3.6 Correlated source coding: distributed compres-
Charlie. This is possible provided QB ≥ H(B)ϕ. The other party, in this case Alice, then implements the fully quantum Slepian-Wolf protocol with Charlie playing the role of Bob. This is possible providedQA≥I(A;R)/2. Looking at the total number of qubits required gives a curious symmetrical formula:
QA+QB≥ 1
2I(A;R)ϕ+H(B)ϕ = H(A)ϕ+H(B)ϕ+H(AB)ϕ
2 . (3.39)
By switching the roles played by Alice and Bob and also time-sharing between the re- sulting two protocols, we find that the region defined by
QA ≥ 12I(A;R)ϕ
QB ≥ 12I(B;R)ϕ QA+QB ≥ H(A)ϕ+H(B)2ϕ+H(AB)ϕ
(3.40)
is contained in the achievable rate region SW(ϕ).
In fact, the region is in some casesequal toSW(ϕ), as we will see by proving a general outer bound on the achievable rate region. Assume that (QA, QB) ∈ SW(ϕ). To begin, fix n > N(²) and let WA and WB be the environments for the Stinespring dilations of the encoding operations EA and EB.
To boundQA, assume that Charlie has received bothCB and WB, that is, all of Bn. Let WC be the output environment for the dilation of Charlie’sD. Again, without loss of generality we can assume that the initial environment state is an unentangled pure state.
First we will make rigorous the intuition that AbBb is almost decoupled from WC. By the fidelity condition,
λmax(ϕRnAbBb) ≥ hψ|AnBnRnϕRnAnBn|viAnBnRn
≥ 1−²,
where λmax(ϕRnAbBb) denotes the maximum eigenvalue of ϕRnAbBb.
Therefore, |ϕiRnAbBWb AWC can be Schmidt decomposed as
|ϕiRnAbBWb AWC =X
i
pλc|iiWAWC|iiWAWC, (3.41)
where λmax≥1−² and
hϕ|RnAbBWb AWC(ϕRnAbBb⊗ϕWAWC)|ϕiRnAbBWb AWC
=
ÃX
i
pλihi|RnAbBbhi|WAWC
! ÃX
j
λj|jihj|RnAbBb ⊗X
k
λk|kXk|WAWC
!
ÃX
`
pλ`|`iRnAbBb|`iWAWB
!
= X
i
λ3i
≥ (1−²)3
≥ 1−3²3. So,
F(|ϕiRnAbBWb AWC, ϕRnAbBb⊕ϕWAWC)≥1− 3
2 ²2 (3.42)
and
D(|ϕiRnAbBWb AWC, ϕRnAbBb⊕ϕWAWC)≤√
3². (3.43)
by the contractivity of distance we have
D(ϕAbBWb C, ϕAbBb⊗ϕWC)≤√ 3² We can now apply the Fannes inequality to yield
¯¯
¯H(ϕAbBWb C)−H(ϕAbBb ⊗ϕWC)
¯¯
¯≤√
3²log(dnAdnBdWC) +η(√
3²) (3.44) for ²≤ √13e.
Since the environment for any quantum operation ρ 7→ ²(ρ) can be modelled as a Hilbert Space of less than d2 dimensions, whered=dim(ρ) we have that
dimWC ≤d2nAd2nB.
Therefore,
|H(ϕAbBWb C)−H(ϕAbBb⊗ϕWC)| ≤n3√
3²log(dAdB) +η(√
3²) (3.45)
for ²≤ √13e.
Next we will make rigorous the intuition that WA nearly purifies WC. Since
F(|ψiAnBnRn, ϕRnAbBb)≥1− ²
2, (3.46)
D(|ϕiAnBnRn, ϕRnAbBb)≤√
² (3.47)
and by Fannes inequality,
H(ϕRnAbBb)≤n√
²log(dAdBdR) +η(√
²), (3.48)
(for √
²≤ 1e). Therefore, n√
²log(dndBdA) +η(√
²)≥H(ϕRnAbBb) = H(WAWC)≥ |H(WA)−H(WC)| (3.49) The first inequality is because RnAbBWb AWC is in a pure state, and the second inequality is the Aracki-Lieb inequality.
Finally we will make rigorous the intuition that Rn nearly purifies AbB.b By the contractiivity of distance,
D(ψAnBn, ϕAbBb)≤D(|ψiAnBnRn, ϕRnAbBb)≤√
² (3.50)
and
D(ψRn, ϕRn)≤D(|ψiAnBnRn, ϕRnAbBb)≤√
² (3.51)
By application of the Fannes inequality,
|H(ϕAbBb)−H(ψAnBn)| ≤n√
²log(dAdB) +η(√
²) (3.52)
and
|H(ϕRn)−H(ψRn)| ≤n√
²log(dAdB) +η(√
²). (3.53)
Therefore,
|H(ϕAbBb)−H(ϕRn)|
≤ |H(ϕAbBb)−H(ψAnBn)|+|H(ψAnBn)−H(ϕAn)|
= |H(ϕAbBb)−H(ψAnBn)|+|H(ψRn)−H(ϕRn)|
≤ n√
²log(dAdBdR) + 2η(√
²). (3.54)
The first inequality is an application of the triangle inequality, while the first equality is because AnBnRn is a pure state.
Putting (3.45),(3.49) and (3.54) together and using the subadditivity of the Von Neumann entropy and the fact that the overall state is pure we have
|H(Bn) +H(Cn)| ≥H(BnCA) =H(wcAbB)b
≥ H(Wc) +H(AbB)b −n3√
3²log(dAdB)−η(√ 3²)
≥ H(WA)−n√
²log(dAdBdR)−η(√
²) +H(Rn)
−n√
²log(dAdBdR)−2η(√
²)−n3√
3 log(dAdB)−η(√ 3²)
= H(WA) +H(Rn)−n√
²log(d2Ad2BD2R)−n3√
3²log(dAdB)−4η(√
²)
≥ H(An)−H(CA) +H(Rn)−n√
²log(d2Ad2Bd2R)−n3√
3²log(dAdB)−4η(√
²).
Therefore,
2nQA ≥ 2nH(CA)
≥ H(An)−H(Bn) +H(Rn)−n√
²log(d2Ad2Bd2n)−n3√
3²log(dAdB)−4η(√
²)
= nI(A;R)−n√
²log(d2Ad2Bd2R)−n3√
3 log(dAdB)−4η(√
²) and so,
QA≥ 1
2I(A;R)−1 2
√²log(d2Ad2Bd2R)− 3√ 3
2 ²log(dAdB)− 2η(√
²)
n . (3.55)
Since this is true for all ² >0 we have that QA ≥ 1
2I(A;R). (3.56)
Switching the roles of Alice and Bob gives the corresponding inequality QB ≥ 1
2I(B;R). (3.57)
To bound QA+QB let us return to the situation where Alive and Bob perform their original encoding. Then,
H(An) = H(CAWA)≤H(WA) +H(CA)≤H(WA) +nQA. (3.58) The first equality follows from the fact that the initial environment is a pure unentangled state and from the unitary invariance of the Von Neumann entropy.
Combining with the analogous inequality forB leads to
n(QA+QB)≥n[H(A) +H(B)]−H(WA)−H(WB). (3.59) By similar arguments as before,
|H(WAWBRn)−H(WAWB)−H(Rn)| ≤n√
3²log(d2Ad2BdR) +η(√
3²), (3.60) for ²≤ √13e.
So,
H(CACB) = H(WAWBRn)
≥ H(WA) +H(WB)−I(WA;WB) +H(Rn)
−n√
3²log(d2Ad2BdR)−η(√ 3²).
Using the purity of overall the overall state, however, gives H(Rn) = nH(AB), which combined with the bound H(CACB)≤n(QA+QB), leads to the inequality
H(WA) +H(WB) ≤ n(ϕA+ϕB)−nH(AB) +I(WA;WB) +n√
3²log(d2Ad2BdR) +n(√
3²). (3.61)
Adding equations (3.59) and (3.61),
2n(QA+QB)≥nH(A) +nH(B) +nH(AB)−I(WA;WB)−n√
3²log(d2Ad2BdR)−η(√ 3²).
(3.62) Thus,
QA+QB ≥ 1 2
"
H(A) +H(B) +H(AB)−I(WA;WB)
n −√
3²log(d2Ad2BdR)− η(√ 3²) n
# . (3.63) Now, let T :Rn →R0 be any CPTP map on Rn
I(WA;WB)−I(WA;WB|R0)
= (H(WA) +H(WB)−H(WAWB))−(H(WA|R0) +H(WB|R0)−H(WAWB|R0))
= H(WA) +H(WB)−H(WAWB)−H(WAR0) +H(R0)
−H(WBR0) +H(R0) +H(WAWBR0)−H(R0)
= H(WA)−H(WAR0) +H(WB)−H(WBR0)−H(WAWB) +H(WAWBR0)
≤ −H(R0) +n√
3²log(d)A2dR) +η(√
3²)−H(B0) +n√
3²log(d2BdR) +η(√
3²) +H(R0)−H(WAWB) +H(WAWBR0)
= H(WAWBR0)−H(WAWB)−H(R0) +n√
3²log(d2AdR) +n√
3²log(d2BdR) + 2η(√ 3²)
≤ n√
3²log(d2Ad2BdR) +n√
3²log(d2AdR) +n√
3²log(d2BdR) + 3η(√ 3²)
= n√
3²log(d4Ad4Bd2R) + 3η(√ 3²),
here we have used similar arguments as before to make rigorous the intuitions that WAR0, WBR0 and WAWBR0 are almost uncorrelated with R0, followed by the Fannes inequality.
We have
I(WA;WB)≤I(WA;WB|R0) +n√
3²log(d4Ad4Bd2R) + 3η(√
3²) (3.64)
By the monotonicity of mutual information under local operations, I(WA;WB)≤I(An;Bn|R0) +n√
3²log(d4Ad4Bd2R) + 3η(√
3²). (3.65)
Therefore,
QA+QB ≥ 1
2[H(A) +H(B) +H(AB)]− 1
2I(A;B|R)
−3√
3²log(d4Ad4Bd2R)− 4η(√ 3²) n −√
3²log(d2Ad2BdR)
= 1
2[H(A) +H(B) +H(AB)]− 1
2I(A;B|R)
−√
3²log(d6Ad6Bd3R)−4η(√ 3²) n
≥ 1
2[H(A) +H(B) +H(AB)]−Esq(ϕAB), (3.66) whereEsq(ϕAB) is the squashed entanglement ofϕAB, defined as the infimum of 12I(A;B|E) over extensions ϕABE of ϕAB [81]. We have used explicitly the fact, proved in the cited paper, that Esq(ϕ⊗n) =nEsq(ϕ).
We have therefore proved the following outer bound on the achievable rate region SW(ϕ):
QA ≥ 12I(A;R)ϕ QB ≥ 12I(B;R)ϕ
QA+QB ≥ H(A)ϕ+H(B)2ϕ+H(AB)ϕ −Esq(ϕ).
(3.67)
In the special case whereϕAB is separable, Esq(ϕ) = 0, which implies that the region defined by Eq. (3.40) is optimal. Under certain further technical assumptions, namely that ϕAB be the density operator of an ensemble of product pure states satisfying a condition called irreducibility, the same conclusion could be found in Ref. [82]. That paper, however, was unable to show that the bound was achievable.
The appearance of the squashed entanglement in (3.67) may seem somewhat mysteri- ous, but a slight modification of the protocols based on fully quantum Slepian-Wolf will
lead to an inner bound on the achievable region that is of a similar form. Specifically, let D0(ϕAB) be the amount of pure state entanglement that Alice and Bob can distill from ϕAB without engaging in any communication. Since this pure state entanglement is decoupled from the reference system R, they could actually perform this distillation process and discard the resulting entanglement before beginning one of their FQSW- based compression protocols. While neither I(A;R) nor I(B;R) would change, each of H(A) and H(B) would decrease by D0(ϕAB). The corresponding inner bound on the achievable rate region SW(ϕ) would therefore be defined by the inequalities
QA ≥ 12I(A;R)ϕ QB ≥ 12I(B;R)ϕ
QA+QB ≥ H(A)ϕ+H(B)2ϕ+H(AB)ϕ −D0(ϕ).
(3.68)
The only gap between the inner and outer bounds, therefore, is a gap between different measures of entanglement.