Squashed Entanglement, $\mathbf {k}$-Extendibility, Quantum Markov Chains, and Recovery Maps

Abstract

Squashed entanglement (Christandl and Winter in J. Math. Phys. 45(3):829–840, 2004) is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information (Fawzi and Renner in Commun. Math. Phys. 340(2):575–611, 2015) greatly extended and simplified in various work since, we show the converse, that a small value of squashed entanglement implies that the state is close to a highly extendible state. As a corollary, we establish an alternative proof of the faithfulness of squashed entanglement (Brandão et al. Commun. Math. Phys. 306:805–830, 2011). We briefly discuss the previous and subsequent history of the Fawzi–Renner bound and related conjectures, and close by advertising a potentially far-reaching generalization to universal and functorial recovery maps for the monotonicity of the relative entropy.

Appendix: Multi-Party Squashed Entanglement

One might wonder if our approach could also be used to prove faithfulness of the multi-party squashed entanglement [50],

$$\begin{aligned} E_{\text {sq}}(\rho ^{A_1\ldots A_n}) = \inf _{\rho ^{A_1\ldots A_n E}} \frac{1}{2}I(A_1:\cdots :A_n|E), \end{aligned}$$

(20)

with $I(A_1\!:\!\cdots \!:\!A_n|E) = \sum _{i=1}^n S(A_i|E) - S(A_1\ldots A_n|E)$ the conditional multi-information. That is, $E_{\text {sq}}(\rho ^{A_1\ldots A_n})$ would vanish iff $\rho$ is n-separable:

$$\begin{aligned} \rho ^{A_1\ldots A_n} = \sum _\lambda p_\lambda \rho _{\lambda |1}^{A_1} \otimes \cdots \otimes \rho _{\lambda |n}^{A_n}. \end{aligned}$$

It seems that with the methods of [7, 28] this cannot be approached.

The idea starts from the identity

$$\begin{aligned} \begin{aligned} I(A_1:\cdots :A_n|E)&= I(A_1:A_2\ldots A_n|E) + I(A_2:\cdots :A_n|E) \\&= \ldots = \sum _{i=1}^{n-1} I(A_i:A_{i+1}\ldots A_n|E), \end{aligned} \end{aligned}$$

showing that $I(A_1:\cdots :A_n|E) \le 2\epsilon$ implies, for all i, $I(A_i:A_{[n]\setminus i}|E) \le 2\epsilon$, and more generally, for all subsets $I\subset [n]$, $I(A_I:A_{[n]\setminus I}|E) \le 2\epsilon$.

In particular, if $\epsilon = 0$, we can use the structure theorem of [19] to find, for each i, a projective measurement $\bigl ( P^{(i)}_{\lambda _i} \bigr )$ on E that commutes with $\rho ^{A_1\ldots A_n E}$, such that for all $\lambda _i$,

$$\begin{aligned} {\text {Tr}}_E \rho ^{A_1\ldots A_n E} P^{(i)}_{\lambda _i} = p_{\lambda _i} \sigma _{\lambda _i}^{A_i} \otimes \tau _{\lambda _i}^{A_{[n]\setminus i}}, \end{aligned}$$

i.e., conditioned on the measurement outcomes $\lambda _i$, $A_i$ and $A_{[n]\setminus i}$ are in a product state. Performing all these measurements in some fixed order, we thus obtain outcomes $\lambda = \lambda _1\ldots \lambda _n$ such that conditioned on $\lambda$, the state is a product state with respect to all partitions $i:[n]\setminus i$, which means that conditioned on $\lambda$, $A_1,\ldots ,A_n$ factorize.

We would like to use the machinery of the recovery maps to extract from E a large number k of approximate copies of each $A_i$, using approximate recovery maps $\widetilde{R}_i:\mathcal {L}(E) \longrightarrow \mathcal {L}(EA_i)$ according to Eq. (6). With $t = \sqrt{8\ln 2}\sqrt{\epsilon }$ and tracing out E, we can indeed get a state $\varOmega ^{A_1 A_2^{[k]}\ldots A_n^{[k]}}$, with $A_i^{[k]} = A_i^1\ldots A_i^k$ consisting of k copies of $A_i$, such that

$$\begin{aligned} \Vert \rho ^{A_1\ldots A_n} - \varOmega ^{A_1 A_2^{j_2}\ldots A_n^{j_n}} \Vert _1 \le (n-1)(k-1)t \le nk\sqrt{8\ln 2}\sqrt{\epsilon }, \end{aligned}$$

for all tuples $(j_2,\ldots ,j_n)$ such that all but at most one $j_i$ equals 1.

We cannot say easily that this holds for all tuples $(j_2,\ldots ,j_n)$, because the different recover maps may interfere with each other. However, if we could conclude that, we would be done: by symmetrizing the k copies of each $A_i$ ($i>1$) we would find, as before, that $\rho$ is $O(\root 4 \of {\epsilon })$-close to a k-extendible state, with $k=\varOmega \left( \frac{1}{\root 4 \of {\epsilon }}\right)$.

We could then again use the results of [30], now extended to the multi-partite case, to see that $\varOmega ^{A_1 A_2^{j_2}\ldots A_n^{j_n}}$ is at trace distance at most $\frac{2}{k}(|A_2|^2+\cdots +|A_n|^2)$ from a fully separable (i.e. n-separable) state. Note that a reasoning along these lines goes through for the – generally larger – multi-party conditional entanglement of mutual information (CEMI) [49, 50]

$$\begin{aligned} E_{\text {I}}(\rho ^{A_1\ldots A_n}) = \inf _{\rho ^{A_1 A_1'\ldots A_n A_n'}} \frac{1}{2}\bigl [ I(A_1A_1':\cdots :A_n A_n') - I(A_1':\cdots :A_n') \bigr ], \end{aligned}$$

as since shown by Wilde [45]. We have to leave the problem of finding an extension of Theorem 1 to $n>2$ parties to the attention of the interested reader.