Vor. UMz 30, NUMszR 10 5 MwRcH 1/73

fusion coefficient on density, magnetic fieM,temperature, and ion mass are consistent withthose of the theoretical diffusion coefficient. Themagnitude of the observed diffusion coefficientis larger by a factor of 3.5 than the theoreticalvalue. The results with the toroidal field maybe explained by considering that the perpendic-ular diffusion damping time is shorter than theparallel phase mixing time.

"%fork supported by the U. S. Atomic Energy Commis-sion under Contract No. AT(04-3)-167, Project Agree-ment No. 38.

T. Ohkawa et al. , Phys. Bev. Lett. 24, 95 (1970).T. Ohkawa et al. , Phys. Bev. Lett. 27, 1179 (1971).T. Ohkawa, J. B. Gilleland, and T. Tamano, Phys.

Bev. Lett. 28, 1107 {1972).T. Ohkawa et a/. , in Plasma Physics and Controlled

¹cleaxelusion Reset ch (International Atomic EnergyAgency, Vienna, 1971), Vol. I, p. 15.

L. C. Johnsons private communication' L. C. John-son and E. Hinnov, Phys. Bev. 187, 143 (1969).

L. C, Johnson and E; Hinnov, Phys. Fluids 12, 1947(1969).

F. C. Fehsenfeld et a/. , J. Chem. Phys. 44, 4087(1966).

M. Yoshikawa et al. , Phys. Fluids 11, 2265 (1968).S. Yoshikawa, in Memational Symposium on Closed

Confinement Systems, 1968 Onternatioual Atamic Ener-gy Agency, Vienna, Austria, 1969).

J. B.Taylor and B. McNamara, Phys. Fluids 14,1492 (1972).

J. M. Dawson, H. Okuda, and B. N. Carlile, Phys.Rev. Lett. 27, 491 (1971).

~H. Okuda and J. M. Dawson, Princeton Plasma Phys-ics Laboratory Report No. PPL-AP 52, 1972 (unpub-lished) .

~3H. Okuda and J. M. Dawson, Phys. Bev. Lett. 28,1625 {1972).

D. Montgomery, C. 8. Liu, and G. Vahala, Phys.Fluids 15, 815 (1972).

'~G. Vahala, Phys. Bev. Lett. 29, 93 (1972).H. C. S. Hsuan, Phys. Fluids 15, 1300 (1972}.

f7J. B. Taylor and %. B. Thompson, ln Proceedingsof the Fifth European Conference on Controlled Fusionand Plasma Physics, Gxegoble, I'vance, 1972 (Serviced'Ionique Generale, Association EUBATOM-Commis-sariat K l'Energie Atomique, Centre d'Etudes Nucle-aires de Grenoble, Grenoble, France, 1972), Vol. I,p 11,

~8M. N. Rosenbluth and C. S. Liu, in Pt'oceedings onthe Fifth European Conference on Controlled Fusiona8d Plasma Physics, GxeÃoble, Ixasce, 1972 (Serviced'Ionique Generale, Association EURATOM-Commis-sariat a 1'Energie Atomique, Centre d'Etudes Nucle-aires de GrenoMe, Grenoble, France, 1972), Vol. I.p 12

J. B. Taylor, in Proceedings of the Fifth EuropeanConference on Controlled fusion and Plasma Physics,Grenohle, France, 1972 (Service d'Ionique Generalc,Association EURATOM-Commissariat a I'EnergieAtomique, Centre d'Etudes Nucleaires de Grenoble,Grenoble, France, 1972), Vol. II, p. 83.

S ~ Ichlmaru and M. N. Rosenbluth, in Plasma Pepys

ics and Controlled Nuclear I'usion Reset"ch {Interna-tional Atomic Energy Agency, Vienna, 1971), Vol. II,p. 373.

A Fundamental Property of Quantum-Mechanical Entropy

Elliott H. Lieb* tEnstitut des IIautes Etudes Scieetifiques, 91 Bm'es-sm -Feette, I"vance

Mary Beth Ruskai*gDepartment of Mathematics, Massachusetts Institute of Technology, Cam&'idge, Massachusetts 09739

{Received 26 December 1972)

%e have proved the strong subadditivity of quantum-mechanical entropy and the sig-ner- Yanase-Dyson conjecture.

There are some properties of entropy, such asconcavity and subadditivity, that are known tobold (in classical and in quantum mecba. nics) ir-respective of any assumptions on the detailed dy-namics of a system. These properties are conse-

quences of the definition of entropy asS(p) = —Trp inp (quantum),

S(p) = — p inp (classical continuous),

S(p) = -g p,. Inp, . (classical discrete),

(la.)

(1b)

(1c)

VOLUME 30, NUMBER 10 PHYSICAL REVIEW LETTERS 5 MARCH 1973

S~(p, E) = 2 Tr[p&, K][p' ~, E],where [A, B]=&f3-gA and 0 ~ p ~1 is fixed. We

(2)

where Tr means trace, p is a density matrix in(1a), and p is a distribution function (usually onA'"} in (1b). In (1c) the p,. are discrete energylevel probabilities.

One such property, strong subadditivity (SSA),was known to hold for classical systems and wasonly conjectured for quantum systems. The ob-servation that cia,ssical entropy has SSA (this, infact, is a theorem in information theory) and thatSSA implies strong results about the thermody-namic limit of entropy per unit volume is due toRobinson and Ruelle. ' Later, Lanford and Robin-son' conjectured that SSA holds for quantum sys-tems as we1.1, and Baumann and Jost" were ableto prove this when p has a special form. Arakiand Lieb' proved a weakened form of SSA, but onewhich held for general p and which was sufficientfor many of the purposes to which SSA had beenput in Ref. l. The physical significance of SSAis explained below [item (f) of Table I].

Prior to these developments, Wigner andYanase' proposed a different definition of entropy(or negative information) which was generalizedby Dyson. ' The conjecture that this generalizedentropy was concave in p was also proved byBaumann and Jost" in special cases, but it wasnot realized that this concavity problem and theSSA problem were related; in fact they are equiv-alent.

Here, we wish to announce that both of theseproblems have been solved affirmatively. Theproofs, which are too long for this note, will begiven in two papers. "

A density matrix is a positive semidefiniteoperator with Trp = i. The signer- Yanase-Dysonp-entropy of p with respect to a self-adjoint op-erator (observable) E. is

can think of (2) as defined for all p ~ 0 and askwhether S (p, E} is concave as a function of p.The term ——,'Trpb' is obviously concave since itis linear, so the problem reduces to that of theconcavity of Trp~gp' ~E. This was proved' when

P = —,'. We have proved the following:Theorem: For each fixed E (not necessarily

self-adjoint), Trp~A' ~p"EC is a concave function ofp for p & 0 whenever p & 0, r ~ 0, and p+ r - 1.This theorem is obviously stronger than neces-sary.

Returning to the conventional entropy (1a}, wesuppose that the Hilbert space of the system is atensor product of three spaces, II=A' g.o' g&'.Thus, the system has three sets of degrees offreedom; for example, these may be thought ofas the degrees of freedom of a gas in three dis-joint regions in space (Q}. Given a density ma-trix p"' on H, we can define a density matrix p"on H' IJ' by partial trace, i.e., p" = Tr,p'". Inlike manner we can form p, p, etc., and foreach of these we have an entropy given by (1a).Denoting S(p"'}by S"', etc. , subadditivity statesthat

S~2 «S~+ 82

while SSA states that

F —= (S' —S"}+(S' —S"}~ 0 (5)

but as I' is convex in p'", it is less than its max-imum value on extremal points, which latter arethose p'" that are pure states. For pure states,y =O.

We also prove some other related theorems,

(4)

We first show that S —S is convex in p '.This implies SSA because, as was pointed outpreviously, ' in the quantum or classical discreteease SSA is equivalent to

TABLE I. Fundamental properties of entropy and their truth (T) orfalsity (F) in three kinds of mechanics.

Classicaldiscrete

Classicalcontinuous Quantum

(a) S(p) is concave in p(b) gi2- ~t+S2(c) ~(O) =o(d) S"-S'(e} s"= Is'- s'I(f) Sj.23 + g2 ( g&2 + g23

(g) S —8 is concave in p

TTTTTTT

TTFFFTT

TTTFTTT

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VOLUME 30, NUMBER 10 PHYSICAL REVIEW LETTERS 5 MARCH 1/73

among which is the following:Theorem: Let & be self-adjoint on a Hilbert

space H and fixed .Then Tr [exp(K+ Inp)] is a con-cave function of p for p &0. Closer inspectionshows that this theorem is a generalization ofthe Golden- Thompson inequality' "to three oper-ators.

To conclude, in Table I we append a list of theknown fundamental entropy [Eq. (1)] inequalitiesand their physical significance. The followingremarks clarify the physical significance ofTable I. The letters (a)-(g) refer to entries inTable I.

(a) states that if two different ensembles areunited, the entropy of the resulting ensemble isgreater than the average entropy of the compo-nent ensembles. '

(b) is a statement of subadditivity and is thebasic tool for proving that the entropy per unitvolume has a thermodynamic limit (which may,however, depend on the particular sequence ofdomains).

(c) expresses a well-known defect of classicalcontinuous statistical mechanics with respect tothe third law of thermodynamics.

(d) expresses an intuitive defect of quantum andclassical continuous statistical mechanics. An

example occurs when p" is a pure state, so thatS"=0. Thus, the entropy of the universe can re-main zero while the entropy of Earth increaseswithout limit.

(e) is a consequence of SSA in the alternativeform of inequality (5) (cf. Ref. 5) and is includedas partial conpensation for (d).

(f) is the statement of SSA. As a technical toolit allows one to prove that the entropy per unitvolume for quantum continuous systems is inde-pendent of shape, at least for rectangular paral-lelepipeds of fixed orientation (cf. Ref. 5). If oneis willing to assume that the entropy of everybounded region is finite [which cannot be proved,as (d) is false quantum mechanically], then thelimit exists for arbitrary regions in the senseof Van Hove. However, (f) ha, s a more heuristicinterpretation. Although the connection betweenentropy and information is hedged with controver-sy, we may suppose, along with the Copenhagenschool, that when we measure a system its den-sity matrix is reduced to that of a pure state andthe entropy is reduced to zero. Thus, entropy

measures the information gain in an experiment.S"—S' can be thought of as the information gainedupon measuring a total system (23) when a sub-system (2) is known. In quantum mechanics itmay be negative because of (d). $"' —$" ~ $"—S' states that this incremental information issmaller when the initial information [(12) asaga. inst (2)] is larger. This, at least, is theinterpretation given in information theory.

(g) states that the incremental information,like the entropy itself, increases when two en-sembles are united.

We thank D. Ruelle for generating and encour-aging our interest in this problem. We are alsograteful to R. Jost, O. Lanford, and D. Robinsonfor their encouragement and for helpful conversa-tions. One author (E.L.) would like to thank theChemistry Laboratory III, University of Copen-hagen, where part of this work was done.

*Work partially supported by U. S. National ScienceFoundation Grant No. GP-31674 X.

$0n leave from Department of Mathematics, Massa-chusetts Institute of Technology, Cambridge, Mass.02139. Work partially supported by a Guggenheim Me-morial Foundation fellowship.

)Presently, National Research Council Fellow at De-partment of Physics, University of Alberta, Edmonton7, Alta. , Canada.

~D. W. Robinson and D. Ruelle, Commun. Math. Phys.5, 288 (1967).

'O. Lanford, IO, and D. W. Robinson, J. Math. Phys.(N. Y.) 9, 1120 (1968).

3F. Baumann, Helv. Phys. Acta 44, 95 (1971).F. Bauman and R. Jost, in I'xoblems of Theoretical

I'kysics; Essays Dedicated to N. N. Bogoriubov (Nauka,Moscow, 1969), pp. 285-293.

H. Araki and E. H. Lich, Commun. Math. Phys. 18,160 (1970).

~E. P. Wigner and M. M. Yanase, Proc. Nat. Acad.Sci. U. S. 49, 910 (1963), and Can. J. Math. 16, 397(1964) .

R. Jost, in "Quanta" —Essays in Theoretical Phys-ics Dedicated to Gregory Wentzel, edited by P. G. O.Freund, C ~ J. Goebel, and Y. Nambu (Univ. of ChicagoPress, Chicago, Ill. , 1970), pp. 13-19.

E. H. Lieb, "Convex Trace Functions and Proof ofthe Wigner-Yanase-Dyson Conjecture" (to be published).

E. H. Lieb and M. B. Ruskai, "Proof of the StrongSubadditivity of Quantum Mechanic~I Entropy" (to bepublished) .

S. Golden, Phys. Rev. B 137, 1127 (1965).C. J. Thompson, J. Math Phys. (N. Y.) 6, 1812 (1965).

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