. T h e w i d e s p r e a d use of this i n d e x has b e e n s u p p o r t e d t h r o u g h several a r g u m e n t s a n d proofs [1, p. 242; 3 - 6 ] . H o w e v e r , it is of i n t e r e s t to n o t e t h a t - e v e n in the same v o l u m e of Foundations of Physics as o n e of these d e m o n s t r a t i o n s [ 4 ] - vigorous criticisms [7] were b e i n g directed against the use of S(p) o n i n f o r m a t i o n - t h e o r e t i c (if n o t o n t h e r m o d y n a m i c [8]) g r o u n d s . T h e m a i n o b j e c t i o n was that S(p), b e i n g b a s e d o n the spectral e x p a n s i o n of p, in effect, does n o t reflect the continuum of all o t h e r possible ( n o n o r t h o g o n a l ) r e p r e s e n t a t i o n s of p. 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
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P.B. Slater / Minimum relative entropies of low-D spin systems
Jaynes [3] had shown that if one expands p as a convex combination of pure states, the entropy of the probability distribution over the pure states formed by these convex weights is minimized in the case of the spectral expansion. (It "provides in the sense of information content, the most economical description of the freedom of choice implied by a density matrix" [3, p. 174]. But employing the spectral expansion for this reason is antithetical to the philosophy of the maximum-entropy formalism that Jaynes was at the time expounding. (It is consistent with this approach, however, to the extent that S(p) is maximized over alternative p - b u t inconsistent to the extent that it is minimized over alternative expansions of a given p.) Jaynes, in essence, asserted that expansions alternative to the spectral one were not permissiblethey "would not be satisfactory because the [probabilities over the pure states] are not in general the probabilities of mutually exclusive events" [3, p. 174]. Band and Park [7] argued that such nonorthogonal expansions ("arrays" in Jaynes' terminology) must be incorporated into an information-theoretic measure of the uncertainty in p. They found daunting, however, the necessity to account for a continuum of possible alternative states and, relatedly, to assign a prior distribution over these s t a t e s - a s statistical practice would dictate in the case of continuous distributions. Band and Park's program appears to have lain dormant since its promulgation, and S(p), for various reasons, including analytic tractability, is widely used. In fact, Park has since co-authored papers [9, 10] (in which S(p) is utilized) with the developers [4] of one of the proofs supporting S(p) and has strongly endorsed this particular demonstration in other articles [11, 12]. The line of research reported h e r e - t h o u g h it was not engendered by Band and Park's a r g u m e n t s - c a n , in retrospect, be viewed as an effort to pursue and further implement their program, if not in its specific framework, at least in its spirit. The initial impetus to this work was, in fact, a remark in an apparently quite removed (mathematical) paper [13], one pertaining to doubly stochastic matrices (nonnegative square n x n matrices with row and column sums all equal to 1). Birkhoff [14] and yon Neumann [15] had demonstrated that the extreme points ("pure s t a t e s " - o n e s not representable as nontrivial combinations of other states) of the convex polytope formed by all n x n doubly stochastic matrices are the n! n x n permutation matrices (having a single 1 in each row and column, and zeros elsewhere). In Birkhoff's proof, a minimumterm (greedy) representation of an arbitrary doubly stochastic matrix ( D ) analogous to the thrifty spectral expansion, as noted by J a y n e s - i s explicitly constructed. Brualdi [13] asked the "altruistic" question, on the other hand, of how to widely spread the probability distribution over the permutation matrices (pure states), so as to yield D, while maximizing (not minimizing- though Brualdi did raise a related question) the resultant entropy of the probability
P.B. Slater / Minimum relative entropies of low-D spin systems
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distribution. This author [16, 17] has investigated Brualdi's query, as well as its generalizations to other convex polytopes, such as stochastic matrices [18, 19] and general nonnegative matrices with given integral row and column sums [20]. In some special cases- such as for a point in an n-dimensional rectangular solid, having 2 ~ extreme p o i n t s - h e has found explicit solutions. The analogous (nonunique) representation of spin-½ states as convex combinations of the pure states lying on the surface of the unit sphere, has been vividly illustrated [21, 22]. The question, thus, naturally arises of representing an arbitrary spin-½ state (lying in the interior of the unit ball) as a convex combination of the pure states, in an entropy-maximizing fashion. (In the rejection of the ignorance interpretation [21, p. 11; 23, p. 144] of quantum states, however, the possibility does not appear to have been considered that all representations of p might not, in some sense, be equiprobable.) Here, however, the extreme p o i n t s - i n contrast to the cases a b o v e - c o m p r i s e a continuum. One cannot simply solve a continuous entropy-maximizing problem by taking the limit of a discrete problem, as the probability measures employed would not be absolutely continuous with respect to each other [24, p. 55]. In the several results presented here, a uniform prior distribution is placed over pure states. Then, given an arbitrary mixed state (v), the form of the probability distribution that has minimum relative entropy [24, section 2.10] with respect to the uniform prior and yields v as its expected value is determined for spin-½, spin-1 and two-photon systems. These results are possible due to parametrizations of the pure states of these systems [21, 22, 25-27]. These findings are reviewed here and some relations between S(p) and the proposed minimum relative entropy measure--based on the duality theory of convex programming [28]-are presented. Finally, some remarks will be made as to how one might view nonunitary (irreversible) evolutions [4, 9, 10, 12, 29] of quantum systems in this light.
2. Minimum relative entropies of spin systems 2.1. Spin- ~
The states of a spin-½ system can be represented by the points in the unit ball in three-dimensional Euclidean space (~3), with the pure states constituting the spherical surface [21,22]. Consider (without any essential loss of g e n e r a l i t y - d u e to the rotatability of the ball) an arbitrary spin-½ state to be described by coordinates (0, 0, x3) , - 1 ~
