Positive maps of low dimensional matrix algebras

Vol. 10 (1976)

REPORTS

POSITIVE

MAPS

ON

MATHEMATICAL

No. 2

PHYSICS

OF LOW DIMENSIONAL

MATRIX

ALGEBRAS

S. L. WORONOWICZ* Centre de Physique Theorique-CNRS

Marseille,

Marseille,

France

(Received January 11, 1976) It is shown that any positive map from MZ into M3 is a sum of completely positive and completely copositive maps. This result does not hold for maps into M4. A generalization of the Kadison inequality is suggested and proved for positive maps defined on MZ or M3.

Introduction A linear mapping from a C*-algebra ~4 into a C*-algebra SJ is called positive if it sends positive elements of d into positive elements of 29. Such a map is said to be normuliced if the image of the unity of d coincides with the unity of G?. In recent years positive maps have become a subject of common interest to mathematicians and physicists. There are several reasons for this: (i) The notion of a positive map representation, conditional expectation

generalizes that of state, representation, and (semi-)spectral measure.

Jordan

(ii) By transposition, a normalized positive map defines an affine mapping between sets of states of C*-algebras. If these algebras coincide with the C*-algebra of observables assigned to a physical system, then this affine mapping corresponds to some “operation” which can be performed on the system. Invertible operations such as those representing the time evolution of a closed system or the symmetries of a system have a very simple mathematical structure. According to [8], they are related to a very special class of positive maps known as Jordan automorphisms. On the other hand, the time evolution of an open system, (i.e. interacting with its surroundings), as well as perturbations of a system caused by a measurement process are described by noninvertible operations. The theory of positive maps is then very useful (cf.

PI, [lOI).

(iii) In our opinion, there is an important purely mathematical reason for investigating continuous affine mappings of sets of states of C*-algebras. To explain it, let us make a few remarks on the Krein-Milman-Choquet theory of convex compact sets. In this * On leave of absence from Department of Mathematical Methods in Physics, Warsaw University, Warsaw, Poland.

11651

S. L. WORONOWICZ

166

theory, a distinguished role is played by sets of probability measures on compact topological spaces. We will call these sets standard. The structure of any compact convex set is investigated

by means

of morphisms

of standard

sets onto the given one.

One may think about the “second order” theory of convex sets in which the role of standard sets will be played by sets of states of C*-algebras. To make the first step towards this new theory, one has to investigate in detail morphisms between new standard objects, i.e. the affine continuous mappings mentioned above. Despite the important role of positive maps, their structure has been well understood only in some special cases. The set of all operators from a Hilbert space K into Hilbert space H will be denoted by B(K, H). B(C”, Cm) will be identified with the set Mmxn of all nz x R complex matrices. We also write B(H) instead of B(H, H) and M, instead of AI,,,,; M,, = B(C”). Let us recall [12] that a linear map @: Jd+B is called n-positive (where

n is a natural

number)

(0.1) if the tensor

product

map

(where id is the identity map in M,) is positive. The map @ is called completely positive if it is n-positive for any natural number n. The structure of completely positive maps is very well understood (see [12]). For example, the set of all completely positive maps from M, into Mm coincides with the convex cone generated by maps of the form M, 3 a H s*as E M,,,,

(0.2)

where s EM,,,. Lindblad [9] indicates that operations which can be performed on a physical system are related to completely positive maps. His argument is based on two assumptions: that any operation can be applied to a part of a composed system, having no influence on the rest (which seems not to be the case for geometric operations like time inversion) and that a C*-algebra of a composed system coincides with the tensor product of c*algebras associated with the components of the system. The latter does not hold if the components contain particles of the same kind. However, for a large class of operations his argument is essentially correct and these operations would be represented by completely positive maps. One may introduce a notion which is in some sense dual to the notion of n-positivity. DEFINITION.

The map

(0.1) is called @@T:

n-copositive if the tensor

product

map

SZ’@M, + SI@M,

(where T denotes the transposition of n x n matrices) is positive. @ is called copositive if it is n-copositive for any natural number n. Let us note that the notions of positivity, I-positivity and I-copositivity

completely coincide.

POSITIVE MAPS OF LOW DIMENSIONAL

All results suitable positive

derived

for n-positive

and completely

MATRIX

positive

ALGEBRAS

maps can be repeated

167 with

modifications for copositive maps. In particular, the set of all completely comaps from M,, into M,,, coincides with the convex cone generated by maps of

the form M n 3 a t-+r*aTr, (0.3) where r E M,,,. It turns out that for some pairs (~4, a), any positive map from d into .%7splits into a sum of completely positive and completely copositive maps. This is the case for (M,, M,), (M2, MJ and (M,, M,). The result for (M,, M,) is in fact contained in [ll] (see also [13]). In the present paper we investigate the case (M,, MS) (see Theorem 1.2). The result for (Mj, M,) follows easily from that for (Mz, MS) by passing to the transposed maps. For other pairs of matrix algebras (M,, Mm) (where n, m 2 2) the statement is not valid. For (M3, MS) a suitable counterexample is given in [3]. The case (M2, MJ is analysed in the present paper (Theorem 1.3). A counterexample for (n/r,, M2) can be obtained from that for (M2, M4) by transposition. The main idea of the paper is based on the duality principle. One finds cones I’ and I”’ dual to the cones of completely positive and completely copositive maps. Then the cone generated by the completely positive and the completely copositive maps is dual to the intersection VnV’. We investigate this intersection in detail and discover that in some cases the dual of VnV’ coincides with the cone of all positive maps. Incidentally, we get a remarkable generalization of the Kadison inequality for positive maps from M, and M3 (see Section 5). 1. Main results Let H be a finite-dimensional Hilbert space. In the following we shall mainly work with operators acting on Hz = H@H. Any operator Q E B(HZ) can be represented by a block matrix

where A, B, C, D E B(H). If u E Hz, d and uz will denote

Q=

I: a]

its first and second

(1.1) component,

respectively:

The vector u will be called simple if u1 and u2 are proportional: ui = I’x, where 3LiE C, x E H (i = 1, 2). The action of the operator (1.1) on the vector (1.2) is defined according to the usual rules of matrix calculus:

S.L. WORONOWICZ

168 In what follows, an important

role will be played by the block transposition

operation:

B(H2) 3 Q H Q E B(H2) which

is defined

by [“c

:1’

= I_:

El.

We shall use Dirac notation: for any x, y belonging to a Hilbert denote an operator acting on K according to the formula

Ix>(YlU = (YlU)X, Let u E H2. By a simple

computation

u

E

space K, Ix) (yj will

K.

one gets:

&‘I b’> @‘II lu’) and

lu’) @‘I lu’>@“I1. Let us note that lu) (~1” is not in general Z 0 if and only if u is a simple vector

positive.

One can easily check that

lu) (~1~

(1.3) where

;I’, ?b2E C, x E H. In this case

lu>(4” = Id (4, where Xix.

u= = Remark.

One should

II

(1.4)

j’x .

note that u’ is defined

up to a phase factor;

the representation of u in the form (1.3). After this preparation, we may state our main technical in Section 2. PROPOSITION 1.1. Let H be a jnite-dimensional statements are equivalent: I. Any positive

result,

(1.4) depends

on

which will be proved

Hilbert space. Then the following

two

map @: M2-+B(H)

is a convex

combination

of maps I$ the form: M,sa

(1.5)

t+S*aSEB(H),

(1.6)

M2 3 a H R*aTR E B(H), where R, S: H -+ C2 are linear maps

(cf. formulae

(0.2) and

(0.3)).

POSITIVE MAPS OF LOW DIMENSIONAL

II. For any operator Q E B(H’) simple vector u E Hz such that u # We shall use this proposition in the statement I holds for dimH =

MATRIX

ALGEBRAS

169

such that Q # 0, Q > 0 and Q’ > 0, there exists a 0, u E Q(H”) and ur E p(H”). both directions. It is known (see Introduction) that 2. Therefore we have:

THEOREM 1.1. Let H be a two-dimensional Hilbert space, Q E B(H’), Q # 0, Q > 0 and p > 0. Then there exists a simple vector u E H2 such that u # 0, u E Q(H’) and u’ E e”(H ‘).

In Section

3 we prove

statement

II for dimH

= 3. Consequently,

THEOREM 1.2.

a positive

Let H be a three-dimensional Then

map.

@(a) = 2

STaSi+

i=l

Hilbert

2

we get

space and @: M, -+ B(H)

Rj*aTRj,

be

(1.7)

j=l

where St, Rj: H -+ C2 are linear maps. In Section

4 we disprove

statement

II for dimH

THEOREM 1.3.

= 4. Thus

Assume that dimH = 4. Then there which cannot be written in the form (1.7).

--f B(H),

In the last section we conjecture a generalization its connection with statements I and II. 2. Positive

exists

of the Kadison

we arrive a positive

inequality

at map

@: M2

and discuss

maps as linear functionals

This section following

is devoted

to the proof

of Proposition

1.1. Its title is motivated

by the

PROPOSITION 2.1. (1)

The formula

w(Q)

=

TrQ

where Q runs over B(H’), dejnes a I-1 correspondence between the set of all linear maps @: M2 -+ B(H) and the set of all linear functionals w: B(H2) -+ C. (2) ds is positive

if and or$y if w(lu) (4) 3 0

for any simple

(3) @ is a convex

combination

of maps of the form

dQ> B 0 for any Q > 0.

(2.2)

vector u E H2. (1.5)

if

and only if (2.3)

S. L. WORONOWICZ

170 (4) @ is a convex

combination

of maps of the form

(1.6) if and only if

dQ> a 0 for any Q E B(H’) such that p > 0. Proof: Ad (1): Clearly, @ determines w. On the other hand, any linear functional can be represented by an operator F E B(H’) in the following sense: m(Q) = Tr QF, Writing

F in the block

matrix

w

Q E B(H’).

form

and setting

@ we find the linear Ad(2):

Letu=

Let us note

that

map

=

~Fl,+BF,z+yFz,+dFzz,

@: M, -+ B(H) . Making

corresponding

to o.

use of (2.1), one may check that

(2.4) Therefore (2.2) holds for positive 0. On the other hand, matrices of the form (2.4) generate the whole cone of positive matrices. Therefore (2.2) implies that (xl @(a)x) B 0 for any positive matrix a. Ad (3): At first we note that any linear map S: H-C= is determined

by a vector

u E HZ in the following

sense: (2.5)

Assume

(cf. (1.5)) that @(a) = S*uS,

where S is given by (2.5). After a simple computation this case

dQ> = (~lQu> This proves

the “only

if” part

of (3).

using

(2.1) and (2.5), one gets in

I

POSITIVE

Conversely,

According

assume

MAPS

OF LOW

DIMENSIONAL

MATRIX

ALGEBRAS

171

(2.3). Then

to the spectral

theorem

one may find vectors

ul, u2, . . . , UN E HZ such that

Then @(a) = f

S,*aS,,

n=l

where

Ad (4): The proof is almost identical to that in Ad (3) and will be omitted. In short Proposition 2.1 provides us with a duality between B(H’) and the space of all linear mappings from M2 into B(H). It turns out that the cone of positive maps is Moreover, the cone generdual to the cone W generated by (1u) ( u ( : u E H2, u-simple). ated by mappings of the form (1.5) is dual to the cone l’= and

the cone generated

by mappings v”=

{QsB(H2):

Q>O}

of the form {QEB(H~):

(1.6) is dual to

PaOo,.

Therefore the cone generated by mappings of the form (1.5) and (1.6) is dual to VnV”. Statement I (see Section 1) can now be expressed in an equivalent dual version: VnV” c w. We shall show that (2.6) is equivalent to statement II: (2.6) 3 II: Let Q E B(H2), Q 2 0 and Q 2 0. This means of (2.6) we have

(2.6)

that Q E VnV”. In virtue

S. L. WORONOWICZ

172 where

u, are simple

vectors.

Then

and for each n we have u, E QW’> p

and

U:, E e”(H’).

II * (2.6): Assume that Q is an extreme element of the cone Vnv”. > 0 and according to II one can find a simple vector u such that ~4E QW2)

Taking into account positive E we have

and

these relations,

Q 2 81~) (4

Then Q 2 0,

UZE e”(H’).

one can easily show that for sufficiently

and

small

@ > &IU) (ul.

This means that Q-E/ U) (U 1E VnV”. This fact contradicts the extremality of Q unless Q is proportional to lu) (~1. Therefore the extreme elements of VnV’ belong to W and (2.6) follows. This completes the proof of Proposition 1.1. 3. Statement

II for dim H = 3

Throughout this section H is a three-dimensional Hilbert space. The relatively simple structure of positive maps from M2 into B(H) discribed in Theorem 1.2 is based on the following PROPOSITION 3.1.

simple

Let Q E B(H2), Q # 0, Q 3 0 and p 2 0. Then there exists vector u E HZ such that u # 0, u E Q(H’) and uT E @(H’).

The present section following special case. LEMMA 3.1.

is devoted

to the proof

of this proposition.

We start

a

with the

Let

PI = where B, C E B(H);

Z is the unity

[;e$

(3.1)

qf B(H). Assume that Q, 2 0

(3.2)

Q; > 0.

(3.3)

and Then there exist a complex

number

t E C and a vector z E H such that z # 0 and

I1 II z lz

E Q,W"),

Z tz

E QI

(HZ).

(3.4)

(3.5)

POSITIVE MAPS OF LQW DIMENSIONAL

Proof:

One can easily check

MATRIX

173

ALGEBRAS

that for any x, y E H we have

([:I IL*“cl [;I)=

I[x+B1’112+(YIC-B*BIy).

This shows

that

(3.2) is equivalent

to C- B*B > 0.

In the same way, (3.3) is equivalent

to C-BB”

Now we shall show that relation :

(3.6)

> 0.

(3.7)

(3.4) and (3.5) are implied

by the following

orthogonality (3.8)

ZlHt, where Ht is the subspace Indeed, (3.8) means

of H spanned that zl

by (B-tZ)Ker(C-

B*B) and (B-

tZ)*Ker(C-

(B-tZ)Ker(C-B*B)

BB*).

(3.9)

and z I (B-

tZ)*Ker(C-

BB”).

(3.10)

(3.9) is equivalent to (B- tZ)*z 1 Ker(C- B*B). For self-adjoint operators, the orthogonal complement of the kernel coincides with the image. Therefore there exists y E H such that (B-tZ)*z Now

and

using

this relation

= (C- B*B)y.

one easily verifies

that

(3.4) follows. In the same way, starting with (3.lO),one proves (3.5). To prove our lemma it is sufficient to show that there exists a complex number t such

that dimH,

< 3.

(3.11)

Let us denote the dimensions of Ker(C-B*B) and Ker(C-BB*) by n+ and n-, respectively. Assume for the moment that n, = 3. Then C-B*B = 0, Tr(C-BB”) = Tr(C-B*B) = 0 and in virtue of (3.7) we have C- BB* = 0 and n_ = 3. In the same way one shows that n_ = 3 implies n, = 3. Therefore (n+ = 3) 0

We shall consider I. n+ +n_

the following

d 2. It is obvious

(n_ = 3).

cases: that

in this case dim Ht < 2 for any t E C.

(3.12)

S. L. WORONOWICZ

174

II. n+ +n_ = 3. In virtue of (3.12) IZ+ # 3 # n_. To be definite assume and n- = 1. Let (e,f) be a basis of Ker(C- B*B) and (g) a basis of Ker(CH, is generated by the vectors e(t)

= (B-tZ)e,

f(t)

= (B-tZ)f,

g(t)

that IZ+ = 2 BB*). Then

= (B-tZ)*g.

(3.13)

Let w(t) = det (e( t),.f(t), g(t)). Clearly, w(t) is a third order polynomial with respect to t and t with leading term of the form - t2L By using standard topological techniques, (the index ([4], p. 226, of the point 0 with respect to the path [0, 2n] 3 cp -+ w(re’p) is 0 for r = 0 and 1 for large r; therefore for some r the path has to pass through 0) one may show that there exists a t such that w(t) = 0. For such t the vectors (3.13) are linearly dependent and (3.11) follows. III. n, +n_ = 4, B*B # BB*. In this case (see (3.12)) n, C- B*B and C- BB* are one-dimensional:

where

= n_ = 2. The operators

C-B*B

= Ix) (xl,

(3.14)

C-BB”

= Iv) (yl,

(3.15)

x, y E H. Therefore B*B-

BB* = jy) (yl-

(3.16)

Ix) (xl.

Computing the traces of both sides of (3.16), we get (x lx) = (y ly). We may assume that x is not proportional to y. Indeed, otherwise lx) (xl = ly) (yl and B would be a normal operator. This case will be considered latter. Let us consider the vectors x, y, Bx, B*y. Since dimH = 3, these vectors are linearly dependent: aB*y+by

(3.17)

= yBx+dx,

where L-X, 8, y, S E C. One may assume that ~1,y are real and non-negative (possible phase factor can always be absorbed by x and y) and that cc+ y > 0 (otherwise a = y = 0 and x would be proportional to y). In virtue of (3.16) (see Appendix) there exists an antiunitary involution 9 such that fB9

= B* and 9.~ = y. Applying

9

to the both

= yB*y+6y.

(3.18)

sZ)x = (B-7Z)*y,

(3.19)

aBx+$x Combining

(3.17) and

sides of (3.17), we get

(3.18), we obtain (B-

where

t = _ d!!?!_ a+y * Let z E H; t, s E C. Taking into account (3.16), (3.19) and the relation one can check by simple computation that jl(B-tZ)z+sy~l*f

I(x/z)+(T-~)s~~

= II(B-ttl)*z+s~J1~+

(XIX)

j(yIz)+(?-i)~/~.

(3.20)

MATRIX

POSITIVE MAPS OF LOW DIMENSIONAL

Now

let us consider

a family

ALGEBRAS

175

of operators

D, = (B-U)+% where t E C and t # z. The determinant detD, is a rational function of t and tends to infinity as t -+ 00. Since any rational function defined on the compactified complex plane takes any complex value, one can find a t such that detD, = 0. Then there exists a nonzero vector z such that D,z = 0. More

explicitly,

my =. t--z 0

(B_tl)z+

This shows that (B- tZ)z is proportional to y. Therefore to Ker(C-BB*), and consequently we get (3.10). Setting

in (3.20)

s = g:

and using

(3.21)

(cf. (3.15)) (B- tl)z is orthogonal

(3.21), we see that

(B- tZ)*z is proportional

to x. Therefore (cf. (3.14)) (B- tZ)*z is orthogonal to Ker(C-B*B) and (3.9) follows. This completes the proof in case III. Let us note that n, +n_ 2 5 iff n, = II_ = 3 (cf. (3.12)). In this case B*B = C = BB*. Therefore, to end the proof of the lemma it is sufficient to consider the following case: IV. B is a normal operator. This is the simplest case. Let t be an eigenvalue let z be the corresponding eigenvector: Bz = tz and B*z = tz. Then

of B and

[:I=[i*“c][J and [rz] =[:,:][a]* Since we have considered all possible cases, the lemma is proved. Now we are ready to prove Proposition 3.1. Assume that the operator

e= fulfils the assumptions consider two cases. I. A is invertible.

of Proposition

1

A B*

Then Q, is of the form

I

3.1. Clearly,

This case can be easily

PI =

B C

A is a positive

reduced

to that

operator.

in Lemma

We shall

3.1. Let

[“o’:’ A:l,2] e yy2 A-4,2] *

(3.1) and satisfies

all the assumptions

of Lemma

3.1. Therefore

176

S. L. WORONOWICZ

there exists a simple vector U, E H2 such that Now, one can easily check that the vector

fulfils our requirements:

P,.

u is simple,

II. A is not invertible. We have

It follows

immediately

u Z 0, u E Q(H”)

Let Ho = A(H).

The projection

and

u; E Q;(H2).

and ut E Q(H”). onto

H,I will be denoted

by

that P,B=

Assume account

u1 # 0, u1 E Q,(H’)

B”P,

= P,B”

= BP,

that CP, # 0. Then there exists a vector xl (3.22), we get

[k]=

Q[Z]

and the vector u = uT =

= 0.

(3.22)

H, such that CX # 0. Taking

into

Qf]

and (“cx] =

satisfies our requirements.

Therefore

we may assume

CP, = P,C=O.

that (3.23)

In short, the relations (3.22) and (3.23) mean that all our operators act in fact in Ho. In that case Proposition 3.1 follows directly from Theorem 1.1 (note that dimH, < 3). Remark. of Theorem

The methods 1.1

presented

in this section

can be used to give a direct

proof

4. Counterexample In this section we present a counterexample showing that, in the case dimH = 4, statement II of Section 1 is not valid. Throughout this section dimH = 4. Elements of B(H) will be represented by 4x 4 complex matrices. Let

POSITIVE MAPS OF LOW DIMENSIONAL

MATRIX

ALGEBRAS

177

where -0101

2;

’

0 10 0010’

c=

0001’ _o 0 0 0I 0°10

BE

0 0 q

$

0

0 0 2;

_

-

With the help of (3.6) and (3.7) one easily checks that Q and Q’ are positive. We claim that 0 is the only simple vector u E HZ such that u E Q(H2) and ZP E p(H’). This fact is very simple, the proof however needs a lot of computations of arithmetic nature (one works with 8 x 8 matrices !). In order not to bore the reader we indicate only the main stages of these computations. (1) Any vector

in Q(H’)

is of the form u = (a, 8, y, 6,5&Y a, B, y+4s)

where c(, fi, y, 6, E E C (to save paper HZ = C”). (2) A vector

of the form

we adopt

(4.1) is simple

(5&Ya, B, yf4E) where

a “horizontal

(4.1) notation”

for

vectors

in

iff either = t(a, B, y, a>,

t E C, or (E, B, y, 6) = 0.

In the first case we have 4 t~,t,1,t-‘+jt2,t3,t~,t,1+.$t3

u=y

(4.2)

(

1

for t # 0 and U = (0, 0, 0, 6,0,0,0,0) for t = 0; in the second

u= (3) The “transposes”

of vectors

(0,0,0,0,5&,0,0,4&), (4.2), (4.3) and

4 uT = y t2, t, 1, t-1+Tt2, ( ZP = (0, 0, 0, 6,0,0,0,

t’i, O),

2.4’= (0, 0, 0, 0, 5E, 0, 0,4&), respectively.

(4.3)

case (4.4) (4.4) are equal

tt, t,t-li++t2i

to (4.5) (4.6) (4.7)

S. L. WORONOWICZ

178 (4) Any vector

in p(H’)

is of the form

ZI = (a’, B’, y’, 8, #&+4&I, y’, 8’, 5&‘), where a’, @‘, y’, 8, I’ E C. (5) Assume that one of the vectors get easily U’ = 0 and u = 0.

(4.8)

(4.5), (4.6), (4.7) is of the form

(4.8). Then

we

5. Remarks on the Kadison inequality Let &, 99 be C*-algebras

and let Q,: &--+a

be a normalized (i.e. @(I) = 1) positive for any self-adjoint element a E d

map.

The famous

Kudison inequality

says that

@(a”) > @(a)“. For qeuality

2-positive [I]

normalized

maps

we have

(5.1)

a stronger

version

of the

Kadison

@@*a) > @(a)*@(u) for any u E G?. For normalized

2-copositive

maps

(5.2)

instead

of (5.2) we have

@(a*@) B @(u)@(a)*. One can easily invent and (5.3).

an inequality

which

is stronger

in-

(5.3) than (5.1) and weaker

than (5.2)

DEFINITION.

such that

We say that @ satisfies the strong Kadison inequality if for any b, c E d c > b”b and c > bb* we have

Q(c) > @(b)*@(b),

(5.4)

@j(c) 2 @(b)@(b)*.

(5.5)

and

In our opinion, Conjecture:

the following

Any normalized

We shall prove

conjecture positive

this conjecture

is very reasonable.

map satisJies the strong Kadison

in some special

inequality.

cases.

THEOREM 5.1. Let @,: RI + g be a normalized positive map. Assume that @ = @I + +@, , where Q1 is 2-positive and a.2 is 2-copositive. Then 0 satisfies the strong Kadison inequality.

Proof:

Let us recall that

D1 is 2-positive

iff

POSITIVE MAPS OF LOW DIMENSIONAL

for all a, b, c, d E ~2. Q2 is 2-copositive

1 b*

20 Since G1 is 2-positive,

and

Ib

I

o

1

>/ 0.

(3.7)) (5.6)

Since a2 is 2-copositive,

I ’ @z(l) >@i,(b) I @l(l), @l(b*) >

and

@l(b)*, @l(c) N we have

@,(b) >@a(c) I’

@z(l),

@z(b)*

these relations

(5.4) and

c

(cf. (3.6) and

we have

@l(l), @l(b) >

and

179

iff

of & such that c > b*b and c > bb”. Then

Let b, c be elements

Combining

MATRIX ALGEBRAS

> o

in a proper

(5.5) follow

and

o

@,(b)> R(c)

’

O,(b)*,

’ ‘-

way and remembering

Q2(c)

that @ is normalized,

we get

immediately.

THEOR M 5.2. Let d = B(H), where dimH < 3. Then any normalized positive map @: d -+ 9l satisfies the strong Kadison inequality. Proof.

Then

Let b, c E ~4. Assume

that c > b”b and c 2 bb”. We introduce

an operator

in H2:

Q acting

(cf. (5.6)) Q>O

and

QaO.

for dim H < 3 (cf. Theorem

Now, we use the relation (2.6) proved osition 3.1). This means that

1.1 and

Prop-

N

Q =

\ where

2 lun)(unl, II=1

ZJ are simple

In an obvious

vectors

notation,

(5.7) is equivalent

1

b

x2

I&J (x.1, G?IX”)

b*

c

a:

IX”) (X”l, xz

to (X”l

IX”> (&I 1

(5.7)

S. L. WORONOWICZ

180

Applying we get

to the both

sides the mapping

Since the tensor product positive and we obtain

(5.4) follows

immediately.

of positive

Replacing

@@id (where

elements

is positive,

id is the identity

all summands

map in M2),

on the r.h.s. are

b by b*, we get (5.5).

Remark I. Theorem 5.2 does not follow directly from Theorem 5.1. One can show (see [13]) that there exists a positive map defined on Mz which cannot be written as a sum of 2-positive and 2-copositive map. Remark

II. Inequality

(5.1) can be proved in a similar way. One has to use the following

THEOREM 5.3. Let SS?be a P-algebra and let ZZ?+denote the cone of positive elements of &‘. Then the convex cone in d@M, generated by all a@q where a E -02, and a runs over all real positive matrices, coincides with

This theorem in turn follows from the known the real 2 x 2 matrices ([2], [ 131).

results

concerning

positive

maps from

Acknowledgement The author wishes to thank Professor A. Grossmann and the other members of C.P.T.-C.N.R.S. in Marseille for illuminating discussions. Special thanks are due to Professor J. E. Roberts for his invaluable remarks and to Professor R. Stora for his kind hospitality during the author’s stay in Marseille. Appendix : Almost normal operators By using the spectral theorem one can easily antiisomorphic to its adjoint B”, i.e. that

show that

any normal

operator

B is

B” = 9B9, where 9 is an antiunitary involution. It turns out that the same result holds for “almost normal” operators B such that [B*, B] is two-dimensional. THEOREM A. Assume that

Let B be an operator B”B-

acting

in a jinite-dimensional

BB” = Iy) (yl-

Ix) (xl.

Hilbert

space H. (A-1)

POSITIVE MAPS OF LOW DIMENSIONAL

Then there exists

an antiunitary

involution

4

9x 9-B4

MATRIX

181

ALGEBRAS

acting in H such that

= y,

64.2)

= B*.

(~4.3)

Proof. We shall constantly meet expressions containing products of many operators B and B* taken in different order acting on vectors x and y. In order to deal with these expressions we have to introduce a convenient notation. In what follows, variables denoted by capital letter B furnished with different subscripts run over the two-element set (B, B*}: B,,B,

,...,

B;,B;

,...,

E{B,B”).

Similarly, variables denoted by small letter z (also with different subscripts) run over the two-element set (x, v}. Moreover, z* denotes x (resp. y) whenever z = y (resp. z = x). For any t E R we put A(t) By direct

computation

We shall prove

one checks

= B+tB*.

that

that (xlA(t)“x)

for any non-negative (YlA(t)“Y)-

integer (xlA(+)

= (YlA(VY)

(A.4)

n. Indeed, = TrA(V{lY)

(VI-IX)

(-4

= ,!$TrA(t)“{A(t)*A(t)-A(t)

= 0.

The last equality follows from the well-known property of the trace: Equation (A.4) can be rewritten in a more sophisticated way: (zIA(t)“zJ In fact (A.5) reduces to (A.4) if z = evidently satisfied. Both sides of (A.5) are polynomials responding coefficients, we get:

where 6 runs

= (zflA(t)“z”). z1 .

If z # z,,

“r=

(A.5) then

z = z:,

of order n with respect

over the set dnk of all k-element B

Tr CD = TrDC.

jB \B*

subsets if if

rEo, rEg.

z1 = z* and

to t. Comparing

of { 1, 2, . . . , n} and

(A.5) is the cor-

S. L. WORONOWICZ

182 Equation

(A.6)

will be used in the following

form:

‘.. &,,z”)) (ZIB0.1 ... &,nZJ-(@I&,” Cr U&“k As we shall see later, the existence of the antiunitary involution (A.3) is equivalent

to the following (z/&B,

= 0. 9 satisfying

. . . B*z,)

. . . B,z,)-(zJB,

= (~14

. . . Bd

= (x(&

.,. B,z*)

(A.2) and

equality = ($I&

. . . B,B,z”).

(A.8)

For m = 0 this relation has been already verified (put n = 0 in (A.5)). that (A.8) holds for all m d n- 1. Then using (A.l), we get (zj B, . . . BkB*BBk+3

(A.7)

. . . BkBB*Bk+3

now

. . . BJ,)

(ylB,c+3 . . . Bnz,)-(~(4

($I&

Assume

. . . B/cx) (xIBk+s

. .. &+,x)-(~14

s.. B,z*) HI&

. . . Bnz,)

. .. &+o)

= (zflBn . . . &+A (xl& . . . B,z*)-@::I&, ... &+JY) (ul& . .. f&z*) = (zfli?,, .. . Bk+3BB*Bk . . . B,z*)-(zi/B,, ...Bk+3B*BBk . . . B,z*). Therefore (z/B1 . . . BkB*BBk+3

. . . B,zJ-@T/B,,

. . . Bk+3BB*Bk

= (z/B1 . . . BkBB*Bk+3 This result

shows

.., B,z*)

. . . B,z,)-(zfIB,,

. . . Bk+JB*BBk

. . . B,z*).

that the difference (z/B,

. . . B,zJ-(z::IB,

. . . B,z*)

= a(k)

is independent of the order of operators in the sequence B1, . . . , B,, . It depends only on z, zl, n and the total number k of entries of B in B,, Bz, . . . . B,,. In particular, all summands in (A.7) are equal to a(k). Therefore cc(k) = 0 and (A.8) is verified for m = n. By the induction principle, (A.8) holds for all non-negative integers nz. Let u = BIB, u* = BfB; Then

it follows

immediately

v = ‘B1’B2 . . . ‘B,,,‘z,

. . . B,,z, . . . B*z* It 9

from

vu* = ‘Bf’B;

. . . ‘B*‘z*. u

(A.8) that (ulv)

= (v*I#*).

(A.9

Let Ho be a subspace of H generated by all vectors of the form B, . . . B,z. Relation (A.9 shows that there exists an antiunitary involution .YO acting on Ho such that YoB,

B, . . . B,,z = B;Bf

If Ho = H, then this involution solves our problem: directly from (A.lO). In the general case H = H,@H,,

(A.lO)

. . . B,*z*. relations

(A.2) and

(A.3) follow

B = C,OC,,

where H, is the orthogonal complement of Ho, Co is the restriction of B to Ho and Cr is the similar restriction to H, . Since x, y E H,,, the operator C, is normal. Making use

POSITIVE

MAPS OF LOW DIMENSIONAL

of the spectral theorem one can find an antiunitary 4, C,4, = CT. Then the involution

satisfies

(A.2) and

MATRIX

involution

ALGEBRAS

9I

183

acting in HI such that

(A.3).

Remark. The theorem remains valid in the ‘case dim H = co if one assumes is a Hilbert-Schmidt operator. The same proof applies.

that

B

REFERENCES [l] [2] [3] [4] [S]

Choi, M.D.: Positive linear maps on C*-algebras, Thesis, University of Toronto, 1972. -: Completely positive linear maps on complex matrices, Linear Algebra and Appl. (to appear). -: Positive semidefinite biquadratic forms, preprint. Dieudonne, J. : Elgments d’analyse. 1. Fondements de l’analyse moderne, Gauthier-Villars. Paris, 1969, Gorini, V., A. Kossakowski, E. C. G. Sudarshan: Completely positive dynamical semigropups of Nlevel systems, The University of Texas at Austin, Preprint ORO-3992-200, CPT 244. [6] Gorini, V., S. E. C. G. Sudarshan: Extreme afine trunJ@mations, Preprint IFUM 165/l? Milan, June 1974. [7] -, -: Irreversibility and dynamical maps of statistical operators, Lecture Notes in Physics 29, Springer-Verlag, Berlin, 1974, 260. [8] Kadison, R. V.: Annals Math. 54 (1951), 325. [9] Lindblad, G.: Commun. Math. Phys. 40 (1975), 147. [IO] -: On the generators of quantum dynamical semigroups, Royal Institute of Technology, Department of Theoretical Physics, Stockholm, Preprint TRITA-TFY-75-l. [Ill Stormer, E.: Acta Math. 110 (1963), 233. 1121 -: Positive linear maps of C*-algebras, Lecture Notes in Physics 29, Springer-Verlag, Berlin, 1974, 85. [13] Woronowicz, S. L.: Nonextendible positive maps, Commun. Math. Phys. (to appear).