Quantum coding theorem from privacy and distinguishability

0.22 MB·English

by  Michal Horodecki

Quantum coding theorem from privacy and distinguishability
Micha l Horodecki
Institute of Theoretical Physics and Astrophysics, University of Gd´ansk, Poland
Seth Lloyd
Massachusetts Institute of Technology, Cambridge, MA 02139 USA
Andreas Winter
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U. K.
We provedirect quantum coding theorem for random quantumcodes. The problem is separated
into two parts: proof of distinguishability of codewords by receiver, and of indistinguishability of
codewords by environment (privacy). For a large class of codes, only privacy has to bechecked.
8
0
0
2 I. INTRODUCTION

n
a The quantumcoding theoremfor transmissionofquantuminformationvia noisychannelis one ofthe fundamental
J achievement of quantum information. First proof, not yet fully rigorous, of direct coding theorem was given in [1].

9 The converse theorem was rigorously proven in [2] (see [3] in this context). An attmept to rigourous proof of direct
1 codingtheoremwassubsequentlydonebyShor,withhisnotespublishedonwebsite[4]. Later,Devetakprovidedfirst

complete proof of coding theorem, using so called random CSS codes. Both for practical as well as for fundamental
2
reasonsitisdesirabletohaveproofofcodingtheoremforfullyrandomcodes,suchasLloyd,orShorones. Inthispaper
v
6 we present a new proof of coding theorem. The main advantage of the present approach, is that we divide problem
0 of coding theorem into two separate problems. One is whether Bob can distinguish signals, the second is whether
0 environmenet cannot distinguish them. It is well known that those two conditions are crucial for sending quantum
2 information reliably. The first one is connected with bit error, and the second with phase error [1, 4, 5, 6]. Actually,
0
Devetak’s coding scheme is a coherent version of cryptographic protocol consisting of two stages: error correction,
7
that ensures distinguishability by Bob and privacy amplification responsible for diminishing Eve’s knowledge.
0
However so far the two problems have not been separated, in the sense, that the proof of coding theorem did not
/
h consist of two completely separate mathematical problems.
p
More specifically, we show that if we have set of N vectorswhich, after crossingthe channelare distinguishable for
-
t Bob(receiver),butnotforEve(controllingenvironment),thenAlice(sender)andBobcansharemaximallyentangled
n
state of rank N. The statement of quantum coding theroem, is that capacity is determined by coherent information
a
I [7]. Thus the question whether for n uses of the channel, the set of vectors (quantum code) that satisfy the two
u coh
q conditions can have 2nIcoh elements.
: In classical case the coding theorem states that capacity is determined by Shannon mutual information I . Since
v M
only distinguishability is needed, one proceeds as follows. One fixes source, and then pick codewordsat random from
i
X the probability distribution of the source. Such a randomly chosen code can have size as large as 2nIM and still be
r distinguishable after passing the channel.
a
Inquantumcasetwoproblemsarise. Firstly,aswehavementioned,weneedtwothings: notonlydistinguishability
byBob,butalsoindistinguishabilitybyEve(onecancallit”privacy”). Secondly,forafixedsource,unlikeinclassical
case, we have many different ensembles which are compatible with the source. Thus ”picking random code” is not
uniquely defined, so that we have to choose an ensemble. The advantage of our approach, is that it allows to see
that various ensembles can do the job. First of all, it turns out that for arbitrary ensemble, if we pick at random
2nIcoh vectors, they will be distinguishable for Bob. This we obtain adapting a one-shotversionof HSW theorem [8].
Thus to know that a given code is good, it remains to check privacy. It is easy to see that this will no longer hold
for arbitrary ensemble. To assure privacy, the ensemble must be reach enough. It turns out that checking privacy
for some ensembles is a fairly easy task. In this paper we will do this for ensemble used by Lloyd, and for ensemble
generated by Haar measure, deformed by source density matrix. Most likely, it also works for Gaussian codes (also
deformed by source).
II. QUANTUM ERROR - ANALOGUE OF CLASSICAL ERROR PROBABILITY
Here we will introduce a quantum parameter analogousto error probability in classical coding theorem. The main
points have been already found in [9], in a slightly different form.
2
We needthe followinglemma,whichsaysthatwhenAliceis productwithenvironment(Eve)thenthere existsBob
decoding, after which he shares with Alice pure state which is purification of Alice’s system
Lemma 1 For any pure tripartite state ψ , if reduced state ρ is product, then there exists unitary operation
ABE AE
UBB′, such that
IAE UBB′ ψABE 0 B′ = ψAB ψEB′ (1)
⊗ | i| i | i| i
Proof. Followsfromthefactthatallpurificationsofafixedstatearerelatedbyaunitarytransformationon(perhaps
extended) ancilla. Here we take the state to be σ and extended ancilla system is BB Without loss of generality
AE ′
we have assumed here that the system B is not smaller than A.
Here is version of the above lemma in approximate case.
Lemma 2 Consider a state ψ and suppose that
ABE
σ σ σ ǫ (2)
AE A E
|| − ⊗ ||≤
Then there exists unitary UBB′ such that
1
F(σA′ B,φAB)≥1− 2ǫ (3)
where σA′ B is reduced density matrix of state IAE ⊗UBB′|ψAEBi|0iB′ and φAB is purification of σA (reduced density
matrix of ψ ); denotes trace norm.
ABE
||·||
Proof. Using inequality F(ρ,σ) 1 1 ρ σ [10] we get F(σ ,σ σ ) 1 ǫ/2. Then, by definition of F,
≥ − 2|| − || AE A⊗ B ≥ −
there exists purification φABB′E such that
1
F(ψABE 0 B′,φABB′E)=F(σAE,σA σB) 1 ǫ (4)
| i| i ⊗ ≥ − 2
From the proof of previous lemma, we see that there exists unitary operation UBB′ which factorizes state φ into B
and B (again we assume here that dimension of the system B is no smaller than that of A). Thus
′
1
F(ψA′ BB′E,φ′AB ⊗φ′EB′)≥1− 2ǫ (5)
where|ψA′ BB′Ei=IAE⊗UBB′|ψABEi⊗|0iB′ andφ′AB⊗φ′EB′ =IAE⊗UBB′|φABB′Ei. Note thatφ′AB ispurification
of σ . From monotonicity of F under partial trace we get:
A
1
F(σA′ B,φAB)≥1− 2ǫ (6)
This ends the proof of the lemma.
Itsaysthatif Alice andEveareapproximatelyproduct,then there existsBob’s decoding,whichrestoreswithhigh
fidelity of entanglement with Alice.
Thus the parameter σ σ σ is what we couldcall”quantumerror”,ananalogueoferrorprobabilityin
AE A E Tr
|| − ⊗ ||
classical coding theorems.
It is however convenient to consider modified quantum error, which, if small, implies that Alice and Bob share a
state close to maximally entangled state of Schmidt rank N. Henceforth in paper we will us this type of quantum
error. It is given by
q = σ τ σ (7)
e AE A E
k − ⊗ k
where τ is normalized projector of rank N. From the above lemma we obtain the main result of this section:
Proposition 1 Consider arbitrary purestateψ . Letσ bereduceddensity matrix of ψ . Then, for arbitrary
ABE AB ABE
ǫ>0 if the quantum error (7) satisfies
q ǫ (8)
e
≤
then there exists Bob’s operation Λ such that
B
1
F(σ ,ψ+ ) 1 ǫ (9)
A′ B AB ≥ − 2
where σ =(I Λ )σ , and ψ+ = 1 N ii .
A′ B A⊗ B AB AB √N i=1| i
P
3
Therefore, the task is to find a bipartite state for Alice, such that if she will send half of it down the channel, then
the quantum error will be small, for N 2nIcoh. We will construct such a state from a code. Given a code α N
≃ {| i}α=1
the state will be
N
1
ψAA′ = ψα α (10)
√N | i| i
α=1
X
with ψ being orthonormalset of vectors. The task is thus to show that, if A is sent down the channel, then q is
α ′ e
{ }
small.
III. DISTINGUISHABILITY AND PRIVACY IMPLY SMALL QUANTUM ERROR
The channel from Alice to Bob implies dual channel to Eve, who represents environment. Thus if Alice sends a
state α , we obtain two kinds of output states: Bob’s output state σα and Eve’s output state σα. In this section we
| i B E
will show that a set of states = α for which Bob’s output states are approximately distinguishable, while Eve’s
C {| i}
output states are not, then the set is a good quantum code. This means that if Alice will create state
C
N
1
ψAA′ = ψα α (11)
√N | i| i
α=1
X
where ψ are orthonormal vectors. and send the system A down the channel, then the quantum error for arising
α ′
tripartite state ψ will be approximately zero. The distinguishability we will quantify by Holevo function.
ABE
Before we formulate suitable proposition, let us express output states of Bob and Eve in terms of the joint state
ψ obtained as described above. If we perform measurement onto basis ψ , the state of the systems BE will
ABE α
collapse to the state ψα . The states σα and σα are then given by
BE B E
σα =Tr (ψ ψ ), σα =Tr (ψ ψ ) (12)
B E | αih α|BE E B | αih α|BE
Let us now define more precisely distinguishability and privacy. For a set of N states = α and channel Λ let
C {| i}
us define distinguishability by Eve and distinguishability as follows
E B
D D
=χ (13)
E E
D
=χ (14)
B B
D
where σα are Eve’s and Bob’s output states defined above and χ = χ( 1,σα ),χ = χ( 1,σα ). Here
E,B E {N E} B {N B}
χ( p ,ρ )=S( p ρ ) p s(ρ ) with S(ρ)=Trρlogρ being von Neumann entropy.
{ i i} i i i − i i i
Proposition 2PThe quantuPm error satisfies the following inequality
q c +(logN ) (15)
e E B
≤ D −D
where c=√2ln2. p
Proof. Let subsystems of ψ are denoted by σ and their entropies by S with suitable subscript. Because σα
ABE X X B
and σα are reduced density matrices of the same state ψα we have
E BE
S(σα)=S(σα) (16)
E B
Thus
χ χ =S S (17)
B E B E
− −
so that we obtain
+(logN )=logN +S S =S(σ τ σ ) (18)
B E E AE AE A E
D −D − | ⊗
where we have used the fact that total state is pure so that S = S , and τ = I/N is maximally mixed state on
AE B A
the system A. Now, using well known relation S(ρσ) 1 ρ σ 2 [11] we obtain the required inequality. This
| ≥ 2ln2k − k
ends the proof of the proposition.
Remark. Essentially, we have merged two facts. First, as obtaind in [9] that when I is close to entropy of the
coh
source, then the error correction condition is approximately satisfied (i.e. q is small). Second, that χ χ =I
e B E coh
−
which was exploited in [5, 6].
4
IV. OVERVIEW
In spirit of Shannon we will consider mental construction: a source ρ⊗An, and a joint state φABE which emerges
from sending half of a purification of ρ⊗An down the channel (as before, E represents environment). We will also use
the state φ projected onto typical subspaces. This will be denoted by φ . The two states φ and φ will be shown
′ABE ′
to be close to each other in trace norm.
A code will be picked at random from ensemble which gives rise to typical version of ρ . Depending on chosen
A
ensemble, we will have different types of codes. For any code α we will consider a bipartite state of Alice of the
{| i}
form ψAA′ = √1N Nα=1|ψαi|αi with {ψα} being orthonormalset of vectors. This is the actual state that will be sent
downthechannel. TheresultingjointstateofAlice,BobandEvewillbedenotedbyψ ( )(shortlyψ ). Again
P ABE C ABE
aversionofthisstate,projectedontotypicalsubspaceswillbe denotedbyψ˜ . We willshowthatforcertaincodes
ABE
(i.e. for certain ensembles) the states ψ and ψ˜ are close to each other in trace norm.
Since the states are close to each other, it is enough to consider privacy and distinguishability for the projected
state. Interestingly, we will show that distinguishability is then merely a consequence of the fact that the ensemble
from which we choose the code gives rise to typical version of ρ . This we obtain by adapting one-shot version of
A
HSW thereom proven by Devetak [8]. Privacy should be checked case by case for different codes.
V. SOURCE, CHANNEL AND TYPICALITY
Let us fix a source ρ⊗An. Half of purification of the source is sent down the channel
Λ⊗n(·)= Ak(·)A†k (19)
k
X
(it is not actual sending but a mental construction, as in classical coding theorem). This creates pure state φ⊗ABnE
shared by Alice, Bob and environment. Explicitly we have
φ = √p i A i k (20)
ABE i A k B E
| i | i | i
i I k K
X∈ X∈
where ρ⊗An = i Ipi|iihi|, I is set of indices of complete eigenbasis of ρ⊗An, and K is set of indices of complete
eigenbasis of ρ⊗EPn.∈Later we will consider typical subspaces, and this we will indicate by omitting I and K.
Subsystems of this state are ρ⊗An, ρ⊗Bn =Λ⊗n(ρ⊗An) and ρ⊗En. Projectors onto typical subspaces of these states are
respectively: ΠA, ΠB and ΠE. The (unnormalized) typical versions of the states we denote by ρˆtAyp = ΠAρ⊗AnΠA,
ρˆtByp =ΠBρ⊗BnΠB, ρˆtEyp =ΠEρ⊗EnΠE. Explicitly we have
ρˆtyp = p i i (21)
A i| ih |
i
X
ρˆtByp =ΠB Akρ⊗AnA†kΠB (22)
k K
X∈
ρˆtEyp = Tr(Akρ⊗AnA†k′)|kihk′| (23)
k,k′
X
In second equation we write k K because the sum runs over all vectors k , while in first and third equation the
∈ | i
sum runs only overthe set of indices correspondingto vectorsfromtypicalsubspace. We haveto explainwhy typical
projector Π is given by k k . This follows form the fact that the Kraus operators can be chosenin such a way
E k| ih |
that the state ρE⊗n is diagPonal in basis {|ki}. We assume that our Kraus operators are such ones (so that the only
diagonal terms of the last equation are in fact nonzero).
We have lemma [12, 13, 14]
Lemma 3 (typical states) For arbitrary 0 <ǫ < 1/2, δ >0, for all n large enough we have that for X =A, X = B
and X =E:
1. The states ρˆtyp are almost normalized i.e.
X
Trρˆtyp 1 ǫ (24)
X ≥ −
5
2. All eigenvalues λ of ρˆtyp satisfy
X
2 n(SX+δ) λ 2 n(SX δ) (25)
− − −
≤ ≤
3. The ranks of states ρˆtyp satisfy
X
rk(ρˆtyp) 2n(SX+δ) (26)
X ≤
Remark. Both in this lemma, and in Prop. 3, ǫ can be exponential in n, i.e. the relations hold with
ǫ=e−cδ2n (27)
where c is a constant.
Let us now consider an unnormalized state
φ˜ =Π Π Π φ (28)
ABE A B E ABE
| i ⊗ ⊗ | i
The state can be written
φ˜ = √p i F i k (29)
ABE i A k B E
| i | i | i | i
i,k
X
where F =Π A .
k B k
The reductions of this state are a bit different than typical states ρˆtyp. We will denote them by ρ˜ , ρ˜ and ρ˜ .
X A B B
We have
ρ˜A = pi|iihi|Tr( Fk|iihi|Fk†) (30)
i k
X X
ρ˜B =ΠB AkρˆtAypA†kΠB = FkρˆtAypFk† (31)
k k
X X
ρ˜E = Tr(FkρˆtAypFk†′)|kihk′| (32)
k,k′
X
Finally, we consider normalized typical Alice’s state ρtyp given by
A
1
ρtyp = ρˆtyp (33)
A Trρˆtyp A
A
We then modify the above states ρ˜ , ρ˜ and ρ˜ into still unnormalized states
A B E
1
ρ′B = Trρˆtyp ρ˜B =ΠB AkρtAypA†kΠB = FkρˆtAypFk†
A k k
X X
1
ρ′E = Trρˆtyp ρ˜E = Tr(FkρtAypFk†′)|kihk′|
A k,k′
X
1
ρ′A = Trρˆtyp pi|iihi|Tr( Fk|iihi|Fk†) (34)
A i k
X X
A. Properties of states ρ′X
The states that we will use most frequently in the proof are ρ . We have the following proposition
′X
Proposition 3 For arbitrary 0<ǫ<1/2, δ >0, for all n large enough we have that for X =A, X =B and X =E:
1. The states ρ are almost normalized i.e.
′X
Trρ 1 ǫ. (35)
′X ≥ −
6
2. The eigenvalues λ of ρ satisfy
′X
λ (1+ǫ)2 n(SX δ) (36)
− −
≤
3. The ranks of states ρ satisfy
′X
rk(ρ′X)≤2n(SX+δ). (37)
4. The following inequality holds
Trρ′X2 ≤(1+ǫ)2(−nSX−δ). (38)
Proof. Wefirstnotethatρ˜ ρˆtyp,whichholds,becauseρ˜ canbeobtainedfromρˆtyp byprojectingontosecond
X ≤ X X X
subsystem of bipartite state, whose reduction is ρ˜ . Using lemma 3, eq. (24) we then obtain that for large n
X
ǫ
ρ′X ≤(1+ 4)ρˆtXyp (39)
This via lemma 3 immediately gives eq. (35) and eq. (36) as well as (37). To prove (38) we note that 0 X Y
≤ ≤
implies TrX2 TrY2.
≤
There follow useful expressions [1]
Tr(FkρtAypFk†′)Tr(Fk′ρtAypFk†)=Trρ′E2 (40)
k,k′
X
Tr(FkρtAypFk†Fk′ρtAypFk†′)=Trρ′B2 (41)
k,k′
X
One also defines matrix
1
ρi/o = Trρˆtyp pi|iiAhi|⊗Fk(|iiBhi|)Fk† (42)
A i,k
X
One finds that reduced density matrices of ρ are ρ and ρ . One can also find by direct checking that
i/o ′A ′B
Trρ2 min(Trρ2,Trρ2). (43)
i/o ≤ ′B ′A
VI. RANDOM CODES
As we have mentioned, in quantum case there is no unique way of drawing codes at random. In this paper we will
use two types of quantum random codes: Lloyd codes defined in [1] and uniform, source-distorted codes For those
codes we will show that the average quantum error is small, if we choose 2nR codewords, with R<I .
coh
Lloyd codes: The codes of [1] are defined as follows. To have a code consisting of N vectors, Alice picks N vectors
according the following distribution
α = √q eiφi i (44)
i
| i | i
i
X
where q and i areeigenvaluesand eigenvectorsof ρtyp,i.e. q =p /Trρˆtyp. The phases φ aredrawnindependently
i | i A i i A i
and uniformly from unit circle. Average over such α’s we will denote by (...)dα. Note that the ensemble defining
Lloyd codes give rise to typical version of ρ⊗An R
α αdα =ρtyp. (45)
| ih | A
Z
Uniform source-distorted codes. Alice picks N vectors according to the following distribution
α = d ρtyp φ (46)
| i A A | i
q
p
7
where φ is taken uniformly from typical subspace of system A (i.e. give by projection Π . The codewords are not
A
| i
of unit length but with high probability they are almost normalized. We prove it by Chebyshev inequality. Compute
variance of αα =d φρtyp φ obtaining
h | i Ah | A | i
d2 d2
Var = A Tr(ρtyp)2+ A Trρtyp 1 Tr(ρtyp)2 (1+ǫ)2 n(SA δ) (47)
d2 +d A d2 +d A − ≤ A ≤ − −
A A A A
which gives
(1+ǫ)
Prob( αα (1 ǫ,1+ǫ)) 1 2 n(SA δ) (48)
− −
h | i 6∈ − ≥ − ǫ2
The last inequality comes from lemma 3, and holds for all n large enough. If we take N = 2nR with R < I , we
coh
get that for arbitrarily fixed ǫ the probability of failure goes exponentially down for all codewords. However we will
not use so strong result. It will be enough to know that a randomly picked codewordwith high probability has norm
close to 1. Finally, note that, again we have
α αdα =ρtyp. (49)
| ih | A
Z
VII. JOINT STATE ψABE FOR A FIXED CODE SENT DOWN THAT CHANNEL.
In this section we discuss properties of the actual state that can be obtained by use of codes. The main result of
this sectionis that with highprobability,the stateψ is close to its versionprojectedonto typicalsubspcaces. We
ABE
willseparatelydiscuss the caseof normalizedandunnormalizedcodes. Fornormalizedcodes, the resultfollowssolely
fromthe factthe ensemble ofthe code givesriseto ρtyp. Forunnormalizedcodesone hasto checkit casebycase(we
A
will check it here for uniform source-distorted codes).
A. Normalized codes
Having fixed a code = α N , Alice creates state
C {| i}α=1
N
1
ψAA′ = ψα A α A′ (50)
√N | i ⊗| i
α=1
X
where ψ N is orthonormal set of N vectors. Then she sends A down the channel Λ n to Bob. The emerging
{| αi}α=1 ′ ⊗
state ψ is the following:
ABE
1
ψ =ψ ( )= ψ A α k (51)
ABE ABE α A k B E
C √N | i | i | i
α k K
XX∈
Subsequently, we consider projected version of the state
ψ˜ =I Π Π ψ . (52)
ABE A E B ABE
| i ⊗ ⊗ | i
This state can be written as
1
ψ˜ = ψ F α k (53)
ABE α A k B E
| i √N | i | i | i
α,k
X
where recall that F =Π A , and we sum only over typical indices k.
k B k
Letusnowprovethatwithhighprobability,the stateψ˜ willbe closetotheactualstateψ sharedbyAlice,
ABE ABE
Bob and Eve.
Proposition 4 For arbitrary 0<ǫ<1, for n large enough we have
ψ ψ ψ˜ ψ˜ ǫ (54)
ABE ABE ABE ABE
||| ih |−| ih |||≤
with high probability for any normalized codes satisfying α αdalpha=ρtyp
| ih | A
R
8
Proof. From lemma 9 it follows that it is enough to prove that for n large enough we have
ψ˜ ψ 2 1 ǫ (55)
ABE ABE
|h | i| ≥ −
We will show now that it is true with high probability. We have
1
|hψ|IE ⊗ΠB ⊗ΠE|ψi|2 = N Tr( Fk†Fk|αihα|) (56)
α k
X X
We note that
|hψ|IE ⊗ΠB ⊗ΠE|ψi|2 α=Tr FkρtAypFk† =Trρ′B (57)
k
(cid:10) (cid:11) X
From proposition 3, Eq. (35) it follows that for n large enough we have Trρ 1 ǫ2. The random variable
1 ψ I Π Π ψ 2 is nonnegative, so we can use Markov inequality obta′Bini≥ng −
E B E
−|h | ⊗ ⊗ | i|
Prob( ψ I Π Π ψ 2 1 2ǫ) ǫ (58)
E B E
|h | ⊗ ⊗ | i| ≤ − ≤
This ends the proof.
Remark. Note that in proof we have only used the fact that codewords are normalized, and picked from an
ensmeble whose density matrix is ρtyp.
A
B. Unnormalized codes
If codewords are not normalized, Alice prepares the following state:
N
1
ψAA′ = ψα A α A′ (59)
N αα α=1| i ⊗| i
α=1h | i X
q
P
The emerging Alice, Bob and Eve state is then
1
ψ = ψ A α k (60)
ABE α A k B E
N αα α k K| i | i | i
α=1h | iX X∈
q
P
We will consider unnormalized version of this state
1
ψA′ BE = √N |ψαiAAk|αiB|kiE (61)
α k K
X X∈
and project it onto typical subspaces, obtaining
1
ψ˜ = ψ F α k . (62)
ABE α A k B E
| i √N | i | i | i
α,k
X
We will now show that for uniform source-distorted codes the state ψ˜ is close in trace norm to the actual state
ABE
ψ shared by Alice and Bob.
ABE
Proposition 5 For arbitrary 0<ǫ<1, for n large enough we have
ψ ψ ψ˜ ψ˜ ǫ (63)
ABE ABE ABE ABE
||| ih |−| ih |||≤
with high probability for uniform source-distorted codes.
Proof. From lemma 9 it follows that we have to show that with high probability we have
ψ ψ˜ 2 1 ǫ (64)
|h A′ BE| ABEi| ≥ −
9
and
ψ ψ 1+ǫ. (65)
h A′ BE| A′ BEi≤
The last inequality reads as
1
αα 1+ǫ (66)
N h | i≤
α
X
We will now compute average and variance of this quantity over codes. Using (47) we get
1
αα = αα =1 (67)
N h | i h | i
α C C
(cid:10) X (cid:11) (cid:10) (cid:11)
and
Var =Var( αα ) (1+ǫ)2 n(SA δ) (68)
− −
h | i ≤
since codewords are picked independently. Thus form Chebyshev inequality we get
1 1+ǫ
Prob( αα 1 ǫ) 2 n(SA δ) (69)
− −
|N h | i− |≥ ≤ ǫ2
α
X
Thus with high probability the inequality (65) is satisfied. To prove the same for inequality (64) we write
N
1 1
|hψ′|IE ⊗ΠB ⊗ΠE|ψ′i|2 = N Tr( Fk†Fk|αihα|)≡ N Xi (70)
α k i=1
X X X
where X defined by the above equality are i.i.d random variables. We note that
i
Xi α=Tr FkρtAypFk† =Trρ′B ≥1−ǫ (71)
k
(cid:10) (cid:11) X
We compute variance.
1
Var( X )=VarX = αY α αY α (Trρ )2 (72)
N i i h | | ih | i α− ′B
i
X (cid:10) (cid:11)
where Y = kFk†Fk. For particular case of uniform source-distortedcodes we obtain
P Var ≤TrYρtAypYρtAyp +(TrYρtAyp)2−(Trρ′B)2 =Trρ′B2 ≤(1+ǫ)2−n(SB−δ) (73)
where we have used Tr( kFk†FkρtAyp)=Trρ′B. Using Chebyshev inequality we get
P 1 ǫ
Prob(|hψ˜ABE|ψABEi|2 ≤1−ǫ)≤ ǫ−2 2−n(SB−δ) (74)
This proves that with high probability, also the inequality (64) is satisfied. This ends the proof.
VIII. CODING THEOREM VIA DISTINGUISHABILITY AND PRIVACY.
In this section we prove the main result of this paper: direct coding theorem by use of distinguishability and
privacy. Our proof will show that for normalized random codes, only two conditions assure that the codes can be
used in coding theorem:
1) The ensemble gives rise to ρtyp.
A
2) Privacy: codewords produce indistinguishable Eve’s output states.
We see that these are quite modest conditions.
10
A. Bob’s and Eve’s output states.
It is useful to write down Alice and Bob’s output states in terms of Kraus operators. We have
σαE = Tr(Ak|αihα|A†k′)|kihk′| (75)
k,k′ K
X∈
σαB = (Ak|αihα|A†k)≡Λ⊗n(|αihα|) (76)
k K
X∈
(77)
Now, we introduce modified states σEα′ and σBα′ given by
σEα′ = Tr(Fk|αihα|Fk†′)|kihk′| (78)
k,k′
X
σBα′ = Fk|αihα|Fk† (79)
k
X
Thesestatesarereductionsofthestate(53)and(62),respectively. Fornormalizedcodes,thestatesaresubnormalized,
while for unnormalized codes, their trace may be both below and above 1. They have the following properties. For
any code satisfying α αdα =ρtyp we obtain
| ih | A
R
σEα′dα =ρ′E, σBα′dα =ρ′B (80)
Z Z
Moreover due to propositions 4 and 5, and monotonicity of trace norm under trace preserving CP maps, they are
close on average to the original states σE, σB, which is stated in the following lemma
α α
Lemma 4 For arbitrary 0<ǫ<1/2 and for n high enough, with high probability we have
1 1
N ||σαE −σEα′||≤ǫ, N ||σαB −σBα′||≤ǫ. (81)
α α
X X
where ǫ is independent on N, and can be taken to be exponential in n.
Recall, that ǫ is exponential in n.
B. Privacy
We first prove that if N = 2nR with R < I then the distinguishability by Eve is arbitrarily small. Let us first
coh
bound χ by average norm distance. We apply Fannes inequality [15]
E
S(ρ) S(σ) σ ρ logd+η( σ ρ ) (82)
| − |≤|| − || || − ||
where ρ and σ are any states satisfying ρ σ < 1/3, η(x) = xlogx, and d is dimension of the Hilbert space.
Denoting 1 σα σ =x, and u|s|ing−co|n|vexity of η we ob−tain
N α || E − E||
∈C
P χ =χ xnlogd+η(x) (83)
E E
≤
where d is input dimension of Hilbert space of ρ (i.e. it is a constant). Thus if we show that 1 σα σ is
A N α || E − E||
exponentially small for rate R<I , then also χ will be exponentially small. ∈C
coh E
P
Proposition 6 (privacy). For the state ψ and N = 2nR where R < I with high probability (over codes) we
ABE coh
have
1
σα σ ǫ. (84)
N || E − E||≤
α
X∈C