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Comments on multiplicativity of maximal p-norms when p = 2

Christopher KingDepartment of Mathematics

Northeastern University, Boston MA [email protected]

Mary Beth Ruskai Department of Mathematics , Tufts UniversityMedford, Massachusetts 02155

[email protected]

January 8, 2004

Dedicated to A.S. Holevo on the occasion of his 60th birthday

Abstract

We consider the maximal p-norm associated with a completely positivemap and the question of its multiplicativity under tensor products. Wegive a condition under which this multiplicativity holds when p = 2, andwe describe some maps which satisfy our condition. This class includesmaps for which multiplicativity is known to fail for large p.

Our work raises some questions of independent interest in matrix the-ory; these are discussed in two appendices.

Partially supported by the National Science Foundation under Grant DMS-0101205.Partially supported by the National Science Foundation under Grant DMS-0314228.

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1 IntroductionIn quantum information theory, noise is modeled by a completely positive andtrace-preserving (CPT) map acting on the states of the quantum system. Ingeneral takes pure states into mixed states, and in order to assess the noisinessof the map one is interested in knowing how close the image states may come topure states. Amosov, Holevo and Werner (AHW) [2] observed that this could bemeasured by the quantity

p() = sup { () p : > 0, Tr = 1} (1)where p = Tr ( ) p

1 /p and 1 p . For any density matrix , p 1with equality if and only if is pure. Hence p() 1 with equality if and onlyif there is a pure state for which () is also pure (since by convexity the supin (1) is achieved on a pure state).

The p-norm dened above can be extended to arbitrary matrices as A p =Tr |A| p

1 /p with |A| = AA. The following useful relationships, which wereestablished in [1], can be readily veried. p() = sup

> 0 ,Tr =1( ) p = sup

A> 0

(A) pTr A

(2)

= supA= A

(A) pTr |A|

= supA= A

(A) pA 1

(3)

supA(A) p

A 1 . (4)

(The equivalence of (2) and (3) follows from the convexity of the p-norm andthe fact that |(A+ A)| = (A+ ) + ( A) when A = A+ A is thedecomposition of a self-adjoint operator into its positive and negative parts.) Itfollows immediately from (3) that for any self-adjoint A

(A) p p() Tr |A|. (5)The representation ( 4) suggests viewing as a map between spaces of com-

plex matrices with different p-norms. As suggested in [1] one can generalize thisby dening

q p = supA(A) p

A q(6)

and let Rq p denote the same quantity when the supremum is restricted tothe real vector space of self-adjoint operators. Then p() is precisely R1 p.2


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In general, Rq p q p and one would expect that the inequality could bestrict for some . However, the second part of Theorem 1 states that equalityholds in all dimensions when p = q = 2; and in Appendix B.3, we show thatequality holds for CPT maps on qubits when q = 1, p 2. This raises thequestion of whether equality always holds and, if not, for what types of mapsstrict inequality is possible.

Amosov, Holevo and Werner conjectured [2] that p is multiplicative on tensorproducts, i.e., that

p( ) = p() p() (7)

This has been veried in a number of special cases, although it is now knownto be false in general. Amosov and Holevo [1] proved (7) when both and are products of depolarizing CPT maps and p is integer. King proved ( 7) for all p 1 and arbitrary under the additional assumption that is a unital qubitCPT map [ 10], is a depolarizing channel in any dimension [11], or is anentanglement breaking map [ 12]. However, Holevo and Werner [19] also showedthat ( 7) need not hold in general by giving a set of explicit counterexamples for p > 4.79 and d 3.Amosov and Holevo conjectured [1] that the quantity in ( 6) should also bemultiplicative for 1 q p, i.e., that

q p = q p q p (8)Beckner [5] established an analogous multiplicativity for commutative systemswhen 1 q p. Curiously, Junge [8] proved (8) for completely positive (CP)maps with p and q in the opposite order, that is for the case 1 p q . However,our main interest is the case q = 1 < p, and Junges result does not seem to shedany direct light on this question.

The conjecture ( 7) is of greatest interest for p near 1 since taking the limitas p 1 yields the von Neumann entropy of , another natural measure of purity, and the validity of ( 7) for p in an interval of the form [1, 1 + ) with > 0would imply additivity of the minimal entropy. Moreover, it has been shown[18] that additivity of minimal entropy is equivalent to several other importantconjectures in quantum information theory, including additivity of Holevo ca-

pacity and additivity of entanglement of formation. Audenaert and Braunstein[3] have also observed a connection between multiplicativity for CP maps, andsuper-additivity of entanglement of formation.

In view of the Holevo-Werner example, it is natural to conjecture that ( 7)holds in the range 1 p 2. This is precisely the range of values of p for whichthe function f (x) = x p is operator convex, and it is also the range for which a

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number of convexity inequalities hold. Verifying ( 7) for the special case of p = 2would suggest its validity for 1 p 2. Unfortunately, even this seeminglysimple case is not as straightforward as one might hope. In this note, we prove(7) when p = 2 for a special class of CP maps. Although this is a rather limitedresult, it gives some insight into the difficulties one encounters in the generalcase.

Note that the multiplicativity for CPT maps follows if it holds for all CPmaps. We consider this more general case, as it does not seem more difficult. Infact, it is not hard to show that multiplicativity ( 7) holds for all p 1 whenever is an extreme CP map. This is because these are precisely the CP maps whichcan be written in the form ( ) = AA, i.e, with one Kraus operator, and onecan then assume without loss of generality that A is diagonal. Thus, the extremeCP maps fall into the diagonal maps considered in Example 1 below, for which

multiplicativity has been proved.Although the rst part of the following theorem is included in Junges result,

we include an elementary argument here.

Theorem 1 22 is multiplicative, i.e., 22 = 22 22 . More-over, 22 = R22Proof: First, recall that the complex n n matrices form a Hilbert space withrespect to the inner product A, B = Tr AB, and let denote the adjoint of the linear operator with respect to this inner product. Since

222

= supATr [(A)](A)

Tr AA = supAA, (

)(A)

A, A , (9)it follows that 22 is the usual operator sup norm on this Hilbert space or,equivalently, the largest singular value of . Thus, 22 is the square root of the largest eigenvalue of ( ). This is the same as the largest eigenvalue of ( ) I ; therefore, I 22 = 22 . The main result then follows fromthe submultiplicativity of the Hilbert space operator norm under compositionsince

22 = ( I ) (I ) 22

( I )

(I

) 22 =

22 22 .

Note that since ( ) has real eigenvalues (in fact, they are non-negative), thesolutions of the eigenvector equation ( )(B) = B are self-adjoint (or can beso chosen if is degenerate). This implies that the supremum in ( 9) is achievedwith a self-adjoint A, which implies the second statement in the Theorem.4


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2 Main TheoremWe now nd it convenient to introduce some notation. When

{e j

} is an or-

thonormal basis for Cd, we will let E jk = |e j ek| denote the matrix with a 1in the j -th row and k-th column and 0s elsewhere. Then the set of operators{E jk } also form an orthonormal basis for the d d matrices with respect to theHilbert-Schmidt inner product. Moreover, if is a matrix on C d C d , we canwrite = jk E jk M jk where M jk = Tr 1 (E jk I ). This is equivalent tosaying that is a block matrix with blocks M jk .

If is a CP map, then ( I )() = jk E jk (M jk ) > 0 which implies that

any 2 2 submatrix(M jj ) (M jk )(M kj ) (M kk )

is positive semi-denite. This implies

in turn that

(M jk ) = ( M jj )1 / 2 R jk (M kk )1 / 2 (10)

where R jk is a contraction. Hence

Tr (M jk )(M jk ) (M jj ) 2 (M kk ) 2 . (11)

Theorem 2 Let and be CP maps one of which (say ) satises the condition

Tr ( E ik )(E j ) 0 i ,j ,k,. (12)Then 2 ( ) = 2 () 2 ().

Proof: Writing an arbitrary density matrix as above, one nds

( )() = jk

(E jk ) (M jk ) (13)

Thus

Tr [( )()]( )()=

ik j

Tr ( E ik )(E j ) Tr (M ik )(M j ) (14)

ik jTr ( E ik )(E j ) (M ik ) 2 (M j ) 2

ik j

Tr ( E ik )(E j ) (M ii ) 2 (M jj ) 2 (M kk ) 2 (M ) 2 [ 2 ()]

2

ik j

Tr ( E ik )(E j ) Tr M ii Tr M jj Tr M kk Tr M (15)5


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where we have used (11) and (5). Now, note that the matrix

N = ik Tr M ii Tr M kk E ik (16)

=

Tr M 11 Tr M 22... Tr M dd

Tr M 11 Tr M 22 . . . Tr M dd (17)

is positive semi-denite and

ik j

Tr ( E ik )(E j )

Tr M ii Tr M jj Tr M kk Tr M

= Tr ( N )(N ) (18)

[ 2 ()]2 (Tr N )2 (19)

= [ 2 ()]2

i

Tr M ii2

= [ 2 ()]2 (Tr ) 2 . (20)

When (12) holds the absolute value bars are redundant in ( 15). One can thensubstitute ( 20) in (15) to yield

Tr [( )()]( )()

[ 2 ()]2 [ 2 ()]2 (Tr ) 2 . (21)

Taking the square root and dividing both sides by Tr , one nds

( )() 2Tr 2 () 2 () 0. (22)

Taking the supremum over gives the desired result.Although condition ( 12) is simple, it is basis dependent. In order that be

multiplicative, it suffices that ( 12) holds for the matrices E jk = |e j ek| associ-ated with some basis for Cd . The question of when such a basis can be foundgives rise to some interesting questions in matrix theory which we remark on

in Appendix A. We remark here only that, although we do not expect ( 12) tohold for all CP maps, we also do not have a counter-example. Thus, one cannotexclude the possibility that the hypothesis of Theorem 2 is actually satised byall CP, or by all CPT, maps.

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3 Special casesIt order to show that our results are not vacuous, we now give some examples of maps which satisfy ( 12).

1. Maps with only diagonal Kraus operators. In this case ( E jk ) = a jk E jkfor some positive matrix A = ( a jk ), and the condition ( 12) follows fromthe orthonormality of the {E jk }. This class of CP maps was studied byLandau and Streater who named them the diagonal maps. In fact a morecomplicated analysis using the Lieb-Thirring inequality can be used to showmultiplicativity for all p 1 for these maps [14].

2. Maps for which (I )(M ) = jk E jk (E jk ) has non-negative elements,where M = jk E jk

E jk is the maximally entangled state. (This is the

block matrix with blocks ( E jk ); it is sometimes called the Choi matrix orJamiolkowski state representative of .) Although this condition is clearlysufficient to satisfy (12), it is not necessary. For example, let A be a positivesemi-denite matrix with some a jk < 0. Then for that particular j, k thecorresponding map in Example 1 has ( E jk ) = a jk E jk with one strictlynegative element.

3. Multiplicativity at p = 2 has been proven for all qubit CPT maps [9].However, we can verify the condition ( 12) only for a subset of qubit CPTmaps. This subset is described using the parametrization of qubit mapsthat was derived in [ 15] and summarized in Appendix B.1.

In terms of that notation, the condition ( 12) is satised when t1 = t 2 = 0(since in this case (E jk ) is diagonal when j = k and skew diagonal when j = k). It is interesting to note that multiplicativity is known to hold forall p 1 under the stronger condition t1 = t2 = t 3 = 0 [10], so this resultsuggests that it may hold for p 1 for a larger class of CPT maps.Another class of qubit maps which satisfy ( 12) are those with 1 2 ,t2 = 0, and t1 0 (again using the notation in Appendix B.1). Thesemaps belong to Example 2 since (E jk ) has non-negative elements for all j, k . Furthermore, in Theorem 3 of [ 9], King proved that multiplicativityholds for these channels for all integer p

1, and later [13] extended this

to all p 2.4. The special class of maps satisfying (23) and discussed below. This class

includes maps for which multiplicativity does not hold for some p > 2.

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Let M denote a d d Hermitian matrix with elements m jk = x jk + iy jk withx jk , y jk real. Let : M (M ) denote a linear map with the following veryspecial properties:

[(M )] jk = d j m when j = k,

a jk (x jk + i jk y jk ) when j = k(23)

where d j 0, a jk are the off-diagonal elements of a xed Hermitian matrix and jk = kj = 1. The map is trace-preserving if and only if the matrix D withelements d j is column stochastic. Not every map of the form ( 23) will necessarilybe CP. However, certain special subclasses can be identied.

a) a jk = 0 j = k. In this case, is a QC map consisting of the projectiononto the diagonal part of M followed by the action of a column stochasticmatrix on the classical probability vector corresponding to the diagonal.

b) A > 0 is a xed positive semi-denite matrix, d j = j a jj and jk = +1.This is exactly the diagonal class described above.

c) d j = 1 j , a jk = 1 j = k and jk = 1. In this case, (M ) =(Tr M )I M T and 1d1 (M ) is the CPT map for which Holevo and Werner[19] showed that ( 7) does not hold for large p.Since King [9, 14] showed that multiplicativity holds for all p 1 for maps of type (a) and (b), multiplicativity at p = 2 may not seem very signicant. How-

ever, maps of type (c) are precisely those used to establish that multiplicativitydoes not hold for sufficiently large p. Moreover, the full class includes convexcombinations of maps of type (a) with one of type (b) or type (c), and Kingsresults do not apply to this class. Thus this class of maps is neither trivial noruninteresting.

Although one can verify that CP maps satisfying ( 23) always satisfy thehypothesis of Theorem 2, we state and prove their multiplicativity as a separateresult. Inequality ( 11) again plays a key role in the proof.

Theorem 3 Let be a CP map satisfying ( 23 ) and let be an arbitrary CP map. Then 2 ( ) = 2 () 2 ().

Proof: As before, let = jk E jk M jk be the matrix with blocks M jk . Then( )() > 0 has blocks

[( )()] jk = d j (M ) when j = k,

a jk (( M jk ) + i jk ( M jk ) when j = k(24)

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where (M jk ) = 12 (M jk + M jk ) and (M jk ) = i2 (M jk M jk ). A straightforwardcalculation givesTr [( )()]( )() (25)

= j n

d j d jn Tr (M )(M nn ) + j = k

|a jk |2 Tr (M jk )(M jk )

j n

d j d jn (M ) 2 (M nn ) 2 + j = k

|a jk |2 (M jj ) 2 (M kk ) 2

[ 2 ()]2

j n

d j d jn Tr M Tr M nn + j = k

|a jk |2 Tr M jj Tr M kk (26)

where we have used the Schwarz inequality for the Hilbert-Schmidt inner productand (11). Now, the term in parentheses in ( 26) is precisely Tr (N )(N ) whereN is the matrix with elements N jk = (Tr M jj Tr M kk )1 / 2 . The desired result thenfollows as in the proof of Theorem 2 since

Tr [( )()]( )() [ 2 ()]2 Tr ( N )(N )

[ 2 ()]2 [ 2 ()]2 (Tr N )2 = [ 2 ()]2 [ 2 ()]2 (Tr ) 2 .

4 Concluding remarksIf one could replace the operator basis {E jk } by a more general orthonormaloperator basis {Gm} for C

d

d

, one could always satisfy the analogue of (12).One need only choose {Gm} to be the basis which diagonalizes the positive semi-denite operator , i.e., for which( )(Gm ) =

2m Gm . (27)

where m are the singular values of . Then Tr ( Gm )(Gn ) = 2n mn 0.Moreover, as noted at the end of the proof of Theorem 1, one can always choosethe basis so that each Gm = Gm is self-adjoint.

Using the orthogonality condition Tr Gm Gn = mn , one can show that adensity matrix on a tensor product space can be written in the form

=m

Gm W m with W m = Tr 1 (Gm I ). (28)

Note Gm self-adjoint implies that W m is also self-adjoint.

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We now try to imitate the proof of Theorem 2. Since ( )() =m (Gm ) (W m ),

Tr [( )()]( )() =mn

Tr ( Gm )(Gn ) Tr( W m )(W n )

=n

2n Tr (W n )(W n ) (29)

[ 2 ()]2

n

2n (Tr |W n |)2 (30)

= [ 2 ()]2 Tr ( N )(N )

[ 2 ()]2 [ 2 ()]2 (Tr |N |)

2 (31)

where N = m Gm Tr

|W n

| (and the rst inequality implicitly used the assump-

tion that Gm is self-adjoint so that W m is).Unfortunately, we can not conclude that Tr |N | Tr as needed to completethe proof. (If we had instead N = m Gm Tr W n , then we would have N =

Tr 2 > 0 and Tr N = Tr |N | = Tr .) This is a real problem. Using the Paulibasis for qubits, for which Gk = 21 / 2 k , consider the maximally entangled Bellstate = G0 G0 + G1 G1 G2 G2 + G3 G3 . Then Tr 2 = 12 I , butN = 1 2

2 2 + i2 i 0

is not positive semi-denite and Tr |N | > 1.Although our results do not prove it, we conjecture that multiplicativity does

hold for all CP maps at p = 2. If this conjecture turns out to be false, then itseems unlikely that any other value of p between 1 and 2 would play a specialrole, and there would probably be counterexamples to multiplicativity all theway down to p = 1. In this case additivity of minimal entropy would be anisolated result and the attempt to prove it using p-norms would probably befutile.

A Comments on positivity condition ( 12):To nd conditions under which ( 12) holds, note that it is equivalent to therequirement that

xik,j = Tr E ik ( )(E j ) 0 i, j, k , (32)which is precisely the condition that the matrix X representing the positive semi-denite linear operator in the orthonormal operator basis {E jk } also hasnon-negative elements xik,j .

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If |f j = U |e j denotes another O.N. basis for Cd, then F jk = |f j f k| =UE jk U is an orthonormal operator basis for Cdd, and the corresponding matrixrepresentative of is (U

T

U )X (U U ). In the proof of Theorem 2 abovewe require only that E jk have the form |e j ek| for some orthonormal basis {e j}for C d so that we can use (11).Therefore, multiplicativity of 2 () will hold if there is a unitary operator

U on Cd such that the matrix ( U T U )X (U U ) has non-negative elements[where X is the matrix with elements dened by ( 32)]. Unfortunately, this is nota very tractable condition.

One way to nd an example of a CP map satisfying ( 12) on Cd is to nd ad2 d2 positive semi-denite matrix X with non-negative elements. Regardingthe d d blocks of X as (E jk ) denes a CP map of the type considered inExample 2. However, as noted before, CP maps satisfying ( 12) need not havethis form.

A d2 d2 matrix X with elements xik,j 0 also denes a positive semi-denitelinear map which one can write as = ( ). The map then satises ( 12),but it need not be CP. We only know that = U for some linear operatorU on C dd which is unitary in the sense Tr U (A)U (B) = Tr AB for all ddmatrices A, B. This implies that must have the form ( A) = U (A)U forsome unitary matrix U .1 Hence, is CP if and only if is; however, this doesnot seem easy to check.

B Qubit maps

B.1 Notation:It will be useful to summarize some basic facts about the representation of matri-ces and CP maps on qubits using the identity and Pauli matrices as bases. Onecan write an arbitrary matrix as A = z 0 I + z with z 0 C , z = w + iu and w , uvectors in R 3 . When z 0 = 0, any norm satises z 0 I + z = |z 0 | I +

1z0 z .Therefore, we will present most results only for z 0 = 1; results for z 0 = 0 are

generally straightforward. The most general CP map has the form

(I + z

) = (1 + s

w + i s

u )I + ( t + T w + iT u )

(33a)

(z) = s zI + ( T z) (33b)1 The fact that U must have the form U (A) = U AU is probably well-known, but was

rst brought to the attention of MBR by Nicolas Boulant, who includes a proof in his paper[6]. If one writes U in Kraus form, one can then use the fact that U U = I to show thatthe Kraus operators can be chosen to be a single unitary.

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where s, t are vectors in R 3 and T is a real 33 matrix. is TP if and only if s = 0; and is unital if and only if t = 0.As observed in [15], one can use the singular value decomposition to assumewithout loss of generality that T is diagonal with real (but not necessarily posi-

tive) elements k . This leads to the canonical form

(I + w) = I +k

(tk + kwk)k (34)

for CPT maps introduced in [15]. Conditions on the parameters tk , k whichguarantee that is CPT are given in [17]; some special cases were consideredearlier in [4].

B.2 Useful formulasWe now restrict attention to CPT maps acting on A = I + z for which (A) =I + ( t + T w + iT u ). Then

AA = (1 + |z|2 )I + 2( w + u w) (35)

with

|w + u w| = |w|2 + |w|

2

|u |2

(u w )2 . (36)

Therefore, the eigenvalues of AA are 1 + |z|2 2|w + u w| or, equivalently,1 + |z|

2

2 |w|2 + |w|2 |u |2 (u w)2 . (37)and those of (A)(A) are = 1 + |t + T w|

2 + |T u |2

2 |t + T w|2 (1 + |T u |2 ) |(t + T w) (T u )|2 . (38)When ( t + T w) (T u ) = 0, ( 38) becomes |t + T w|+ 1 + |T u |2 2 .We now wish to evaluate and bound A 21 = (Tr |A|)2 Note that ( 37) impliesthat the eigenvalues of

|A

| = AA are

1 + |z|2 2 |w|2 + |w|2 |u|2 (u w)2 , (39)and observe that their product can be written as

(1 + |z|2 )2 4(|w|

2 + |w|2

|u |2

(u w)2 ) = (1 |w|

2 + |u |2 )2 + 4( u w)

2 .

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Therefore,

(Tr |A|)2

= 2 1 + |z|2

+ (1 |w|2

+ |u |2

)2

+ 4( u w )2

(40)

2 1 + |w|2 + |u |

2 + 1 |w|2 + |u |

2 (41)

4(1 + |u|2 ) if |w|2 1 + |u |24|w|2 if |w|2 > 1 + |u |2 .

(42)

B.3 Equality for CPT maps when p 2:We now show that R1 p = 1 p for CPT maps on qubits when p 2. ForA = I + z we prove the somewhat stronger result that

[I + ( w + iu )] 2 pI + ( w + iu )] 21 [I + w )

2 p

I + w 21

(43)

where w is the unit vector dened by w = |w| w . Our argument will use thefollowing easily veried results. When a 0 and m 1,f (x) = |x + a|m + |x a|m is increasing for x > 0 (44)g(x) =

[f (x)]2 /m

x2 is decreasing for x > 0. (45)

Note that f (x) is symmetric in x and a from which it follows that the expressionon the right in ( 44) is also increasing in a.

It follows from (42) and (38) that

(A) 2 pA 21

(+ ) p/ 2 + || p/ 22 /p

4max(|w|2 , 1 + |u |2 ) (46)

Since the numerator has the form of f in (44) with m = p/ 2, the monotonicityin a implies that dropping the dot product terms in ( 38) increases the right sideof (46). Hence

(A) 2 pA 21

|t + T w|+ 1 + |T u |2 p + |t + T w| 1 + |T u |

2 p2 /p

4max(|w|2 , 1 + |u |2 ) (47)

Since |T u | |u |, (44) again implies that the right side of ( 47) increases when|T u| is replaced by |u | in the numerator. Also we can only increase the ratio

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on the right side of (47) by choosing t T w to be positive. Therefore, we canconclude from (44) that this ratio is increasing in |w| for |w|2 1 + |u |2 , andfrom (45) that it is decreasing in |w| for |w|

2

1 + |u |2

. Hence this ratio ismaximized when |w|2 = 1 + |u |2 . Therefore the ratio in ( 47) is less than

|t + T w|+ |w| p + |t + T w| |w|

p 2 /p

4|w|2 , (48)

which we want to show is smaller than the RHS of ( 43). Since |w|2 = 1+ |u |2 1,|t + T w| |w|t + T w = |w| |t + T w|. (49)Using (44) again to replace the |t + T w| term in (48), we nd

(A) 2 pA 21

|t + T w|+ 1 p + |t + T w| 1

p2 /p

4 (50)

=(I + w )

2 p

I + w 21

(51)

R1 p2

= p()2 . (52)

We next consider the case z 0 = 0, for which A = z, |A| = |z |I and (A) =(T z). Then (A) p = |T z| for all p so that(A) p

A 1=

(z

) p

z 1=

|T z

||z| max

k

k

p() (53)

where the last inequality follows from the fact that [ 2 ()]2 12 (1 + |t + T w|)2 |T w|2 which can be made equal to max k 2k for some w with |w| = 1.To complete the proof, recall that for z 0 = 0, z 0 I + z p = |z 0 | I + 1z0 z pand note that the factor |z 0 | will cancel in any ratio of norms. Therefore, if wetake the supremum over all complex matrices A, we can use (52) and (53) toconclude that 1 p p() = R1 p when p 2. The reverse inequality R1 p 1 p always holds; therefore, we must have equality for p 2.

B.4 RemarksSuppose that both p() = R1 p and 1 p are multiplicative for some p, .Suppose also that R1 p = 1 p in dimension d (e.g., d = 2.) Then indimension d2 (e.g., d = 4),

R1 p = R1 p

2= 1 p

2= 1 p. (54)

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Thus, equality also holds in dimension d2 for maps of the form .The argument in Section B.3 breaks down for 1 p < 2. Although one does

not expect ( 43) to hold, the weaker inequality (A) p A1

p() might stillhold, and this is all that is needed to show 1 p R1 p. However, evenfor p = 1, we have been unable to verify (or nd a counter-example to) this.For the general CP form ( 33a) with s = 0, 1 () = 1 + |s| > 1 is achievedwith w = s|s |, and the eigenvalues of (A)(A) are

(S 2 + |t + T w|2 + |T u |

2

2 |t + T w|2 [(S 2 + |T u |2 ] |(t + T w ) (T u )|2with S 2 = (1 + s w )2 + ( s u )2 . If one tries to use the argument in the previoussection, the RHS of (47) becomes

|t + T w|+ S 2 + |T u |2 p + |t + T w| S 2 + |T u |2

p2

/p

4max(|w|2 , 1 + |u|2 ) . (55)

This does not have the form |x + a|m + |x a|m because a = S 2 + |T u|2 andS 2 depends on w .References

[1] G. G. Amosov and A. S. Holevo, On the multiplicativity conjecture forquantum channels, Theor. Probab. Appl. 47 , 143146 (2002).

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