International Journal of Quantum Information
c World Scientific Publishing Company
CALCULATING A MAXIMIZER FOR QUANTUM MUTUAL
INFORMATION
T.C. DORLAS∗ and C. MORGAN†
Dublin Institute for Advanced Studies,
School of Theoretical Physics,
10 Burlington Road, Dublin 4,
Ireland.
Received Day Month Year
Revised Day Month Year
We obtain a maximizer for the quantum mutual information for classical information
sent over the quantum amplitude damping and depolarizing channels. This is achieved
by limiting the ensemble of input states to antipodal states, in the calculation of the
product state capacity for the channels. We also use these results to demonstrate that
the product state capacity of a convex combination of two channels is not given by the
minimum of their respective capacities.
Keywords: product state capacity; maximizing ensemble; memoryless channels.
1. Introduction
Holevo 1 has introduced a measure of the amount of classical information remaining
in a state that has been sent over a noisy quantum channel. The capacity of a
channel is given by the maximization of this Holevo quantity over an ensemble
of input states, and can be interpreted as the amount of information that can be
sent reliably over the channel. We focus on obtaining the maximizer for classical
information transmitted in the form of product states over noisy quantum channels.
In this case the fact that the capacity is given by the maximum of the Holevo
quantity is known as the Holevo-Schumacher-Westmoreland (HSW) Theorem. Here
we consider in particular the problem of determining this maximizer in the case
of the amplitude damping channel. It is known in general that the maximizing
ensemble can always be assumed to consist of at most d2 pure states if d is the
dimension of the state space, but we show that in the case of the amplitude damping
channel, the maximum is in fact obtained for an ensemble of two pure states.
Moreover, these states are in general not orthogonal as in the example considered
by Fuchs 2 . We also consider a convex combination of a depolarizing channel and
∗ [email protected]
† [email protected]
1
2
T.C. Dorlas and C. Morgan
an amplitude damping channel and show that the corresponding product state
capacity, which was shown in Ref. 3 to be given by the supremum of the minimum
of the corresponding Holevo quantities, is not equal to the minimum of their product
state capacities.
2. Classical information over a quantum channel
The transmission of classical information over a quantum channel is achieved by
encoding the information as quantum states. To accomplish this, a set of possible
input states ρj ∈ S (H) with probabilities pj are prepared, describing the ensemble
P
{pj , ρj }. The average input state to the channel is expressed as ρ = j pj ρj . For a
channel given by a completely positive map E : S(H) → S(K), the average output
P
state is ρ˜ = j pj E(ρj ) 4 .
When a state is sent though a noisy quantum channel, the amount of information
about the input state that can be inferred from the output state is called the
accessible information. The Holevo bound, Ref. 5, Ref. 6, provides an upper bound
on the accessible information and is given by


X
X
pj S (E(ρj )) .
(1)
pj E(ρj ) −
H(X : Y ) ≤ S 
j
j
The right-hand side of equation (1) is called the Holevo-χ-quantity. The second
term in the Holevo bound is often referred to as the output entropy.
2.1. The Holevo-Schumacher-Westmoreland (HSW) theorem
If the possible input states to a channel are prepared as product states of the form
ρ1 ⊗ ρ2 ⊗ · · · , then the associated capacity is known as the product state capacity.
This implies that the input states have not been entangled over multiple uses of the
channel. The capacity for channels with entangled input states has been studied
7
, and it has been shown that for certain channels the use of entangled states can
enhance the inference of the output state and increase the capacity (e.g. Ref. 8).
We concentrate here on the product state capacity for noisy quantum channels.
The HSW theorem, proved independently by Holevo 1 and by Schumacher and
Westmoreland 9 , provides an expression to calculate the product state capacity for
classical information sent through a quantum channel, E, and can be calculated
using the following expression,
  


X
X
pj S (E(ρj ))
χ∗ (E) = max S E 
(2)
pj ρj  −
{pj ,ρj }
j
j
where S is the von Neumann entropy, S(ρ) = −trace (ρ log ρ). If ρ has eigenvalues
P
λi , then S(ρ) = − i λi log(λi ). The capacity is given by the maximum mutual
information calculated over all ensembles {pj , ρj } 6 .
Calculating a maximizer for quantum mutual information
3
The maximum is always attained for an ensemble of pure states ρj . Indeed, we can decompose each ρj as convex combinations of pure states: ρj =
P
not change the first term of (2), but by concavity
k qk |ψj,k ihψj,k |. This does
P
of the entropy, S(E(ρj )) ≥ k qk S(E(|ψj,k ihψj,k |)).
Moreover, it follows from Carath´eodory’s theorem (see Ref. 10, Ref. 11, Ref. 12),
that the ensemble can always be assumed to contain no more than d2 pure states,
where d = dim (H).
2.2. Convex combination of memoryless channels
In Ref. 3 the product state capacity of a convex combination of memoryless channels
was determined. Given a finite collection of memoryless channels E1 , . . . , EM with
common input Hilbert space H and output Hilbert space K, a convex combination
of these channels is defined by the map
M
X
γi Ei⊗n (ρ(n) ),
E (n) ρ(n) =
(3)
i=1
where γi , (i = 1, . . . , M ) is a probability distribution over the channels E1 , . . . , EM .
Thus, a given input state ρ(n) ∈ S(H⊗n ) is sent down one of the memoryless
channels with probability γi . This introduces long-term memory, and as a result
the capacity of the channel E (n) is no longer given by the maximum of the Holevo
quantity. Instead, it was proved in Ref.3 that it is given by
#
"M
^
(n)
χi ({pj , ρj }) ,
(4)
C(E ) = sup
{pj ,ρj }
i=1
where χi is the Holevo quantity for the i-th channel Ei .
3. The amplitude damping channel and the Holevo-χ-quantity.
The amplitude damping channel models the loss of energy in a qubit quantum
system and is described by the following operation elements,
√ 0 γ
1
0
√
, E1 =
E0 =
(5)
0 1−γ
0 0
for 0 ≤ γ ≤ 1 6 . The amplitude damping channel can therefore be expressed as
Eamp (ρ) = E0 ρ E0† + E1 ρ E1† .
(6)
a b
Acting on the general qubit state ρ, given by ρ = ¯
, the channel Eamp
b 1−a
√
a + (1 − a)γ
b 1−γ
is given by Eamp (ρ) =
¯b√1 − γ (1 − a)(1 − γ) . The eigenvalues of Eamp (ρ)
are easily found to be
q
1
2
2
1 ± (1 + 2a(γ − 1) − 2γ) − 4|b| (γ − 1) .
(7)
λamp± =
2
4
T.C. Dorlas and C. Morgan
The Holevo-χ-quantity is defined as

 
X
X
pj S (E(ρj ))
χ(E)({pj , ρj }) = S E 
p j ρj   −
j
(8)
j
and in the case of the amplitude damping channel,


p
X pj (aj + (1 − aj )γ) pj bj (1 − γ) 
p
χ(Eamp )({pj , ρj }) = S 
pj ¯bj (1 − γ) pj (1 − aj )(1 − γ)
j
−
X
j
pj S
√
aj + (1 − aj )γ
bj 1 − γ
√
.
¯bj 1 − γ (1 − aj )(1 − γ)
To maximize Eq. (8) we show that the first term is increased, while keeping the
second term fixed if each pure state ρj is replaced by itself and its mirror image
a
bj
in the real axis, i.e. if we replace ρj = ¯j
associated with probability
b (1 − aj )
j
aj
bj
aj
−bj
′
pj , with the states ρj =
and ρj =
, both with
b¯j (1 − aj )
−¯bj (1 − aj )
probabilities pj /2.
As remarked above, the maximum in Eq. (2) can be achieved by a pure state
ensemble of (at most) d2 states, where d is the dimension of the input to the
channel. In fact, we do not need to use the bound on the number of states here.
However, it follows that we can take each ρj to be pure for the maximization of
the Holevo quantity. In general, the states (3) must lie inside the Poincar´e sphere
2
a − 21 + |b|2 ≤ 41 and so, the pure states will lie on the boundary
2
1
1
+ b2 = ⇒ b2 = a(1 − a).
a−
2
4
We first show that the second term in (8) remains unchanged when the states are
replaced in the way described above. Indeed, since the eigenvalues (7) depend only
on |b|, we have S (E(ρj )) = S E(ρ′j ) and therefore,
X pj X
S E(ρj + ρ′j ) =
pj S (E(ρj )) .
2
j
j
P
p
ρ
is in fact increased by replacing each
Secondly, we prove that S E
j j j
state with itself and its mirror image, each with half their original weight. Indeed,
as S is a concave function,

  


 

X pj
X
X
1
S
E(ρj + ρ′j ) ≥ S E 
pj ρj  + S E 
pj ρ′j 
2
2
j
j
j
Calculating a maximizer for quantum mutual information
5
P
P
and again, since S E( j pj ρj ) = S E( j pj ρ′j ) ,



 
X pj
X
S
pj ρj  .
E ρj + ρ′j  ≥ S E 
2
j
j
We can conclude that the first term in Eq. (8) is increased with the second term
fixed if each state ρj is replaced by itself together with its mirror image.
3.1. Convexity of the output entropy
We concentrate here on proving that, in the case of the amplitude damping channel,
the second term in the equation for the Holevo-χ-quantity is convex p
as a function
of the parameters aj when ρj is taken to be a pure state, i.e. bj = aj (1 − aj ).
Thus S (E(ρj )) is a function of one variable only, i.e. S(aj ). It is given by,
S(aj ) = S(Eamp (ρaj )),
p
a
a(1 − a)
where ρa = p
, that is,
a(1 − a) 1 − a
p
√
a
+
(1
−
a)γ
a(1 − a) 1 − γ
√
σ(a) = Eamp (ρa ) = p
.
a(1 − a) 1 − γ (1 − a)(1 − γ)
Given the eigenvalues for the amplitude damping channel
p
1
1 ± 1 − 4γ(1 − γ)(1 − a)2 ,
λamp± =
2
p
denote x = 1 − 4γ(1 − γ)(1 − a)2 .
Then
1+x
1−x
1−x
1+x
log
−
log
.
S(a) = −
2
2
2
2
(9)
(10)
(11)
We prove that S ′′ (a) is positive. A straightforward calculation yields
2γ(1 − γ)
4γ(1 − γ)
1+x
S ′′ (a) ln 2 =
−
ln
3
x
1−x
x2
1
1+x
2γ(1 − γ)
−
2
.
ln
=
x2
x
1−x
As the first term in the above equation is positive, the problem of proving the
convexity of S(a) reduces to proving that,
1+x
ln
≥ 2x.
1−x
This is an easy exercise. We conclude that S ′′ (a) is positive and therefore S(a) is
convex.
6
T.C. Dorlas and C. Morgan
Now notice that if ρ¯ =
P
j
pj ρj then a
¯=
P
j
χ({pj , ρj }) = S(Eamp (¯
ρ)) −
pj aj and hence
X
pj S(aj )
j
≤ S(Eamp (¯
ρ)) − S(¯
a).
The capacity is therefore given by
1
χ(Eamp ) = max S
(σ(a) + σ ′ (a)) − S(σ(a)) ,
2
a∈[0,1]
(12)
where σ(a) is given by
σ(a) = Eamp (ρa ) =
and hence
p a + (1 −√a)γ
a(1 − a) 1 − γ
1
(σ(a) + σ(a)′ ) =
2
p
√
a(1 − a) 1 − γ
,
(1 − a)(1 − γ)
a + (1 − a)γ
0
0
(1 − a)(1 − γ)
.
We have proved that S(a) is convex. Therefore −S(a) is concave. On the other hand,
it follows from the concavity of S that the first
term is also a concave function of a.
It follows that χAD (a) = S 21 (σ(a) + σ ′ (a)) − S(σ(a)) is a concave function, and
its maximum is achieved at a single point. The maximizing value of a is given by
(a)
the transcendental equation dχAD
= 0 and can only be computed numerically.
da
The maximizing a for fixed γ ∈ [0, 1] is plotted as a function of γ in Figure 1. Note
that amax ≥ 0.5 for all γ. This is easily proved: The determining equation is
χ′ (a) = −(1 − γ) ln
a + γ(1 − a)
4γ(1 − γ)(1 − a) 1 + x
+
ln
= 0.
(1 − γ)(1 − a)
2x
1−x
(13)
′
′ 1
Since χ(a) is concave, the
p statement follows if we show that χ (0) > 0 and χ ( 2 ) > 0.
But, if a = 0 then x = 1 − 4γ(1 − γ) = |1 − 2γ| so
2γ(1 − γ) 1 + |1 − 2γ|
1−γ
1−γ
1−γ
+
ln
=
ln
> 0.
γ
|1 − 2γ|
1 − |1 − 2γ|
1 − 2γ
γ
p
we have x = −1γ + γ 2 and
χ′ (0) = (1 − γ) ln
For a =
1
2
χ′ (0.5) = −(1 − γ) ln
γ(1 − γ) 1 + x
1+γ
+
ln
.
1−γ
x
1−x
This is also positive because x > γ and the function
1
1+x
tanh−1 (x)
ln
=
2x 1 − x
x
is increasing.
Calculating a maximizer for quantum mutual information
7
0.62
0.6
0.58
0.56
0.54
0.52
0.2
Fig. 1.
0.4
0.6
1
0.8
Γ
Maximising a’s for 0 ≤ γ ≤ 1 for the amplitude damping channel.
4. Convex combinations of two memoryless channels
4.1. The depolarizing channel
The depolarizing channel for qubits has the form
I
EDep (ρ) = (1 − α)ρ + α
.
2
Explicitly, the channel EDep (ρ) can be written as,
b(1 − α)
(1 − α)a + α2
EDep (ρ) =
¯b(1 − α) (1 − α)(1 − a) +
α
2
.
The product state
capacity of this channel is well-known of course, and given by
α
∗
χDep = 1 − H 2 , where H(p) = −p log p − (1 − p) log(1 − p) is the binary entropy.
(In fact, it was proved by King 13 , that this is also the capacity of the channel.)
Note, that the above argument for the amplitude damping channel also applies
here: replacing each ρj as described for the amplitude damping channel the Holevo
quantity increases. The second term of the Holevo quantity is then independent of
the input ensemble, since for pure states, the corresponding eigenvalues are λDep+ =
α
α
2 and λDep− = 1 − 2 . We can therefore again maximize over a minimal ensemble
of two mirror image states with equal probability 21 . The maximum is attained at
a = 21 .
5. Convex combinations of two channels
Let us now consider a convex combination of two memoryless channels. It was shown
in Ref 3 that the product-state capacity is given by (4). In case both memoryless
8
T.C. Dorlas and C. Morgan
channels are depolarizing channels with parameters α1 and α2 , we can write
) = χ∗ (α1 ) ∧ χ∗ (α2 ) = χ∗ (α1 ∨ α2 ).
C(Eα(n)
1 ,α2
(14)
Indeed, we always have
C(E (n) ) ≤ χ∗1 ∧ χ∗2
(15)
and on the other hand, since the maximizing ensemble for both channels is the same,
namely two projections onto orthogonal states, this also maximizes the minimum
χ1 ∧ χ2 .
A convex combination of amplitude damping channels is similar. In that case,
the maximizing ensemble does depend on the parameter γ, but as can be seen from
Figure 2, for any a, χAD (a) decreases with γ, so χ(γ1 ) ∧ χ(γ2 ) = χ(γ1 ∨ γ2 ) and we
have again,
In fact, for γ ≤
Clearly, if
1
2
) = χ∗ (γ1 ) ∧ χ∗ (γ2 ) = χ∗ (γ1 ∨ γ2 ).
C(Eγ(n)
1 ,γ2
(16)
this can be seen as follows. The derivative w.r.t. γ is given by:
a + γ(1 − a)
(2γ − 1)(1 − a)2 1 + x
∂χ
= −(1 − a) ln
+
ln
.
∂γ
(1 − γ)(1 − a)
x
1−x
(17)
x ≥ (1 − 2γ)(1 − a)
(18)
a
1−a
> 1 − 2γ both terms are negative. Otherwise, we remark that
so that it suffices if
x > y = 1 − 2γ − 2a(1 − γ) > 0.
This is easily checked.
In case γ > 0.5, we need to show that
f (a, γ) = ln
a + γ(1 − a)
(2γ − 1)(1 − a) 1 + x
−
ln
≥ 0.
(1 − γ)(1 − a)
x
1−x
Now, if a = 0, then f (0, γ) = 0, and the derivative is easily computed to be
∂f (a, γ)
1−γ
1
2γ − 1 1 + x 2(1 − a)2 (2γ − 1)
ln
.
=
+
+
−
∂a
a + γ(1 − a) 1 − a
x3
1−x
x2
This is positive since the first two terms are positive and the other two are bounded
by
2γ − 1 1 1 + x
2γ − 1 1 + x 2(1 − a)2 (2γ − 1)
ln
≥
−
ln
−
2
≥ 0.
x3
1−x
x2
x2
x 1−x
We now investigate the product-state capacity of a convex combination of an
amplitude damping and a depolarizing channel. Let χ1 and χ2 denote the Holevo
quantity of the amplitude damping and depolarizing channels respectively.
They are plotted in Figure 2 for 0 ≤ γ, α ≤ 1. The plot in Figure 2 indicates
that, for certain values of γ and α the maximizer for the amplitude damping channel
lies to the right of the intersection of χ1 (a) and χ2 (a) for the depolarizing channel,
Calculating a maximizer for quantum mutual information
9
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
a
Fig. 2. The Holevo χ quantity for the amplitude damping channel and the depolarizing channel
plotted as a function of a. The amplitude damping channel is represented in bold.
whereas that for the depolarizing channel lies to the left. This proves that the
maximum of the minimum of the channels does not equal the minimum of the
maximum of the channels.
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