Quantum Conditional Mutual Information, Reconstructed States, and State
Redistribution
Fernando G.S.L. Brandão1 , Aram W. Harrow2 , Jonathan Oppenheim3 , Sergii Strelchuk4
1
4
Department of Computer Science, University College London
2
Institute for Theoretical Physics,
Massachusetts Institute of Technology
3
Department of Physics, University College London
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge
using Lemma 4.2.2 of [4]. For V an isometry from X →
Y Z, define
SUPPLEMENTAL MATERIAL
A.
Auxiliary lemmas
dX
1 X
|σ(V )iXY Z = √
|ii V |ii
dX i=1
Lemma 1. If π ≤ 2λ σ, then S(ρ||π) ≥ S(ρ||σ) − λ.
The proof of the lemma follows directly from the operator monotonicity of the log function.
⊓
⊔
Lemma 2 is due to Audenaert and Eisert:
Lemma 2 (Theorem 3 of [1]). For all states ρ and σ on a
d-dimensional Hilbert space, with T = kρ − σk1 and β =
λmin (σ),
1
T log β
S(ρ||σ) ≤ T log(d) + min −T log T,
−
. (1)
e
2
The next lemma is due to M. Piani. It will suffice to
state it for the case where M is the set of all measurements.
Lemma 3. [Theorem 1 of [2]] Consider two systems X and
Y with joint Hilbert space HX ⊗ HY , and a convex reference
set K. Suppose the reference set K is such that for all POVM
elements Mi and all σXY ∈ K, trX (MiX σXY ) ∈ P (up to
normalization). Then for every ρXY ,
min S(ρXY ||σXY )
(2)
σXY ∈K
≥ min MS(ρX ||σX ) + min S(ρY ||σY ).
σX ∈K
σY ∈K
The following Lemma is due to Fawzi and Renner [3]
who stated it in a slightly more general form. It was
used in their proof of Eq. (4) in the main text. Below is a
very similar, but somewhat shorter, proof.
Lemma 4. Suppose ρX n Y n satisfies ρX n = τX n and
[ρ, PXY (π)] = 0 for all π ∈ Sn . Then there exists a measure
µ over states σXY (independent of ρ) with each σX = τY and
Z
2 2
ρX n Y n ≤ nO(dX dY ) σ ⊗n µ(dσ),
(3)
where dX , dY are the dimensions of X and Y .
Proof. First purify ρ to a state |ρiX n Y n Z n ∈ Symn (XY Z)
(the symmetric subspace in (XY Z)n ) with dZ = dX dY
(4)
and σ(V ) = |σ(V )i hσ(V )|. Observe that σ(V )X = τX .
We will show that
Z
2 2
|ρi hρ| ≤ nO(dX dY ) σ(V )⊗n µ(dσ),
(5)
which will imply Eq. (3).
Our strategy will be to expand both sides of Eq. (5) in
the Schur basis. Schur duality uses the following notation:
M
Ud ×Sn
ˆ λ.
Qdλ ⊗P
(6)
(Cd )⊗n ∼
=
λ∈Par(n,d)
This is explained in detail in [5], but briefly, Par(n, d) denotes the set of partitions of n into ≤ d parts, Qdλ is an
irrep of the unitary group Ud , Pλ is an irrep of the symˆ means that we interpret the tensor
metric group Sn , ⊗
U ×S
n
d
product as an irrep of Ud × Sn , and ∼
= means that the
isomorphism respects this representation structure. Let
USch denote the unitary transform realizing the isomorphism in Eq. (6). We can write
X
YZ
(USch
⊗ USch
) |ρi =
X
cλ1 ,λ2 |λ1 iX |λ2 iY Z |χλ1 ,λ2 i |θλ1 ,λ2 i , (7)
λ1 ∈Par(n,dX )
λ2 ∈Par(n,dY dZ )
P
where λ |cλ |2 = 1, |χλ1 ,λ2 i , |θλ1 ,λ2 i are arbitrary unit
vectors in QdλX
⊗QλdY2 dZ and Pλ1 ⊗Pλ2 respectively. How1
ever, the permutation invariance and Schur’s Lemma
mean that (following arguments along the lines of Section 6.4.1 of [5]) the only nonzero terms have λ1 = λ2
and |θλ,λ i =: |Φλ i is the unique permutation-invariant
state in Pλ ⊗Pλ . Thus we can (using dX ≤ dY dZ ) rewrite
Eq. (7) as
X
X
YZ
cλ |λiX |λiY Z |χλ i |Φλ i .
(USch
⊗ USch
) |ρi =
λ∈Par(n,dX )
(8)
2
To calculate cλ we use the fact that ρX n = τX n . Thus
measuring the irrep label should yield outcome λ with
probability dim Pλ dim QdλX /dnX , and we have
s
dim Pλ dim QdλX
.
(9)
cλ =
dnX
A similar argument implies that
⊗n
X
YZ
(USch
⊗ USch
) |σ(V )i =
X
cλ |λiX |λiY Z |χλ (V )i |Φλ i , (10)
λ∈Par(n,dX )
for some states |χλ (V )i. The coefficients cλ are the same
⊗n
as in Eq. (9) because σ(V )⊗n
An = τAn . Averaging σ(V )
over all isometries V yields a state that commutes with
(UX ⊗IY Z )⊗n and (IX ⊗UY Z )⊗n for all UX ∈ UdX , UY Z ∈
UdY dZ . Thus
(11)
It follows from (8) and (11) that
We now turn to proving the measured entropy lower
bound:
Proposition 6 (Eq. (17) in the main text). For every state
ρBCR one has
lim
1
⊗n
S(ρ⊗n
BCR ||Λ ⊗ idRn (ρBR ))
n
MS(ρBCR ||Λ ⊗ idR (ρBR )).
(15)
min
n→∞ Λ:B n →B n C n
min
Λ:B→BC
Λ̃(ω) :=
1 X
PBC (π)Λ(PB† (π)ωPB (π))PBC (π)† ,
n!
π∈Sn
(16)
with PX (π) a representation of a permutation π from
Sn (symmetric group of order n) in X ⊗n
such that
PX (π) |a1 , . . . , an i = aπ−1 (1) , . . . , aπ−1 (n) . Let Sym
be the set of all permutation-invariant quantum operations, i.e. all Λ such that Λ = Λ̃.
Using Proposition 3 in the main text and the fact that
the relative entropy is doubly convex we obtain [7]
|ρi hρ| ≤ (max dim QdλX dim QλdY dZ ) E[σ(V )⊗n ]
V
λ
dX + n − 1 dY dZ + n − 1
=
E[σ(V )⊗n ]
V
n
n
≤ ndX ndY dZ E[σ(V )⊗n ]
V
=n
and the fact that trB n ((π T ⊗IB n C n )JΛ ) = Λ(π)/dim(B)n
and trB n ((π T ⊗ IB n C n )JE⊗n ) = E ⊗n (π)/dim(B)n .
⊓
⊔
Proof. For Λ : B n → B n C n , define
λ
λ∈Par(n,dX )
d2X d2Y +dX
(14)
trB n ((π T ⊗ IB n C n )JΛ )
Z
≤ poly(n) trB n ((π T ⊗ IB n C n )JE⊗n )µ(dE),
≥
X
YZ
X
YZ †
(USch
⊗ USch
) E[σ(V )⊗n ](USch
⊗ USch
) =
V
X
|cλ |2 |λ, λi hλ, λ|⊗τQdX ⊗τQdY dZ ⊗|Φλ i hΦλ | .
λ
and each σB̄ = τB̄ . This latter condition means that each
σ can be also thought of as JE for some E : B → BC. We
complete the proof using the relation:
E[σ(V )⊗n ].
V
⊓
⊔
B. The measured entropy lower bound
To prove the inequality stated in the Eq. (17) of the
main text, we first prove the following lemma which is a
version of the postselection technique of [6] for quantum
operations.
Lemma 5. For every permutation-invariant quantum operation Λ : B n → B n C n and every state π ∈ B n ,
Z
Λ(π) ≤ poly(n) E ⊗n (π)µ(dE),
(12)
where µ is a measure over quantum operations E : B → BC.
Proof. Since Λ : B n → B n C n is permutation-invariant, it
n
follows that its Jamiolkowski state JΛ ∈ D((B ⊗ B n ⊗
C n ) (with B ∼
= B) is permutation-invariant. We now
apply Lemma 4 in the Supplemental Material to find a
distribution µ over σ ∈ D(B ⊗ B ⊗ C) with
Z
JΛ ≤ poly(n) σ ⊗n µ(dσ),
(13)
lim
min
≥ lim
min
n→∞ Λ:B n →B n C n
n→∞ Λ:B n →B n Rn
Λ∈Sym
1
⊗n
S(ρ⊗n
BCR ||Λ ⊗ idRn (ρBR ))
n
1
⊗n
S(ρ⊗n
BCR ||Λ ⊗ idRn (ρBR )).
n
Lemma 5 gives that for every Λn : B n → B n C n ∈
Sym,
(Λn ⊗ idRn )(ρ⊗n
BR )
Z
≤ poly(n) (E ⊗ idR (ρBR ))⊗n µn (dE),
(17)
with µ(dE) a measure over quantum operations E :
B → BC. Using the previous equation and the operator monotonicity of the log (see Lemma 1 above),
1
⊗n
(18)
S(ρ⊗n
BCR ||Λ ⊗ idRn (ρBR ))
n
Z
1
⊗n
⊗n
≥ lim min S ρBCR k (E ⊗ idR (ρBR )) µn (dE) .
n→∞ µn n
lim
min
n→∞ Λ:B n →B n C n
To complete the proof we make use of Lemma 3
above. Consider the state ρ⊗n
BCR and let X be the first
3
copy of ρBCR in the tensor product and Y the remaining ρ⊗n−1
BCR . Define
[
K=
conv{(E ⊗ idR )(ρBR )⊗k : E : B → BC} ,
Iterating the equation above n times gives Eq. (15).
⊓
⊔
k∈N
(19)
i.e. the convex hull of tensor products of reconstructed
states. It is easy to check that K satisfies the assumption
of Lemma 3 above. Therefore:
Z
⊗n
⊗n
(20)
min S ρBCR k (E ⊗ idR (ρBR )) µn (dE)
µn
Z
≥ min MS ρBCR k (E ⊗ idR (ρBR ))µ(dE)
µ
Z
⊗n−1
⊗n−1
µn−1 (dE) .
+ min S ρBCR k (E ⊗ idR (ρBR ))
µn−1
[1] K. M. R. Audenaert and J. Eisert. Continuity bounds on
the quantum relative entropy. J. Math. Phys., 46(10):–, 2005,
{\ttfamilyarXiv:quant-ph/0503218}.
[2] M. Piani. Relative entropy of entanglement and restricted
measurements. Phys. Rev. Lett., 103:160504, Oct 2009,
{\ttfamilyarXiv:0904.2705}.
[3] O. Fawzi and R. Renner. Quantum conditional mutual information and approximate markov chains, 2014,
{\ttfamilyarXiv:1410.0664}.
[4] R. Renner. Security of quantum key distribution. PhD
thesis, ETHZ, Zurich, 2005, {\ttfamilyarXiv:quant-ph/
0512258}.
[5] A. W. Harrow. Applications of coherent classical communica-
tion and Schur duality to quantum information theory. PhD
thesis, M.I.T., Cambridge, MA, 2005, {\ttfamilyarXiv:
quant-ph/0512255}.
[6] M. Christandl, R. Koenig, and R. Renner. Post-selection
technique for quantum channels with applications to
quantum cryptography. Phys. Rev. Lett., 102:020504, 2009,
{\ttfamilyarXiv:0809.3019}.
⊗n
[7] In more detail:
S(ρ⊗n
=
Eπ∈Sn
BCR ||Λ(ρBR ))
⊗n
π
π
†
π
π
π
π
†
S(PCBR ρBCR (PCBR ) ||PCBR Λ((PBR )† ρ⊗n
P
)(P
CBR ) )
BR BR
⊗n
π
π † ⊗n π
π
†
=
≥
Eπ∈Sn S(ρBCR ||PCB Λ((PB ) ρBR PB )(PCB ) )
π
π † ⊗n π
π
†
with
S ρ⊗n
BCR || Eπ∈Sn PCB Λ((PB ) ρBR PB )(PCB ) ,
π
:= PCBR (π).
PCBR