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arXiv:quant-ph/0212057v1

10Dec2002

Maximal p-norms of entanglement breaking

channels

Christopher KingDepartment of Mathematics

Northeastern UniversityBoston MA 02115

[email protected]

January 14, 2004

Abstract

It shown that when one of the components of a product channel is

entanglement breaking, the output state with maximal p-norm is always

a product state. This result complements Shors theorem that both min-

imal entropy and Holevo capacity are additive for entanglement breaking

channels. It is also shown how Shors results can be recovered from thep-norm results by considering their behavior for p close to one.

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Holevo [1] introduced the following class of channels:

() =Kk=1

Rk Tr

Xk

(1)

where each Rk is a density matrix and where the {Xk} form a POVM, thatis Xk 0 and

Xk = I. As Shor pointed out [2], channels of this form are

entanglement breaking, meaning that the state ( I)(12) is separable for anybipartite state 12. For this reason these channels are now known as entangle-ment breaking (EB) channels. Shor proved additivity of the minimal entropyand the Holevo capacity for EB channels [2], thereby settling the question oftheir classical information-carrying capacity.

The purpose of this note is to show that EB channels also satisfy anotheradditivitytype property involving the maximal p-norm. This notion was in-troduced by Amosov, Holevo and Werner [3], and involves the following non-commutative version of the usual lp norm for p 1:

||A||p =

Tr |A|p1/p

(2)

The maximal p-norm of a channel is defined to be

p() = sup

||()||p (3)

where the sup runs over density matrices in the domain of .

Theorem 1 Let be an entanglement breaking channel, and let be an arbi-trary channel. Then for any p 1,

p( ) = p() p() (4)

The proof of Theorem 1 relies on an intermediate bound which we statebelow as Lemma 2. To set up the notation, consider the action of the channel

(1) on a bipartite state 12:

( I)(12) =Kk=1

Rk Tr1 [(Xk I)(12)] (5)

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Define

xk = Tr [ (Xk I)(12)] (6)

Gk = x1k Tr1 [(Xk I)(12)]

Then (5) reads

( I)(12) =Kk=1

xkRk Gk (7)

where now {Rk, Gk} are all density matrices, and xk 0 with

xk = 1. Also,writing 1 = Tr2(12) for the reduced density matrix it follows from (7) that

(1) =

Kk=1

xkRk (8)

Define the following 1 K block row vector:

R =

(x1R1)1/2 (xKRK)

1/2

(9)

Then R is a K 1 block column vector, and (8) can be rewritten as

(1) = R R (10)

Lemma 2 For all p 1,

Tr

( I)(12)p

K

k=1

Tr [(RR)p]kk Tr[(Gk)p] (11)

where [(RR)p]kk is the kth diagonal block of the K K block matrix (RR)p.

Proof of Theorem 1: let 12 = (I )(12) so that

( I)(12) = ( )(12) (12)

Then from (6) it follows that

Gk =

x1k Tr1[(Xk I)(12)]

=

Gk

(13)

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where Gk = Tr1[(Xk I)(12)] is a density matrix. Therefore (3) implies that

Tr[(Gk)p] p()p (14)

Together with (11) and (12) this implies

Tr

( )(12)p

p()p

Kk=1

Tr [(RR)p]kk (15)

= p()p Tr[(RR)p]

= p()p Tr[(RR)p]

= p()p Tr[(1)

p]

where we used the fact that the matrices RR and RR share the same nonzerospectrum (and where Tr changes its meaning several times). Using again thedefinition of maximal p-norm (3) we deduce

Tr

( )(12)p

p()p p()

p (16)

Since this bound holds for all 12 it follows that

p( ) p() p() (17)

and this implies the Theorem since the right side of ( 4) can be achieved with a

product state. QED

Proof of Lemma 2: this is an application of the Lieb-Thirring inequality [4],which states that for positive matrices A and B, and any p 1,

Tr

A1/2BA1/2p

Tr

Ap/2BpAp/2

= Tr

ApBp

(18)

If B 0 and C is a general (non-positive) matrix, then CBC has the samenonzero spectrum as the matrix (CC)1/2B(CC)1/2, so the Lieb-Thirring in-equality also implies that in this case

TrCBCp

Tr(CC)pBp (19)Recall (7), and define

Fk = (xkRk)1/2 I (20)

Hk = I Gk (21)

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Then (7) can be rewritten as

( I)(12) = ( F1 FK )

H1 0...

. . ....

0 HK

F1...

FK

= F H F (22)

where F is the 1K block row vector indicated, and H is the KK diagonalblock matrix. Applying (19) gives

Tr

( I)(12)p

Tr

(FF)pHp

(23)

Comparing with (9) shows that

(FF)p = (RR)p I, Hpk = I Gpk (24)

and hence the result follows. QED

As a further comment we note that Shors results about additivity of minimalentropy and Holevo capacity [2] for EB channels can also be derived easily fromLemma 2. Taking the derivative of (11) at p = 1 gives

S

( I)(12) S

(1)

+

Kk=1

xkS(Gk) (25)

Again letting 12 = (I )(12) and using (13) it follows that

S(Gk) = S

(Gk)

(26)

where Gk is a density matrix. Using the definition of minimal entropy

Smin() = inf

S

()

(27)

it follows from (25) that

Smin( ) S

(1)

+Kk=1

xkSmin() Smin() + Smin(), (28)

which immediately implies the additivity of Smin.

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The additivity of Holevo capacity also follows easily from (25). It is con-

venient to first introduce a new quantity, the minimal average entropy of anoutput ensemble from the channel, for a fixed average input state :

Sav(; ) = inf {pk,k}

k

pkS

(k)

:

pkk =

(29)

As Matsumoto et al point out [5], the Holevo capacity of a channel can beexpressed in terms of this average output entropy:

() = sup

S

() Sav(; )

(30)

Lemma 3 Let be an entanglement breaking channel, and let be an arbitrarychannel. Then for any bipartite state 12,

Sav( ; 12) Sav(; 1) + Sav(; 2) (31)

Lemma 3 follows easily from (25), by taking the average input state to be12 =

pk12

(k) and applying the bound to each term in the sum

k

pkS

( )((k)12 )

(32)

Then combining (31) and (30) with the subadditivity bound

S

(12) S

(1)

+ S

(2)

(33)

immediately implies that

( ) () + (), (34)

which establishes the additivity result for .

Acknowledgements This work was partially supported by National Sci-

ence Foundation Grant DMS0101205. Part of this work was completed at aworkshop hosted by the Mathematical Sciences Research Institute, and the au-thor is grateful to the workshop organisers and the Institute for the invitationto participate.

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References

[1] A. S. Holevo, Quantum coding theorems, Russian Math. Surveys, 53,12951331 (1999).

[2] P. Shor, Additivity of the classical capacity of entanglement-breaking chan-nels, Journal of Mathematical Physics, 43, no. 9, 4334 4340 (2002).

[3] G. G. Amosov, A. S. Holevo, and R. F. Werner, On Some Additivity Prob-lems in Quantum Information Theory, Problems in Information Transmis-sion, 36, 305 313 (2000).

[4] E. Lieb and W. Thirring, Inequalities for the Moments of the Eigenvalues

of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities,in Studies in Mathematical Physics, E. Lieb, B. Simon, A. Wightman eds.,pp. 269303 (Princeton University Press, 1976).

[5] K. Matsumoto, T. Shimono and A. Winter, Remarks on additivity of theHolevo channel capacity and of the entanglement of formation, preprintlanl:quant-ph/0206148.

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http://lanl.arxiv.org/abs/quant-ph/0206148http://lanl.arxiv.org/abs/quant-ph/0206148

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