Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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levels consist of very many contributions of different

kinds; dropping quantum numb er indices for sim-

plicity, the energy levels can be written as an

expression of the form

E ¼

ðZÞ





þðZÞ

þ



þ E

rad

þ E

rec

þ E

nucl

þ ½5

Let us observe that it is convenient to write

explicitly the Z factors even when Z = 1 for a better

bookkeeping of the various corrections. As usual, m

is the reduced mass of the electron, m

= m

þ M) the mass of the nucleus being M; the first

term in the square bracket, 1=n

, the familiar

Balmer term, is by and far the dominant one, giving

for the n = 1 level in the Z = 1 case an energy of

about 13.6 eV or a corresponding frequency of

3.3 10

Hz. The other terms in the square

bracket, f

and f

, are known coefficients (depend-

ing also on the small parameter m

=M; f

essentially the fine structure).

The term E

rad

, is the bulk of the radiative QED

corrections; it can be written as a multiple expan-

sion on (Z),  and L = ln [1=(Z)

], which turns

out to have the following explicit form:

E

rad

¼m

ðZÞ









h

L þA

þðZÞA

þðZÞ

þA

L þA



þ









þðZÞB

þðZÞ

þB

L þB



þ









þðZÞC

þ½þ



½6

The first index of the coefficients refers to the power

of (Z), the second to the power of L; as a rule,

there are three powers of (Z) due to the normal-

ization of the wave function and one power of (Z)

for each interaction with the nucleus (in the leading

term of eqn [5] one must subtract two powers of

(Z) due to the long-range nature of the Coulomb

interaction), while the terms in L = ln [1=(Z)

] are

related to the infrared divergences of the scattering

amplitude, with the binding energy acting as infra-

red cutoff. The A-coefficients refers to order (=)

or one-loop virtual co rrection (we do not distinguish

here between one-loop self-mass and vacuum-

polarization contribution, as usually done in the

literature), the B-coefficients to two loops, etc. The

coefficients are pure numbers, entirely determined

within QED, even if their actual calculation is an

extremely demanding task. One of the first results

obtained in 1947 was A

= (4=3)

, contributing to

the 2S but not to the 2P states (quite in genera l,

most corrections are much bigger for l = 0 states

than for higher-angular-momentum states), which is

sufficient to give the right order of magnitude of the

(2S

1=2

–2P

1=2

) Lamb shift (about 1000 MHz). The

other coefficients are now known, thanks to the

strenuous an d continued efforts (Eides et al. 2001)

since then, which is impossible to refer properly here

in any detail. The current frontier of the theoretical

calculation (around the dots in the previous for-

mula) corresponds to 8–9 total powers of (=) and

(Z) or some kHz for the 1S state.

The next term in eqn [5], E

rec

contains

contributions of order m

(Z)

=M) or smaller

(some care must be done for classifying the

contributions of order m

=M, which can be

accounted for by proper use of m

rather than m

and genuine m

=M contributions), and are suffi-

ciently known for practical purposes; the same is

true for many other contributions discussed in Eides

et al. (2001) and skipped in eqn [5]. A troublesome

contribution comes however from E

nucl

; at leading

order, one has

E

nucl

2ðZÞ

mcR

h





where R

is the so-called root-mean-square charge

radius of the proton, which is not well known

experimentally (in the literature, there are indeed

two direct measurements, R

= 0.805(11) fm and

= 0.862(12) fm, in poor agreement with each

other; a new independent measurement is strongly

needed).

The hyperfine splitting The effect of the interac-

tion of the electron with the spin of the positive

particle introduces the so-called hyperfine splitting

of all the levels. The order of magnitude of the

hyperfine splitting of the 1S state is given by the

Fermi energy

ðZÞ

where g

’ 5.586 is the g-factor of the proton,

which gives ’1.42 GHz. It was dubbed hyperfine

because it is smaller than the fine structure terms by

the factor m

. Many classes of corrections can

be worked out, with patterns similar to those of the

previous subsection, and also in this case the nuclear

contributions (this time mainly due to the theoreti-

cally unknown magnetic form factor and the

172 Quantum Electrodynamics and Its Precision Tests

so-called polarizability of the proton) prevent from

obtaining predictions with an error less than 1 kHz

(or a relative precision better than 1 10

6

The comparison with the experiments Experimen-

Experimentally, one measures transition frequencies

among the various levels. For many years the

precision record was given by the hyperfine splitting

of the ground states of hydrogen 

hfs

(1S) was

measured long ago (see Hellwig et al. (1970) and

Essen et al. (1971)),



hfs

ð1SÞ¼1 420 405:751 766 7ð9ÞkHz ½7

with a relative error 6 10

13

. The current record in

the optical range is the value of the (1S–2S)

hydrogen transition frequency, obtained by means

of two-photon Doppler-free spectroscopy Niering

et al. (2000),

ð1S–2SÞ¼2 466 061 413 187:103ð46ÞkHz ½8

with a relative precision 1.9 10

14

; other optical

transitions, such as (2 S –8D), (2S–12D) are measured

with precision of about 1 10

11

The measurement of the Lamb shift was repeated

several times, with results in nice agreement with the

original value, such as Lundeen and Pipkin (1986),

1057.845(9) MHz. The most preci se value,

1057.8514  0.0019 MHz was given in Palchikov

et al. (1985) (the result depends, however, on the

theoretical value of the lifetime, and should be

changed into 1057.8576  0.0021 according to

subsequent analysis (see Karshenboim (1996)). The

experimental (2S

1=2

–2P

1=2

) Lamb shift was also

obtained as the difference between the measured

fine structure separation (2P

3=2

–2S

1=2

) and the

theoretical value of the (2P

3=2

–2P

1=2

) frequency,

and the radiative corrections E

rad

to any level are

now referred to as the Lamb shift of that level.

As a some what deceiving conclusion, the wonder-

ful experimental results of eqns [7] and [8] cannot

be used as a high-precision test of the theory or to

obtain precise values of many fundamental con-

stants, as the theoretical calculations depend, unfor-

tunately, on hadronic quantities which are not

known accurately. Combining theoretical predic-

tions, the above transitions and Lamb shift data, and

the available values of  and m

, one can indeed

obtain a measure of R

= 0.883  0.014,

according to Melnikov and van Ritbergen (2000))

and the value of R

already quoted above.

Muonium

The muonium is the bound state of a positive m

meson and an electron. At variance with the proton,

the m

lepton has no strong interactions, the m



system can be studied theoretically within pure

QED, with the weak interactions giving a known

and small perturbation. Further, the ratio of the

masses m

’ 4.8 10

3

is small, so that the

external field approximation holds. However, the m

is unstable (lifetime ’2.2 m s), which makes experi-

ments more difficult to carry out. The best measured

quantity is the hyperfine splitting of the 1S ground

state (see Liu et al. (1999))



hfs

ðme; 1SÞ¼4 463 302 765ð53ÞHz

with a relative precision of 12 10

9

. The theore-

tical treatment is similar to the case of hydrogen,

with the important advantage that nuclear interac-

tions are absent and everything can be evaluated

within QED, so that the bulk of the contribution is

given by a formula with the structure of eqn [6]. But

the prediction depends, in any case, on the m

mass, which is not known with the required

precision. Indeed, a recent theoretical calculation

(Czarnecki et al. 2002) (which includes also a

contribution of 0.233(3) kHz from hadronic

vacuum polarization) gives 4 463 302 680(510)

(30)(220) Hz, where the first (and biggest) error

comes from m

, the second from , and the third

is the theoretical error (an estimation of higher-

order contributions not yet evaluated).

Positronium

The positronium is the bound state of an electron

and a positron. Theoretically, it is an ideal system to

study, as it can be described entirely within QED,

without any unknown parameter of non-QED

origin. As the masses of the two constituents,

positron and electr on, are strictly equal, the reduced

mass of the system is exactly equal to half of the

electron mass, m

= m

=2, and the energy scale of

the bound states is half of R

At variance with the muonium case, the external

field approximation is not valid, so that positron ium

must be treated with the full two-body bound-state

machinery of QFT, of which it provides an excellent

test (Karshenboim 2004).

Experimentally, radioactive positron sources are

available, so that positronium is easier to produce

than muonium. It is, however, unstable; states with

total spin S equal 0 (also called parapositronium

states) annihilate into an even number (mainly two)

of gammas, and states with S = 1 (orthopositronium)

into an odd number (mainly three) of gammas, with

short lifetimes (which make precise measurements

difficult). Further, as positronium is the lightest

Quantum Electrodynamics and Its Precision Tests 173

atom, Doppler-broadening effects are very impor-

tant, reducing the precision of spectroscopical

measurements.

Positronium decay rates There has been a long-

time discrepancy between theory and experiment in

decay rate of ground-state orthopositronium, which

prompted thorough theoretical investigations look-

ing for errors in the calculations or flaws in the

formalism, but it turned out that the flaw was on the

experimental side. The current theoretical prediction

for the ground state S = 1 decay is (Adkins et al.

2002)

ð1S; orthoÞ¼

1 þA









ln 



þB









3

2

 þ C





ln  þ



¼ 7:039979ð11Þms

1

where 

= 2(

9)m



=(9) = 7.2111670(1),

A= 10.286606(10),B =45.06(26),C= 5.517, in

nice agreement with the less precise experimental

result of Karshenboim (2004, ref. 38) 7.0404

(10)(8)ms

1

. As a curiosity, the coefficients A, B

above are among the greatest coefficients so far

appeared in QED radiative corrections.

The agreement between theory and experiment for

the ground-state parapositronium decay rate has

always been good; the current status of Karshenboim

(2004, ref. 41) is 7990.9(1.7) ms

1

for the experimental

result and of Karshenboim (2004,ref.43)

7989.64(2) ms

1

for the theoretical prediction.

Positronium levels The quantum number structure

of the levels is similar to muonium, with the

important difference, however, that the hyperfine

splitting (which in hydrogen or muonium is small

because it is proportional to the ratio of the masses of

the two components) is in fact of the same order as

the fine structure. The theoretical evaluation of the

energy levels provides a very stringent check of QED

and of the overall treatment of the bound-state

problem. Corrections have been evaluated, typically,

up to order mc



. The best-known quantities are

the ground state (hyper)fine splitting, experimental

value (Ritter et al. 1984) 203.38910(74) GHz

(3.6 10

6

relative error), theoretical (Karshenboim

2004) 203.3917(6), and the 1S–2S transition for

orthopositronium, experiment (Fee et al. 1993) 1233

607 216.4 (3.2) MHz, theory 1 233 607 222.2(6).

The general agreement is good; the precisions

achieved are, however, not yet sufficient to allow a

determination of R

or  competitive with other

measurements.

The Anomalous Magnetic Moments

of Leptons

The precision of the measurements requires, for both

the e and  leptons, to also take into account graphs

with contributions from the other leptons as virtual

intermediate states and those of hadronic and weak

origin. Quite in general, if the mass of the virtual

particle, say m

, is smaller than the mass of the

external lepton, say m

, one can have an ln (m

)

behavior of the contributions; that is the case of the

virtual electron contributions to the muon magnetic

anomaly a

, which can be enhanced by powers of

ln (m

). In the opposite case, m

> m

, the

contribution has the behavior (m

)

; that is the

case of the (m



)

contributions to a

from 

loops and of the (m

)

contributions from 

loops to th electron magnetic anomaly, a

. As strong

and weak interactions are in general associated with

heavy-mass particles, they are expected to be more

important for a

than a

; further, a given heavy

particle contribution to a

is smaller by a factor

)

than the corresponding contribution to a

The Magnetic Anomaly a

of the m

The a

has been reviewed in Passer a (2005). The

present (2005) world average experimental value is

ðexpÞ¼116 592 080ð60Þ10

11

with a relative error 0.5 10

6

Theoretically, one can write

¼ a

ðQEDÞþa

ðhadÞþa

ðEWÞ½9

where the three terms stand for the contributions

from pure QED, strong interacting hadrons and

electroweak interactions. In turn, one can expand

(QED) in powers of  as

ðQEDÞ¼







ðlÞ

þ A

ðlÞ



þ A

ðlÞ



þA

ðlÞ

;









½10

The coefficients A

(l)

involve only the photon and

the external lepton as virtual states, are identically

the same as in a

; they are known up to l = 4

included (but, strictly speaking, the contribution of

(4)

is smaller than the experimental error of a

)

and will be discussed later for the electron. The

(l)

) are very large, being enhanced by

powers of ln (m

), and are required and known

up to l = 5; A

(l)

) starts with A

(2)

) ’

174 Quantum Electrodynamics and Its Precision Tests

1=45(m

)

, contributing 4.2 10

11

to a

,so

that the A

(l)

)withhighervaluesofl are not

needed. A

(l)

, m

), finally, starts from

l = 3, and gives a negligible contribution

0.7 10

11

. Summing up, one finds C

= 1=2,

= 0.765 857 410(27) (the error is from the experi-

mental errors in the lepton masses) C

= 24.050

509 64(43), C

= 131.011(8), and C

= 677(40). As

already observed, the coefficients are large due to the

presence of ln (m

) factors. The last term C

contributes 4.6(0.3) 10

11

to a

, and the total QED

contribution is

ðQEDÞ¼116 584 718:8ð0:3Þð0:4Þ10

11

where the first error is due to the uncertainties in the

coefficients C

, C

, and C

and the second from the

value of  coming from atom interferometry

measurements (see below).

The hadronic contributions are of two kinds,

those due to vacuum polarization, a

(vac.pol),

which can be evaluated by sound theoretical

methods by using exis ting experimental data, and

those due to light-by-light hadronic scattering,

(lbl), whose evaluation relies on much less firmer

grounds and are entirely model-dependent. The

value of a

(vac.pol) varies slightly among the

various authors (see Passera (2005) for reference to

original work), let us take as a typical value

(vac.pol) = 6834(92) 10

11

(based on e



scat-

tering data and including also first-order radiative

corrections). The model -dependent value of the

light-by-light contribution changed several times in

the years (also in sign!) but now there is a general

consensus that it should be positive; let us take,

somewhat arbitrarily, a

(lbl) = 136(25) 10

11

,so

that the total hadronic contributi on becomes

ðhadÞ¼6970ð92Þ10

11

The electroweak contrib ution, finally, is

ðEWÞ¼154ð2Þ10

11

which accounts for a one-loop purely weak

contribution and a two-loop electromagnetic and

weak contribution, which turns out to be very large

(42 10

11

) for the presence of logarithms in the

masses (the error is due to the uncertainty in the

Higgs boson mass).

Summing up, eqn [9] gives a

= 116 591 842

(92) 10

11

, so that

ðexpÞa

¼ 138ð60Þð90Þ10

11

The substantial agreement can be considered to be a

good overall check of QED and electroweak inter-

actions. But another attitude is often adop ted in

the scientific community: the validity of QED and

electroweak models is taken for granted, and a

disagreement, if any, is considered to be an indica-

tion of new physics. To obtain significant informa-

tion in that direction, however, the experimental

and the theoretical errors (dominated in turn by the

experimental error in e



scattering data) should

be significantly reduced.

The Magnetic Anomaly a

of the Electron

Experimentally, one has the 1987 value (Kinoshita

2005, ref. 1).

ðexpÞ¼1 159 652 188:4ð4:3Þ10

12

½11

with a relative error 3.7 10

9

and the preliminary

Harvard (2004) measurement (Kinoshita 2005,ref.3).

ðHarvardÞ¼1 159 652 180:86ð0:57Þ10

12

½12

with 0.5 10

9

relative error, that is, an increase in

precision by a factor 7.

Theoretically, eqns [9] and [10] apply also to the

electron; given the smallness of the electron mass,

the relevant terms up to the precision of the

experimental data are

¼A

ð1Þ







þ A

ð2Þ







þA

ð3Þ







þ A

ð4Þ







þþA

ð2Þ









þ a

ðhadÞþa

ðEWÞ½13

The explicit calculation gives

ð1Þ

Passera 2005; ref: 1ðÞ

ð2Þ

197

144



ln2 þ

ð3Þ

¼0:328478965579 ...

Passera 2005; ref: 17ðÞ

ð3Þ



ð 3Þ

215

ð5Þ

100

2 







239

2160



139

ð 3Þ

298



ln2

17101

810



28259

5184

¼1:181 241456... Laporta and Remiddi 1996ðÞ

ð4Þ

¼1:7283ð35Þ Kinoshita 2005ðÞ

Quantum Electrodynamics and Its Precision Tests 175

and

ð2Þ









’









’ 2:72 10

12

ðhadÞ¼1:67ð0:02Þ10

12

ðEWÞ¼0:03 10

12

½14

For obtaining a meaningful prediction, one needs

now a precise value of . The most precise value

available at present is that of Passera (2005, ref. 49)



1

ðaifÞ¼137:036 000 3ð10Þ

with relative error 7 10

9

, obtained by the atom

interferometry method (which is independent of

QED, depending only on the kinematics of the

Doppler effect). With that value of , the theoretical

prediction for a

becomes

¼ 1 159 652 175:9ð8:5Þð0:1Þ10

12

where the first error comes from  and the second

from C

; conversely, one can us e the QED predic-

tion for a

and a

(Harvard) for obtaining ; one

obtains in that way



1

ðQED; a

Þ¼137:035 999 708ð12Þð67Þ

where the first uncertainty is from C

and the

second from the experiment. We see that theory and

experiment are in good agreement.

As a concluding remark, another independent and

more precise (or analytic!) evaluation of C

contribu-

tion would be welcome. The five-loop term is not

known; but as (=)

 0.07 10

12

,ifC

is, say,

not greater than 2, its contribution to a

becomes

equal to the contribution of the error C

of C

and

is not yet required to match the current precision of

( exp ). The ultimate theoretical limit, the error of

the hadronic contribution, a

(had) = 0.02 10

12

is still smaller, corresponding to a change

C

= 0.0007 of C

or C

= 0.3 of C

See also: Abelian and Nonabelian Gauge Theories Using

Differential Forms; Anomalies; Effective Field Theories;

Electroweak Theory; Quantum Field Theory: A Brief

Introduction; Standard Model of Particle Physics.

Further Reading

Adkins GS, Fell RN, and Sapirstein J (2002) Two-loop correction

to the orthopositronium decay rate. Annals of Physics (NY)

295: 136.

Czarnecki A, Eidelman SI, and Karshenboim SG (2002) Muonium

hyperfine structure and hadronic effects. Physical Review D

65: 053004.

Dirac PAM (1927) The quantum theory of emission and absorp-

tion of radiation. Proceedings of the Royal Society A 114: 243.

Eides MI, Grotch H, and Shelyuto VA (2001) Theory of light

hydrogenlike atoms. Physics Reports 342: 63 (arXiv:hep-ph/

0002158).

Essen L, Donaldson RW, Bangman MJ, and Hope EG (1971)

Frequency of the hydrogen maser. Nature 229: 110.

Fee MS et al. Measurement of the positronium 1

–2

interval

by continuous-wave two-photon excitation. Physical Review

Letters 70: 1397.

Hellwig H, Vessot RFC, Levine MW, Zitzewitz PW, Allan DW,

and Glaze DJ (1970) Measurement of the unperturbed

hydrogen hyperfine transition frequency. IEEE Transactions

on Instrument Measurement IM19: 200.

Karshenboim SG (1996) Leading logarithmic corrections and

uncertainty of muonium hyperfine splitting calculations.

Z. Physics D 36: 11.

Karshenboim SG (2004) Precision study of positronium: testing

bound state QED theory. International Journal of Modern

Physics A 19: 3879 (e-print hep-ph/0310099).

Kinoshita T (2005) Improved 

term of the electron anomalous

magnetic moment (hep-ph/0507249).

Kusch P and Foley HM (1947) Precision measurements of the

ratio of the atomic ‘g values’ in the

3=2

and

1=2

states of

gallium. Physical Review 72: 1256(L).

Lamb WE Jr. and Retherford RC (1947) Fine structure of the

hydrogen atom by a microwave method. Physical Review 72:

241.

Laporta S and Remiddi E (1996) The analytical value of the electron

(g  2) at order 

in QED. Physics Letters B 379: 283.

Liu W et al. (1999) High precision measurements of the ground

state hyperfine structure interval of muonium and of the muon

magnetic moment. Physical Review Letters 82: 711.

Lundeen SR and Pipkin FM (1986) Separated oscillatory field

measurement of Lamb shift in atomic hydrogen, n = 2.

Metrologia 22(1): 9.

Melnikov K and van Ritbergen T (2000) Three-loop slope of the

Dirac form factor and the 1S Lamb shift in hydrogen. Physical

Review Letters 84: 1673 (hep-ph/9911277).

Niering M et al. (2000) Measurement of the hydrogen 1s–2s

transition frequency by phase coherent comparison with a

microwave cesium fountain clock. Physics Review Letters 84:

5496.

Pal’chikov VG, Sokolov YL, and Yakovlev VP (1985) Measure-

ment of the Lamb shift in H, n = 2. Metrologia 21: 99.

Passera M (2005) The standard model prediction of the muon

anomalous magnetic moment. Journal of Physics G 31: R75

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176 Quantum Electrodynamics and Its Precision Tests

Quantum Entropy

D Petz, Budapest University of Technology

and Economics, Budapest, Hungary

In the past 50 y ears, entropy has broken out of

thermodynamics and statistical mechanics and

invaded communication theory, ergodic theory

mathematical statisti cs, and even the social and

life sciences. The favorite subjects of entropy

concern macroscopic phenomena, irreversibility,

and incomplete knowledge. In the strictly mathe-

matical sense entropy is related to the asymptotics

of probabilities or concerns the asymptotic beha-

vior of probabilities.

This review is organized as follows. First the

history of entropy is discussed generally and then we

concentrate on the von Neumann entropy again

somewhat historically following the work of von

Neumann. Umegaki’s quantum relative entropy is

discussed both in case of finite systems and in the

setting of C



-algebras. An axiomatization is pre-

sented. To show physical applicat ions of the concept

of entropy, the statistical thermodynamics is

reviewed in the setting of spin chains. The relative

entropy shows up in the asymptotic theory of

hypothesis testing and data compression.

General Introduction to Entropy: From

Clausius to von Neumann

The word ‘‘entropy’’ was created by Rudolf Clausius

and it appeared in his work Abhandlungen u¨ ber die

mechanische Wa¨rmetheorie publi shed in 1864. The

word has a Greek origin, its first part reminds us of

‘‘energy’’ and the second part is from ‘‘tropos,’’

which means ‘‘turning point.’’ Clausius’ work is the

foundation stone of classical thermodynamics.

According to Clausius, the change of entropy of a

system is obtained by adding the small portions of

heat quantity received by the system divided by the

absolute temperature during the heat absorption.

This definition is satisfactory from a mathematical

point of view and gives nothing oth er than an

integral in precise mathematical terms. Clausius

postulated that the entropy of a closed system

cannot decrease, which is generally referred to as

the second law of thermodynamics.

The concept of entropy was really clarified by

Ludwig Boltzmann. His scientific program was to

deal with the mechanical theory of heat in connec-

tion with probabilities. Assume that a macroscopic

system consists of a large number of microscopic

ones, we simply call them particles. Since we have

ideas of quantum mechanics in mind, we assume

that each of the particles is in one of the energy

levels E

< E

< < E

. The number of particles

in the level E

is N

,so

= N is the total

number of particles. A macrostate of our system is

given by the occupation numbers N

, N

, ..., N

The energy of a macrostate is E =

. A given

macrostate can be realized by many configurations

of the N particles, each of them at a certain energy

level E

. These configurations are called microstates.

Many microstates real ize the same macrostate. We

count the number of ways of arranging N particles

in m boxes (i.e., energy levels) such that each box

has N

, N

, ..., N

particles. There are

; N

; ...; N



:¼

!...N

½1

such ways. This multinomial coefficient is the

number of microstates realizing the macrostate

, N

, ..., N

) and it is proportional to the

probability of the macrostate if all configurations

are assumed to be equally likely. Boltzmann called [1]

the thermodynamical probability of the macrostate,

in German ‘‘thermodynamische Wahrscheinlichkeit,’’

hence the letter W was used. Of course, Boltzmann

argued in the framework of classical mechanics and

the discrete values of energy came from an approxi-

mation procedure with ‘‘energy cells.’’

If we are interested in the thermodynamic limit N

increasing to infinity, we use the relative numbers

:= N

=N to label a macrostate and, instead of the

total energy E =

, we consider the average

energy pro particle E=N =

. To find the most

probable macrostate, we wish to maximize [1] under

a certain constraint. The Stirling approximation of

the factorials gives

log

; N

; ...; N



¼ Hðp

; p

; ...; p

ÞþOðN

1

log NÞ½2

where

Hðp

; p

; ...; p

Þ:¼

p

log p

½3

If N is large then the approximation [2] yields that

instead of maximizing the quantity [1] we can

maximize [3]. For example, maximizing [3] under

the constraint

= e, we get

E

½4

Quantum Entropy 177

where the constant  is the solution of the equation

E

¼ e

Note that the last equation has a unique solution if

< e < E

, and the distribution [4] is now known

as the discrete Maxwell–Bol tzmann law.

Let p

, p

, ..., p

be the probabilities of different

outcomes of a random experiment. According to

Shannon, the expression [1] is a measure of our

ignorance prior to the experiment. Hence it is also

the amount of information gained by performing the

experiment. The quantity [1] is maximum when all

the p

’s are equal. In information theory, logarithms

with base 2 are used and the unit of information is

called bit (from binary digit). As will be seen below,

an extra factor equal to Boltzmann’s constant is

included in the physical definition of entropy.

The comprehensive mathematical formalism of

quantum mechanics was first presented in the famous

book Mathematische Grundlagen der Quantenme-

chanik published in 1932 by Johann von Neumann.

In the traditional approach to quantum mechanics, a

physical system is described in a Hilbert space:

observables correspond to self-adjoint operators and

statistical operators are associated with the states. In

fact, a statistical operator describes a mixture of pure

states. Pure states are really the physical states and

they are given by rank-1 statistical operators, or

equivalently by rays of the Hilbert space.

von Neumann associated an entropy quantity to a

statistical operator in 1927 and the discussion was

extended in his book (vo n Neumann 1932). His

argument was a gedanken experiment on the

grounds of phenomenological thermodynamics. Let

us consider a gas of N( 1) molecules in a box.

Suppose that the gas behaves like a quantum system

and is described by a statistical operator ! which is a

mixture



j’

ih’

j, where j’

i’

are orthogonal

state vectors. We may take 

N molecules in the pure

state ’

for every i. The gedanken experi ment gave



j’

ih’





Sðj’

ih’

jÞ  



log 

½5

where  is Boltzmann’s constant and S is certain

thermodynamical entropy quantity (relative to the

fixed temperature and molecule density).

After this, von Neumann showed that S(j’ih’j)is

independent of the state vecto r j’i, so that



j’

ih’



¼



log 

½6

up to an additive constant, which could be chosen to

be 0 as a matter of normalization. Equation [6] is

von Neumann’s celebrated entropy formula; it has a

more elegant form

Sð!Þ¼ tr ð!Þ½7

where the state ! is identified with the correspond-

ing statistical operator, and  : R

!R is the

continuous function (t) = t log t.

von Neumann solved the maximization problem

for S(!) under the constraint tr !H = e. This means

the determination of the ensemble of maximal

entropy when the expectation of the energy operator

H is a prescribed value e . It is convenient to rephrase

his argum ent in terms of conditional expectations.

H = H



is assumed to have a discrete spectrum and

we have a conditional expectation E determined by

the eigenbasis of H. If we pass from an arbitrary

statistical operator ! with tr !H = e to E(!), then the

entropy is increasing, on the one hand, and the

expectation of the energy does not change, on the

other, so the maximizer should be searched among

the operators commuting with H. In this way we are

(and von Neumann was) back to the classical

problem of statistical mechanics treated at the

beginning of this article. In terms of operators, the

solution is in the form

expðHÞ

tr expðHÞ

½8

which is called Gibbs state today.

The von Neumann Entropy

von Neumann was aware of the fact that statistical

operators form a convex set whose extreme points

are exactly the pure states. He also knew that

entropy is a concave functional, so







Sð!

Þ½9

for any convex combination. To determine the

entropy of a statistical operator, he used the

Schatten decomposition, which is an orthogonal

extremal decomposition in our present language.

For a statistical operator ! there are many ways to

write it in the form

! ¼



if we do not require the state vectors to be

orthogonal. The geometry of the statis tical opera-

tors, that is, the state space, allows many extremal

decompositions and among them there is a unique

orthogonal one if the spectrum of ! is not

178 Quantum Entropy

degenerate. Nonorthogonal pure states are essen-

tially nonclassical. They are between identical and

completely different. Jaynes recognized in 1956 that

from the point of view of information the Schatten

decomposition is optimal. He proved that

S ð!Þ¼sup 



log 

: ! ¼



½10

where the supremum is over all convex combina-

tions ! =



statistical operators. This is Jaynes

contribution to the von Neumann entropy. By the

way, formula [10] may be used to define von

Neumann entropy for states of an arbitrary



-algebra whose states cannot be described by

statistical operators.

Certainly the highlight of quantum entropy theory

in the 1970s was the discovery of subadditivity. This

property is formulated in a tripartite system whose

Hilbert space H is a tensor product H

H

A statistical operator !

ABC

admits several reduced

densities, !

, !

, and others. The strong

subadditivity is the inequality due to Lieb and

Ruskai in 1973:

Theorem 1

Sð!

ABC

ÞþSð!

ÞSð!

ÞþSð!

Þ½11

The strong subadditivity inequality [11] is con-

veniently rewritten in terms of the relative entropy.

For statistical operators  and !,

Sðk!Þ¼tr ð log   log !Þ½12

if supp   supp !, otherwise S(k!) = þ1. The

relative entropy ex presses statis tical distinguishabil-

ity and therefore it decreases under stochastic

mappings:

Sðk!ÞSðEðÞkEð!ÞÞ ½13

for a completely positive trace-preserving mapping E.

The strong subadditivity is equivalent to

Sð!

;’ !

ÞSð!

ABC

;’ !

Þ½14

where ’ is any state on B(H

) of finite entropy. This

inequality is a consequenc e of monotonicity of the

relative entropy, since !

= E(!

ABC

) and ’  !

E(’  !

), where E is the partial trace over H

Clearly, the equality in [11] is equivalent to equality

in [14].

Theorem 2 The equality holds in [11] if and only

if there is an orthogonal decomposition p

H

, p

= sup p !

, such that the density

operator of !

ABC

satisfies

ABC

ðp

Þ!

 !

½15

where !

2 B(H

)  B(H

) and !

2 B(H

) 

B(H

) are density operators and p

2 B(H

) are

the orthogonal projections H

H

Quantum Relative Entropy

The quantum relative entropy is an information

measure representing the uncertainty of a state with

respect to another state. Hence it indicates a kind of

distance between the two states. The formal defini-

tion [12] is due to Umegaki.

Now we approach quantum relative entropy

axiomatically. Our crucial postulate includes the

notion of conditional expectation. Let us recall that

in the setting of operator algebras conditional

expectation (or projection of norm 1) is defined as

a positive unital idempotent linear mapping onto a

subalgebra.

Now we list the properties of the relative entropy

functional which will be used in an axiomatic

characterization:

1. Conditional expectation property. Assume that A

is a subalgebra of B and there exists a projection of

norm 1 E of B onto A,suchthat’  E = ’. Then

for every state ! of B S(!, ’) = S(!jA, ’jA) þ

S(!, !  E)holds.

2. Invariance property. For every automorphism 

of B we have S(!, ’) = S (!  , ’  ).

3. Direct sum property.AssumethatB= B

B

.Let

’

(a b) = ’

(a) þ(1 )’

(b)and!

(a b) =

!

(a) þ(1 )!

(b) for every a 2B

, b 2B

and

some 0 <<1. Then S(!

,’

)= S (!

,’

)þ

(1 )S(!

,’

4. Nilpotence property. S(’, ’) = 0.

5. Measurability property. The function (!, ’) 7!

S(!, ’) is measurable on the state space of the

finite dimensional C



-algebra B (when ’ is

assumed to be faithful).

Theorem 3 If a real valued functional R(!, ’)

defined for faithful states ’ and arbitrary states !

of finite quantum systems s hares the properties

[1]–[5], then there exists a constant c 2 R such

that

Rð!; ’Þ¼c Tr D

ðlog D

 log D

’

The relative entropy may be defined for linear

functionals of an arbitrary C



-algebra. The general

definition may go through von Neumann algebras,

normal states and the relative modular operator.

Another possibility is based on the monotonicity.

Let ! and ’ be states of a C



-algebra A. Consider

finite-dimensional alge bras B and completely posi-

tive unital mappings  : B!A. Then the supremum

Quantum Entropy 179

of the relative entropies S(!  k’  ) (over all )

can be defined as S(!k’).

Theorem 4 The relative entropy of states of



-algebras shares the following properties.

(i) (!, ’) 7!S(!k’) is convex and weakly lower-

semicontinuous.

(ii) k’  !k

 2S(!, ’).

(iii) For a unital Schwarz map  : A

the

relation S(!  k’  )  S(!k’) holds.

Property (iii) is Uhlmann’s monotonicity theorem,

which we have already applied above.

The relative entropy appears in many concepts

and problems in the area of quantum information

theory (Nielsen and Chuang 2000, Schumacher and

Westmoreland 2002).

Statistical Thermodynamics

Let an infinitely extended system of quantum spins

be considered in the simple cubic lattice L = Z



where  is a positive integer. The observables

confined to a lattice site x 2 Z



form the self-adjoint

part of a finite-dimensional C



-algebra A

which is a

copy of the matrix algebra M

(C). It is assumed that

the local observables in any bounded region   Z



are those of the finite quantum system



x2

It follows from the definition that for   

have A



= A



A



n

, where 

n is the comple-

ment of  in 

. The algebra A



and the subalgebra



 CI



n

of A



have identical structure and we

identify the element A2A



with A  I



n

in A



If   

then A



A



and it is said that A



isotonic with respect to . The definition also

implies that if 

and 

are disjoint then elements

of A



commute with those of A



. The quasilocal



-algebra A is the norm completion of the normed

algebra A

= [



, the union of all local algebras



associated with bounded (finite) regio ns   Z



We denote by a

the element of A

corresponding

to a 2A

(x 2 Z



). It follows from the definition

that the algebra A

consists of linear combinations

of terms a

(1)

a

(k)

where x

, ..., x

and a

(1)

, ..., a

(k)

run through Z



and A

, respectively. We define 

to be the linear transformation

ð1Þ

a

ðkÞ

7! a

ð1Þ

þx

a

ðkÞ

þx



corresponds to the space translation by x 2 Z



and it extends to an automorphism of A. Hence  is

a representation of the abelian group Z



automorphisms of the quasilocal algebra A. Clearly,

the covariance condition



ðA



Þ¼A

þx

holds, where  þ x is the space-translate of the

region  by the displacement x.

Having described the kinematical structure of

lattice systems, we turn to the dynamics. The local

Hamiltonian H() is taken to be the total potential

energy between the particles confined to . This

energy may come from many-body interactions of

various orders. Most generally, we assume that there

exists a global function  such that for any finite

subsystem  the local Hamiltonian takes the form

HðÞ¼

X

ðXÞ½16

Each (X) represents the interaction energy of the

particles in X. Mathematically, ( X) is a self-adjoint

element of A

and H () will be a self-adjoint

operator in A



. We restrict our discussion to

translation-invariant interactions, which satisfy the

additional requirement



ððXÞÞ ¼ ðX þ xÞ

for every x 2 Z



and every region X  Z



.An

interaction  is said to be of finite range if () = 0

when the cardinality (or diameter) of  is large

enough, d()  d



. The infimum of such numbers is

called the range of .

If ’ is a state of the quasilocal algebra A then it

will induce a state ’



on A(), the finite system

comprising the spin in the bounded region  of Z



The (local) energy, entropy, and free energy of this

finite system are given by the following formulas:



ð’Þ :¼ tr



HðÞ



ð’Þ :¼tr



log !



½17





ð’Þ :¼ E



ð’Þ





ð’Þ

Here !



denotes the density of ’



with respect to the

trace tr



of A



,and denotes the inverse temperature.

The functionals E



, S



,andF





are termed local. It is

rather obvious that all three local functionals are

continuous if the weak



topology is considered on the

state space of the quasilocal algebra. The energy is

affine, the entropy is concave and consequently, the

free energy is a convex functional.

The free energy functional F





is minimized by the

Gibbs state (see [8] with H = H(  )), and the

minimum value is given by



log tr



HðÞ

½18

180 Quantum Entropy

Our aim is to explain this variational principle after

the thermodynamic limit is perfor med.

The thermodynamic limit ‘‘ tends to infinity’’

may be taken along lattice parallelepipeds. Let a 2 Z



with positive coordinates and define

ðaÞ¼fx 2 Z



: 0  x

< a

; i ¼ 1; 2; ...;g½19

When a !1, (a) tends to infinity in a manner

suitable for the study of thermodynamic limit: the

boundary of the parallelepipeds is getting more and

more negligible compared with the volume. The

notion of limit in the sense of van Hove makes this

idea more precise and physically more satisfactory.

For the sake of simplicity, we restrict ourselves to

thermodynamic limit along parallelepipeds.

Denoting by j j the volume of  (or the number

of points in ), we may define the global energy,

entropy, and free energy functionals of translation-

ally invariant states to be

eð’Þ :¼ lim

!1



ð’Þ=jj½20

sð’Þ :¼ lim

!1



ð’Þ=jj½21



ð’Þ :¼ lim

!1





ð’Þ=jj½22

The existence of the limit in [21] is guaranteed by

the strong subadditivity of entropy, while that of the

limits in [20] and [22] is assumed if the interaction

is suitably tempered, as it certainly does if the

interaction is of finite range.

Theorem 5 If ’ is a translationally invariant state of

the quasilocal algebra A, then the limit [21] exists and

sð’Þ¼inffS

ðaÞ

ð’Þ=jðaÞj: a 2 Z



g½23

Moreover, the von Neumann entropy density functional

’ 7!s(’) is affine and upper-semicontinuous when the

state space is endowed with the weak



topology.

Let  be an interaction of finite range. Then the

thermodynamic limit [20] exists and the energy

density is given by

eð’Þ¼’ðE



Þ and E



02

ðÞ

jj

Furthermore, e(’) is an affine weak



continuous

functional of ’.

It follows that the free energy density f (’) exists

and it is an affine lower-semicontinuous function of

the translation-invariant stat e ’.

For 0 <<1 the thermodynamic limit

lim

!1

jj

log tr



HðÞ

 pð; Þ

exists.

In accordance with the lattice-gas interpretation

of our model, the global quantity p is termed

pressure.

In the treatment of quantum spin systems, the set



of all translation-invariant states is essen tial. The

global entropy functional s is a continuous affine

function on S



and physically it is a macroscopic

quantity which does not have microscopic (i.e.,

local) counterpart. Indeed, the local entropy func-

tional is not an observable because it is not affine on

the (local) state space. The local internal energy



(’) is microscopic observable and the energy

density functional e of S



is the corresponding

global extensive quantity.

As an analog of the variational principle for finite

quantum systems, the global free-energy functional f



attains an absolute minimum at a translationally

invariant state, and the minimum value of f



is equal

to the thermodynamic limit of the canonical free-

energy densities of the local finite systems. In the next

theorem, this global variational principle will be

formulated in a slightly different but equivalent way.

Theorem 6 Wh en  is an interaction of finite

range, then

pð; Þ¼supfsð!Þeð!Þg

holds, when the supremum is over all translationally

invariant states ! on A.

The minimizers of the right-hand side are called

equilibrium states and they have several different

characterizations.

Asymptotical Properties

We keep the notation of the previous section but we

consider one-dimensional chains,  = 1. Let ! be

translation-invariant state on A and we fix a positive

number "<1. We have in our mind that " is small and

say that a sequence of projection Q

[1, n]

is of high

probability if !(Q

)  1  ". The size of Q

,the

cardinality of a maximal pairwise orthogonal family of

projections contained in Q

, is given by tr

.(The

subscript n in tr

indicates that the algebraic trace

functional on A

is meant here.) The theorem below

says that the entropy density of ! governs asymptoti-

cally the rank of the high-probability projections.

Theorem 7 Assume that ! is an ergodic translation-

invariant state of A. Then the limit relation

lim

n!1

infflog tr

g¼sð!Þ

holds, when the infimum is over all proje ctions

[1, n]

such that !

)  1  ".

Quantum Entropy 181