The Erwin Schrodinger International Pasteurgasse

ESI Institute for Mathematical Physics A Wien Austria

Geometries of Quantum States

Denes Petz

Csaba Sudar

Vienna Preprint ESI March

Supp orted by Federal Ministry of Science and Research Austria

Available via WWWESIACAT

Geometries of Quantum States

Denes Petz and Csaba Sudar

Department of Mathematics Faculty of Chemical Engineering

Technical University Budap est

H Budap est XI Szto czek u Hungary

Abstract

The quantum analogue of the Fisher information met

ric of a probability simplex is searched and several Rie

mannian metrics on the set of p ositive denite density

matrices are studied Some of them app eared in the lit

erature in connection with CramerRao typ e inequali

ties or the generalization of the Berry phase to mixed

states They are shown to b e sto chastically monotone

here All sto chastically monotone Riemannian metrics

are characterized by means of op erator monotone func

tions and it is proven that there exist a maximal and

a minimal among them A class of metrics can b e

extended to pure states and the FubiniStudy metric

shows up there



Also Mathematical Institute of the Hungarian Academy of Sciences

H Budap est PF Hungary



Email PETZCHBMEHU



Email SUDARCHBMEHU

I Intro duction

The state space of a classical system with n alternatives is the simplex of proba

bility distributions on the np ointspace The probability simplex is an n dimen

sional manifold with b oundary and its ane structure is fairly trivial The extreme

b oundary consists of n discrete p oints In quantum mechanics the state space of an

n level system is identied with the set of all n n p ositive semidenite complex

matrices of trace They are called density matrices The case n is easily

visualized as the unit ball in the space

x y iz

x y z IR x y z

x iy x

The b oundary consists of noninvertible matrices and it is an innite set The case

n is simple but for higher n the structure of the top ological b oundary is rather

complicated The extreme b oundary consists of the density matrices of rank one and

for n it is much smaller than the top ological b oundary As far as dimensionality

concerned the top ological b oundary is n and the extreme one is n The

extreme states are usually called pure and they are describ ed in the textb o oks by

nonzero vectors of a complex Hilb ert space of linear dimension n The same state is

describ ed by a vector as well as where is any complex numb er dierent from

This means that pure states are in onetoone corresp ondence to rays f

n

Cg The rays form a smo oth manifold called complex pro jective space CP

On the level of convex structure the dierence b etween the classical and quantum

state space is wellundersto o d The classical one is a Cho quet simplex and dierent

axiomatizations of the quantum one are available in the literature the reader may

b e referred to the works for example Our main concern here is the p ossible

Riemannian structure in the quantum case Before turning to that sub ject we review

briey the classical case that is the Riemannian structure on the space of measures

From the viewp oint of information geometry the spherical representation of the

probability simplex is adequate b ecause the squared length of the tangent vector of

p

p a curve equals the Fisher information Indeed intro duce the parameters z

i i

P P

where i n and p Then z and the probability simplex is

i

i

i i

parametrized with a p ortion of the nsphere Let xt b e a curve on the sphere The

square of the length of the tangent is

X X

p t log p t x h x xi

i t i t i t t

i i

which is the Fisher information The geo desic distance b etween two probability distri

butions Q and R can b e computed along a great circle and it is a simple transform

of the Hellinger distance The lecture notes contains further details as well as

statistical applications of this geometric approach To the b est of our knowledge

Riemannian metric on quantum states was rst considered by Helstrom in connec

tion with state estimation theory Since Helstroms work several other metrics

app eared in the literature see for example and Uhlmann approached

Helstroms metric in a dierent way

The present pap er is organized as follows In Section II we survey the work

of Chentsov b oth in the probabilistic and in the quantum case We explain how

he arrived at the study of invariant metrics on the space of probability measures

motivated by decision theory and how far he could go towards the quantum gen

eralization after his unicity result ab out the Fisher information in the probabilistic

context Section IV reviewes dierent approaches to Riemannian metric on the quan

tum state space The relation of Uhlmanns and Helstroms work to Chentsovs idea

is enlightened and a concise description of the complex pro jective space is given The

main results are contained in Sections IV and V We construct monotome metrics

by means of op erator monotone functions and prove that all monotone metrics are

obtained in this way Our result completes the program initiated by Chentsov It

turns out that the symmetric logarithmic derivative metric of Helstron which is

the same as the metric studied by Uhlmann is monotone Furthermore this metric

is minimal among all monotone metrics The sub ject of Section V is the extension

of monotone metrics to pure states We prove that if the extension exists then it

coincides with the standard metric of pure states up to a constant factor

I I The viewp oint of Chentsov

Chentsov was led by decision theory when he considered a category whose ob jects

are probability spaces and whose morphisms are Markov kernels Although he worked

in with arbitrary probability spaces his idea can b e demonstrated very well on

nite ones In this case a morphism from the probability nsimplex S to an m

n

simplex S is an n m sto chastic matrix If is such a matrix and P S then P

m n

S is considered more random than P Generally sp eaking the parametrized family

m

Q is more random than the parametrized family P with the same parameter

i i

set if there exists a sto chastic matrix such that P Q for every value of the

i i

parameter i Two parametric families P and Q are equivalent in the theory of

i i

statistical inferences if there are two sto chastic matrices and such that

P Q and Q P

i i i i

for every i Chentsov said a numerical function f dened on pairs of measures to b e

invariant if

P P Q Q implies f P P f Q Q

and monotone if

f P P f P P

for every sto chastic matrix A monotone function f is obviously invariant Statis

tics and information theory know a lot of monotone functions relative entropy and

its generalizations are so If a Riemannian metric is given on all probability sim

plexes then this family of metrics is called invariant resp ectively monotone if the

corresp onding geo desic distance is an invariant resp ectively monotone function

Chentsovs greate achievement was that up to a constant factor the Fisher informa

tion yields the only monotone family of Riemannian metrics on the class of nite

probability simplexes see also A decade later Chentsov turned to the quan

tum case where the probability simplex is replaced by the set of density matrices

A linear mapping b etween two matrix spaces sends a density matrix into a density

if the mapping preserves trace and p ositivity ie p ositive semidenitness By now

it is wellundersto o d that completely p ositivity is a natural and imp ortant require

ment in the noncommutative case Therefore we call a trace preserving completely

p ositive mapping sto chastic One of the equivalent forms of the completely p ositivity

of a map T is the following

n n

X X

 

a T b b a

i i

i i

i j

for all p ossible choice of a b and n A completely p ositive mapping T satises the

i i

 

Schwarz inequality T a a T a T a

Chentsov recognized that sto chastic mappings are the appropriate morphisms in

the category of quantum state spaces The monograph contains more informa

tion ab out sto chastic mappings see also The ab ove denitions of invariance

and monotonicity make sense when sto chastic matrices are replaced by sto chastic

mappings Chentsov with Morozova aimed to nd the invariant or monotone

Riemannian metrics in quantum setting as well They obtained the following result

Assume that a family of Riemannian metrics is given on all spaces of den

sity matrices which is invariant then there exist a function cx y and a constant

C such that the squared length of a tangent vector A A at a diagonal p oint

ij

D Diag p p p is of the form

n

n

X X

cp p jA j A C p

j k j k

k k

k

j k k

Furthermore the function cx y is symmetric and cx y cx y This

result of Morozova and Chentsov was not complete Although they had prop osals

for the function cx y they did not prove monotonicity or invariance of any of the

corresp onding metrics A complete result will b e given here but now a few comments

on are in order

Both the function cx y and the constant are indep endent from the matrix size

n Restricting ourselves to diagonal matrices which is in some sense a step back to

the probability simplex we can see that there is no ambiguity of the metric Lo osely

sp eaking the unicity result in the simplex case survives along the diagonal and the

odiagonal provides new p ossibilities for the denition of a sto chastically invariant

metric

I I I Riemannian metrics on quantum states

The demand for Riemannian structure on the whole quantum state space or on

a parametrized family of density op erators app eared in mathematical physics a long

time ago and in rather dierent contexts

In the parametric problem of quantum statistics a family D of states of a

systems is given and one has to decide b etween several alternative values of the

parameter by using measurements The set of outcomes of the applied measurements

m

is the parameter set and we assume that it is a region in IR So an estimator

measurement M is a p ositiveop erator valued measure on the Borel sets of and

its values are observables of the given quantum system The probability measure

B B Tr D M B B represents the result of the measurement

M when the true state is D The choice of the estimators has to b e made by

taking into account the exp ected errors The aim of an optimal desision pro cess is

to search estimators with small error To an error one can attribute several sizes

For example one can seek a measurement such that its value is approximately

equal to the true parameter value If this holds in the mean then the estimator

is free of destorsion and such estimator is commonly called unbiased The accuracy

of an unbiased measurement is describ ed by the total meansquare deviation which

should b e small on the parameter space if we want to cho ose an eective estimator

measurement

The quantum state estimation was initiated by Helstrom in the s

see also He followed the CramerRao pattern of mathematical statistics and

intro duced the concept of symmetric logarithmic derivative Let M b e a p ositive

n

op erator valued measure on IR The corresp onding measurement is an unbiased

estimator of the parameter if

m

Z

dTr D M t

i t i

m

IR

for every i m The integration is taken with resp ect to the measure B

i

are observables dened as Tr D M B The symmetric logarithmic derivatives L

t

Tr D A

i i

Tr L D D L A

i

for every observable A The measurement has two characteristic matrices the co

variance matrix C C and the information matrix J J They

ij ij

are determined as follows

Z

t t dTr D M t C

i i j j ij

m

IR

i j

J Tr D L L

ij

A quantum version of the CramerRao inequality due to Helstrom says that

C J

for an unbiased measurement The inequality means that the dierence is p ositive

semidenite The information matrix J may b e regarded the metric tensor on

the parameter space

From the p oint of view of the statistical state estimation problem the numb er

n of the real parameters is much smaller than the dimension of the whole state

space However we can parametrize the whole state space as well Assume that the

parametrization is ane

X

D I n a

i i

i

where a are traceless selfadjoint matrices D is p ositive denite if is in a certain

i

2 2

n n

op en subset of IR and the mapping D IR yields atlas of a single

chart We refer to as the ane parametrization of invertible density matrices

D

n

i

The symmetric logarithmic derivative L is given by the equation

i i

D L L D A

i

When A is regarded as a tangent vector at D its squared length equals to

i

i i

Tr D L Tr L A

i

P

p is the sp ectral decomp osition of D then the solution of may If

j j

j

b e written in the form

X

i

L p A p

k i j

k j

k j

To show an example we consider the case and cho ose a with the three

i i

Pauli spin matrices that is

i

p p p

i

Then

k k i

i

D

if the fo otp oint D is diagonal Diag A convenient ane co ordinate system ex

ists also for the whole set M of invertible n n density matrices and the Riemannian

n

metric of the symmmetric logarithmic derivative may b e written in the form

n

X X

A kA k p jA j

ij j k

k k D

k

p p

j k

j k k

where D Diag p p p So the value of the constant C is and the

n

MorozovaChentsov function of the metric of the symmetric logarithmic derivative

is

cx y

x y

Before we show how Uhlmann obtained essentially the same Riemannian metric

in a completely dierent approach we review shortly the complex pro jective space

n

CP It was explained in the intro duction that the extreme b oundary of the

n

state space of an nlevel quantum system is CP

n

CP is equipp ed with an atlas containing n charts Let U b e the set of the

i

equivalence classes of all ntuples z z z of complex numb ers that z

n i

and set

z z z z

i i n

pz z z

i n

z z z z

i i i i

n

The standard Riemannian metric of CP is given by considering the n

sphere

jz j jz j jz j C

n

which is parametrized now by complex numb ers S as the group of complex numb ers

of mo dulus one has a natural isometric action on the n sphere The orbits are

n

CP The homeomorphic to circles and the space of orbits may b e identied with

orbits may b e given a metric by taking that obtained by pro jecting the metric on

n

S orthogonally to the orbits This metric is invariant under the natural action

of the unitary group U n and called sometimes the FubiniStudy metric Strictly

sp eaking the FubiniStudy metric is a Kaehler metric on CP viewed as a Kaehler

manifold

One of the key issues of quantum mechanics compared with classical one is that

a subsystem of a system in pure state can b e in a mixed state More precisely if D

is any density op erator on the Hilb ert space H then one can nd a vector in the

enlarged Hilb ert space H H such that

Tr DA h A I i

for every observable A of the smaller system ie acting on H The vector is not

determined uniquely and called the purication of D It is worthwile to regard H H

as the Hilb ertSchmidt op erators acting on H Then the observable A of the small

system corresp onds to the multiplication op erator L X AX on H H So

A

condition reads as

Tr DA hW L W i

A



when W is written instead of Since hW L W i Tr W WA conditions

A

simply b ecomes



W W D



Among all lifts of D into the bration W W W there is a canonical one which

satises the socalled parallelity or horizontality condition

 

W W W W

Uhlmann arrived at this condition from the following minimization problem related

to the generalization of the Berry phase to mixed states Let D t b e a

smo oth curve of density matrices with purication W t If the arclength of W t

with resp ect to the standard FubiniStudy metric is minimal then the paralellity

n

 

CP such that W Y YW are condition is satised The vectors Y T

W

n

0

CP called horizontal Any vector X T M admits a horizontal lift X T

D W

and Uhlmann prop osed the Riemannian metric

0 0 FS B

X X X X g g

W D



for any W with W W D If DG GD D then

B

g D D Tr GD

B

In G one recognize the symmetric logarithmic derivative and g is the corresp ond

B

ing metric up to a factor one half The letter B in g refers to Bures b ecause the

B

geo desic distance in the metric g coicides with the one intro duced by Bures many

years earlier The Bures distance is

q

Tr D D D d D D

B

It is worthwile to mention that Dittmann computed several geometric characteristics

of the space of density matrices endowed with the ab ove metric For example

this space is not lo cally symmetric and all sectional curvatures are greater than

Braunstein and Caves obtained recenly the same metric by optimizing over all gener

alized quantum measurements that can b e used to distinguish neighb oring quantum

states D and D dD

IV Monotone metrics

If a distance b etween density matrices expresses statistical distinguishability then

this distance must decrease under coarsegraining A go o d example of coarsegraining

arises when a density matrix is partitioned in the form of a blo ck matrix and

the coarsegraining forgets ab out the odiagonal

A B A



B C C

In the mathematical formulation a coarsegraining is a completely p ositive map

ping which preserves the trace and hence sends density matrix into density matrix

Such mapping will b e called sto chastic b elow A Riemannian metric is dened to

b e monotone if the dierential of any sto chastic mapping is a contraction If the

ane parametrization is considered then D D tA is a curve for an invert

t

ible density D and for a selfadjoint traceless A Under a sto chastic mapping T this

curve is trasformed into TD TD tTA provided that TD is an invertible

t

density and the real numb er t is small enough The monotonicity condition for the

Riemannian metric g on M reads as

n

TA TA g A A g

D

TD

where D is an invertible density A is traceless selfadjoint and T is sto chastic Our

goal is to show many examples of monotone metrics and to give their characterization

in terms of op erator monotone functions

Let us recall that a function f IR IR is called op erator monotone if the

relation K H implies f K f H for any matrices K and H of any

order The theory of op erator monotone functions was established in the s

by Lowner and there are several reviews on the sub ject for example are

suggested

Let us intro duce some sup erop erators as

L A D A R A AD A M C

D D n

Theorem Let f IR IR b e an op erator monotone function such that

f t tf t for every t and set a sup erop erator

K R f L R R

D D

D D D

acting on matrices Then the relation

AB g A B Tr K

D

D

determines a monotone Riemannian metric on M

n

Proof Since an op erator monotone function is analytic the bilinear form is

A is selfad smo oth in D The condition f t tf t on f makes sure that K

D

joint whenever A is so Hence the bilinear form is real For an invertible D the

sup erop erator K is invertible and p ositive denite So is really a nondegen

D

erate metric and its monotonicity is to b e checked

In the pap er the following inequality was obtained

f L R R TR f L R R T R

E

TE

F F F

TF TF TF

if E F are p ositive denite matrices T is a sto chastic mapping and T denotes its

adjoint with resp ect to the Hilb ertSchmidt inner pro duct Putting E F D

b ecomes

TK T K

D

TD

which is equivalent to

T K T K

D

TD

The latter condition is exactly the monotonicity of the metric ut

It is in order to make a comment on the relation of the function f in Theorem

and the MorozovaChentsov function cx y in Given f we have cx y

y f xy and conversely f t ct Some examples of functions f satisfying

the hyp othesis of Theorem are the following

p

x x x x x x

x log x log x x log x x

where The lattest function f gives the MorozovaChentsov function

and we obtain that the metric of the symmetric logarithmic derivative is

monotone

The metrics on M provided by Theorem are rotation invariant they dep end

p

x y z and split into radial and tangential comp onents only on r

r

ds dn where g t dr g

r r r f t

The radial comp onent is indep endent of the function f In case of the metric of the

symmetric logarithmic derivative the tangential comp onent is indep endent of r

Theorem Every monotonone metric is provided by Theorem

Proof A monotone metric is invariant in the sense of Section II and due to the

result of Chentsov and Morozova the metric is of the form Set a function f

as f t ct where c is the function of two variables from By means

of this function the monotone metric can b e written in terms f exactly in the form

describ ed in Theorem see and What we have to prove is that f is

op erator monotone This will b e shown following

We cho ose a particular sto chastic mapping T

X X A B X A

T X

A B X X B X

With this choice the monotonicity condition yields that

Y f L R R

Y Y

Y

is a concave mapping or equivalently

t t

Y f Y Y I Y

is concave for a p ositive denite density matrix Y The concavity extends to all

p ositive denite matrices obviously We write for a blo ck matrix

Y

Y

Y

then we observe that concavity of implies the concavity of the mapping

t t

Y Y f Y Y I Y

Now the choice Y I gives that the mapping Y f Y must b e concave What

we have arrived at is the op erator concavity of f which is known to b e equivalent to

the op erator monotonicity of f cf ut

Let f and f b e functions satisfying the hyp othesis of Theorem and let

K and K b e the corresp onding sup erop erators dened by If f f then

A A for the A A g The inverse changes this ordering hence g K K

D D D D

corresp onding metrics The relation b etween op erator monotone functions and mono

tone metrics established by Theorems and resp ects ordering in the sense that

biger function gives a smaller metric Comparison of dierent metrics is meaningful

only under some normalization The most natural is

g A A Tr D A whenever DA AD

D

which corresp onds to f It is known see that among all op erator mono

tone functions with f and f t tf t there is a minimal and a maximal

They are

t t

f t f t

max min

t

So we obtain

Theorem Under the normalization the metric of the symmetric loga

rithmic derivative is minimal among all monotone metrics

Proof One has to verify that the function f yields the stated metric From

max

and we have

g A A hL R A Ai

D D D

and L L R is exactly the solution of equation Hence

D D

matches

We have to emphesize that the theorem states the minimality of the logarithmic

derivative metric only under the essential condition that the whole state space of a

spin is parametrized If this is not the case then no information is provided by the

theorem The largest monotone metric is the metric of the socalled left logarithmic

derivative That app eared in the literature in connection with CramerRao typ e in

equalities Its monotonicity was established in The fact that the left logarithmic

metric is larger than the symmetric one is elementary and it has b een known for

example p

The metric corresp onding to the MorozovaChentsov function

log x log y

x y

is the Kub o or Mori or Bogoliub ov inner pro duct which showed up in and was

studied in In particular it was proved that the Kub o pro duct is monotone under

more general assumption than a nite spin and a conjecture was made Namely

the scalar curvature of the Kub o metric is monotone as well Monotonicity of the

Kub o metric is not surprising b ecause this result is a kind of reformulation of the

Lieb convexity theorem However the monotonicity of the scalar curvature

seems to b e an inequality of new typ e provided that the cunjecture is really true

Concerning details we refer to and

In Hasegawa intro duced a family of metrics They can b e obtained by the

ab ove construction of monotone metrics however we are unable to prove that the

auxiliary functions are op erator monotone Numerical computations supp ort the

monotonicity of Hasegawas metric

V Extension to pure states

The ob jective of this section is to discuss the extension of monotone metrics

n

CP Since pure states form a low dimensional part of of M to pure states

n

the top ological b oundary of M it should b e wellsp ecied how the extension is

n

understo o d



Let M denote the set of all elements of M whose eigenvalues are distinct and

n

n

n



dene a pro jection M CP as follows Let D b e the onedimensional

n



eigenspace corresp onding to the largest eigenvalue of D M This map is smo oth

n

n



see I I and M is a smo oth bre bundle over CP with pro jection

n

see I The structure group of this bundle is U U n where U k is

the group of k k unitary matrices The bre space is e where e is the ray

n

C generated by the vector

Let T b e the dierential of at D and let H b e the orthogonal complement

D D



of Ker T in T M with resp ect to a xed monotone Riemannian metric g

D D D

n

Since T is surjective the restriction of T gives a linear isomorphism b etween

D D

n n

CP If v T CP then we can dene a unique liftv H H and T

D D

D D

of v such that T v v Using this lift we can dene the following inner pro duct

D

n

D

k on T CP

D

D

n

D

CP k u v g u v u v T

D

D

D

n



We say that a sequence D M is radial at p CP if D p for every n

n n

n

and D is convergent to p when p is considered as a density matrix that is a one

n

dimensional pro jection op erator Now we can dene the radial extension of g

n

A smo oth metric k on CP is called the radial extension of g if for

n n

every p CP u v T CP and for every radial sequence D at p

p n

D

n

lim g u v k u v

p

p

n!1

holds In the next theorem we give a necessary and sucient condition for the exis

tence of the radial extension

Theorem Let g b e a monotone Riemannian metric on M and let f

n

IR IR b e the corresp onding op erator monotone function describ ed in Theorem

The radial extension k of the given metric g of M exist if and only

n

if f In the case of existence

h i k

f

n

CP where h i is the standard Riemannian metric on

D

Proof The pro of is based on the direct computation of k For any unitary

D



matrix U and D M we have

n

UDU U D

which implies



1

T UXU UT X X T M

D D

UDU

n

by dierentiation Since k is unitary invariant

U Ker T U Ker T 1 and UH U H 1

D D

UDU UDU

n

CP hence we get Moreover U v U U v for any v T

D

1

UDU D

u v g g U u U v

D U D

D

From this equality it follows that it is sucient to compute k if D is diagonal



and let t and D is the pro jection onto e Assume these and let X T M

D

n

and v t b e the largest eigenvalue and the unit eigenvector corresp onding to t of



D tX where t IR For suciently small t D tX M and t and v t are

n

smo oth functions of t For D t D tX we have

D t tv t

0

Dierentiating this expression we obtain that x and

x x

n

0

T X v

D

n

where are the eigenvalues of D and X x If X Ker T

n ij D

then the expression of T X gives

D

x

x x

B C

n

B C

X

A

x x

n nn

R as in Since D is diagonal f L R Let K

D D

D D

x

ij

K X

D ij

f

j i j

hence we get K Ker T Ker T If V H then the last equation gives

D D D D

v v

n

v

B C

B C

V

A

v

n

n

where v C for i n If v v v T CP then and

i n

e

give

v v

n n

v

B C

B C

v

A

v

n n

D

Now we can express g

n

X

i

i i D

u v g u v Re

f

i

i

n

where u v T CP

e

Let us consider now the general case Let D b e a radial sequence at p and let

m

n n

m

u v T CP Let B b e linear op erators on T CP such that

p p

p

D m

m

g u v hB u v i

p

p p

n

where h i is the inner pro duct on T CP induced by the standard metric Let

p p

p is diagonal and D U D U U b e unitary op erators such that D

m m m

m m m

with p e Using we have

m m

B U B U

m

p m p

0

m m

Since lim and lim for i n by

m!1 m!1

i

m

lim kB cI k c f

p p

p 0 0

0

m!1

n

CP and k k is the op erator norm induced where I is the identity map on T

p p

0 0

by h i It follows from that

m m

kB cI k kU B U cU I U k

p m p m

p m p m 0

0

m m

kU B cI U k kU k kB cI k kU k

p m p m

m p 0 m p 0

0 0

n n

CP to T CP kU k and by we Since U are isometries from T

p m m p

0

obtain

m

lim kB cI k c f

p

p

m!1

So we have proved that the radial extension exists if f

The sp ecial case n is very transparent from and it explains the termi

nology radial extension The case shows also that the condition f is

necessary to sp eak ab out extension ut

Acknowledgement

DP thanks to the Erwin Schrodinger Institut on Mathematical Physics Vienna

for an invitation and CS acknowleges supp ort of the Hungarian National Foundation

for Scientic Research grant no OTKA

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