1
An Infinite Sequence of Additive Channels:
the Classical Capacity of Cloning Channels
arXiv:0903.1638v3 [quant-ph] 17 Sep 2009
Kamil Brádler
Abstract— We introduce a novel way of proving the additivity
of the Holevo capacity. The proof comes from the analysis of
an infinite-dimensional channel which appears in the context
of quantum field theory in curved spacetime – the Unruh
channel. The Unruh channel decomposes into a sequence of finitedimensional channels where the additivity of the first channel
of the class induces the additivity of another one resulting in
the domino effect. Surprisingly, the channel series is very closely
related to the quantum channels arising from universal quantum
cloning machines for which the additivity of the optimized
coherent information has recently been proven too. In addition,
this method also provides an easy way of proving the additivity of
the Holevo capacity for the original Unruh channel for which the
quantum capacity is already known. Consequently, we present
not only an infinite series of finite-dimensional channels but also
a nontrivial example of an infinite-dimensional channel for which
the classical and quantum channel capacities are easily calculable.
Index Terms— additivity question, classical and quantum capacity, degradable channels
I. I NTRODUCTION
Recently, a notorious open problem in quantum information
theory known as the additivity of the Holevo capacity was
finally resolved [1] with the negative answer. The article
culminated a long period of waiting for the answer to the
question (later a conjecture) which appeared shortly after
people started to ask about the role of quantum correlations for
information theory [2]. The former conjecture states that entangled states do not improve the classical capacity of quantum
channels. Quantumchannel Φ is a completely positive (CP)
(O)
(I)
→ F Hout . F H (K) is the state
map Φ : F Hin
space for a K−dimensional Hilbert space H (K) occupied by
Hermitean operators of trace one. The ultimate formula for the
classical capacity is C = limn→∞ n1 CHol (Φ⊗n ). CHol (Φ) is
the Holevo capacity of a channel Φ [3] defined as
X
X
df
pi S Φ(̺i ) ,
pi Φ(̺i ) −
CHol (Φ) = sup S
{pi ̺i }
i
i
(1)
where {pi ̺i } is the input ensemble ̺ = i pi ̺i and S(σ) =
− Tr σ log σ is the von Neumann entropy 1 . The calculation
of C appears to be an intractable problem. Hence, the conjecture claimed that CHol (Φ ⊗ Ψ) = CHol (Φ) + CHol (Ψ) for
arbitrary channels Φ, Ψ. This condition is slightly stronger
(strong additivity) than if Ψ = Φ (weak additivity). One can
immediately see how the calculation of C might have been
much simpler if the conjecture had been correct. Let us stress,
however, that even if the conjecture doesn’t hold in general
P
School of Computer Science, McGill University, Montreal, Canada
gives the logarithm to base two.
1 log
there are important classes of channels for which it holds [9],
[10].
The final disproof of the conjecture wouldn’t be possible
without many important intermediate results. First, it was
shown that the additivity of the Holevo capacity is globally
(that is, not for a particular channel) equivalent to another
additivity questions [4], particularly to the additivity of the
minimum output entropy (MOE) [5]. The MOE belongs to
the more general class of entropies known as the minimum
output Rényi entropy (MORE). The MORE of a channel Φ is
defined
df
Spmin (Φ) = min {Sp (Φ(̺))},
̺
̺ ∈ F (Hin ),
(2)
where Sp (̺) = (1 − p)−1 log Tr ̺p is the Rényi entropy (for
p → 1+ we get the von Neumann entropy). The MORE
conjecture was disproved for various intervals of p (for p > 1
in [6] and for p → 0 in [7]) and, as indicated, at last also
for p = 1 [1]. Note that by the concavity of entropy we may
restrict ourselves to the minimization over input pure states.
In section II we briefly recall the properties of the Unruh
channel [19] and present its decomposition into a sequence
of finite-dimensional channels and relate this sequence with
the channels arising from universal quantum cloning machines
(UQCM) for qubits [13]. Section III contains the main result
of the paper. We show here that it is sufficient to prove the
additivity of the Holevo capacity for the first channel of the
sequence and consequently (by induction) the whole class is
additive. This strategy of proving the additivity has not been,
to our knowledge, presented elsewhere. As an interesting consequence, the additivity of the Holevo capacity for the Unruh
channel is proved and thus provides us with an example of an
infinite-dimensional channel for which both the classical and
quantum capacity are known and easily calculable. The Unruh
channel occupies an important place in the field of relativistic
quantum information and quantum field theory [19].
The additive property of the new channel class is not the
only interesting property. Ref. [14] introduced a class of channels called degradable by virtue of existence of a degrading
map transforming the output of the channel to its complementary output (degradable channels were further studied in [15]).
The importance of degradability lies in the fact that for such
channels there exists a single-letter formula for the optimized
coherent information [8] enabling to calculate their quantum
capacity. The family of channels whose additivity of the
Holevo capacity is studied here belongs to a slightly different
class of so-called conjugate degradable channels. In a separate
paper [22] we found that conjugate degradable channels enjoy
the property of having the optimized coherent information
2
additive so for the family of channels investigated in this paper
both capacities can be calculated in a easy way.
Even though the additivity of the optimized coherent information is thus settled we conclude the present paper with
Section IV where we ask the question whether the infinite
sequence of channels is also degradable. We did not bring
the complete answer to this question but we show that a few
lowest-dimensional channels in the class are indeed degradable.
II. U NRUH CHANNEL
In this section we briefly review the definition and properties
of the Unruh channel [19]. The channel naturally appears
if we ask what is the transformation of a photonic qubit
prepared by a stationary Minkowski observer if it is detected
by a uniformly accelerated observer. It is well known that
an inertial and non-inertial observers cannot agree on the
notion of particle with the most dramatic example being
Minkowski vacuum seen by an non-inertial observer as a
thermally populated state [20].
In the same spirit, a pure qubit prepared in the Hilbert space
of a Minkowski observer is seen as an infinite-dimensional
mixed state in the Hilbert space of the accelerated observer.
In the language of equations, we transform a pure qubit |ψi =
(αb† + βa† ) |vaci as
Uabcd (r) =
×e
† †
† †
1
etanh r(a c +b d )
cosh2 r
− ln cosh r(a† a+b† b+c† c+d† d) − tanh r(ac+bd)
e
. (3)
For an input state |ψi we can further simplify |φi =
Uabcd (r) |ψi as
|φi = 1/ cosh3 r(αb† + βa† ) exp [tanh r(a† c† + b† d† )] |vaci .
(4)
From the physical point of view, the modes c, d appear beyond
the event horizon of the accelerated observer and are therefore
unobservable. Tracing over them we get a state with an
interesting structure further investigated in [19]. If we reorder
the basis according to the total number of incoming photons
in modes a and b we obtain an infinite-dimensional blockdiagonal density matrix
σ = 1/2(1 − z 3 )
2
∞
M
ℓ=2
ℓ(ℓ − 1)z ℓ−2 εℓ ,
(5)
where 0 ≤ z < 1, z = tanh r and r is the proper acceleration
of the inertial observer. The states εℓ will be introduced
and analyzed in the next subsection. We should better say
reintroduced because they were first identified in [11]. The authors studied the black hole stimulated emission of impinging
photonic qubits. The stimulated emission process is governed
by exactly the same Hamiltonian as the one leading to our
unitary operator Uabcd . The reason for this formal similarity
lies in the linear relations known as Bogoliubov transformation [12]. Bogoliubov transformation connects the creation and
annihilation operators of the Hilbert space of a Minkowski
observer and a uniformly accelerating observer in our case
and similarly the Hilbert space of a freely falling observer
and an observer in a distant future in case of Ref. [11]. In the
former case the physical parameter of the evolution operator
is the proper acceleration r, in the latter it is the black hole
surface gravity 2 . Even more interestingly, as observed in [11],
the same Hamiltonian is very closely related to the N → M
universal cloning machine for qubits [13] (ℓ = M +1). In other
words, if an observer throws an N −qubit photonic state into a
black hole (the state is already symmetrized due to the bosonic
nature of the photons) another observer in a distant future gets
M approximate copies depending on the total number M of
photons he measures. This can explicitly be seen in Eq. (5)
for the case of 1 → ℓ − 1 cloning machines. The output state
σ is a block-diagonal, normalized density matrix where every
block εℓ correspond to an ℓ−qubit state.
Note that we will refer to 1 → ℓ − 1 cloning machines
simply as 1 → ℓ − 1 cloning channels in this paper.
III. A N
INFINITE SEQUENCE OF ADDITIVE CHANNELS
The previous section served as a physical motivation for the
appearance of N → M UQCM for qubits. Now comes the
main part of the paper where we observe that for N = 1 the
cloning channels ‘constitute’ the corresponding Unruh channel
in a very specific way enabling us to show that the additivity
of the Holevo capacity for the 1 → 2 cloning channel implies
the additivity of the Holevo capacity for all 1 → ℓ − 1 cloning
channels (that is for all ℓ > 3). Another consequence will be
the proof of additivity of the Holevo capacity for the Unruh
channel itself.
We first recall the definition of unitarily covariant channels
introduced in [16].
Definition 1 Let G be a unitary compact group of Lie type
and let r1 (g) ∈ Hin, r2 (g) ∈ Hout
be irreps of g ∈ G. A
channel N : F Hin → F Hout is unitarily covariant if
(6)
N r1 (g)̺r1 (g)† = r2 (g)N (̺)r2 (g)†
holds for all ̺.
In the following text, by covariant we mean unitarily covariant.
It has been proved that for covariant channels the equivalence
condition holds locally
CHol (N ) = log f − S min (N ),
(7)
where f = dim Hout . Nevertheless, for Eq. (7) to hold
the conditions in Definition 1 are not necessary and can be
relaxed [17].
Important notice: Let us stress that we will leave the domain
of the Fock space and adopt new notation. From now on, |ni
represents a qudit living in an abstract Hilbert space H and
not a Fock state of n photons like in Section II. The reason is
that the Fock space formalism is a bit clumsy for the quantum
information considerations which will follow. We will make
occasional connections from one formalism to another to avoid
possible confusion.
Let W be the Hilbert space isometry W : A ֒→ EH such that
(K)
(K)
W (ϕ) = UEH (|ϕiA , |0i) where UEH is a K−dimensional
2 Note that the particle-antiparticle basis used in [11] exactly corresponds
to the dual-rail encoding in which an input state |ψi is written.
3
unitary transformation defined by its action on an input pure
state |ϕi = α |0i +β |1i and an ancilla |0i. This results in a
bipartite entangled state
s
(K)
UEH
2
|ϕiA |0ianc −−−→
(k + 1)(k + 2)
×
k
X
√
α k − n + 1 |k − n + 1iE |niH
n=0
!
√
+ β n + 1 |k − niE |niH , (8)
(thus K = 2(k + 1)). This unitary operation induces a class
of CP maps Sℓ (ϕ) = TrH [W (ϕ)] which we identify with
1 → ℓ − 1 cloning channels (ℓ = k + 2). The explicit output of
a 1 → ℓ − 1 cloning channel and its complementary channel
c
Sℓ−1
(ϕ) = TrE [W (ϕ)] for an input qubit ϕ = 1/2 + ~n · J~(2)
read
X
2
(ℓ)
(9)
ni Ji
εℓ =
1(ℓ) (ℓ − 1)/2 +
ℓ(ℓ − 1)
i=x,y,z
X
2
(ℓ−1)
κℓ−1 =
, (10)
1(ℓ−1) ℓ/2 +
ñi Ji
ℓ(ℓ − 1)
i=x,y,z
(ℓ)
where Ji are the ℓ−dimensional generators of the su(2)
(1)
algebra (Ji = 0), nx = αβ̄ + ᾱβ, ny = −i(αβ̄ − ᾱβ), nz =
|α|2 − |β|2 and ñx = nx , ñy = −ny , ñz = nz . For the
purposes of this paper we consider only input pure states
k~nk2 = 1.
States in Eq. (9) are exactly those from Eq. (5) but stripped
off all optical interpretations or relations to UQCM. However,
one could get the same matrix form from the 1 → ℓ − 1
UQCM if the channel output was rewritten in the completely
symmetric (fixed) basis of ℓ − 1 qubits.
Comparing an input state ϕ with εℓ we see that the
transformation preserves the Stokes parameters ni even as the
dimension of the representation changes. We also observe that
Sℓ is a covariant channel (all UQCM’s are covariant) and so is
c
Sℓ−1
. For ℓ = 2, S2 is an identity map and the complementary
map S1c is just an ordinary trace map.
Some interesting things start to happen for ℓ = 3 where
S2c (ϕ) = 1/3(ϕ̄ + 1) 3 . This is an instance of the transpose
depolarizing channel (alias the optimal transposition map
for qubits) whose Holevo capacity is known to be strongly
additive [10]. It follows that its complement S3 is strongly
additive too [18].
Before we state our first theorem, let us note that in
the remaining sections by additivity we always mean strong
additivity.
Theorem 1 1 → ℓ − 1 cloning channels Sℓ are additive for
all ℓ.
We want to argue that if Sℓ is additive then Sℓc is additive
too. Looking at Eqs. (9) and (10) we notice that κℓ ∝ ε̄ℓ + 1(l) .
Written in this form, the mapping εℓ 7→ κℓ is not generally
3 Bar
over an operator means its entry-wise complex conjugation which
results in transposition for density matrices.
even a CP map (after the normalization) despite Sℓc being
definitely a legitimate CP map. The reason is the occurrence
of transposition, which is not a CP map. Fortunately for us, all
of this is completely irrelevant. The action of Sℓc is effectively
ε̄ℓ 7→ κℓ = λε̄ℓ + (1 − λ)1(ℓ) /ℓ what can be written with
the help of Sℓ as κℓ = λSℓ (ϕ̄) + (1 − λ)1(ℓ) /ℓ. This is the
depolarizing channel [9] composed with Sℓ and the choice of λ
for all ℓ complies with the CP requirement. We just redefined
the input state as |ϕ̄i = ᾱ |0i +β̄ |1i and thus avoided any
problems with forbidden maps.
Lemma 1 Let Pℓ (̺) = λ̺+(1−λ)1(ℓ) /ℓ be the depolarizing
channel. Then if Sℓ is additive, the channel Sℓc = Pℓ ◦ Sℓ is
additive.
Proof: Pℓ is strongly additive [9], hence the MOE satisfies S min (Pℓ ⊗ T ) = S min (Pℓ ) + S min (T ) for any channel T . By definition we first have S min (Pℓ ◦ Sℓ ⊗ T ) ≤
S min (Pℓ ◦ Sℓ ) + S min (T ). The channel Pℓ ◦ Sℓ is evidently
covariant (as the complement of a covariant channel). Because
S min (N ) = S(N (ϕ)) holds for all covariant channels [16] we
have
S min (Pℓ ◦ Sℓ ) = S(Pℓ ◦ Sℓ )(ϕ̄).
(11)
The opposite inequality direction (≥) follows from
S min (T1 ◦ T2 ) ≥ minϕ′ {S(T1 (ϕ′ ))} ≡ S min (T1 )((T2 )) (the
last equality is for notational purposes) for any two channels
T1 , T2 and an arbitrary pure state ϕ′ considering the concavity
of entropy. Hence
S min ((Pℓ ⊗ T ) ◦ (Sℓ ⊗ 1)) ≥ S min (Pℓ ⊗ T )((Sℓ ⊗ 1))
= S(Pℓ (Sℓ (ϕ̄))) + S min (T ),
(12)
where the first term on the rhs is equal to Eq. (11).
Fig. 1.
Numbers indicate the dimension of the particular Hilbert space
H for both subsystems E, H. Note a slight abuse of notation for the case
dim E = 2, dim H = 1 when the input and output spaces coincide (ℓ = 2).
However, as noted above, nothing interesting happens there.
Proof: [Proof of Theorem 1] For the proof we take a look
at Fig 1. We know that S3 is additive and we have just shown
that S3c is thus additive too. But that means its complement, S4 ,
is additive and the whole sequence of channels Sℓ is additive
by induction.
Corollary 1 The previous theorem allows us to explicitly
write down the formula for the classical capacity. Since all
Sℓ are covariant we suitably choose the coefficients α, β such
that states εℓ from Eq. (9) are diagonal (α = 1, β = 0). Then
ℓ−1
εℓ =
1 X
|kihk|,
∆
k=0
4
where ∆ = ℓ(ℓ−1)/2. Hence, considering f = log ℓ in Eq. (7),
we get
A
Sℓ
-
E
Ďℓ
ℓ−1
-
1 X
C(Sℓ ) = 1 − log (ℓ − 1) +
k log k.
∆
First a lemma.
Lemma 2 Let A, B be additive and covariant but otherwise arbitrary finite-dimensional channels whose input Hilbert
spaces are of the same dimension.
Then a channel G :
F H → F HA ⊕ HB is additive for any ensemble
{qA , qB }.
Proof: The channel output is unitarily equivalent to
G
̺−
→ qA ̺A ⊕ qB ̺B ≡ |0ih0| ⊗ qA ̺A + |1ih1| ⊗ qB ̺B . (13)
Defining T to be an arbitrary channel we see that for any input
pure state ω of the channel G ⊗ T the output state is a blockdiagonal matrix σ = qA (A ⊗ T )(ω) ⊕ qB (B ⊗ T )(ω). Thus,
S(σ) = S({qA , qB })+ qA S((A⊗ T )(ω))+ qB S((B ⊗ T )(ω)).
Hence S min (G ⊗ T ) = S({qA , qB }) + qA minω {S((A ⊗
T )(ω))} + qB minω′ {S((B ⊗ T )(ω ′ ))} = S({qA , qB }) +
qA S(A(ϕ)) + qB S(B(ϕ)) + S min (T ) ≡ S(G(ϕ)) + S min (T )
using the properties of A and B.
Proof: [Proof of Theorem 2] The proof is a direct
application of the previous
since the Unruh channel
Llemma
∞
happens to be U(ϕ) =
ℓ=2 pℓ Sℓ (ϕ) where pℓ = (1 −
z 3 )z ℓ−2 (ℓ − 1)ℓ/2, 0 ≤ z < 1 and Sℓ is a 1 → ℓ − 1 cloning
channel. The channel U(ϕ) is the same channel as in Eq. 3
but written in the Fock space representation.
We show using the block-diagonal structure of the output
state and the unitary covariance of the channel that the
inductive process described above approximates the channel
output with an arbitrary precision for
PKany input qubit. Namely,
let us denote a partial sum cK = ℓ=2 pℓ and we get
.
....
....
....
......
-
Theorem 2 The infinite-dimensional Unruh channel studied
in [19] is additive
E′
Dℓ
c
Sℓ−1
k=0
?
C
H
Fig. 2. In the diagram, Sℓ is a 1 → ℓ − 1 cloning channel with a (conjugate)
degrading map Dℓ (Ďℓ ). The dotted line signalizing a non-CP map is complex
conjugation C : κℓ−1 ↔ κ̄ℓ−1 . The situation captured in the diagram holds
for ℓ = 3 . . . 7 (and trivially for ℓ = 2) so these channels are both degradable
and conjugate degradable. We conjecture that it holds for ℓ > 7 as well.
A single-letter quantum capacity formula exists for conjugate
degradable channels. The Unruh channel is manifestly conjugate degradable [19] but it does not imply the conjugate
degradability of its cloning channel constituents. However, in
the separate paper where the concept of conjugate degradability has been introduced [22] we also proved that all N → M
cloning channels for qubits are conjugate degradable. Hence,
there exists a single-letter formula for the quantum capacity
of these channels.
It is conceptually interesting to know whether the channels
associated with universal qubit cloning machines are also
degradable 4 . We will not bring the answer to the question
but we will attempt to construct degrading maps for several
low-dimensional 1 → ℓ − 1 cloning channels we studied in
the previous section.
Looking at Eqs. (9) and (10) we see that the complementary
output of every Sℓ is effectively conjugated with respect to
the channel output so one could suspect that Sℓ might not be
degradable. Let us first analyze the case ℓ = 3. If the channel
is degradable the eventual degrading map D3 is covariant.
The covariance property of the channel leads to the condition
similar to Eq. (6)
D3 (r1 (g)̺ r1 (g)† ) = r2 D3 (̺) r2† ,
(16)
1
2+2(K 2 −1)z K −K(K +1)z K−1 −K(K −1)z K+1
2
(14)
and so limK→∞ cK = 1 for all 0 ≤ z < 1.
where the presence of bars is the result of conjugation. In
this case, r2 and r1 is the two- and three-dimensional irrep of
g ∈ G = SU (2), respectively. This is, however, the same as
the contravariance condition
Remark 1 Note that the channel input for otherwise infinitedimensional Unruh channel is naturally energy constrained
since the set of input states is limited to qubits.
D3 (r1 ̺ r1† ) = r2 (g) D3 (̺) r2 (g)T .
cK =
IV. (C ONJUGATE ) DEGRADABILITY
We start with the definition of conjugate degradability [22]
resembling the one of degradability [14].
Definition 2 A channel N is conjugate degradable if there
exists a map Ď called a conjugate degrading map which
degrades the channel to its complementary channel N c up
to complex conjugation C
Ď ◦ N = C ◦ N c .
(15)
(17)
By rephrasing this condition within the Choi-Jamiołkowski
isomorphism [21] we get
(18)
RD3 , r2 ⊗ r1 = 0
when RD3 is a positive semidefinite matrix corresponding
L to a
CP map D3 . One of Schur’s lemmas dictates RD3 = i ci Πi
where ci ≥ 0 and Πi are projectors into the subspaces of the
split product [2] ⊗ [3] = [2] ⊕ [4] and
thus RD3 ≡ RD
3 . We
insert RD3 into D3 (ε3in ) = Trin (1out ⊗ ε̄3in ) RD3 since
4 The existence of conjugate degradable but not degradable channels is
conjectured [22].
5
we are looking for such RD3 that κ2 = D3 (ε3 ) where (index
in omitted)
√


2
2αβ̄ √ 0
2|α|
1 √
ε3 =
(19)
2ᾱβ √ 1
2αβ̄  ,
3
0
2ᾱβ 2|β|2
1 |α|2 + 1
ᾱβ
κ2 =
.
(20)
αβ̄
|β|2 + 1
3
In other words, we maximize the fidelity between these
two states checking whether it reaches one for
some
c 1 , c2
considering the constraints c1,2 ≥ 0 and Trout RD3 = 1(3) .
Because we are dealing with mixed states, we use the fidelity
expression due to Bures which simplifies for two-dimensional
matrices [23] as
p
F (D3 (ε3 ), κ2 ) = Tr[D3 (ε3 )κ2 ] + 2 Det[D3 (ε3 )] Det[κ2 ].
(21)
Perhaps surprisingly, the fidelity reaches one so S3 is also
degradable. The whole situation can be seen in Fig. 2 and
the corresponding calculation for another channel in the row,
S4 , leads to the same conclusion. Unfortunately, the optimization for ℓ ≥ 5 gets progressively intractable mainly
because the advantageous fidelity form (21) does not hold
anymore (starting already with S4 ). The good news is that an
ansatz can be made. Following the lowest-dimensional exact
solutions for the degradability of S3 and S4 we observe that
the only surviving coefficient ci from the expression for the
Jamiołkowski matrices is the one accompanying the highest
irrep of the SU (2) tensor product. Indeed, applying this guess
on a few more 1 → ℓ − 1 cloning channels (ℓ = 5, 6, 7) always
yields the maximal possible fidelity. Hence, for now we have
a conjecture.
Conjecture 1 All 1 → ℓ − 1 qubit cloning channels are
degradable.
V. C ONCLUSIONS
The general non-additivity result [1] is in some sense very
satisfactory. Not only did entanglement prove to be useful for
the transmission of classical information but it will spark even
more effort to find out what makes a channel (non-)additive.
Also, some novel strategies may be found to prove (non)additivity for particular channels as it is now known that
there is no general proof. This paper presented such a strategy
which, to our knowledge, has not been shown elsewhere. We
took an infinite-dimensional channel called the Unruh channel
whose output is block-diagonal and it is known that every
block is actually an output form 1 → ℓ − 1 universal quantum
cloning machine with the block labeled by ℓ = 2 . . . ∞. The
structure of the channel enabled us show that the additivity
of the simplest channel (nontrivially) guarantees the additivity
of the next one in row and so forth. With this result in hand
we were also able to present the additivity proof of the Unruh
channel which otherwise seems intractable. This result might
find an important future application in quantum field theory
in curved spacetime considering the prominent role the Unruh
channel has in this branch of physics.
Using the recent results showing that 1 → ℓ − 1 universal
quantum cloning machines posses a single-letter formula also
for the quantum capacity [22] we brought by this work into the
light an infinite sequence of finite-dimensional channels whose
classical and quantum capacities are easy to calculate. The
infinite-dimensional Unruh channel is now a member of this
family since the existence of a single-letter quantum capacity
formula has been proven elsewhere.
ACKNOWLEDGMENTS
The author is very grateful to Patrick Hayden for comments
and discussions. The work was supported by QuantumWorks
and in part by a grant from the Office of Naval Research
(N000140811249).
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