Classical information capacity of a class of quantum channels

M.M. Wolf1 and J. Eisert2,3,4

1 Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany 2 Blackett Laboratory, Imperial College London, Prince Consort Rd, London SW7 2BW, UK 3 Institute for Mathematical Sciences, Imperial College London, Exhibition Rd, London SW7 2BW, UK 4 Institut f ur Physik, Universit at Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany (Dated: September 15, 2005) We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters [0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify several instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.

arXiv:quant-ph/0412133 v3 14 Apr 2005

I.

INTRODUCTION

The study of capacities is at the heart of essentially any quantitative analysis of the capabilities to store or transmit quantum information. This includes the case of transmission of quantum states through noisy channels modeling decohering transmission lines, such as bers or waveguides in quantum optical settings. Capacities and entropic quantities characterizing the specics of a given quantum channel come in several avors: for each resource that is allowed for, one may dene a certain asymptotic rate that can be achieved. A question that is of key interest here and a notoriously difcult one is whether the respective quantities are generally additive. In other words: if we encode quantum information before transmitting it through a quantum channel, can it potentially be an advantage to use entangled inputs over several invocations of the channel? This question is particularly interesting for two central concepts characterizing quantum channels: the minimal output entropy and the classical information capacity. The classical information capacity species the capability of a noisy channel to transmit classical information encoded in quantum states [1, 2]. The question of the classical information capacity is then the one of the asymptotic efciency of sending classical information from sender to receiver, assuming the capability of encoding data in a coherent manner. This capacity is one of the central notions in the study of quantum channels to assess their potential for communication purposes. The minimal output entropy in turn is a measure for the decoherence accompanied with invocations of the channel. It species the minimal entropy of any output that can be achieved by optimizing over all channel inputs [3]. The conjectures on general additivity of both quantities have been linked to each other, in that they are either both true or both false [4, 5, 6]. It is the purpose of this paper to investigate the additivity properties of a class of quantum channels for which the output norm is maximized if the output state is (up to normalization) a projection. For such channels we prove additivity of the minimal output -entropies in the interval [0, 2]. This further exploits an idea going back to Alicki and Fannes in Ref. [7] and Matsumoto and Yura in Ref. [8]. For all weakly covariant instances of the considered channels the additivity is shown to extend to the classical information capacity. Both additivity results are proven for the case of an arbitrary number of different channels. So on the one hand, this paper provides several new instances of channels for which the additivity of the minimal output entropy and the classical information capacity is known. On the other hand, it further substantiates the conjecture that this additivity might be generally true. Finally, following the ideas of Ref. [5], we relate the obtained additivity results to the additivity of the entanglement of formation for instances of bipartite quantum states. We will begin with an introduction of basic notions and related results in Sec. II and the characterization of the considered class of quantum channels in Sec. III.

2
II. PRELIMINARIES

Consider a quantum channel, i.e., a completely positive trace-preserving map T : S ( d ) S ( d ) taken to have input and output Hilbert spaces of dimension d. The minimal output entropy of the channel, measured in terms of the Renyi -entropy [9], is given by (T ) := inf (S T )(),

S () :=

1 log tr [ ] , 1

(1)

0. The -Renyi entropies are generalizations of the von-Neumann entropy dened as S () = tr [ log ], which is obtained in the limit 1. Therefore we consistently dene S1 () := S (). Physically, can be interpreted as a measure of decoherence induced by the channel when acting on pure input states. The minimal output -entropy is said to be additive [10] if for arbitrary N 1 T N = (T ). N (2)

It is known that additivity of does not hold in general for > 4.79 [11]. For smaller values of , however, no counterexample is known so far and in particular in the interval [1, 2], where the function x x becomes operator convex, additivity is conjectured to hold in general. The classical information capacity of a quantum channel, can be inferred from its Holevo capacity [1]. The Holevo capacity of the channel T is dened as
n n

C (T ) := sup S
i=1

pi T (i )
i=1

pi (S T )(i ) ,

(3)

n d2, where the supremum is taken over pure states 1 , ..., n S ( d ) and all probability distributions (p1 , ..., pn ). The classical information capacity is according to the Holevo-SchumacherWestmoreland theorem [1, 2] given by CCl (T ) := lim 1 C T N , N (4)

so as the asymptotic version of the above Holevo capacity. Unfortunately, as such, to evaluate the quantity in Eq. (4) is intractable in practice, being in general an innite-dimensional non-convex optimization problem. However, in instances where one can show that 1 C T N = C (T ), N (5)

for all N , then Eq. (3) already gives the classical information capacity. That is, to know the single-shot quantity in Eq. (3) is then sufcient to characterize the channel with respect to its capability of transmitting classical information. A stronger version of the additivity statements in Eqs. (2, 5) is the one where equality is not only demanded for N instances of the same channel but for N different N channels i=1 Ti . We will refer to this form of additivity as strong additivity. The additivity of the Holevo capacity in the sense of the general validity of Eq. (5) or the additivity of the minimal output entropy is one of the key open problems in the eld of quantum information theory despite a signicant research effort to clarify this issue. In the case = 1 the two additivity statements in Eq. (2) and Eq. (5) were shown to be equivalent in their strong version in the sense that if one is true for all channels (including those with different input and output dimensions), then so is the other [4, 5, 6]. For a number of channels, additivity of the minimal output entropy for = 1 [7, 12, 13, 14, 15, 17, 18, 19] and additivity of the Holevo capacity [16, 17, 18, 19, 20, 21] are known. For integer , the minimal output -entropy is more accessible than for values close to one [22, 23]. Notably, for the case = 2, a number of additivity statements have been derived [24], and the minimal output entropy can be assessed with relaxation methods from global optimization [25]. For covariant

3 channels, one can indeed infer the additivity of the Holevo capacity from the additivity of the minimal output von Neumann entropy [26]. In fact, as we will discuss in Sec.V, a much weaker assumption already sufces for this implication. A paradigmatic and well known representative of the class of channels we consider in this paper is the Werner-Holevo channel [11], which is of the form T () =

d T
d1

(6)

This channel serves as a counter-example for the additivity of the minimal output -entropy for > 4.79. However, for with [1, 2] and for the Holevo capacity additivity have been proven in Refs. [7, 8]. In the following we will generalize these additivity results to a much larger class of channels.
III. CHARACTERIZATION OF THE CLASS OF QUANTUM CHANNELS

We will consider a class of channels with a remarkable property: for this class of quantum channels one can relate the problem of additivity of the minimal output entropy to that of another Renyi- entropy. The rst key observation is the following: Lemma 1 (Basic property) Let T be a quantum channel for which (T ) = (T ), > 0. (7)

Then the additivity of the minimal output -entropy implies the additivity for the minimal output entropy. Proof. This statement follows immediately from the fact S () S () for all S ( 0 [27], and the inequality chain (T ) = (T ) = 1 1 T N = inf (S T N )() N N (8)
d

) and all

1 inf (S T N )(), N

for N . Since on the other hand (T ) T N /N equality has to hold in Eq. (8). Surprisingly, the property required in Eq. (7) does not restrict the channels to the extent that only trivial examples can be found. Quite to the contrary, a fairly large class of channels has this property. A simple example of a class of channels for which condition (7) is satised is the generalization of the Werner-Holevo channel: Example 1 Consider a channel T : S (
d

) S(

) of the form , (9)

T () =

d M ()
d1

where M : S ( d ) S ( d ) is a linear, trace-preserving positive map (not necessarily a channel) which has the property that there exists an input state leading to a pure output state. Then for all > 0 (T ) = log(d 1). (10)

4 Proof. Let us rst note that tr [(d ) ] is convex for any 1 and concave for 0 < 1. Hence, the sought extremum over the convex set of all states is attained at an extreme point, i.e., a pure state. Moreover, all pure states will give the same value. Exploiting this together with the fact that there exists an output under M which is pure and inserting into S () = (log tr [ ])/(1 ) yields Eq. (10). The class of channels in Example 1 has the property that (T ) is independent of and therefore condition (7) is trivially satised. However, it is not yet the most general class of channels for which is constant. In fact, all quantum channels fullling this condition can easily be characterized. This will be the content of the next theorem, which will make use of a Lemma that we state subsequently. The following channels are the ones investigated in this paper: Theorem 1 (Characterization of channels) Let T : S ( the following three statements are equivalent:
d

) S(

) be a quantum channel. Then

1. The minimal output -entropy is independent of . That is, for all > 0 we have (T ) = (T ). 2. The channel is of the form T () =

d mM ()
dm

(11)

where M is a positive, linear and trace-preserving map for which there exists an input state 0 such that mM (0 ) is a projection of rank m. 3. The maximal output norm sup ||T ()|| is attained for an output state being a normalized projection. Proof. 1 2 : Since in general + S () is a non-increasing function for all S ( d ), there exists a state 0 which gives rise to the minimum in for all values of . Then, by Lemma 2, T (0 ) has to be a projection except from normalization. In particular sup ||T ()|| ||T (0 )|| = 1/m0 where m0 := rank(T (0 )). This means that the map M0 : S ( d ) S ( d ) dened as M0 () := 1 d T () m0 (12)

is positive and has the property that M0 (0 ) is except from normalization a projection of rank m = d m0 . Due to the fact that T is trace-preserving, the map M : S ( d ) S ( d ), M () := m0 M0 (), d m0 (13)

is also trace-preserving. Hence, the channel T has indeed a representation of the form claimed above. 2 3 : We want to argue that sup || mM ()|| is attained if R := mM () is a projection. To this end note that R is an element of the convex set C := {r 0 | tr [r] = m, r }, (14)

whose extreme points are projections of rank m. Remember further that the maximum of a convex function (as the largest eigenvalue of a positive matrix) over a closed convex set is attained at an extreme point. When optimizing over the entire set C, the maximum is thus attained for R being a projection of rank m, which is indeed accessible due to the assumed property of M . 3 1 : This follows immediately from + S being a non-increasing function together with the fact that for any normalized projection out , S (out ) = log rank out is independent of .

5 Lemma 2 Let S ( d ) be a state for which S () = S () for some > 0. Then is except from normalization a projection and for all 0 we have S () = log rank(). (15)

Proof. The function + S () is a convex and non-increasing function [27]. Hence, the assumption in the Lemma immediately implies that S () = S () =: c for all , i.e., tr = 2c(1 ) , (16)

for all . Taking the th root on both sides and then the limit leads to 2c = |||| and thus tr /||||
1 = |||| .

(17)

Considering again the limit yields that the multiplicity of the largest eigenvalue of is equal 1 to |||| , such that has indeed to be a normalized projection.

IV.

ADDITIVITY OF THE MINIMAL OUTPUT ENTROPY

For a class of channels of the form in Thm.1 we nd the additivity of the minimal output -Renyi entropy for [0, 2]. We exploit Lemma 1 for these channels in the simple case where = 2 and [0, 2]. What then remains to be shown is the additivity of the minimal output 2-entropy. This can, however, be done in the same way as has been done in Ref. [7] for the specic case M () = T , except that more care has to be taken due to the fact that the involved projections are not necessarily one-dimensional. Theorem 2 (Strong additivity of the minimal output entropy) Consider channels T1 , . . . , TN of the form in Eq. (11) such that N i=1 Mi is a positive map. Then the minimal output -entropy is strongly additive for all [0, 2], i.e.,
N N N

i=1

Ti

=
i=1

Ti

=
i=1

log(di mi )

(18)

for Ti : S (

di

) S(

di

) as in Eq. (11).

Proof. We can express with Ti () = channel T := N i=1 Ti as

N

di mi Mi () /(di mi ) the action of the tensor product

( C )
{1,...,N } k

T () =

1 d mi i=1 j

(mk ),

(19)

6 where C denotes the complement of , := (M1 ... MN )(), and denotes the reduced density matrix of with respect to the systems labeled with . Hence, we obtain tr T ()
2

1 ( d mi )2 i i=1 1 ( d mi )2 i i=1
j C \\ N N

, {1,...,N } k l 2 tr {1,...,N }

2 (mk )(ml )tr

dk
k( )C

(mk )
C C \ k

(ml )
l

dj

1 ( d mi )2 i i=1

2 tr {1,...,N } k

m2 k
j C

(dj 2mj ).
1 m i and thus

2 Now, exploiting the subsequently stated Lemma 3, we have tr 2

tr

T ()

1 . d mi i i=1

(20)

Together with the fact that 2 (Ti ) = log(di mi ) this means nally that we obtain
N N

2 (T )
i=1

log(di mi ) =
i=1

2 (Ti ) 2 (T ),

(21)

implying by Lemma 1 the claimed additivity in the entire interval [0, 2]. Lemma 3 Let Mi : S (Cdi ) S (Cdi ), i = 1, . . . , N be trace preserving linear maps, for which there exist positive numbers mi N such that di tr [] mi Mi () is completely positive. If in addition N i=1 Mi is a positive map, then 2
N N

S (

i di

Proof. Let Mi be the adjoint map dened by tr [Mi (A)B ] = tr [AMi (B )]. Then the complete positivity condition is equivalent to the validity of Mi di (P12 )
2

) : tr

Mi ()

i=1

1 m i .

(22)

i=1

di
mi

tr1 P12

(23)

for all positive operators P12 S ( di ). In order to apply this inequality we exploit some of the properties of the ip operator Fd : | | | | for | , | d . Recall that tr A2 =

7
2 tr [(A A)Fd ] and FT d =

d i,j =1

|i, i j, j |. Hence,
i

tr

Mi ()

i=1

= tr = tr

Mi ()
i T

Mi di (Fdi )
2 Mi di (FT di )

(24)

Mi () Mi ()
i

(25)

i T j

tr =
j

mj 1

(26) (27)

mj 1 .

Lemma 3 and therefore Thm. 2 require the assumption that i Mi is a positive map. Although the presented proof depends on this property, we do at present not know of any channel of the form in Eq. (11) for which Eq. (22) is not valid. In fact, all the following examples are such that Mi = i , where each i is completely positive and is the transposition. For all these cases i Mi is evidently positive. Obviously, Thm. 2 implies in particular that for any channel T : S ( d ) S ( d ) of the considered form we have for all [0, 2] 1 T N = (T ). N (28)

As mentioned earlier the most prominent example of channels in the considered class is the WernerHolevo channel itself for which M () = T . For this channel, the additivity of the minimal output entropy has been shown in Ref. [8], and with inequivalent methods in Refs. [12] and [28]. The following list includes further instances of channels for which we nd additivity of the minimal output entropy as a consequence of Thm. 2. As stated above all examples are such that the corresponding M is a concatenation of a completely positive map and the transposition. Example 2 (Stretching) For being a pure state consider M () = T + (1 ) , m=1. (29)

Complete positivity is a consequence of this channel being a convex combination of the completely positive Werner-Holevo channel and the channel (d )/(d 1). Obviously, 0 = T leads to a normalized projection at the output. Example 3 (Weyl shifts) Consider the set of unitaries Wi = M () = 1 d
d d j =1

|j + i mod d j | and take (30)

Wi T Wi ,
i=1

m=1.

Complete positivity of the respective channel T follows from the fact that it is a composition of the Werner-Holevo channel with another completely positive map. The state 0 with i|0 |j = 1/d for all i, j = 1, ..., d is an example for an appropriate pure input state for which M (0 ) = 0 . Example 4 (Pinching) Let {Pi } be a set of orthogonal projections yielding a resolution of the identity, i.e., i Pi = d . Then take M () =
i

Pi T Pi ,

m=1.

(31)

8 Again the respective channel T is a composition of two completely positive maps and thus itself completely positive. Moreover, any pure state 0 for which T 0 is in the support of any Pi gives rise to a normalized projection at the output of T . So far the examples were restricted to the case m = 1. The following examples show explicitly that all larger values of m are possible as well: Example 5 (Casimir channel for a reducible representation) This example is based on a Casimir channel T : S ( 4 ) S ( 4 ) (see Section V) for a reducible representation of SU (2),
3

T () =
i=1

Ai A i,

(32)

where Ai = (4/3)1/2 (Ji ), with (J1 ) = i (|2 3| + |4 1| |1 4| |3 2|) , 2 i (J2 ) = (|3 1| + |4 2| |1 3| |2 4|) , 2 i (J3 ) = (|1 2| + |4 3| |2 1| |3 4|) . 2 (33) (34) (35)

The operators (J1 ), (J2 ), (J3 ) form generators of a four-dimensional reducible representation of the Lie algebra of the group SU (2). As an example for m = 2, consider the channel T () = 3T () + . 4 (36)

This map is clearly completely positive by construction. We nd M to be given by M () = 4 /2 T (). (37)

An appropriate input 0 for which the output is a two-dimensional projection M (0 ) = (|3 3| + |4 4|)/2 up to normalization is given by 0 = (|1 1| + i|1 4| i|4 1| + |4 4|) /2. (38)

Finally, M is a positive map, as it can actually be written as a transposition , followed by a completely positive map , that is, M = . To show that this is indeed the case, consider (M id)(T1 ) =

4
2

4
4

1 3 (T id)(T1 ) T1 0, 4 4
1 2 4 i=1

(39)

where is the maximally entangled state with state vector | =

|i, i .

Example 6 (Shifts and pinching) Let Wk be dened as in Example 3 and K {1, . . . , d}: 1 M () = |K |
d |i i| Wk Wk |i i| , kK i=1

m = |K | .

(40)

In fact, T is an entanglement-breaking channel (cf. [13, 20]) which can be written as T () = 1 d |K |

d

i||i
i=1 k{1,...,d}\K

Wk |i i|Wk .

(41)

9 Example 7 (Coarse graining) For M () =

U (D ) d

= dU

, consider
n

U T
i=1 i=1

m=D,

(42)

where the integration is with respect to the Haar measure. The averaging operation in M may physically be interpreted as a coarse graining of an operation which is only capable of resolving n blocks of size D within a d = n D dimensional system. In order to prove that the above M leads to an admissible and for n > 1 not entanglement-breaking channel, let us rst note that we may after a suitable reshufe equivalently write M () = dU

n U T n U

= T n

D
D

(43)

T where the tensor product is that of d = n D and T n is the reduction of with respect to the n rst tensor factor . Obviously, M is positive, trace-preserving and for 0 with i|0 |j = 1/d we obtain a normalized projection of rank D. Complete positivity of T is equivalent to

(T id)() 0,
d

(44)

1 where | = i=1 |i, i is again the state vector of a maximally entangled state . Exploiting again d that the latter is related to the ip operator F|i, j = |j, i via partial transposition, i.e., T2 = F/d, we obtain

(T id)() =

d2

1 Fn D2 d d

/(d D) ,

(45)

where Fn is the ip operator on n n . Since the latter has eigenvalues 1, the channel dened as above is indeed completely positive. In order to prove that T is not entanglement breaking it is sufcient to show that the partial transpose of Eq. (45) is no longer positive, which is true since the negative term picks up an additional factor n. Finally, additivity of the minimal output entropy holds for any channel for which there exists a pure output state, leading to a vanishing output entropy. In this case additivity of the minimal output entropy in the form of Eq. (2) is evident. However, strong additivity within the considered class of channels is still a non-trivial result. This applies in particular to instances of the 3-and 4-state channels of Ref. [29] and the class of so-called diagonal channels, for which strong additivity was proven recently in Ref. [30]: Example 8 (Diagonal channels) Consider T : S (
K d

) S(

) with (46)

T () =
k=1

Ak A k,

where Ak , k = 1, ..., K , are all diagonal in a distinguished basis.

V. CLASSICAL INFORMATION CAPACITY

So far we have considered the minimal output entropy of quantum channels and their additivity properties. It turns out that for a large subset of the considered channels, including all the discussed Examples 3-8, one can indeed infer the additivity of the Holevo capacity as well. On the one hand, for each covariant instance of a quantum channel from which we know that the minimal output entropy is additive, we can conclude that the Holevo capacity is also additive [26]. For example, this argument

10 applies to the Werner-Holevo channel itself. One the other hand, a quantum channel does not necessarily have to be covariant for a very similar argument to be valid. Subsequently, we will restate the result of Ref. [26] using weaker assumptions. The main difference is that for a given channel, one may exploit properties of the state for which the output entropy is minimal. This is particularly useful in our case at hand, where these optimal input states can always be identied in a straightforward manner. We will rst state the modied proposition in a general way, and then apply it to the channels at hand of the form as in Thm. 1. Theorem 3 (Strong additivity for the classical information capacity) Let T : S ( d ) S ( d ) be a quantum channel for which the minimal output von-Neumann entropy is additive, and let {i } be a set of input states for which the minimal output entropy is achieved. If for any probability distribution {pi } and := i pi i we have that (S T )() = sup(S T )()

(47)

holds, then the Holevo capacity C (T ) is additive and the classical information capacity is given by CCl (T ) = (S T )() 1 (T ). (48)

Moreover, if the assumptions are satised by an arbitrary number of different channels {Tk } among which we have strong additivity of the minimal output entropy, then C k C (Tk ). k Tk = Proof. Let us rst consider the Holevo capacity of a single channel. Obviously, C (T ) is always upper bounded by the maximal minus the minimal output entropy. Due to the assumed properties of the set {i } this bound is, however, saturated and we have In other words the supremum in C (T ) can be calculated separately for the positive and the negative part. Now consider the expression C k Tk . If we again separate the two suprema, then by the assumed strong additivity the maximum of the negative part is attained for product inputs. The same is true for the positive part, since the entropy satises the sub-additivity inequality S (AB ) S (A ) + S (B ). Hence, by evaluating the suprema separately we obtain an upper bound which coincides with the sum of the achievable upper bounds for the single channels. In practice, one is often in the position to have a channel which is weakly covariant on an input state 0 which minimizes the output entropy. That is, there are unitary (not necessarily irreducible) representations and of a compact Lie group or a nite group G such that for all g G T (g )0 (g ) = (g )T (0 )(g ) ; (50) C (T ) = sup S pj T (j )
j j

pj (S T )(j ) = (S T )() 1 (T ).

(49)

in addition the image of the group average of 0 under T is the maximally mixed state. That is, in case of a nite group 1 |G| (g )T (0 )(g ) =
g G

d
d

(51)

where we have to replace the sum by an integral with respect to the Haar measure if G is a compact Lie group. The optimal set of states {j } in Thm. 3 is then taken to be the set of equally distributed states { (g )0 (g ) } (i.e., pg = |G|1 for all g G for a nite group). In fact, the discussed Examples 3-8 are of this weakly covariant form.

11 Obviously, quantum channels which are covariant with respect to an irreducible representation of a compact Lie group always have the required properties. For instance for the d-dimensional WernerHolevo channel, one may take for the group G = SU (d), the dening representation , and the conjugate representation . Note, however, that the property of the channel required by Thm.3 is signicantly weaker than covariance. To construct new instances of quantum channels for which the additivity of the classical information capacity is found, let us consider the above mentioned examples. To start with Example 3, we know that the state 0 with elements i|0 |j = 1/d for i, j = 1, ..., d is an optimal input. To construct an appropriate group G, consider the set of unitaries, Uj := j = 1, ..., d. It is straightforward to show that
T (Uj 0 Uj ) = Uj T (0 )Uj , d 1 l=0

2ilj d

|l l|,

(52)

(53) (54)

1 d

d Uj T (0 )Uj = j =1

d
d

That is, by virtue of Thm. 3 the channel in Example 3 has a classical information capacity of CCl (T ) = log(d) log(d 1). (55)

Example 4 can be treated in a similar fashion. Let us choose the basis in which the projections are diagonal, and take 0 = |1 1|. Obviously, we have that T (Wi 0 Wi ) = Wi T (0 )Wi , i = 1, ..., d, 1 d
d

(56) (57)

Wi T (0 )Wi =
i=1

d
d

where the Wi are again the unitary shift operators, again forming an appropriate nite group G. The classical information capacity is given by CCl (T ) = log(d) log(d 1). Note that the same argument using shift operators, leading to a classical information capacity of CCl (T ) = log(d), can be applied to the class of diagonal channels of Example 8. This result of a maximal classical information capacity is no surprise, however, as one can encode classical information in a way such that information transmission through the channel is entirely lossless. Then, Example 5 is another example of a channel with additive Holevo capacity. This becomes manifest as a consequence of the fact that every Casimir channel [31] based on some representation of SU (2) is covariant under the respective representation. Such Casimir channels are convenient building blocks to construct a large number of channels with additive Holevo capacity. So let us consider for G = SU (2) a d-dimensional representation of G [32]. The generators of the associated Lie algebra are denoted with Jk , k = 1, 2, 3. In a mild abuse of notation, we will denote with (Jk ) the generators of the Lie algebra of the group SU (2) in the representation . The respective Casimir channel is given by T () = 1
3

(Jk ) (Jk ).
k=1

(58)

where normalization follows from the Casimir operator

3

(Jk )2 = d .

(59)

k=1

12 For irreducible representations of SU (2) we have that = (d 1)(d + 1)/4. The covariance of the resulting quantum channels can be immediately deduced from the structural constants of the Lie algebra specied as [Ji , Jj ] = ii,j,k Jk , i, j, k {1, 2, 3}. (60)

by making use of the exponential mapping into the group SU (2). Casimir channels T : S ( d ) S ( d ) with respect to a d-dimensional representation as in Eq. (58) are covariant in the sense that T ( (g ) (g ) ) = (g )T ()(g ) (61)

for all states , where is either the dening or the conjugate representation of SU (2). For d = 3, for example, we reobtain the Werner-Holevo channel. Then, in Example 5 as an example of a Casimir channel with respect to a reducible representation we nd that the channel is covariant with respect to this reducible representation. This channel is covariant with respect to the chosen representation of SU (2). Moreover, we may start from the optimal input state 0 as specied in the example, leading to an output T (0 ) = (|3 3| + |4 4|)/2. We can generate then an ensemble of states that averages to the maximally mixed state, assuming the Haar measure. That is, we have that dg (g )T (0 ) (g ) =
gSU (2)

4
4

(62)

To be very specic, with Ux := exp(ix2 (J2 )) exp(ix1 (J1 )) exp(ix3 (J3 )), x = (x1 , x2 , x3 ) 3, this average amounts to
4 2

dx1
0 0

dx2
0

dx3

sin(x2 ) 4 Ux T (0 )Ux = . 16 2 4

(63)

Therefore, we again conclude that the classical information capacity is given by CCl (T ) = log(4) log(2) = 1. In a similar way, the above coarse graining channel can be shown to exhibit an additive Holevo capacity. Here, M () can be written as in Eq. (42). Therefore, the reducible representation of SU (n) corresponding to V D , V SU (n) (64)

can be taken as the group appropriately twirling the output resulting from the optimal input. This argument leads to an additive Holevo capacity such that the classical information capacity becomes CCl (T ) = log(d) log(d D). (65)

These examples give substance to the observation that quite many channels of the above type can be identied for which the classical information capacity can be evaluated. At this point, indeed, one may be tempted to think that all of the above channels have an additive Holevo capacity. While we cannot ultimately exclude this option, it is not true that Thm. 3 can be applied to all channels of the form as in Thm. 1. A simple counterexample is provided by Example 2, where only a single optimal input state exists, namely = T , such that Thm. 3 cannot be applied.
VI. NOTE ON THE ENTANGLEMENT COST OF CONCOMINANT BI-PARTITE STATES

Finally, we remark on the implications of the results for the additivity of the entanglement of formation. In Ref. [5], the additivity of weakly covariant channels has been directly related to the additivity of the entanglement of formation [33]
n

EF () = inf
i=1

pi (S trB )(i )

(66)

13 where the inmum is taken over all ensembles such that turn, is the asymptotic version, EC () = lim
N n i=1

pi i = . The entanglement cost, in

1 EF (N ). N

(67)

This entanglement cost quanties the required maximally entangled resources to prepare an entangled state: it is the rate at which maximally entangled states are asymptotically necessary in order to prepare a bi-partite state using only local operations and classical communication. In contrast to the asymptotic version of the relative entropy of entanglement [34], which is known to be different from the relative entropy of entanglement, for the entanglement of formation no counterexample for additivity is known. Moreover, additivity of the entanglement of formation for all bi-partite states has been shown to be equivalent to the strong additivity of the minimal output entropy and that of the Holevo capacity [4]. For the channels considered above, the construction in Ref. [5] can readily be applied, yielding further examples of states for which the entanglement cost is known, beyond the examples in Refs. [5, 8, 35]. The construction is as follows: from the quantum channel T : S ( d ) S ( d ) one constructs a Stinespring dilation, via an isometry U : d d K for appropriate K . For any bi-partite state S ( d K ) with carrier on K := U d which achieves C (T ) = (S tr1 )() EF () we know that EC () = EF () = 1 (T ). (69) (68)

The following state is an example of a state with known entanglement cost constructed in this manner. Example 9 (State with additive entanglement of formation) Let the state vectors from K 4 be dened as K = span(|1 , ..., |4 ), with |1 |2 |3 |4 = = = = i(|1, 4 + |2, 3 |3, 2 ) + |4, 1 /2 i(|1, 3 + |2, 4 + |3, 1 ) + |4, 2 /2 i(|1, 2 |2, 1 + |3, 4 ) + |4, 3 /2 i(|1, 1 |2, 2 |3, 3 ) + |4, 4 /2.
4

(70) (71) (72) (73)

Then EC () = EF () = 1, where = (|1 1 | + ... + |4 4 |)/4. (74)

In just the same fashion, a large number of examples with known entanglement cost can be constructed from the above quantum channels.
VII. SUMMARY AND CONCLUSIONS

In this paper, we investigated a class of quantum channels for which the norm of the output state is maximized for an output being a normalized projection, with respect to their additivity properties. We introduced three equivalent characterizations of this class of quantum channels. For all channels of this type, which satisfy an additional (presumably weak) positivity condition, one can infer the additivity of the minimal output von Neumann entropy from the respective additivity in case of the 2-entropy. Several examples of channels of this type were discussed in quite some detail, showing that a surprisingly large number of quantum channels is included in the considered class. Finally, we investigated instances of this class of quantum channels with a weak covariance property, relating the minimal output entropy to both the classical information capacity. This construction gives indeed rise to a large class of channels with a known classical information capacity.

14
VIII. ACKNOWLEDGEMENTS

We thank M.B. Ruskai and A.S. Holevo for valuable comments. One of us (JE) would like to thank David Gro for interesting discussions. This work was supported by the DFG (SPP 1078), the European Commission (QUPRODIS IST-2001-38877), and the European Research Councils. This work beneted from discussions during an A2 meeting, funded by the DFG (SPP 1078).

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[26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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