A stronger subadditivity relation? With applications to squashed entanglement, sharability and separability Andreas Winter1 and Ke Li2 Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.∗ Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Dated: Presented at BIRS workshop, 27 February – 2 March 2012) 1

2

1. The question. It is well-known that the relative entropy D(ρkσ) = Tr ρ(log ρ − log σ) is monotonic under quantum channels. I.e., for a cptp map T : A → B, D(ρkσ) ≥ D(T ρkT σ). This is well-known to be equivalent to the strong subadditivity of the von Neumann entropy. We would like to know whether one can strengthen this: More precisely, for given cptp map T : A → B and states ρ, σ on A, does there exist a cptp map R : B → A with RT σ = σ and D(ρkσ) − D(T ρkT σ) ≥ D(ρkRT ρ) ?

(1)

Furthermore, can R be chosen to depend only on T and σ, R = R(T, σ), and with the following “functorial” properties? 1. For any state σ, R(id, σ) = id; 2. For any maps T1 and T2 , and states σ1 and σ2 , R(T1 ⊗ T2 , σ1 ⊗ σ2 ) = R(T1 , σ1 ) ⊗ R(T2 , σ2 ). 2. Motivation. First of all, the l.h.s. of (1) is non-negative, while the r.h.s. is strictly positive unless ρ = RT ρ, so the inequality would be a systematic improvement of the monotonicity. Furthermore, it would have to extend in a certain sense the case of equality, D(ρkσ) = D(T ρkT σ), which has been treated by Petz: Indeed, defining R = Tσ0 explicitly as the “transpose channel” √ −1 √ −1  √ √ σ, (2) Tσ0 (β) = σ T ∗ T σ β T σ where T ∗ is the adjoint (cpup) map, Petz showed that D(ρkσ) = D(T ρkT σ) iff ρ = Tσ0 T ρ. Note that the transpose channel has all the properties above, including functoriality. 3. Affirmative answer in the classical case. (I.e., diagonal states ρ and σ corresponding to probability vectors and T maps diagonal to diagonal matrices, being described by a classical channel n n aka conditional probability.) We have two probability vectors Pm p~ = (pj )j=1 and ~q = (qj )j=1 , and m,n a stochastic matrix T = [tij ]i,j=1 (meaning that for all j, i=1 tij = 1). The adjoint of cptp map translates into the linear map given by the transpose matrix T t . Then,  m  m X X T p~ =  tij pj  , T ~q =  tij qj  , j

∗

[email protected]

i=1

j

i=1

2 and R = Tq~0 = [rji ]n,m q )i = tij qj , i.e. j,i=1 with rji (T ~ 
m n R~b = qj T t bi /(T ~q)i i=1 j=1 !n X bi . = qj tij P k tik qk i

j=1

Thus, 
m n RT p~ = qj T t (T p~)i /(T ~q)i i=1 j=1 !n X P tik pk = qj tij Pk , k tik qk i

j=1

leading to the following expressions for the three relative entropies concerned: X pj D(~ pk~q) = pj log , qj j   P X X tik pk   , D(T p~kT ~q) = tij pj log Pk k tik qk i j   X pj 1 . P D(~ pkRT p~) = pj log  P t p qj tij Pk ik k j

i

k tik qk

The claimed inequality can be rearranged, and simplifies to ! P X X P tik pk X X tik pk k pj log ≥ . tij P pj tij log Pk k tik qk k tik qk j

i

j

i

However, this is true for each term j individually, due to the concavity of log and

P

i tij

= 1.

t u

4. Other example cases. An important special case is where the original map T is just a partial trace, T = TrC : ABC −→ AB for a composite system AB, with a state ρABC and σ ABC = ρA ⊗ ρBC . Then the left hand side of eq. (1) is I(A : BC) − I(A : B) = I(A : C|B), the conditional mutual information, and the transpose channel, having the functoriality property, reads Tσ0 = idA ⊗ R, with p −1 p −1 p p R(β) = ρBC ρB β ρB ⊗ 11C ρBC , where we take the usual step of substituting the generalised inverse in case ρB hasn’t full rank – or we simply restrict our attention to the support of ρB . Petz’s theorem tells us that Tσ0 recovers ρABC from ρAB if and only if it is a quantum Markov chain, i.e. I(A : C|B) = 0. In general, it maps ρAB to q q q q −1 −1 AB C ABC A B→BC AB A A A BC B B ω = (id ⊗ R )ρ = 11 ⊗ ρ 11 ⊗ ρ ρ 11 ⊗ ρ ⊗ 11 11A ⊗ ρBC .

3 Looking at this, it is fairly clear that if the operators ρAB , ρBC and ρB (suitably extended with identities) all commute, then indeed D(ρkω) = I(A : C|B)ρ , i.e. the conjectured inequality is saturated with equality! This may not be too impressive, since we suspect that the required commutation relations make ρABC look rather classical. Let us consider instead the case of the extended d×d-antisymmetric projector: A = C = Cd , B = (Cd )⊗k−1 , and   1 d 1 X ABC ABC ρ = (sgn π)π, dk = . Pk+1 , Pk = dk+1 k! k π∈Sk

1 This is motivated by [5], where the above states are considered as extensions of ρAC = d(d−1) (11 − AC F ) and shown to give very small, and presumably the best upper bounds on Esq (ρ ). Noting that

ρAB =

1 AB P , dk k

ρBC =

1 BC P , dk k

ρB =

1 dk−1

B Pk−1 ,

the recovered state ω is easy to write down: ω ABC =

  dk−1 A B B ) (11A ⊗ PkBC ). )PkAB (11A ⊗ Pk−1 (11 ⊗ PkBC ) (11A ⊗ Pk−1 2 dk

Now, we seem to be stuck, until we realise that in the relative entropy D(ρkω) = Tr ρ(log ρ − log ω) ABC . Furthermore, due to the we only need to understand the spectrum of log ω restricted to Pk+1 general relation Pk+` ≤ Pk ⊗ P` (and P1 = 11), Pk+1 actually commutes with ω. Thus, Pk+1 log ωPk+1 = log Pk+1 ωPk+1 dk−1 = 2 Pk+1 , dk which immediately leads to D(ρkω) = log

d2k = I(A : C|B), dk−1 dk+1

so again the conjectured inequality is satisfied with equality! 5. Unsuitability of the transpose channel. We subjected eq. (1) with the transpose channel to a numerical assault. For example, one could plot the l.h.s. vs. r.h.s. for practically all points in the manifold of triples (T, ρ, σ) of cptp maps T and states ρ and σ in Cd . In fact, the number of parameters of T is d4 − d2 , those of ρ are d2 − 1, and σ has d − 1 free parameters since it may be assumed diagonal, leading to d4 + d − 2 real parameters. (One could reduce that even a little further since we may assume that T σ is diagonal in the same basis.) For d = 2 this evaluates to 16 (for d = 3 to 82), and the plot in Fig. 1 shows the result of about a million randomly chosen triples of qubit states and maps. So it appears that the inequality is violated, albeit only rarely and by a little – although the latter may be an artifact of the small dimension. But there is still hope: First, perhaps a modified inequality, with a different r.h.s., is true — even just introducing a factor of 12 would be consistent with the numerics. Secondly, maybe the recovery map R should not be the transpose channel but something else.

4

FIG. 1. One million randomly chosen triples of a map and two states on qubits: the horizontal axis is the l.h.s., the vertical axis the r.h.s. of eq. (1).

6. A possible application to squashed entanglement. We feel that there will be a number of applications, but one that motivated the whole questions orginally is the following: Recall that squashed entanglement is defined as Esq (ρAB ) =

inf

ρABE extension of ρAB

1 I(A : B|E). 2

The optimisation is an infimum because the size of E is unbounded, and so far nobody has been able to show a bound on the size of E given ρAB . It is easy to see that for separable states ρAB , Esq (ρ) = 0. For quite a while it was open to prove is the converse of this statement, that Esq (ρ) = 0 implies separability of ρAB . It’s clear that we seem to need a statement of the form that I(A : B|E) ≤  implies that the distance of ρAB from the separable set is bounded by some function of  and possibly |AB|, but not of |E|. Here is an idea, that avoids the problem of trying to find the nearest separable state directly (which, in the light of [7] and [5] might not be such a good approach). Namely, assume that a functorial map R satisfying eq. (1), so that for states ρAEB and σ AEB = ρA ⊗ ρEB , and the partial trace map TrB it can be written R = idA ⊗ RE→EB . Let’s apply R to the system E a number k of times, to extract k sort-of-copies of B from E. Thus we get a state ω AEB1 B2 ...Bk with the property √ that for all j = 1, . . . , k, kρAB − ω ABj k1 ≤ kδ, with δ = 2 ln 2  because of Pinsker’s inequality D(ρkσ) ≥

1 kρ − σk21 . 2 ln 2

[Note that we cannot use this reasoning to get an analogous statement when we include system E, i.e. for ρAEB , even for I(A : B|E) = 0. The reason is that the recovery map might sever the R quantum correlations possibly present in the states ρej C , between E and Bj−1 , when we create system Bj .] In fact, by averaging over all permutations of the systems B1 , . . . , Bk , and tracing over E, we actually get a permutation-invariant state ΩAB1 ...Bk such that ∀j

kρAB − ΩABj k1 ≤ kδ.

5 The state ΩAB1 ...Bk is a witness of the fact that ΩAB1 is “k-sharable” or “k-extendable” [10], and it is proved in these references that as k → ∞, ΩAB1 goes closer and closer to a separable state. At the same time, ρAB is kδ-close to ΩAB1 , so if we let k diverge slower than δ −1 , we get a bound on how close ρAC is to the separable set. This quantitative relationship already exists in the literature: from [4, Theorem II.7’] we get in fact kΩ − SEP(A : B)k1 ≤

2|C|2 . k

j q k So, choosing k = |C| 2δ , we conclude that √ kρAB − SEP(A : B)k1 ≤ 4|C| 4 . Re-inserting the definition of squashed entanglement, this yields the faithfulness of the latter entanglement measure: q kρAB − SEP(A : B)k1 ≤ 4|C| 4 2Esq (ρAB ), or equivalently, 1 kρAB − SEP(A : B)k41 . 512|C|4

Esq (ρAB ) ≥

[A small variation is to apply the recovery map to extract A a few times, too. Thus we actually get a state ΩA1 ...Ak B1 ...B` such that for all i, j, the reduced states ΩAi Bj are the same and (k + `)δ-close to ρAB . Not sure if that yields anything better than the above.] This approach could possibly explain the dimension factor in the d × d-antisymmetric state example of [5]: after all, that state is immensely sharable, in fact (d − 1)-fold. On the other hand it is at constant distance from the set of separable states, for all d. So, it would seem that it is rather the distance of sharable to separable states that hurts us... 6 12 . Faithfulness of multi-party squashed entanglement. It seems that – unlike [3] – our approach can be used to prove faithfulness of even the multi-party squashed entanglement [1] Esq (ρA1 ...An ) =

inf

ρA1 ...An E

1 I(A1 : · · · : An |E), 2

(3)

where the n-party conditional information is defined as I(A1 : · · · : An |E) =

n X

S(Ai |E) − S(A1 . . . An |E).

i=1

This is because we have the identity I(A1 : · · · : An |E) = I(A1 : A2 . . . An |E) + I(A2 : · · · : An |E) =

n−1 X

I(Ai : Ai+1 . . . An |E),

i=1

which shows that I(A1 : · · · : An |E) ≤  implies that for all i, I(Ai : A[n]\i |E) ≤ . Hence we an use the above machinery of extracting from E a large number√k of approximate of each Ai ,  copies γ  using approximate recovery maps Ri : E −→ Ai . With δ = 2 ln 2 and k = nδ , tracing out E

6 k

k

we get a state ΩA1 ...An which is symmetric under the action on Skn by permuting the systems Aji (j = 1, . . . , k) for every i = 1, . . . , n independently, and such that 1

1

kρA1 ...An − ΩA1 ...An k1 ≤ γ. I.e. ρ is γ-close to the k-extendable states (meaning that each of the n parties has k copies), where γ can be derived from suitable multi-party versions of the de Finetti inequality used for n = 2 [8].

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