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

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This result is strongly related to data compression.

When ! is interpreted as a stationary quantum source

(with possible memory), then efficient and reliable

data compression needs a subspace of small dimension

and the range of Q

can play this role. The entropy

density is the maximal rate of reliable compression.

It is interesting that one can impose further

requirements on the high-probability projections

and the statement of the theorem remains true.

1. The partial trace of Q

nþ1

over A

nþ1

is Q

;

2. e

n(s")

 tr Q

 e

n(sþ")

if n is large enough; and

3. if q  Q

is a minimal projection (in A

[1, n]

), then

!(q)  e

n(s")

if n is large enough.

In (2) and (3) s stands for s(!). Let D

be the density

matrix of the restriction of ! to A

[1, n]

. It follows

that for an eigenvalue  of Q

the inequality

s  " 

log 

holds.

From the point of view of data compression, it is

important if the sequence Q

[1, n]

works uni-

versally for many states. Indeed, in this case the

compression algorithm can be universal for several

quantum sources.

Theorem 8 Let R > 0. There is a projection

[1, n]

such that

lim sup

log trQ

 R ½24

and for any ergodic state ! on A such that s(!) < R

the relation

lim

!ðQ

Þ¼1 ½25

holds.

In the simplest quantum hypothesis testing prob-

lem, one has to decide between two states of a

system. The state 

is the null hypothesis and 

the alternative hypothesis. The problem is to decide

which hypothesis is true. The decision is performed

by a two-valued measurement {T, I  T}, where

0  T  I is an observable. T corresponds to

the acceptance of 

and I  T corresponds to the

acceptance of 

. T is called a test. When the

measurement value is 0, the hypothesis 

accepted, otherwise the alternat ive hypothesis 

accepted. The quantity [T] = tr

(I  T) is inter-

preted as the probability that the null hypothesis is

true but the alternative hypothesis is accepted. This

is the error of the first kind. Similarly, [T] = tr

is the probability that the alternative hypothesis is

true but the null hypothesis is accepted. It is called

the error of the second kind.

Now we fix a formalism for an asymptotic theory

of the hypothesis testing. Suppose that a sequence

) of Hilbert spaces is given, (

(n)

)and(

(n)

)are

density matrices on H

. The typical example we have

in mind is 

(n)

= 

 



and 

(n)

= 







. A positive contraction T

2B(H

)is

considered as a test on a composite system. Now the

errors of the first and second kind depend on n:



] = tr

(n)

(I  T

)and

] = tr

(n)

Set





ðn;"Þ¼infftr

ðnÞ

g½26

where the infimum is over all A

2B(H

) such that

0  A

 I and tr

(n)

(I  A

)  ". In other words,

this is the infimum of the error of the second kind

when the error of the first kind is at most ". The

importance of this quantity is in the customary

approach to hypothesis testing .

The following result is the quantum Stein lemma.

Theorem 9 In the above setting, the relation

lim

n!1

log 



ðn;"Þ¼Sð

k

holds for every 0 <"<1.

Bibliographic Notes

A book about entropy in several contexts is Greven

et al. (2003). The role of von Neumann in the

mathematization of the quantum entropy is

described in Petz et al. (2001); this work contains

also his gedanken experiment.

The first proof of the strong subadditivity appeared

in Lieb and Ruskai (1973) and a didactical elementary

approach is in Nielsen and Petz (2005). The structure

of the case of equality was obtained in Hayden et al.

(2004). The quantum relative entropy was introduced

by Umegaki in 1962 and their properties and its

axiomatization are contained in the monograph by

Ohya and Petz (1993). The monotonicity of the

relative entropy is called Uhlmann’s theorem, see

Uhlmann (1977) and Ohya and Petz (1993).

The rigorous and comprehensive treatment of

quantum lattice systems was one of the early

successes of the algebraic approach to quantum

statistical thermodynamics. The subject is well

summarized in Bratteli and Robinson (1981). The

book by Sewell (1986) contains more physics and

has less in mathematical technicalities. For many

interesting entropy results concerning mean field

systems, see, for example, Ragio and Werner (1991).

The high probability subspace theorem is due to

Ohya and Petz – Petz (1992) and Ohya and Petz

(1993) for product states – and was extended to

182 Quantum Entropy

some algebraic and Gibbs states by Hiai and Petz.

The application to data compression was first

observed by Schumacher (1995). The chained

property of the high-probability subspaces was

studied in Bjelakovic

et al. (2003) and the univers-

ality is from Kaltchenko and Yang (2003).

A weak form of the quantum Stein lemma was

proved in Hiai and Petz (1991) and the stated form

is due to Nagaoka and Ogawa (2000). An extension

to the case where 

(n)

is not a product was given in

Bjelakovic

and Siegmund-Schultze (2004).

Other surveys about quantum entropy are Petz

(1992) and Schumacher and Westmor eland (2002).

See also: Asymptotic Structure and Conformal Infinity;

Capacities Enhanced by Entanglement; Channels in

Quantum Information Theory; Entropy and Qualitative

Transversality; Positive Maps on C



-Algebras; von

Neumann Algebras: Introduction, Modular Theory and

Classification Theory; von Neumann Algebras: Subfactor

Theory.

Further Reading

Bjelakovic

I, Kru¨ ger T, Siegmund-Schultze R, and Szko

la A

(2003) Chained typical subspaces – a quantum version of

Breiman’s therem. Preprint.

Bjelakovic

I and Siegmund-Schultze R (2004) An ergodic theorem

for the quantum relative entropy. Communications in Math-

ematical Physics 247: 697–712.

Bratteli O and Robinson DW (1981) Operator Algebras and

Quantum Statistical Mechanics. 2. Equilibrium States. Models

in Quantum Statistical Mechanics, Texts and Monographs in

Physics, (2nd edn., 1997). Berlin: Springer.

Greven A, Keller G, and Warnecke G (2003) Entropy. Princeton

and Oxford: Princeton University Press.

Hayden P, Jozsa R, Petz D, and Winter A (2004) Structure of

states which satisfy strong subadditivity of quantum entropy

with equality. Communications in Mathematical Physics 246:

359–374.

Hiai F and Petz D (1991) The proper formula for relative entropy

and its asymptotics in quantum probability. Communications

in Mathematical Physics 143: 99–114.

Kaltchenko A and Yang E-H (2003) Universal compression of

ergodic quantum sources. Quantum Information and Compu-

tation 3: 359–375.

Lieb EH and Ruskai MB (1973) Proof of the strong subadditivity

of quantum mechanical entropy. Journal of Mathematical

Physics 14: 1938–1941.

Nielsen MA and Petz D (2005) A simple proof of the strong

subadditivity inequality. Quantum Information and Computa-

tion 5: 507–513.

Ogawa T and Nagaoka H (2000) Strong converse and Stein’s

lemma in quantum hypothesis testing. IEEE Transactions on

Information Theory 46: 2428–2433.

Ohya M and Petz D (1993) Quantum Entropy and Its Use, Texts

and Monographs in Physics, (2nd edn., 2004). Berlin:

Springer.

Petz D (1992) Entropy in quantum probability. In: Accardi L (ed.)

Quantum Probability and Related Topics VII, pp. 275–297.

Singapore: World Scientific.

Petz D (2001) Entropy, von Neumann and the von Neumann

entropy. In: Re´dei M and Sto¨ltzner M (eds.) John von Neumann

and the Foundations of Quantum Physics. Dordrecht: Kluwer.

Raggio GA and Werner RF (1991) The Gibbs variational

principle for inhomogeneous mean field systems. Helvetica

Physica Acta 64: 633–667.

Schumacher B (1995) Quantum coding. Physical Review A 51:

2738–2747.

Schumacher B and Westmoreland MD (2002) Relative entropy in

quantum information theory. In: Quantum Computation and

Information, (Washington, DC, 2000), Contemp. Math. vol.

305, pp. 265–289. Providence, RI: American Mathematical

Society.

Sewell GL (1986) Quantum Theory of Collective Phenomena,

New York: Clarendon.

Uhlmann A (1977) Relative entropy and the Wigner–Yanase–

Dyson–Lieb concavity in an interpolation theory. Commu-

nications in Mathematical Physics 54: 21–32.

von Neumann J (1932) Mathematische Grundlagen der Quanten-

mechanik

. Berlin: Springer. (In English: von Neumann J,

Mathematical Foundations of Quantum Mechanics. Princeton:

Princeton University Press.)

Quantum Ergodicity and Mixing of Eigenfunctions

S Zelditch, Johns Hopkins University, Baltimore,

MD, USA

Quantum ergodicity and mixing belong to the field

of quantum chaos, which studies quantizations of

‘‘chaotic’’ classi cal Hamiltonian systems. The basic

question is: how does the chaos of the classical

dynamics impact on the eigenvalues/eigenfunctions

of the quantum Hamiltonian

H and on long-time

dynamics generated by

These problems lie at the foundations of the

semiclassical limit, that is, the limit as the Planck

constant h !0 or the energy E !1. More generally,

one could ask what impact any dynamical feature of a

classical mechanical system (e.g., complete integrabil-

ity, KAM, and ergodicity) has on the eigenfunctions

and eigenvalues of the quantization.

Over the last 30 years or so, these questions have

been studied rather systematically by both mathe-

maticians and physicists. There is an extensive

literature comparing classical and quantum

dynamics of model systems, such as comparing the

geodesic flow and wave group on a compact (or

finite-volume) hyperbolic surface, or comparing

classical and quantum billiards on the Sinai billiard

or the Bunimovich stadium, or comparing the

Quantum Ergodicity and Mixing of Eigenfunctions 183

discrete dynamical system generated by a hyperbolic

torus automorphism and its quantization by the

metaplectic representation. As these models indicate,

the basic problems and phenomena are richly

embodied in simple, low-dimensional examples in

much the same way that two-dimensional toy

statistical mechanical models already illustrate com-

plex problems on phase transitions. The principles

established for simple models should apply to far

more complex systems such as atoms and molecules

in strong magnetic fields.

The conjectural picture which has emerged from

many computer experiments and heuristic argu-

ments on these simple model systems is roughly

that there exists a length scale in which quantum

chaotic systems exhibit universal behavior. At this

length scale, the eigenvalues resemble eigenvalues of

random matrices of large size and the eigenfunctions

resemble random waves. A small sample of the

original physics articl es sugges ting this picture is

Berry (1977), Bohigas et al. (1984), Feingold and

Peres (1986),andHeller (1984).

This article reviews some of the rigorous mathe-

matical results in quantum chaos, particularly those

on eigenfunctions of quantizations of classically

ergodic or mixing systems. They support the

conjectural picture of random waves up to two

moments, that is, on the level of means and

variances. A few results also exist on higher

moments in very special cases. But from the

mathematical point of view, the conjectural links

to random matrices or random waves remain very

much open at this time. A key difficulty is that the

length scale on which universal behavior should

occur is far below the resolving power of any known

mathematical techniques, even in the simplest model

problems. The main evidence for the random

matrix and random wave connections comes from

numerous computer experiments of model cases in

the physics literature. We will not review numerical

results here, but to get a well-rounded view of the

field, it is important to understand the computer

experiments (see, e.g., Ba¨ cker et al. (1998a, b) and

Barnett (2005)).

The model quantum systems that have been most

intensively studied in mathematical quantum chaos

are Laplacians or Schro¨ dinger operators on com-

pact (or finite-volume) Riemannian manifolds, with

or without boundary, and quantizations of sym-

plectic maps on compact Ka¨hl er manifolds. Similar

techniques and results apply in both settings, so for

thesakeofcoherenceweconcentrateonthe

Laplacian on a compact Riemannian manifold

with ‘‘chaotic’’ geodesic flow and only briefly

allude to the setting of ‘‘quantum maps.’’

Additionally, two main kinds of methods are in

use: (1) methods of semiclassical (or m icrolocal)

analysis, which apply to general Laplacians (and

more general Schro¨ dinger operators), and (2)

methods of number theory and automorphic

forms, which apply to arithmetic models such as

arithmetic hyperbolic manifolds or quantum cat maps.

Arithmetic models are far more ‘‘explicitly solvable’’

than general chaotic systems, and the results obtained

for them are far sharper than the results of semiclassi-

cal analysis. This article is primarily devoted to the

general results on Laplacians obtained by semiclassical

analysis; see Arithmetic Quantum Chaos for results by

J Marklov. For background on semiclassical analysis,

see Heller (1984).

Wave Group and Geodesic Flow

The model quantum Hamiltonians we will discuss

are Laplacians  on compact Riemannian mani-

folds (M, g) (with or without boundary). The

classical phase space in this setting is the cotangent

bundle T



M of M , equipped with i ts canonical

symplectic form

^d

. The metric defines

the Hamiltonian

Hðx;Þ¼jj

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ij¼1

ðxÞ



on T



M, where

¼ g

;



] is the inverse matrix to [g

]. We denote the

volume density of (M, g) by dVol and the corre-

sponding inner product on L

(M)byhf , g i. The unit

(co-) ball bundle is denoted B



M = {(x, ):jj1}.

The Hamiltonian flow 

of H is the geodesic

flow. By definition, 

(x, ) = (x

, 

), where (x

, 

)is

the terminal tangent vector at time t of the unit

speed geodesic starting at x in the direction . Here

and below, we often identify T



M with the tangent

bundle TM using the metric to simplify the

geometric description. The geodesic flow preserves

the energy surfaces {H = E} which are the co-sphere

bundles S



M. Due to the homogeneity of H,

the flow on any energy surface {H = E} is equivalent

to that on the co-sphere bundle S



M = {H = 1}.

(This homogeneity could be broken by adding a

potential V 2 C

(M) to form a semiclassical

Schro¨ dinger operator h

 þ V, whose underlying

Hamiltonian flow is generated by jj

þ V(x).) See

h-Pseudodifferential Operators and Applications.

184 Quantum Ergodicity and Mixing of Eigenfunctions

The quantization of the Hamiltonian H is the

square root

ﬃﬃﬃﬃ



of the positive Laplacian

 ¼

ﬃﬃﬃ

i;j¼1

of (M, g ). Here, g = det [g

]. We choose to work

with

ﬃﬃﬃﬃ



rather than  since the former generates

the wave

¼ e

ﬃﬃﬃ



which is the quantization of the geodesic flow 

By the last statement we mean that U

is related to



in several essentially equivalent ways:

1. singularities of waves, that is, solutions U

the wave equation, propagate along geodesics;

2. U

is a Fourier integral operator (= quantum

map) associated to the canonical relation defined

by the graph of 

in T



M  T



M;and

3. Egorov’s theorem holds.

We only define the latter since it plays an important

role in studying eigenfunctions. As with any quantum

theory, there is an algebra of observables on the

Hilbert space L

(M, dvol

) which quantizes T



Here, dvol

is the volume form of the metric. The

algebra is that 



(M) of pseudodifferential operators

DO’s of all orders, though we often restrict to the

subalgebra 

of DO’s of order zero. We denote by



(M) the subspace of pseudodifferential operators of

order m. The algebra is defined by constructing a

quantization Op from an algebra of symbols a 2



M) of order m (polyhomogeneous functions on



Mn0) to 

. The map Op is not unique. In the

reverse direction is the symbol map 

: 



M) which takes an operator Op(a)tothe

homogeneous term a

of order m in a.

Egorov’s theorem for the wave group concerns the

conjugations



ðAÞ:¼U



; A 2

ðMÞ½1

Such a conjugation defines the quantum evolution of

observables in the Heisenberg picture, and, since the

early days of quantum mechanics, it was known to

correspond to the classical evolution

ðaÞ:¼a  

½2

of observables a 2 C



M). Egorov’s theorem is

the rigorous version of this correspondence: it states

that 

defines an order-preserving automorph ism of





(M), that is, 

(A) 2

(M)ifA 2

(M), and

that





ðx;Þ¼

ð

ðx;ÞÞ:¼V

ð

Þ;

ðx;Þ2T



Mn0 ½3

This formula is almost universally taken to be the

definition of quantization of a flow or map in the

physics literature.

The key difficulty in quantum chaos is that it

involves a comparison between long-time dynamical

properties of 

and U

through the symbol map and

similar classical limits. The classical dynamics

defines the ‘‘principal symbol’’ behavior of U

and

the ‘‘error’’ U



 Op(

 

) typically grows

exponentially in time. This is just the first example

of a ubiquitous ‘‘exponential barrier’’ in the subject.

Eigenvalues and Eigenfunctions of 

The eigenvalue problem on a compact Riemannian

manifold

’

¼ 

’

; h ’

;’

i¼

is dual under the Fourier transform to the wave

equation. Here, {’

} is a choice of orthonormal basis

of eigenfunctions, which is not unique if the

eigenvalues have multiplicities > 1. The individual

eigenfunctions are difficult to study directly, and so

one generally forms the spectral projections kernel,

Eð; x; yÞ¼

j : 



’

ðxÞ’

ðyÞ½4

Semiclassical asymptotics is the study of the  !1

limit of the spectral data {’

, 

}orofE(, x, y). The

(Schwartz) kernel of the wave group can be

represented in terms of the spect ral data by

ðx; yÞ¼

it

’

ðxÞ’

ðyÞ

or equivalently as the Fourier transform

it

dE(, x, y) of the spectral projections. Hence,

spectral asymptotics is often studied through the

large-time behavior of the wave group.

The link between spectral theory and geometry,

and the source of Egorov’s theorem for the wave

group, is the construction of a parametrix (or WKB

formula) for the wave kernel. For small times t, the

simplest is the Hadamard parametrix,

ðx; yÞ

iðr

ðx;yÞt

k¼0

ðx; yÞ

ððd3Þ=2Þk

d

ðt < injðM; gÞÞ ½5

where r(x, y) is the distance between points,

(x, y) =

1=2

(x, y) is the volume 1/2-density,

inj(M, g) is the injectivity radius, and the higher

Hadamard coefficients are obtained by solving

transport equations along geodesics. The parametrix

is asymptotic to the wave kernel in the sense of

Quantum Ergodicity and Mixing of Eigenfunctions 185

smoothness, that is, the difference of the two sides of

[5] is smooth. The relation [5] may be iterated using

= U

to obtain a parametrix for long times.

This is obviously complic ated and not necessarily

the best long-time parametrix construction, but it

illustrates again the difficulty of a long-time

analysis.

Weyl Law and Local Weyl Law

A fundamental and classical result in spectral

asymptotics is Weyl’s law on counting eigenvalues:

NðÞ¼#fj : 

 g

ð2Þ

VolðM; gÞ

þ Oð

n1

Þ½6

Here, jB

j is the Euclidean volume of the unit ball

and Vol(M, g) is the volume of M with respect to the

metric g. An equivalent formula which emphasizes

the correspondence between classical and quantum

mechanics is

trE



Volðjj

 Þ

ð2Þ

½7

where Vol is the symplectic volume measure relative

to the natural symplectic form

j = 1

^ d



M. Thus, the dimension of the space where

H =

ﬃﬃﬃﬃ



is   is asym ptotically the volume where

its symbol jj

 .

The remainder term in Weyl’s law is sharp on the

standard sphere, where all geodesics are periodic, but

is not sharp on (M, g) for which the set of periodic

geodesics has measure zero (Duistermaat–Guillemin,

Ivrii) (see Semiclassical Spectra and Closed Orbits).

When the set of periodic geodesics has measure zero

(as is the case for ergodic systems), one has

NðÞ¼#fj : 

 g

ð2Þ

VolðM; gÞ

þ oð

n1

Þ½8

The remainder is then of smaller order than the

derivative of the principal term, and one then has

asymptotics in shorter intervals:

Nð½;  þ 1Þ ¼ #fj : 

2½;  þ 1g

¼ n

ð2Þ

VolðM; gÞ

n1

þ oð

n1

Þ½9

Physicists tend to write   h

1

and to average over

intervals of this width. Then mean spacing between

the eigenvalues in this interval is  C

Vol(M, g)

1





(n1)

, where C

is a constant depending on the

dimension.

An important generalization is the ‘‘local Weyl law’’

concerning the traces trAE(), where A 2 

(M).

It asserts that





hA’

;’

ð2Þ





dx d

þ Oð

n1

Þ½10

There is also a pointwise local Weyl law:





j’

ðxÞj

ð2Þ

j

þ Rð; xÞ½11

where R(, x) = O(

n1

) uniformly in x. Again,

when the periodic geodesics form a set of measure

zero in S



M, one could average over the shorter

interval [,  þ 1]. Combining the Weyl and local

Weyl law, we find the surface average of 

is a

limit of traces:

!ðAÞ:¼

ðS



MÞ





d

¼ lim

!1

NðÞ





hA’

;’

i½12

Here,  is the ‘‘Liouville measure’’ on S



M, that is,

the surface measure d = dx d=dH induced by the

Hamiltonian H = jj

and by the symplectic volume

measure dx d on T



Problems on Asymptotics Eigenfunctions

Eigenfunctions arise in quantum mechanics as

stationary states, that is, states for which the

probability measure j (t, x)j

dvol is constant in time

where (t, x) = U

(x) is the evolving state. This

follows from the fact that

’

¼ e

it

’

½13

and that je

it

j= 1. They are the basic modes of the

quantum system. One would like to know the

behavior as 

!1 (or h !0 in the semiclassical

setting) of invariants such as:

1. matrix elements hA’

, ’

i of observables in this

state;

2. transition elements hA’

, ’

i between states;

3. size properties as measured by L

norms k’

;

4. value distribution as measured by the distribution

function Vol{x 2 M : j’

(x)j

> t}; and

5. shape properties, for example, distribution of

zeros and critical points of ’

Let us introduce some problems which have

motivated much of the work in this area.

186 Quantum Ergodicity and Mixing of Eigenfunctions

Problem 1 Let Q denote the set of ‘‘quantum

limits,’’ that is, weak



limit points of the sequence

{

} of distributions on the classical pha se space



M, defined by

a d

:¼hOpðaÞ’

;’

where a 2 C



M).

The set Q is independent of the definition of Op.

It follows almost immediately from Egorov’s theo-

rem that QM

, where M

is the convex set of

invariant probability measures for the geodesic flow.

Furthermore, they are time-reversal invariant, that

is, invariant under (x, ) !(x, ) since the eigen-

functions are real valued.

To see this, it is helpful to introduce the linear

functionals on 



ðAÞ¼hOpðaÞ’

;’

i½14

We observe that 

(I) = 1, 

(A)  0ifA  0,

and that





¼

ðAÞ½15

Indeed, if A  0 then A = B



B for some B 2 

and we can move B



to the right-hand side.

Similarly, [15] is proved by moving U

to the right-

hand side and using [13]. These properties mean

that 

is an ‘‘invariant state’’ on the algebra 

More precisely, one should take the closure of 

the operator norm. An invariant state is the analog

in quantum statistical mechanics of an invariant

probability measure.

The next important fact about the states 

is that

any weak limit of the sequence {

}on

is an

invariant probability measure on C(S



M), that is,

a positive linear functional on C(S



M) rather than

just a state on 

. This follows from the fact that

hK’

, ’

i!0 for any compact operator K, and so any

limit of hA’

, ’

i is equally a limit of h(A þ K)’

, ’

Hence, any limit is bounded by inf

kA þ Kk (the

infimum taken over compact operators), and for any

A 2 

, k

= inf

kA þ Kk.Hence,anyweak

limit is bounded by a constant times k

and is

therefore continuous on C(S



M). It is a positive

functional since each 

, and hence any limit, is a

probability measure. By Egorov’s theorem and the

invariance of the 

,anylimitof

(A) is a limit of



(Op(

 

)) and hence the limit measure is

invariant.

Problem 1 is thus to identify which invariant

measures in M

show up as weak limits of the

functionals 

or equivalently the dist ributions d

The weak limits reflect the concentration and

oscillation properties of eigenfunctions. Here are

some possibilities:

1. Normalized Liouville measure. In fact, the func-

tional ! of [12] is also a state on 

for the

reason explained above. A subsequence {’

}of

eigenfunctions is considered diffuse if 

!!.

2. A periodic orbit measure 



defined by





ðAÞ¼





where L



is the length of . A sequence of

eigenfunctions for which 

!



obviously con-

centrates (or strongly ‘‘scars’’) on the closed

geodesic.

3. A finite sum of periodic orbit measures.

4. A delta-function along an invariant Lagrangian

manifold   S



M. The associated eigenfunctions

are viewed as ‘‘localizing’’ along .

5. A more general invariant measure which is

singular with respect to d.

All of these possibilities can and do happen in

different examples . If d

!!, then in particular

we have

VolðMÞ

j’

ðxÞj

dVol !

VolðEÞ

VolðMÞ

for any measurable set E whose boundary has

measure zero. Interpreting j’

(x)j

dVol as the

probability density of finding a particle of energy



at x, this result means that the sequence of

probabilities tends to uniform measure.

However, d

!! is much stronger since it says

that the eigenfunctions become diffuse on the energy

surface S



M and not just on the configuration space

M. As an example, consider the flat torus R

An orthonormal basis of eigenfunctions is furnished

by the standard exponentials e

2ihk, xi

with k 2 Z

Obviously, je

2ihk, xi

= 1, so the eigenfunctions are

already diffuse in configuration space. On the other

hand, they are far from diffuse in phase space, and

localize on invariant Lagrangian tori in S



M. Indeed,

by definition of pseudodifferential operator,

2ihk, xi

= a(x, k)e

2ihk, xi

, where a(x, k) is the com-

plete symbol. Thus,

hAe

2ihk;xi

; e

2ihk;xi

i¼

aðx; kÞdx





jkj



A subsequ ence e

2ihk

, xi

of eigenfunctions has a weak

limit if and only if k

=jk

j tends to a limit vector 

the unit sphere in R

. In this case, the associated

Quantum Ergodicity and Mixing of Eigenfunctions 187

weak



limit is



(x, 

)dx, that is, the delta-

function on the invariant torus T



 S



M defined

by the constant momentum condition  = 

. The

eigenfunctions are said to localize on this invariant

torus for 

The flat torus is a model of a completely

integrable system on both the classical and quantum

levels. Another example is that of the standard

round sphere S

. In this case, the author and

D Jakobson showed that absolutely any invariant

measure  2M

can arise as a weak limit of a

sequence of eigenfunctions. This reflects the huge

degeneracy (multiplicities) of the eigenvalues.

On the other hand, if the geodesic flow is ergodic,

one would expect the eigenfunctions to be diffuse in

phase space. In the next section, we will discuss the

rigorous results on this problem.

Off-diagonal matrix elements



ðAÞ¼hA’

;’

i½16

are also important as trans ition amplitudes between

states. They no longer define states since 

(I) = 0,

are positive, or invariant. Indeed, 



) =

it(



)



(A), so they are eigenvectors of the

automorphism 

of [1]. A sequence of such matrix

elements cannot have a weak limit unless the

spectral gap 

 

tends to a limit  2 R. In this

case, by the same discussion as above, any weak

limit of the functionals 

will be an eigenmeasure

of the geodesic flow which transforms by e

it

under

the action of 

. Examples of such eigenmeasures

are orbital Fourier coefficients



it



ð

ðx;ÞÞdt

along a periodic orbit. Here,  2 (2=L



)Z.We

denote by Q



such eigenmeasures of the geodesic

flow. Problem 1 has the following extension to off-

diagonal elements:

Problem 2 Determine the set Q



of ‘‘quantum

limits,’’ that is, weak



limit points of the sequence

{

} of distributions on the classical phase space



M, defined by

ad

:¼hOpðaÞ’

;’

where 

 

=  þ o(1) and where a 2 C



M), or

equivalently of the functionals 

As will be disc ussed in the section ‘‘Quantum

weak mixing, ’’ the asymp totics of off-diag onal

elements depends on the weak mixing properties of

the geodesic flow and not just its ergodicity.

Matrix elements of eigenfunctions are quadratic

forms. More ‘‘nonlinear’’ problems involve the

-norms or the distribution functions of eigenfunc-

tions. Estimates of the L

-norms can be obtained

from the l ocal Weyl law [10].Sincethejumpin

the lef t-hand side at  is

j : 

= 

j’

(x)j

and the

jump in the right-hand side is the jump of R( , x),

this im plies

j:

¼

j’

ðxÞj

¼ Oð 

n1

Þ¼)jj’

¼ Oð

n1

Þ½17

For general L

-norms, the following bounds were

proved by C Sogge for any compact Riemannian

manifold:

k’

k’k

¼Oð

ðpÞ

Þ; 2  p 1 ½18

where

ð pÞ¼







;

2ðn þ 1Þ

n  1

 p 1

n  1





; 2  p 

2ðn þ 1Þ

n  1

½19

These estimates are sharp on the unit sphere S



nþ1

. The extremal eigenfunctions are the zonal

spherical harmonics, which are the L

-normalized

spectral projection kernels 

(x, x

)=k

(  , x

centered at any x

. However, they are not sharp

for generic (M, g), and it is natural to ask how

‘‘chaotic dynamics’’ might influence L

-norms.

Problem 3 Improve the estimates k’

=k’k

O(

(p)

) for (M, g) with ergodic or mixing geodesic

flow.

C Sogge and the author have proved that if a

sequence of eigenfunctions attains the bounds in

[17], then there must exist a point x

so that a

positive measure of geodesics starting at x

in S



returns to x

at a fixed time T. In the real analyt ic

case, all return so x

is a perfect recurrent point. In

dimension 2, such a perfect recurrent point cannot

occur if the geodesic flow is ergodic; hence

k’

= o(

(n1)=2

) on any real analytic surface

with ergodic geodesic flow. This shows that none

of the L

-estimates above the critical index are sharp

for real analytic surfaces with ergodic geodesic flow,

and the proble m is the extent to which they can be

improved.

The random wave model (see the section ‘‘Random

waves and or thonormal bases’’) p redicts that e igen-

functions of Riemannian manifolds with chaotic

geodesic flow should have the bounds k’



= O(1)

for p < 1 and that k’



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

log 

.Butthereare

188 Quantum Ergodicity and Mixing of Eigenfunctions

no rigorous estimates at this time close to such

predictions. The best general estimate to date on

negatively curved compact manifolds (which are

models of chaotic geodesic flow) is just the logarithmic

improvement

jj’

¼ O



n1

log 



on the standard remainder term in the local Weyl

law. This was known for compact hyperbolic

manifolds from the Selbe rg trace formula, and

similar estimates hold manifolds without conjugate

points (P Be´rard). The exponential growth of the

geodesic flow again causes a barrier in improving

the estimate beyond the logarithm. In the analogous

setting of quantum ‘‘cat maps,’’ which are models of

chaotic classical dynamics, there exist arbitrarily

large eigenvalues with multiplicities of the order

O(

n1

=log ); the L

-norm of the L

-normalized

projection kernel onto an eigenspace of this multi-

plicity is of order of the square root of the

multiplicity (Faure et al. 2003). This raises doubt

that the logarithmic estimate can be improved by

general dynamical arguments. Further discussion of

-norms, as well as zeros, will be given at the end

of the next section for ergodic systems.

Quantum Ergodicity

In this section, we discuss results on the problems

stated above when the geodesic flow of (M, g)is

assumed to be ergodic. Let us recall that this means

that Liouville measure is an ergodic measure for 

This is a spectral property of the operator V

of [2]

on L



M,d), namely that V

has 1 as an

eigenvalue of multiplicity 1. That is, the only

invariant L

-functions (with respect to Liouville

measure) are the constant functions. This implies

that the only invariant sets have Liouville measure 0

or 1 and (Birkhoff’s ergodic theorem) that time

averages of functions are constant almost every-

where (equal to the space average).

In this case, there is a general result which

originated in the work of Schnirelman and was

developed into the following theorem by Zelditch,

Colin de Verdie` re, and Sunada (manifolds without

boundary), and Ge´rard–Leichtnam and Zelditch–

Zworski (manifolds with boundary). The following

discussion is based on the articles (Zelditch

1996b, c, Zelditch and Zworski 1996), which

contain further references to the literature.

Theorem 1 Let ( M, g) be a compact Riemannian

manifold (possibly with boundary), and let {

, ’

}

be the spectral data of its Laplacian . Then the

geodesic flow G

is ergodic on (S



M,d) if and only

if, for every A 2 

(M), we have:

(i) lim

 !1

N()





j(A’

, ’

)  !(A)j

= 0.

(ii) (8)(9) lim sup

 !1

N()

j6¼k : 

,

j



j<

j(A’

, ’

<.

This implies that there exists a subsequence {’

}

of eigenfunctions whose indices j

have counting

density 1 for which hA’

, ’

i!!(A). We will call

the eigenfunctions in such a sequence ‘‘ergodic

eigenfunctions.’’ One can sharpen the results by

averaging over eigenvalues in the shorter interval

[,  þ 1] rather than in [0, ].

There is also an ergodicity result for boundary values

of eigenfunctions on domains with boundary and with

Dirichlet, Neumann, or Robin boundary conditions

(Ge´rard–Leichtnam, Hassell–Zelditch, Burq). This cor-

responds to the fact that the billiard map on B



is ergodic.

The first statement (i) is essentially a convexity

result. It remains true if one replaces the square by

any convex function ’ on the spectrum of A,

NðEÞ



E

’ðhA’

;’

i!ðAÞÞ!0 ½20

Before sketching a proof, we point out a some-

what heuristic ‘‘picture proof’’ of the theorem.

Namely, ergodicity of the geodesic flow is equivalent

to the statement that Liouville measure is an

extreme point of the compact con vex set M

.In

fact, it further implies that ! is an extreme point of

the compact convex set E

of invariant states for 

of eqn [1]; see Ruelle (1969) for background. But

the local Weyl law says that ! is also the limit of the

convex combination

NðEÞ



E



An extreme point cannot be written as a convex

combination of other states unless all the states in the

combination are equal to it. In our case, ! is only a

limit of convex combinations so it need not (and does

not) equal each term. However, almost all terms in the

sequence must tend to !, and that is equivalent to [1].

Sketch of Proof of Theorem 1(i) As mentioned

above, this is a convexity result and with no

additional effort we can consider more general

sums of the form. We then have



E

’ðhA’

;’

i!ðAÞÞ



E

’ðhhAi

 !ðAÞ’

;’

iÞ ½21

Quantum Ergodicity and Mixing of Eigenfunctions 189

where

hAi

T



We then apply the Peierls–Bogoliubov inequality

j¼1

’ððB’

;’

ÞÞ  tr ’ðBÞ

with B =

[hAi

 !(A)]

to get



E

’ðhhAi

 !ðAÞ’

;’

iÞ

 tr ’ð

½hAi

 !ðAÞ 

Þ½22

Here, 

is the spectral projection for

H corre-

sponding to the interval [0, E]. From the Berezin

inequality we then have (if ’(0) = 0):

NðEÞ

tr ’ð

½hAi

 !ðAÞ



NðEÞ

tr 

’ð½hAi

 !ðAÞÞ

! !

ð’ðhAi

 !ðAÞÞÞ; as E !1

As long as ’ is smooth, ’(hAi

 !(A)) is a

pseudodifferential operator of order zero with

principal symbol ’(h

 !(A)). By the assump-

tion that !

!! we get

lim

E!1

NðEÞ



E

’ðhA’

;’

i!ðAÞÞ



fH¼1g

’ðh

 !ðAÞÞd

where

h

T



 

As T !1 the right-hand side approaches ’(0) = 0

by the dom inated convergence theorem and by

Birkhoff’s ergodic theorem. Since the left-hand side

is independent of T, this implies that

lim

E!1

NðEÞ



E

’ðhA’

;’

i!ðAÞÞ ¼ 0

for any smooth convex ’ on Spec(A) with ’(0) = 0.

As mentioned above, the statement of Theorem 1(i)

is equivalent to saying that there is a subsequence

{’

} of counting density 1 for which 

!!.The

above proof does not and cannot settle the question

whether there exist exceptional sparse subsequences

of eigenfunctions of density zero tending to other

invariant measures. To see this, we observe that

the proof is so general that it applies to seemingly very

different situations. In place of the distributions

{

}wemayconsidertheset



of periodic orbit

measures for a hyperbolic flow on a compact manifold

X.Thatis,





ðf Þ¼



f for f 2 CðXÞ

where  is a closed orbit and T



is its period.

According to the Bowen–Marguli s equidistribution

theorem for closed orbits of hyperbolic flows, we

have

ðTÞ

:T



T

jdetðI  P



Þj





!

where as above  is the Liouvi lle measure, where P



is the linear Poincare´ map and where (T) is the

normalizing factor which makes the left side a

probability measure, that is, defined by the integral

of 1 against the sum. An exact repetition of the

previous argument shows that up to a sparse

subsequence of ’s, 



! individually. Yet clearly,

the whole sequence does not tend to d: for

instance, one could choose the sequence of iterates



of a fixed closed orbit.

Quantum Ergodicity in Terms of Operator Time

and Space Averages

The first part of the result above may be reformu-

lated as a relation between operator time and space

averages.

Definition Let A 2 

be an observable and define

its time average to be:

hAi :¼ lim

T!1

T



and its space average to be scalar operat or

!ðAÞI

Here, the limit is taken in the weak operator

topology (i.e., one matrix ele ment at a time). To see

what is involved, we consider matrix elements with

respect to the eigenfunctions. We have

T



dt’

;’



sin Tð

 

Tð

 

ðA’

;’

from which it is clear that the matrix element tends

to zero as T !1 unless 

= 

. However, there is

no uniformity in the rate at which it goes to zero

since the spacing 

 

could be uncontrollably

small.

190 Quantum Ergodicity and Mixing of Eigenfunctions

In these terms, Theorem 1(i) states that

hAi¼!ðAÞI þ K; where lim

!1



ðK



KÞ!0 ½23

where !



(A) = tr E()A. Thus, the time average

equals the space average plus a term K which

is semiclassically small in the sense that its

Hilbert–Schmidt norm square kE



in the span

of the eigenfunctions of eigenvalue  is o(N()).

This is not exactly equivalent to Theorem 1(i)

since it is independent of the choice of orthonormal

basis, while the previous result depends on the

choice of basis. However, when all eigenvalues have

multiplicity 1, then the two are equivalent. To see

the equivalence, note that hAi commutes with

ﬃﬃﬃﬃ



and hence is diagonal in the basis {’

} of joint

eigenfunctions of hAi and of U

. Hence, K is the

diagonal matrix with entries h A’

,’

i!(A). The

condition is therefore equivalent to

lim

E!1

NðEÞ



E

jhA’

;’

i!ðAÞj

¼ 0

Since all the terms are positive, no cancellation is

possible and this condition is equivalent to the

existence of a subset SN of density 1 such that

:= {d

: k 2S} has only ! as a weak



limit

point. As above, one says that the sequence of

eigenfunctions is ergodic.

One could take this restatement of Theorem 1(i)

as a semiclassical defi nition of quantum ergodicity.

Two natural questions arise. First:

Problem 4 Suppose the geodesic flow 

of (M, g)

is ergodic on S



M. Is the operator K in

hAi¼!ðAÞþK

a compact operator? In this case,

ﬃﬃﬃﬃ



is said to be

quantum uniquel y ergodic (QUE). If ergodicity is

not sufficient for the QUE property, what extra

conditions need to be added?

Compactness would imply that hK’

,’

i!0,

hence hA’

,’

i!!(A) along the entire sequence.

Quite a lot of attention has been focused on this

problem in the last decade. It is probable that

ergodicity is not by itself sufficient for the QUE

property of general Riemann manifold. For instance,

it is believed that there exis t modes of asymptotic

bouncing ball type which concentrate on the

invariant Lagrangian cylinder (with bounda ry)

formed by bouncing ball orbits of the Bunimovich

stadium (see e.g., Heller (1984) for more on such

‘‘scarring’’). Further, Faure et al. (2003) have shown

that QUE does not hold for the hyperbolic system

defined by a quantum cat map on the torus. Since

the methods applicable to eigenfunctions of

quantum maps and of Laplacians have much in

common, this negative result shows that there

cannot exist a universal structural proof of QUE.

The principal positive result available at this time

is the recent proof by Lindenstrauss of the QUE

property for the orthonormal basis of Laplace–

Hecke eigenfunctions on arithmetic hyperbolic sur-

faces. It is genera lly believed that the spectrum of

the Laplac e eigenvalues is of multiplicity 1 for such

surfaces, so this should imply QUE completely for

these surfaces. Earlier partial results on Hecke

eigenfunctions are due to Rudnick–Sarnak, Wolpert,

and others. For references and further discussion onf

Hecke eigenfunctions, see Rudnick and Sarnak

(1994) (see Arithmetic Quan tum Chaos).

So far we have not mentioned Theorem 1(ii). In

the next section, we will describe a similar but more

general result for mixing systems and the relevance

of (ii) will become clear. An interesting open

problem is the extent to which (ii) is actually

necessary for the equivalence to classical ergodicity.

Problem 5 Converse QE: What can be said of the

classical limit of a quantum ergodic system, that is, a

system for which hAi= !(A) þ K, where K is

compact? Is it necessarily ergodic?

Very little is known on this converse problem at

present. It is known that if there exists an open set in



M filled by periodic orbits, then the Laplacian

cannot be quantum ergodic (see Marklof and

O’Keefe (2005) for rece nt results and references).

But no proof exists at this time that KAM systems,

which have Cantor-like positive measure invariant

sets, are not quantum ergodic. It is known that there

exists a positive proportion of approximate eigen-

functions (quasimodes) which localize on the invari-

ant tori, but it has not been proved that a positive

proportion of actual eige nfunctions has this localiza-

tion property.

Further Problems and Results on Ergodic

Eigenfunctions

Ergodicity is also known to have an impact on

the distribution of z eros. The complex zeros in

Ka¨ hler phase spaces of ergodic eigenfunctions of

quantum ergodic maps become uniformly distrib-

uted with respect to the Ka¨hler volume form

(Nonnenmacher–Voros, Shiffman–Zelditch). An inter-

esting problem is whether the real analog is true:

Problem 6 Ergodicity and equidistribution of

nodal sets. Let N

’

 M denote the nodal set (zero

set) of ’

, and equip it with its hypersurface volume

form dH

n1

induced by g. Let (M, g) have ergodic

geodesic flow, and suppose that {’

} is an ergodic

Quantum Ergodicity and Mixing of Eigenfunctions 191