Simplifying additivity problems using direct sum constructions
Simplifying additivity problems using direct sum constructions
Motohisa Fukuda1, Michael M. Wolf2
1Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge
2 Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany.
(Dated: August 20, 2021)
We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy
and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds
for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to
imply strong additivity for any convex entanglement monotone. The implications are obtained by considering
direct sums of channels (or states) for which we show how to obtain several information theoretic quantities
from their values on the summands. This provides a simple and general tool for lifting additivity results.
I. INTRODUCTION
A central question in classical and quantum information
theory is, how much information can be transmitted through
a given noisy channel. For classical channels the maximal
asymptotically achievable rate—the capacity—was derived in
the seminal work of Shannon [1]. For quantum channels,
however, the matter is complicated by the existence of entan-
glement and the possibility of exploiting it in the encoding to
protect information against decoherence. If one excludes this
possibility, a capacity formula for the transmission of clas-
sical information through quantum channels was proven by
Holevo [2] and Schumacher and Westmoreland [3] (HSW).
Since then, considerable effort was devoted to the question
whether (or in which cases) entangled inputs can lead to rates
beyond the HSW capacity. This issue—the additivity problem
for the HSW capacity—is still undecided, although for sev-
eral classes of channels additivity has been shown to be true,
i.e., entanglement does not seem to help in any case (see, e.g.,
[4, 5, 6, 7] and references therein). Instead, other additiv-
ity problems appeared which are similar in spirit but concern
very different quantities like the minimal output entropy and
the entanglement of formation, an entanglement measure for
bipartite states for which in addition strong super-additivity
has been conjectured.
A major conceptional insight was then gained in [8, 9, 10,
11] where it was shown that all these additivity problems are
globally equivalent in the sense that if additivity holds for one
of these quantities in general, then it does so for all of them.
Here ‘in general’ means that it has to be true for arbitrary pairs
of channels (or states), a condition we will call strong additiv-
In this work we present a further conceptional simplifica-
tion of these and related additivity problems. We show that
strong additivity is implied by weak additivity, meaning ad-
ditivity for arbitrary pairs of equal channels or states. More-
over, based on [12] we argue that it suffices to consider pairs
of identical unital channels only. This observation may be
a small step on a notorious path but it might guide future
research as it for instance underlines recent attempts to un-
derstand the asymptotic structure of tensor powers of unital
channels [13]. Moreover, one may think of other additivity
questions than the ones stated above for which our techniques
could be of use. In particular, we think of regularized quanti-
ties (like quantum capacities (cf. [14, 15]) or certain entangle-
ment measures) for which weak additivity holds by definition.
Our main tool is the use of direct sums of channels or
states. For the latter case similar constructions appeared in
[16, 17, 18]. We begin with a discussion of direct sum chan-
nels. This will contain more than what is needed for the sub-
sequent additivity results as we think that these tools might be
of independent interest.
II. DIRECT SUMS OF QUANTUM CHANNELS
We consider direct sums of channels, i.e., completely posi-
tive and trace preserving maps of the form T = ⊕iTi, where
each Ti is a channel in its own right. Our aim in this section is
to express information theoretic functionals of T in terms of
their values for the Ti’s. The definition of the quantities ap-
pearing in the following proposition will be given in the proof.
Proposition 1 (Direct sums) Consider a direct sum T =
⊕ni=1Ti, n ∈ N of arbitrary finite dimensional channels. Then
1. Minimal output α-Renyi entropy (α ≥ 1):
Smin,α
= min
Smin,α(Ti)
, (1)
2. Coherent information:
= max
J(Ti)
, (2)
3. Mutual information:
= max
S({λi}) +
λiI(Ti), (3)
= log
2I(Ti), (4)
where {λi} is a probability distribution and S({λi}) its
entropy.
4. HSW capacity:
= max
S({λi}) +
λiχ(Ti) (5)
= log
2χ(Ti). (6)
http://arxiv.org/abs/0704.1092v2
Remark: Let us briefly comment on the interpretation of
the above formulas. Concerning the HSW capacity, classical
information can either be sent through the channelsTi or it can
be encoded in the choice of blocks i = 1, . . . , n. Eq.(5) shows
exactly the competition between these two ways of communi-
cating classical information. For the quantum mutual infor-
mation, which gives the entanglement assisted capacity [19],
we obtain the same interpretation (note that the ’2’ comes
from the fact that we take log in base 2). The coherent infor-
mation is related (via regularization) to the quantum capacity
[20]. In this case encoding information in the choice of blocks
is not possible—this would be purely classical as all the co-
herences get lost. Similarly, for the minimal output entropies
the minimum is obtained by putting all the weight into the
least noisy channel.
Proof. 1. The α-Renyi entropy is defined as
Sα(ρ) =
log tr[ρα] =
log ‖ρ‖α, (7)
for 0 ≤ α ≤ ∞. Here ‖ · ‖p is the Schatten p-norm.
When α = 1 the functional is defined by its limit which is
the minimal output entropy Smin(T ) = infρ S(T (ρ)) with
S(ρ) = −trρ log ρ the von Neumann entropy. Let us consider
this case first.
As the direct sum ⊕iTi erases the off-diagonal blocks so
that all possible outputs can be obtained upon block-diagonal
inputs we can restrict to ρ = ⊕iρ̃i. Here ρ̃i is not necessarily
normalized so that the weights tr[ρ̃i] =: λi form a probability
distribution. Writing ρi := ρ̃i/λi and using the concavity of
von Neumann entropy we get
⊕i Ti(λiρi)
λiS(Ti(ρi)). (8)
This leads to Eq.(1) when α = 1. For α > 1 the minimization
of Sα(T (ρ)) amounts to a maximization of ‖T (ρ)‖α and the
result follows from convexity of ‖ · ‖α in a similar way.
2. The coherent information is defined as
J(T ) = sup
T (ρ)
T ⊗ id(Ψ)
, (9)
where Ψ is a purification of ρ such that ρ = trBΨ. Since
T and T ⊗ id erase the off-diagonal blocks we can replace ρ
and Ψ by their diagonal blocks: ⊕iλiρi and ⊕iλiΨi. Here,
Ψi is an extension of ρi. Since the conditional entropy
S(ρAB)−S(ρA) is concave in ρAB [21] considering a convex
decomposition of each Ψi into pure states shows Eq.(2) in a
similar way as above.
3. The mutual information defined as
I(T ) = sup
S(ρ) + S
T (ρ)
T ⊗ id(Ψ)
is concave in ρ so the maximum will be achieved by a block
diagonal ρ = ⊕iλiρi for T = ⊕iTi. To see this, let V =
⊕j exp{2πij/n}Ij and average V kρV ∗k over k = 1, . . . , n.
Take a purification Ψ of ρ = ⊕iλiρi, and then replace Ψ by
its diagonal blocks: ⊕iλiΨi as before. However, each Ψi
is a purification of ρi in this case. Indeed, suppose ρi =
pij |ij〉〈ij|, where {|ij〉}j is an orthonormal basis in the
ith subspace. Then, Ψ is
λiλkpijpkl|ij〉〈kl| ⊗ |ij〉〈kl|,
and its ith diagonal block λiΨi is
pijpil|ij〉〈il| ⊗ |ij〉〈il| = λi|Ψi〉〈Ψi|.
Here, |Ψi〉 =
pij |ij〉 ⊗ |ij〉. Exploiting this together
with S(λρ) = λ(S(ρ) − logλ) then gives Eq.(3). Eq.(4) fol-
lows then from determining the optimal λi via Lagrange mul-
tipliers in the following way. The maximization problem of
S({λi})+
λici for a probability distribution {λi} amounts
then to maximizing
S({λi}) +
λici + Λ
λi − 1
, (11)
where Λ is the Lagrange multiplier. Taking partial derivatives
we obtain for extremal {λ̃i}:
− log λ̃i −
+ ci + Λ = 0 ∀i (12)
λ̃i − 1 = 0. (13)
Hence (12) shows ci − log λ̃i is a constant, say, C for ∀i, and
by (13) we get C = log
ci . Therefore
S({λ̃i}) +
λ̃ici =
λ̃iC = log
2ci . (14)
As this is lower bounded by mini[I(Ti)] it must be the maxi-
4. The HSW capacity χ is given by
χ(T ) = sup
S(T (ρ))−HT (ρ), (15)
where HT (ρ) = inf
pkρk=ρ
pkS(T (ρk)). (16)
Here, HT (ρ) is the convex closure of the output entropy; {pk}
is a probability distribution and ρk are density matrices. The
r.h.s. of Eq.(15) for a fixed average input state ρ is a constraint
HSW capacity which we will denote by χ(T, ρ). Since the
inputs can again be assumed to be block-diagonal we have:
χ (⊕iTi,⊕iλiρi) = S(⊕iλiTi(ρi))−
λiHTi(ρi)
= S({λi}) +
λiχ(Ti, ρi). (17)
The first equality is explained by the fact that since the von
Neumann entropy is concave there is an optimal decompo-
sition of ⊕iλiρi for which each state has its support in one
of the diagonal blocks. The second equality comes from
S(λiρi) = λi(S(ρi) − logλi). Taking the supremum over
all states {ρi} then leads to Eq.(5). Again, Eq.(6) is obtained
by using Lagrange multipliers as above. In fact, for unital
channels (6) has been obtained in [30].
III. SIMPLIFYING ADDITIVITY PROBLEMS
Let us now turn to the additivity conjectures and exploit
Prop.1 in order to show that in several cases weak additivity
(for equal channels or states) implies strong additivity (i.e., for
different ones).
Proposition 2 (Reduction for channels) The following (in-)
equalities hold for arbitrary pairs of different channels T1 and
T2 iff they hold for arbitrary equal pairs T1 = T2.
1. Smin,α(T1 ⊗ T2) = Smin,α(T1) + Smin,α(T2), for any
α ≥ 1.
2. χ(T1 ⊗ T2) = χ(T1) + χ(T2).
3. HT1⊗T2(ρ) ≥ HT1(ρ1) +HT2(ρ2) for all states ρ with
respective subsystems ρ1, ρ2.
4. HT1⊗T2(ρ1⊗ρ2) = HT1(ρ1)+HT2(ρ2) for all product
states ρ1 ⊗ ρ2.
Remark: The conjectured equality in 1. is the additiv-
ity of the minimal output entropy when α = 1 [22], and
it becomes the multiplicativity of maximal output p-norms
for p = α > 1. This was conjectured to be true for all
α ∈ [1,+∞] before a counterexample was found [23] ruling
out all values α > 4.79. The equation in 2. is the conjec-
tured additivity of the HSW capacity, which gives the clas-
sical capacity as long as entangled states are not allowed to
be used in the encoding [2, 3]. The additivity would show
that the HSW capacity itself is the unconstrained classical ca-
pacity of quantum channels. The conjectures 3. and 4. are
called strong superadditivity and additivity of the convex clo-
sure of the output entropy. When T1, T2 are partial traces they
become strong superadditivity and additivity of entanglement
of formation, respectively, which we discuss in greater detail
below.
We note that Prop.2 remains valid in the case where ‘arbi-
trary channels’ refers to a restricted set of channels which is
closed under direct sums and tensor products.
Proof. 1. Let σ1 and σ2 be optimal output states for T1 and
T2 respectively. Then, form the following two channels:
T ′1(ρ) = T1(ρ)⊗ σ2, T ′2(ρ) = σ1 ⊗ T2(ρ). (18)
It is not difficult to see that T1 ⊗ T2 and T ′1 ⊗ T ′2 share the
additivity property. Hence we can assume that T1 and T2 have
the same optimal output: Smin,α(T1) = Smin,α(T2). If we
apply first weak additivity and then Prop. 1.1. we obtain:
Smin,α(((T1 ⊕ T2)⊗ (T1 ⊕ T2))) (19)
= 2Smin,α(T1 ⊕ T2) = Smin,α(T1) + Smin,α(T2). (20)
On the other hand, if we first apply Prop. 1.1. and then weak
additivity, we obtain that (19) and thus (20) is upper bounded
by Smin,α(T1 ⊗ T2). The converse inequality is trivial.
2. Consider
T1 ⊕ T2
T1 ⊕ T2
= 2 log
2χ(T1) + 2χ(T2)
= log
22χ(T1) + 22χ(T2) + 2χ(T1)+χ(T2)+1
This follows from first applying weak additivity and then the
proposition 1.4. On the other hand, applying them in reverse
order we have
T1 ⊕ T2
= log
2χ(T1⊗T1) + 2χ(T2⊗T2) + 2 · 2χ(T1⊗T2)
= log
22χ(T1) + 22χ(T2) + 2χ(T1⊗T2)+1
Together they prove the claimed equality.
For 3. we obtain by weak superadditivity,
HT1⊗T2(ρ) = H(T1⊕T2)⊗(T1⊕T2)(0⊕ ρ⊕ 0⊕ 0)
≥ HT1⊕T2(ρ1 ⊕ 0) +HT1⊕T2(0⊕ ρ2)
= HT1(ρ1) +HT2(ρ2). (21)
Here, ρ1, ρ2 are reduced states of ρ. This proves 3. and the
statement 4. follows in a similar way when replacing ρ by a
product state.
Proposition 3 (Unital channels) Proving one of the conjec-
tures in proposition 2 for all pairs of identical unital channels
would show the conjecture is true for arbitrary channels.
Proof. In [12] a unital channel T̃ is constructed for a given
channel T so that these two channels T̃ and T share the fol-
lowing additivity properties: additivity of minimal output α-
Renyi entropy, and strong superadditivity and additivity of the
convex closure of the output entropy. Hence these conjectures
can be restricted to products T̃1 ⊗ T̃2 for all channels T1, T2.
As for the HSW, we have the same reduction but for a differ-
ent reason (See the remark below). Finally, for the above two
unital channels T̃1, T̃2 we can construct the direct sum T̃1⊕ T̃2
which is again a unital channel. Then the result follows from
the proof of proposition 2.
Remark: We explain local relation between minimal out-
put entropy and the HSW capacity, which was implicitly writ-
ten but not clear in [12]. Since the unital extension T̃ sort of
mixes up outputs of T we have the following formula.
χ(T̃1 ⊗ T̃2) = log d1d2 − Smin(T̃1 ⊗ T̃2), (22)
where d1, d2 are the dimensions of the output spaces of T̃1
and T̃2 respectively. Hence the additivity of HSW capacity
is equivalent to the additivity of the minimal output entropy
for products of those extensions T̃1 ⊗ T̃2 by Eq.(22). Hence
the additivity conjecture of the HSW capacity can also be re-
stricted to products T̃1 ⊗ T̃2 for all channels T1, T2 by using
global equivalence [8, 9, 10, 11].
Finally, we will discuss additivity issues for entanglement
measures. The one already mentioned is the entanglement of
formation which was introduced in [24]. Since then the fol-
lowing conjectures have been considered:
EF (ρ) ≥ EF (ρ1) + EF (ρ2) (23)
EF (ρ1 ⊗ ρ2) = EF (ρ1) + EF (ρ2). (24)
In fact, both are again globally equivalent to the additivity of
the HSW capacity and the minimal output entropy. Moreover,
additivity would imply that EF equals an important opera-
tionally defined entanglement measure, the entanglement cost
Ec, since Ec(ρ) = limn→∞ EF (ρ
⊗n) [25].
The entanglement of formationEF (ρ) is the convex closure
of output entropy HT (ρ) when T is a partial trace.
Following a similar strategy as above we will now show
that strong additivity in the sense of Eq.(24) is again implied
by weak additivity (i.e., Eq.(24) with ρ1 = ρ2). In fact, this
will not only hold for EF but for any convex entanglement
monotone [24, 26, 27]. The main reason behind is that every
such functional satisfies [28]:
f(⊕iλiρi) =
λif(ρi), (25)
where {λi} is a probability distribution and ρi are states as
before.
Proposition 4 (Convex entanglement monotones) [29]
Suppose f is a convex entanglement monotone which is
weakly additive, i.e., f(ρ1 ⊗ ρ2) = f(ρ1) + f(ρ2) for all
ρ1 = ρ2. Then f is strongly additive in the sense that this
holds also for all ρ1 6= ρ2.
Proof. Let ρ = 1
(ρ1 ⊕ ρ2). Then
f(ρ⊗ ρ) = 2f(ρ) = f(ρ1) + f(ρ2). (26)
Here, we applied the weak additivity and then (25). Applying
them in reverse order we get
f(ρ⊗ ρ) =
i,j=1
f(ρi ⊗ ρj)
(f(ρ1) + f(ρ2) + f(ρ1 ⊗ ρ2)). (27)
Using similar ideas, it has recently been shown that for reg-
ularized entanglement measures like Ec or the asymptotic rel-
ative entropy of entanglement, monotonicity (i.e., essentially
Eq.(25)) and strong additivity are equivalent [18].
Acknowledgement M.F. would like to thank his supervisor
Y.M.Suhov for constant encouragement and numerous discus-
sions. M.W. thanks K.G. Vollbrecht for discussions and J.I.
Cirac for support. Both authors thank M. B. Ruskai for bring-
ing [30] to their attention.
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