Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels

0.16 MB·English

by  Shun Watanabe

Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels
Shun Watanabe∗
Department of Information Science and Intelligent Systems, University of Tokushima,
2-1, Minami-josanjima, Tokushima, 770-8506 Japan
(Dated: October26, 2011)
Twonewclassesofquantumchannels,whichwecallmore capableandless noisy,areintroduced.
Themorecapableclassconsistsofchannelssuchthatthequantumcapacitiesofthecomplementary
channelstotheenvironmentsarezero. Thelessnoisyclassconsistsofchannelssuchthattheprivate
capacities of the complementary channels to theenvironment are zero. For themore capable class,
itisclarifiedthattheprivatecapacityandquantumcapacitycoincide. Forthelessnoisyclass, itis
clarified that the privatecapacity and quantumcapacity can be single letter characterized.
2
1 PACSnumbers: 03.67.-a
0
2
I. INTRODUCTION terizationsofthe private capacityandquantumcapacity

n
were established [3, 12].
a
J One of the mostimportant problemin quantuminfor- A quantum channel is called conjugate degradable if
5 mation theory is to determine the quantum capacity of thereexistsanotherdegradingchannelsuchthatthecon-
2 a noisy quantum channel. The capacity is defined as the junction of the channel to the legitimate receiver and
transmissionrateoptimizedoverallpossiblequantumer- the degrading channel coincide with the complementary

] ror correcting codes such that decoding errors vanish in channel to the eavesdropper up to complex conjugation.
h
p the limit of asymptotically many uses of the channel. In such a case, the single letter characterizations were
- Mathematically, a quantum channel can be described also established [13].
t
n bythetracepreservingcompletelypositive(TPCP)map To date, all quantum channel whose capacities are
a from the input system to the output system. By using single letter characterized are degradable or conjugate
u the Stinespring dilation of the TPCP map, we can natu- degradable1,anditisimportanttoclarifyabroaderclass
q
rally define a complementary channel to an environment of quantum channels such that the single letter charac-
[
system,andwecanregardthenoisyquantumchannelas terizations are possible.

3 awire-tapchannel[1,2]fromthesendertothelegitimate Aside from the possibility of the single letter charac-
v receiver and the eavesdropper who can observe the envi- terizations, there is also another interesting problem. In
6
ronmentsystemofthechannel(eg.see[3]). Thenwecan thequantuminformationtheory,theprivateinformation
4
7 define the private capacity of the noisy quantum chan- transmission and the quantum information transmission
5 nel as the transmission rate optimized over all possible are closely related [4, 14–16], and the possibility of the
. wire-tapcodessuchthatdecodingerrorsandinformation latter implies the possibility of the former. However,the
0
1 leakage vanish in the limit of asymptotically many uses privateinformationtransmissionandthe quantuminfor-
1 of the channel. mation transmission are not exactly equivalent. Indeed,
1 The private capacity and quantum capacity of noisy although the private capacity and quantum capacity co-
: quantumchannelswereestablishedin[4–7]. Howeverun- incide for degradable quantum channels [17], the former
v
i like the capacity formula of a classical noisy channel or can be strictly larger than the latter in general. Espe-
X the private capacityformulaofa classicalwire-tapchan- cially the private capacity can be positive even if the
r nel, the private capacityandquantumcapacityformulae quantum capacity is zero [18]. Thus it is important to
a
are not single letter characterized, i.e., the formulae in- clarify a condition on quantum channels such that the
volve the limit with respect to the number of channel private capacity and quantum capacity coincide or not.
uses, and they are not computable. Indeed, some nu-
Toshedlightontheabovementionedtwoproblems,we
mericalevidencesclarifiedthattheexpressionsintheca-
introducetwonewclassesofquantumchannels,whichwe
pacity formulae are not additive [8–11], and the single
call more capable and less noisy. The more capable class
letter characterization is not possible in general at least
consists of channels such that the quantum capacities
by using the same expressions.
of the complementary channels are zero. The less noisy
A quantum channelis called degradableif there exists classconsistsofchannelssuchthattheprivatecapacities
another degrading channel such that the conjunction of of the complementary channels are zero. Later, these
the channel to the legitimate receiver and the degrading definitionsturnouttobe naturalanalogiesofthe partial
channel coincide with the complementarychannel to the
eavesdropper. In such a case, the single letter charac-
1 Therearealsochannelscalledanti-degradableorconjugateanti-
degradable. Thecapacitiesofthosechannelsarealsosingleletter
∗Electronicaddress: [email protected] characterized, butthecapacities areequaltozero.
2
whichisusuallycalledthecomplementarychannelofN .
B
Although the Stinespring dilation is not unique, the fol-
More Capable
lowing arguments do not depend on the choice of the
dilation because two dilations can be converted to each
otherbyapplyingalocalunitarytotheenvironmentsys-
tems.
Less Noisy
Throughout the paper, we basically follow the no-
tations from [3, 20]. The von Neumann entropy of a
density matrix ρ is defined by H(ρ) = −Trρlogρ, and
Degradable the quantum relative entropy between ρ and σ is de-
or fined by D(ρkσ) = Trρ(logρ−logσ). For input state
Conjugate ρA to the channel NB, the coherent information is de-
Degradable fined by Ic(AiB)ρ = H(NB(ρA))−H(NE(ρA)). When
the input state is clear from the context, we omit the
subscript and denote I (AiB). The quantum mutual
c
information of ρ on the joint system is defined by
XB
I(X;B)=H(ρ )+H(ρ )−H(ρ ). Especially, when
X B XB
FIG. 1: The inclusive relation of the degradable, the conju- ρ is classical with respect to X, i.e., ρ is of the
XB XB
gate degradable, the less noisy, and the more capable classes form
of quantum channels.
ρ = P (x)|xihx|⊗ρx,
XB X B
Xx
orderings,more capable and less noisy, between classical
then the quantum mutual information can be written as
channels [19].
Theinclusiverelationofthe degradable,theconjugate
I(X;B)=H(ρ )− P (x)H(ρx).
degradable, the less noisy, and the more capable classes B X B
are summarized in Fig. 1. In this paper, we show that Xx
the private capacity and quantum capacity coincide for When the legitimate receiver can observe the output
channelsinthemorecapableclass. Furthermore,wealso of N and the eavesdropper can observe the output of
B
showthattheprivatecapacityandquantumcapacitycan N , the private capacity is characterizedby [4, 5]
E
besinglelettercharacterizedforchannelsinthelessnoisy
class. Theseresultsprovidepartialsolutionstotheabove 1
C (N )= lim C(1)(N⊗n), (1)
mentioned two problems. p B n→∞n p B
The rest of the paper is organized as follows. In Sec-
where
tion II, we review some known results on the private ca-
pacity and quantum capacity of quantum channels. In C(1)(N ):= max [I(U;B)−I(U;E)],
Section III, the more capable and less noisy classes are p B PU,{ρuA}
introduced,andwestateourmainresults. InSectionIV,
we summarize certain properties implied by more capa- where{ρuA}arestates(notnecessarilypurestates)onHA
ble and less noisy, and show proofs of our main results. indexed by u ∈ U, and PU is a probability distribution
We finalize the paper with conclusion in Section IV. on a finite set U.
On the other hand, when the sender want to transmit
quantuminformationtothereceiverthroughthechannel
N , the quantum capacity is characterized by [4, 6]
II. PRELIMINARIES B
1
Q(N )= lim Q(1)(N⊗n), (2)
Let NB be a quantum channel from an input system B n→∞n B
H to an output system H . By using the Stinespring
A B
where
dilation (eg. see [3]), there exist an environment system
H and an isometry U from H to the joint system
E BE A Q(1)(N ):=maxI (AiB) .
HB ⊗HE such that B ρA c ρ
N (ρ)=Tr [U ρU∗ ] Both Eqs. (1) and (2) cannot be single letter charac-
B E BE BE
terized, i.e., C (N ) > C(1)(N) and Q(N) > Q(1)(N)
p B p
for every input ρ, where Tr is the partial trace with in general [8–11]. Furthermore, the private capacity
E
respect to the environment system. By using this repre- can be strictly larger than the quantum capacity, i.e.,
sentation, we can naturally define another channel C (N )>Q(N ) for some channels [18].
p B B
ThechannelN issaidtobedegradableifthereexists
B
N (ρ)=Tr [U ρU∗ ], a TPCP map D such that D ◦ N = N . This is a
E B BE BE B E
3
quantum analogue of degraded broadcast channel [21]. In [19], alternative description of less noisy, less di-
WhenN isdegradable,thenitisknown[3,12]thatthe vergence contracting, was introduced, and we can also
B
singleletterformulaehold,i.e., C (N )=C(1)(N )and extend such a description to the quantum channel. The
p B p B
Q(N ) = Q(1)(N ). Furthermore, it is also known that quantum channel NB is said to be less divergence con-
B B
tracting if
C (N )=Q(N ) for degradable channel N [17].
p B B B
speLcetttCodaenfioxteedebnatrsyis-woifseHco.mTplheexnc,otnhjeugcahtaionnnewliNth reis- D(NB⊗n(ρAn)kNB⊗n(ρˆAn))
E B
said to be conjugate degradable if there exists a TPCP ≥ D(NE⊗n(ρAn)kNE⊗n(ρˆAn)), ∀ρAn,ρˆAn (8)
map D such that D◦N = C ◦N . When N is con-
B E B holds for every n ≥ 1. Later, we will show that the
jugate degradable, it is known that Q(N ) = Q(1)(N )
B B quantumchannelislessnoisyifandonlyiflessdivergence
[13]. Later, it will turn out that C (N ) = Q(N ) =
p B B contracting(Proposition4). This alternativedescription
Q(1)(N ) also holds.
B plays a crucial role when we prove Theorem 2.
The followings are our main results.
III. MAIN STATEMENTS
Theorem 1 Suppose that the quantum channel N is
B
more capable. Then, we have
In this section, we introduce two new classes of quan-
tum channels, and show our main results. C (N )=Q(N ).
p B B
Definition 1 The quantum channel NB is said to be Theorem 2 Suppose that the quantum channel NB is
morecapablethantheenvironment,orjustmorecapable, less noisy. Then, we have
ifthequantumcapacityofthecomplementarychannelto
C (N )=Q(N )=Q(1)(N ).
the environment is zero, i.e., p B B B
When N is conjugate degradable, we can show that
Q(N )=0. (3) B
E C (N ) = 0 as follows. Suppose that the sender sends
p E
a state ρ that corresponds to the message i, and the
Definition 2 ThequantumchannelN issaidtobeless i
B eavesdropper2 uses a POVM {M }. Then, for the entry-
noisy than the environment, or just less noisy, if the pri- wise complex conjugate POVM {iM¯ }, we have
i
vate capacity of the complementary channel to the envi-
ronment is zero, i.e., Tr[M¯ D⊗n◦N⊗n(ρ )] = Tr[M¯ C⊗n◦N⊗n(ρ )]
i B i i E i
= Tr[M N⊗n(ρ )],
Cp(NE)=0. (4) i E i
where the last equality follows because M¯T = M and
SinceC (N )≥Q(N ),lessnoisyimpliesmorecapable. i i
p E E (C⊗n◦N⊗n(ρ ))T = N⊗n(ρ ). Thus, the legitimate re-
By using the eigenvalue decomposition E i E i
ceivercangetexactlythe sameinformationastheeaves-
dropper and the private information transmission to the
ρAn = PXn(xn)|ψxnihψxn| eavesdropper is impossible. From this argument, conju-
xnX∈Xn gate degradable implies less noisy.
When the quantum capacity of the channel is 0 but it
of ρAn, we can rewrite the coherent information as
canbeusedtoshareboundentanglement,thenthechan-
I (AniBn)=I(Xn;Bn)−I(Xn;En), (5) nel is called a binding entanlement channel [22]. Espe-
c
ciallywhenthechannelproducesapositivepartialtrans-
where we set |X| = dimH . Thus, by noting Eq. (2), pose (PPT) bound entanglement, the channel is called
A
thequantumchannelN beingmorecapablecanbealso PPT binding entanglement channel. If a complemen-
B
described as tary channel is a binding entanglement channel, then
the main channel obviously belongs to the more capable
I(Xn;Bn)≥I(Xn;En), ∀(PXn,{|ψxni}) (6) class. Since the complementary channelofthe conjugate
degradablechannelcanonlyproducePPTbipartitestate
holds for every n ≥ 1. Furthermore, by noting Eq. (1), [13], if there exists a conjectured negative partial trans-
the quantum channel NB being less noisy can be also pose (NPT) binding entanglement channel, the comple-
described as mentary of such a channel belongs to the more capable
class but not to the conjugate degradable class.
I(Un;Bn)≥I(Un;En), ∀(PUn,{ρuAnn}) (7)
holds for every n ≥ 1. Eqs. (6) and (7) resemble the
definitions of more capable and less noisy for classical
2 The role of the legitimate receiver and the eavesdropper is in-
channels[19],anditisjustified tocallquantumchannels
terchanged because we are considering the private capacity of
satisfying Eqs. (3) or (4) more capable or less noisy. NE.
4
It is known that there exists a channel such that the where
quantum capacity is zero (PPT binding entanglement
channel)butthe privatecapacityis strictly positive[18]. ρA = PU(u)PX˜|U(u,x|u)|ψu,xihψu,x|.
Let the complementary channel NE be such a channel. Xu,x
ThenthechannelN ismorecapablebutnotlessnoisy3.
B Since P and {ρu} are arbitrary, we have
Thus, the morecapable class is strictly broaderthan the U A
less noisy class. However, it is not yet clear whether the
C(1)(N )≤Q(1)(N ).
lessnoisy classis strictlybroaderthanthe degradableor p B B
conjugate degradable classes.
Theoppositeinequalityisobviousfromthedefinitionsof
C(1)(N ), Q(1)(N ), and Eq. (5). (cid:3)
p B B
IV. PROOF OF THEOREMS
A. Properties of Cp(1)(NB) and Q(1)(NB) Proposition 2 Suppose that Eq. (6) does not hold for
n=1, and the density operator ρ∗ maximizing I (AiB)
In this section, we summarize the properties of A c
is full rank. Then, we have
C(1)(N ) and Q(1)(N ) when Eqs. (6) or (7) hold for
p B B
n = 1. For n ≥ 2, we can also show similar properties C(1)(N )>Q(1)(N ).
p B B
of C(1)(N⊗n) and Q(1)(N⊗n) when Eqs. (6) or (7) hold
p B B
for each n by considering n times extension of NB. The Especially when dimHA = 2 and Cp(1)(NB) > 0, the
following properties can be regarded as quantum exten- sufficient and required condition on
sionsofthepropertiesshownforclassicalchannelsinthe
literatures [2, 19, 23, 24] C(1)(N )=Q(1)(N )
p B B
Proposition 1 Suppose that Eq. (6) holds for n = 1. is that Eq. (6) holds for n=1.
Then we have
Proof. SinceEq.(6)doesnotholdforn=1,thereexists
Cp(1)(NB)=Q(1)(NB). ρˆA such that
Proof. For any PU and {ρuA}, let Ic(AiB)ρˆ<0.
ρuA = αu,x|ψu,xihψu,x| Since ρ∗ is full rank, there exists 0 < λ < 1 such that
A
Xx ρ∗ −λρˆ is positive semidefinite. We construct P and
A A U
be an eigenvalue decomposition. Let X˜ be the random {ρuA} as follows. Let
variable on U ×X such that
ρˆ = Pˆ (x)|ψˆ ihψˆ |,
A X x x
P (u′,x|u)= αu,x if u=u′ . Xx
X˜|U (cid:26)0 else
ρ∗ −λρˆ = β |φ ihφ |
A A x x x
Then, we have Xx
I(U;B)−I(U;E)=[I(X˜;B)−I(X˜;E)] be eigenvalue decompositions. Let U = {0}∪X. Then,
−[I(X˜;B|U)−I(X˜;E|U)]. (9) we set PU(0) = λ, PU(u) = βu for u ∈ X, ρ0A = ρˆA, and
ρu =|φ ihφ | for u∈X. Let X˜ be the random variable
A u u
Since Eq. (6) holds for n=1, we have on U ×X such that
I(X˜;B|U =u)−I(X˜;E|U =u)≥0 Pˆ (x) if u=u′ =0
X
for every u, which means that the second bracket of PX˜|U(u′,x|u)= 1 if x=u=u′ 6=0 .
 0 else
Eq. (9) is non-negative. Furthermore, by noting that
{|ψ i} are pure, we have 
u,x Then, we have
I(X˜;B)−I(X˜;E)=I (AiB),
c I(U;B)−I(U;E)
= I(X˜;B)−I(X˜;E)−[I(X˜;B|U)−I(X˜;E|U)]
= I(X˜;B)−I(X˜;E)
3 Notethat the privateandquantum capacities ofthis channel is
strictlypositive,whichcanbecheckedasfollows. IfQ(NB)=0, −λ[I(X˜;B|U =0)−I(X˜;E|U =0)]
then the complementary channel NE is more capable. Then, > I(X˜;B)−I(X˜;E) (10)
Theorem 1 implies that Q(NE) = Cp(NE), which contradict
withthefactCp(NE)>Q(NE)=0. = Ic(AiB)ρ∗, (11)
5
where Eq. (10) follows from Then, we have
I(X˜;B|U =0)−I(X˜;E|U =0)=I (AiB) <0 I(U;B)−I(U;E)=[I(X˜;B)−I(X˜;E)]
c ρˆ
−[I(X˜;B|U)−I(X˜;E|U)]. (13)
and Eq. (11) follows from
We also have
PU(0)PX˜|U(0,x|0)|ψˆxihψˆx| I(X˜;B)−I(X˜;E)=Ic(AiB)ρ
Xx
and
+ P (u)P (u,x|u)|φ ihφ |=ρ∗.
U X˜|U x x A
uX,x∈X m
I(X˜;B|U)−I(X˜;E|U)= p I (AiB) .
i c ρi
Next, we showthe latter statementofthe proposition. Xi=1
ThesufficientconditionfollowsfromProposition1. Sup-
Thus, from Eq. (13), Eq. (7) holds for n=1 if and only
pose that if the coherent information I (AiB) is concave. (cid:3)
c
I (AiB)≤0, ∀ρ (12)
c A
holds. Since C(1)(N ) > 0, there exists P and {ρu} Proposition 4 The following two conditions are equiv-
p B U A
such that alent.
(i) Eq. (7) holds for n=1.
I(U;B)−I(U;E)>0,
(ii) Eq. (8) holds for n=1.
which implies the required condition. Next, we consider
Proof. We first show that (i) implies (ii). For any ρ ,
the case such that Eq. (12) does not hold. In this case, A
ρˆ , and 0≤λ≤1, let U ={0,1},P (0)=λ, P (1)=
we have A Uλ Uλ
1−λ, ρ0 =ρ , and ρ1 =ρˆ . Then, let
A A A A
maxI (AiB)>0.
c f(λ)=I(U ;B)−I(U ;E)
ρA λ λ
= λD(N (ρ )kN (ρ¯ ))+(1−λ)D(N (ρˆ )kN (ρ¯ ))
B A B A B A B A
Then, since dimH = 2, ρ∗ must be full rank. Thus,
A A − λD(N (ρ )kN (ρ¯ ))−(1−λ)D(N (ρˆ )kN (ρ¯ )),
therequiredconditionfollowsfromtheformerstatement E A E A E A E A
of the proposition. (cid:3) where
ρ¯ =λρ +(1−λ)ρˆ .
A A A
By elementary calculation (cf. [3, Ex. 1.4]), we have
Proposition 3 Eq.(7)holds forn=1if andonly ifthe
coherent information is concave4, i.e., f′(0)=D(N (ρ )kN (ρˆ ))−D(N (ρ )kN (ρˆ )).
B A B A E A E A
m Obviously, we have f(0) = 0. Since Eq. (7) holds for
Ic(AiB)ρ ≥ piIc(AiB)ρi, n=1, f(λ)≥0 for any 0≤λ≤1. Thus, f′(0) must be
Xi=1 non-negative, which means that Eq. (8) holds for n=1.
Next, we show that (ii) implies (i). For any P and
where ρ= mi=1piρi. {ρu}, we have U
A
P
Proof. Let
I(U;B) = P (u)D(N (ρu)kN (ρ¯ )),
U B A B A
Xu
ρ = p |ψ ihψ |
i i,x i,x i,x I(U;E) = P (u)D(N (ρu)kN (ρ¯ )),
Xx U E A E A
Xu
be an eigenvalue decomposition. Then, let U =
where
{1,...,m}, P (u) = p , X˜ be the random variable on
U i
U ×X such that ρ¯ = P (u)ρu.
A U A
Xu
p if u′ =u
PX˜|U(u′,x|u)=(cid:26) 0i,x else . Since Eq. (8) holds for n=1, we have
I(U;B)≥I(U;E).
(cid:3)
4 Itshouldbenotedthatthecoherentinformationisknowntobe
concaveifthequantumchannelNB isdegradable[3,Eq.(9.89)].
6
B. Proof of Theorem 1 Thus, Eq. (14) can be proved by induction. (cid:3)
It is a straight forward consequence of Proposition 1.
Since N is more capable,Eq. (6) holds for every n≥1.
B V. CONCLUSION
Thus, we have C(1)(N⊗n)=Q(1)(N⊗n) for every n≥1,
p B
and the statement of the theorem follows from Eqs. (1)
and (2). (cid:3) In this paper, we introduced two new classes of quan-
tumchannels,whichwecallmorecapableandlessnoisy.
For the more capable class, we showed that the private
C. Proof of Theorem 2 capacity and quantum capacity coincide. For the less
noisy class, we showed that the private capacity and
quantum capacity can be single letter characterized.
Since less noisy implies more capable, by Theorem 1,
it suffice to show Q(N ) =Q(1)(N ). For this purpose, Our results shed light on further understanding of
B B
we will show that the private and quantum capacities of quantum chan-
nels. However, the conditions such that a certain chan-
maxI (AniBn)≤nmaxI (AiB) (14) nel belongs to the more capable class or the less noisy
c c
ρAn ρA class are hard to verify in general, and we do not yet
know whether there exists a channel that belongs to the
holds for every n ≥ 1. For any input state ρAkAℓ on less noisy class but not to the degradable or conjugate
HA⊗(k+ℓ),letρAk andρAℓ be thepartialtraces. Then,we degradableclasses,whichisanimportantfutureresearch
have
agenda.
I (AkiBk)+I (AℓiBℓ)−I (AkAℓiBkBℓ)
c c c
= D(NB⊗(k+ℓ)(ρAkAℓ)kNB⊗k(ρAk)⊗NB⊗ℓ(ρAℓ))
Acknowledgment
−D(NE⊗(k+ℓ)(ρAkAℓ)kNE⊗k(ρAk)⊗NE⊗ℓ(ρAℓ)).
SinceEq.(7)holdsforn=(k+ℓ),by(ntimesextension This research is partly supported by Grand-in-Aid
of)Proposition4,Eq.(8)alsoholdsforn=(k+ℓ),which for Young Scientists(B):2376033700 and Grant-in-Aid
implies for Scientific Research(A):2324607101. The author also
would like to thank an anonymous reviewer for helpful
I (AkAℓiBkBℓ)≤I (AkiBk)+I (AℓiBℓ). comments.
c c c
[1] A.D. Wyner,Bell Syst.Tech. J. 54, 1355 (1975). arXiv:0909.3297.
[2] I. Csisz´ar and J. K¨orner, IEEE Trans. Inform. Theory [14] B. Schumacher and M. D. Westmoreland, Phys. Rev.
24, 339 (1979). Lett. 80, 5695 (1998), arXiv:quant-ph/9709058.
[3] M. Hayashi, Quantum Information: An Introduction [15] P. W. Shor and J. Preskill, Phys. Rev. Lett. 85, 441
(Springer,2006). (2000), arXiv:quant-ph/0003004.
[4] I. Devetak, IEEE Trans. Inform. Theory 51, 44 (2005), [16] B. Schumacher, Phys. Rev. A 54, 2614 (1996),
arXiv:quant-ph/0304127. arXiv:quant-ph/9604023.
[5] N.Cai, A.Winter,andR.W.Yeung,ProblemsofInfor- [17] G. Smith, Phys. Rev. A 78, 022306 (2008),
mation Transmission 40, 26 (2004). arXiv:0705.3838v1.
[6] S. Lloyd, Phys. Rev. A 55, 1613 (1997), [18] K. Horodecki, L. Pankowski, M. Horodecki, and
arXive:quant-ph/9604015. P. Horodecki, IEEE Trans. Inform. Theory 54, 2621
[7] P.W.Shor,inLecture Notes, MRSIWorkshoponQuan- (2008), arXiv:quant-ph/0506203.
tum Computation (2002). [19] J. K¨orner and K. Marton, Keszthely Colloquium on In-
[8] G. Smith, J. M. Renes, and J. A. Smolin, Phys. Rev. formation Theory pp. 411–423 (1975).
Lett.100, 170502 (2008), arXiv:quant-ph/0607018. [20] M. A. Nielsen and I. L. Chuang, Quantum Computation
[9] P. W. Shor and J. A. Smolin (1996), arXiv:quant- andQuantumInformation (CambridgeUniversityPress,
ph/9604006. 2000).
[10] D. P. DiVincenzo, P. W. Shor, and J. A. Smolin, Phys. [21] T.M.CoverandJ.A.Thomas,ElementsofInformation
Rev.A 57, 830 (1998), arXiv:quant-ph/9706061. Theory (John Wiley & Sons, 2006), 2nd ed.
[11] G.Smith and J. A.Smolin, Phys.Rev.Lett.98, 030501 [22] P. Horodecki, M. Horodecki, and R. Horodecki, Jour-
(2007), arXiv:quant-ph/0604107. nal of Modern Optics 47, 347 (2000), arXiv:quant-
[12] I.DevetakandP.W.Shor,Comm.Math.Phys.256,287 ph/9904092.
(2005), arXiv:quant-ph/0311131. [23] M.vanDijk,IEEETrans.Inform.Theory43,712(1997).
[13] K. Br´adler, N. Dutil, P. Hayden, and A. Muhammad, [24] O. Ozel and S. Ulukus, in Proc. IEEE Int. Symp. Inf.
Journal of Mathematical Physics 51, 072201 (2010), Theory 2011 (Saintpeterburg, Russia, 2011), pp. 627–
7
631, arXiv:1110.4613.