Extendibility of bosonic Gaussian states

TL;DR

This paper characterizes the extendibility of bosonic Gaussian states, providing efficient criteria, bounds on their distance to separable states, and implications for quantum steerability and entanglement.

Contribution

It introduces a semidefinite program for deciding $k$-extendibility of Gaussian states and derives Gaussian de Finetti theorems with universal scaling.

Findings

Efficient semidefinite program for $k$-extendibility

Universal bounds on distance to separable states

Upper bounds on entanglement of formation for Gaussian states

Abstract

Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is $k$-extendible if and only if it has a Gaussian $k$-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide $k$-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a $k$-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and R\'enyi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian $k$-extendible states, which has no analogue in the finite-dimensional setting.

Equations

$$V_{AB}=\left(\begin{array}{cc}{V}_A& X\\ X^{\mathsf{T}}& V_B\end{array}\right).$$

$${\tilde{V}}_{A{B}_1\dots B_k}=\left(\begin{array}{ccccc}{V}_A& X& X& \dots & X\\ X^{\mathsf{T}}& V_B& Y& \dots & Y\\ X^{\mathsf{T}}& Y& V_B& \ddots & ⋮\\ ⋮& ⋮& \ddots & \ddots & Y\\ X^{\mathsf{T}}& Y& \dots & Y& V_B\end{array}\right),$$

$${\omega }_{A_1{B}_{11}\dots B_{mk}}^{(m)}:=(U_1\otimes \dots \otimes U_k)\left(\underset{\ell =1}{\overset{m}{⨂}}{\tilde{\rho }}_{A_{\ell }{B}_{\ell 1}\dots B_{\ell k}}\right)(U_1\otimes \dots \otimes U_k{)}^{†}.$$

$$V_{AB}\ge \mathrm{i}{\Omega }_A\oplus \left(\left(1-\frac{1}{k}\right){\Delta }_B+\frac{1}{k}\mathrm{i}{\Omega }_B\right).$$

$$V_{AB}\ge \mathrm{i}{\Omega }_A\oplus \left(-\left(1-\frac{2}{k}\right)\mathrm{i}{\Omega }_B\right).$$

$$M=\left(\begin{array}{cc}P& Z\\ Z^{†}& Q\end{array}\right)\ge 0\iff P\ge 0,M/P:=Q-Z^{†}{P}^{-1}Z\ge 0,$$

$$\mathrm{i}{\Omega }_B\le {\Delta }_B\le \frac{k}{k-1}\left(V_B-X^{\mathsf{T}}(V_A-\mathrm{i}{\Omega }_A{)}^{-1}X\right)-\frac{1}{k-1}\mathrm{i}{\Omega }_B.$$

$$V_B-X^{\mathsf{T}}(V_A-\mathrm{i}{\Omega }_A{)}^{-1}X\ge -\left(1-\frac{2}{k}\right)\mathrm{i}{\Omega }_B.$$

$$(\text{mathds}{1}_k-|+⟩⟨+|)\otimes (V_B-Y-\mathrm{i}{\Omega }_B)$$

$$+|+⟩⟨+|\otimes \left(V_B+(k-1)Y-k{X}^{\mathsf{T}}{\left(V_A-\mathrm{i}{\Omega }_A\right)}^{-1}X-\mathrm{i}{\Omega }_B\right)\ge 0.$$

$$\frac{k}{k-1}\left(V_B-X^{\mathsf{T}}(V_A-\mathrm{i}{\Omega }_A{)}^{-1}X\right)-\frac{1}{k-1}\mathrm{i}{\Omega }_B\ge \pm \mathrm{i}{\Omega }_B,$$

$$\mathrm{i}\Omega \le \Delta \le \frac{k}{k-1}\left(Y+\mathrm{i}X\Omega X^{\mathsf{T}}\right)-\frac{1}{k-1}\mathrm{i}\Omega .$$

$$\sqrt{\det Y}\ge 1-\frac{1}{k}+\left|\det X-\frac{1}{k}\right|.$$

$${\Vert {\rho }_{AB}^{\mathrm{G}}-\mathrm{SEP}(A:B)\Vert }_1$$

$$E_{R,\alpha }({\rho }_{AB}^{\mathrm{G}})\le n\ln \left(1+\frac{{\eta }_{k,\alpha }}{k-1}\right)$$

$$E^{\mathrm{\scriptscriptstyle G}}_{F,\alpha}\left(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}\right)=\inf\big{\{}S_{\alpha}(\gamma_{A}):\ \text{$\gamma_{AB}$ pure QCM and $\gamma_{AB}\leq V_{AB}$}\big{\}}\,,$$

$$E_{F,2}^{\mathrm{G}}\left({\rho }_{AB}\right)=\min \left\{M({\gamma }_A):\text{γAB pure QCM and γAB≤VAB}\right\},$$

$$[r,r^{\mathsf{T}}]=\mathrm{i}\Omega =\mathrm{i}{\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)}^{\oplus n}.$$

$$V\ge \mathrm{i}\Omega .$$

$$V_{AB}=\left(\begin{array}{cc}{V}_A& X\\ X^{\mathsf{T}}& V_B\end{array}\right),$$

$$V=S\left(\underset{j=1}{\overset{n}{⨁}}{\nu }_j\text{mathds}{1}_2\right)S^{\mathsf{T}},$$

$$\rho =U_S^{†}\left(\underset{j=1}{\overset{n}{⨂}}{\rho }^{\mathrm{G}}({\nu }_j)\right)U_S,$$

$${\rho }^{\mathrm{G}}(\nu ):=\frac{2}{\nu +1}\sum _{\ell =0}^{\infty }{\left(\frac{\nu -1}{\nu +1}\right)}^{\ell }|\ell ⟩⟨\ell |.$$

$${\mathcal{L}}_{\lambda }(\cdot ):={\mathrm{Tr}}_B\left[U_{\lambda }((\cdot {)}_A\otimes {|0⟩⟨0|}_B)U_{\lambda }^{†}\right],$$

$$U_{\lambda }:=e^{-\arccos \left(\sqrt{\lambda }\right)(a{b}^{†}-a^{†}b)}=e^{-\sqrt{\frac{1-\lambda }{\lambda }}a{b}^{†}}{\lambda }^{\frac{a^{†}a-b^{†}b}{2}}{e}^{\sqrt{\frac{1-\lambda }{\lambda }}{a}^{†}b},$$

$${\mathcal{L}}_{\lambda }(\cdot )=\sum _{j=0}^{\infty }\frac{{\left(\frac{1}{\lambda }-1\right)}^j}{j!}{a}^j{\lambda }^{a^{†}a/2}(\cdot ){\lambda }^{a^{†}a/2}(a^{†}{)}^j.$$

$$\mathrm{Tr}\left[r_{B_j,\beta }^2{\tilde{\rho }}_{A{B}_1\dots B_k}\right]=\mathrm{Tr}\left[r_{B_j,\beta }^2{\tilde{\rho }}_{A{B}_j}\right]=\mathrm{Tr}\left[r_{B,\beta }^2{\rho }_{AB}^{\mathrm{G}}\right]<\infty .$$

$${\left({\tilde{V}}_{A{B}_1\cdots B_k}\right)}_{A,\alpha ;B_j,\beta }$$

$$=\mathrm{Tr}[(r_{A,\alpha }{r}_{B_j,\beta }+r_{B_j,\beta }{r}_{A,\alpha }){\tilde{\rho }}_{A{B}_j}]$$

$$=\mathrm{Tr}[(r_{A,\alpha }{r}_{B_j,\beta }+r_{B_j,\beta }{r}_{A,\alpha }){\rho }_{AB}]$$

$$=X_{\alpha ,\beta }.$$

$$Y_{i,\ell }$$

$$=\mathrm{Tr}[(r_{B_1,i}{r}_{B_2,\ell }+r_{B_2,\ell }{r}_{B_1,\ell }){\tilde{\rho }}_{B_1{B}_2}]$$

Full text

Entanglement is the mainspring of modern quantum technologies. To tally the performance of such technologies, a comprehensive characterization and quantification of entanglement is needed. One of the defining features of entanglement is its monogamy Terhal (2004); Coffman et al. (2000); Fannes et al. (1988); Raggio and Werner (1989); Doherty et al. (2002, 2004); Lancien et al. (2016), the fact that entangled states cannot be shared among arbitrarily many subsystems. Exploring the middle ground of partially shareable states or, precisely, partially extendible states, offers a rich and practically meaningful lookout into the virtues of entanglement as a resource.

A bipartite quantum state $\rho_{AB}$ of systems $A$ and $B$ is called $k$-extendible on $B$ if there exists a quantum state $\widetilde{\rho}_{AB_{1}\dotsb B_{k}}$ on $A$ and $k$ copies $B_{1},\dotsc,B_{k}$ of $B$ that is permutation-invariant with respect to the systems $B_{i}$ and satisfies $\operatorname{Tr}_{B_{2}\dotsb B_{k}}\left[\,\widetilde{\rho}_{AB_{1}\dotsb B_{k}}\right]=\rho_{AB}$, where $B_{1}\equiv B$. It is well-known that a state $\rho_{AB}$ is separable if and only if it is $k$-extendible for all $k\geq 2$ Fannes et al. (1988); Raggio and Werner (1989); Doherty et al. (2002, 2004). The nested sets of $k$-extendible states can thus be used to approximate the set of separable states, which has resulted in work on quantum de Finetti theorems Hudson and Moody (1976); Caves et al. (2002); Christandl et al. (2007); König and Renner (2005); König and Wolf (2009); Brandao and Harrow (2013); Chen et al. (2016) and other studies of entanglement Navascués et al. (2009); Brandão and Christandl (2012). Extendibility also arises in the contexts of security of quantum key distribution Moroder et al. (2006); Myhr et al. (2009); Khatri and Lütkenhaus (2017), capacities of quantum channels Nowakowski and Horodecki (2009); Kaur et al. (2018); Berta et al. (2018), Bell’s inequalities Terhal et al. (2003); Kumari et al. (2017), and other information-theoretic scenarios Lancien (2016); Li and Winter (2018). More broadly, the extendibility problem is a special case of the QMA-complete quantum marginal problem Klyachko (2006); Eisert et al. (2008); Carlen et al. (2013); Schilling (2014); Tyc and Vlach (2015); Liu (2006); Liu et al. (2007), which has been referred to in quantum chemistry as the $N$-representability problem Tredgold (1957); Coulson (1960); Coleman (1963). For fixed $k$, the extendibility problem can be formulated as a semidefinite program (SDP), making it efficiently solvable for low-dimensional systems $A$ and $B$ Doherty et al. (2002, 2004). Analytic conditions for $k$-extendibility in finite-dimensional systems are known only for particular values of $k$ and/or for special classes of states Ranade (2009); Johnson and Viola (2013); Chen et al. (2014); Fung et al. (2014); Kumari et al. (2017).

In the infinite-dimensional case, of central relevance for quantum-optical realizations, the theory of Gaussian entanglement has been explored thoroughly in the past two decades Adesso et al. (2014); Serafini (2017); Lami et al. (2018). However, more general extendibility questions have been approached sparingly. The only work that we are aware of is Rajarama Bhat et al. (2017), where it was shown that a Gaussian state is separable if and only if it is Gaussian $k$-extendible for all $k$.

Here we study and characterize the full hierarchy of extendibility for quantum Gaussian states. After showing that any Gaussian state is $k$-extendible if and only if it is Gaussian $k$-extendible, we derive a simple SDP in terms of the state’s covariance matrix in order to decide its $k$-extendibility. The size of our SDP scales linearly with the number of local modes. We also provide an analytic condition that completely characterizes the set of $k$-extendible states in the case of the extended system containing one mode only, generalizing the well-known positive partial transpose (PPT) criterion Peres (1996); Horodecki et al. (1996); Simon (2000). We then discuss several applications of this result, deriving along the way: (i) analytic conditions for $k$-extendibility for all single-mode Gaussian channels; (ii) a tight de Finetti-type theorem bounding the distance between any $k$-extendible Gaussian state and the set of separable states; tight upper bounds on (iii) Rényi relative entropy of entanglement and (iv) Rényi entanglement of formation for any $k$-extendible Gaussian state. Our results reach unexplored depths in the ocean of continuous-variable quantum information.

Gaussian states.

We recall the basic theory of quantum Gaussian states Wang et al. (2007); Serafini (2017); Adesso et al. (2014); Weedbrook et al. (2012). Let $x_{j}$ and $p_{j}$ ($1\leq j\leq n$) denote the canonical operators of a system of $n$ harmonic oscillators (modes), arranged as a vector $r\coloneqq(x_{1},p_{1},\ldots,x_{n},p_{n})^{\scriptscriptstyle\mathsf{T}}$. The canonical commutation relations can be compactly written as $[r,r^{\scriptscriptstyle\mathsf{T}}]=\mathrm{i}\Omega$, where $\Omega\coloneqq\left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right)^{\oplus n}$ is the standard symplectic form. Given any (not necessarily Gaussian) $n$-mode state $\rho$, its mean or displacement vector is $s\coloneqq\operatorname{Tr}[r\,\rho]\in\mathds{R}^{2n}$, while its quantum covariance matrix (QCM) is the $2n\times 2n$ real symmetric matrix $V\coloneqq\operatorname{Tr}\left[\{r-s,(r-s)^{\scriptscriptstyle\mathsf{T}}\}\,\rho\right]$. Gaussian states $\rho^{\mathrm{\scriptscriptstyle G}}$ are (limits of) thermal states of quadratic Hamiltonians and are uniquely identified by their displacement vector $s$ and QCM $V$. We shall often assume $s=0$, since the mean can be adjusted by local displacement unitaries that do not affect $k$-extendibility. Physically legitimate QCMs $V$ satisfy the Robertson–Schrödinger uncertainty principle $V\geq\mathrm{i}\Omega$, hereafter referred to as the bona fide condition Simon et al. (1994). Any matrix obeying this condition can be the QCM of a Gaussian state.

Extendibility of Gaussian states.

Let $\rho_{AB}$ be a (not necessarily Gaussian) state of a bipartite system of $n=n_{A}+n_{B}$ modes. We assume that $\rho_{AB}$ has vanishing first moments and finite second moments, so that we can construct its QCM

[TABLE]

It can be shown 111See the Supplemental Material for proofs and additional technical derivations. that every $k$-extension $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$ of $\rho_{AB}$ also has (a) vanishing first moments and (b) finite second moments, arranged in a QCM of the form

[TABLE]

where $Y$ is a symmetric matrix. A similar structure had already been identified in Rajarama Bhat et al. (2017); however, there the crucial fact that $Y$ needs to be symmetric was not observed. We are now concerned with the $k$-extendibility of Gaussian states. Our first result indicates that Gaussian states are in some sense a closed set under $k$-extensions:

Theorem 1**.**

A Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is $k$-extendible if and only if it has a Gaussian $k$-extension.

Proof.

Let $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$ be a (not necessarily Gaussian) $k$-extension of $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$. Consider $m$ identical copies of it across the systems $A_{\ell}B_{\ell 1}\ldots B_{\ell k}$, where $1\leq\ell\leq m$. For $1\leq j\leq k$, let $U_{j}$ be a passive unitary that acts on the annihilation operators $b_{\ell j}$ of the systems $B_{\ell j}$ so that $U_{j}^{\dagger}b_{1j}U_{j}=\frac{b_{1j}+\ldots+b_{mj}}{\sqrt{m}}$. Set

[TABLE]

By the quantum central limit theorem Cushen and Hudson (1971); Petz (1992), the reduced state $\omega^{(m)}_{A_{1}B_{11}\ldots B_{1k}}$ satisfies $\lim_{m\to\infty}\left\|\omega^{(m)}_{A_{1}B_{11}\ldots B_{1k}}-\widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}}\right\|_{1}=0$, where $\widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}}$ is the Gaussian state with the same first and second moments as $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$, and $A_{1}\equiv A$, $B_{1j}\equiv B_{j}$ Note (1).

We now show that $\widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}}$ is indeed a Gaussian $k$-extension of $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$. First, it is symmetric under the exchange of any two $B$ systems, say $B_{1}\leftrightarrow B_{2}$. In fact, (i) the state in (3) is invariant under the exchange $(B_{11},\ldots,B_{m1})\leftrightarrow(B_{12},\ldots,B_{m2})$; (ii) consequently, the reduced state $\omega^{(m)}_{A_{1}B_{11}\ldots B_{1k}}$ is invariant under the exchange $B_{11}\leftrightarrow B_{12}$; (iii) symmetry is preserved under limits. Finally, to show that $\widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}}=\rho^{\mathrm{\scriptscriptstyle G}}_{AB}$ under the identification $B_{1}\equiv B$, we observe that the QCM of $\widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}}$, which is the same as that of $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$, is as in (2). Since its upper-left $2\times 2$ corner corresponds to the QCM of $\rho^{\mathrm{\scriptscriptstyle G}}_{AB}$, we conclude that $\widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}}$ and $\rho^{\mathrm{\scriptscriptstyle G}}_{AB}$ have the same first and second moments; being Gaussian, they must coincide. ∎

By virtue of Theorem 1, we can confine the search of $k$-extensions of Gaussian states to the same Gaussian realm. The next result shows that this reduces to an efficiently solvable SDP feasibility problem, with the size of the SDP scaling linearly in the number of modes of the $B$ system. In the case of $B$ being composed of one mode only, we find an analytic solution in the form of a simple necessary and sufficient condition for $k$-extendibility.

Theorem 2**.**

Let $\rho_{AB}$ be a $k$-extendible (not necessarily Gaussian) state of $n_{A}+n_{B}$ modes with QCM $V_{AB}$. Then there exists a $2n_{B}\times 2n_{B}$ quantum covariance matrix $\Delta_{B}\geq\mathrm{i}\Omega_{B}$ such that

[TABLE]

Moreover, the above condition is necessary and sufficient for $k$-extendibility when $\rho_{AB}=\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is Gaussian. If in addition $n_{B}=1$, then $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is $k$-extendible if and only if

[TABLE]

In the proof of Theorem 2, we employ the following characterization of positive semidefiniteness of Hermitian block matrices (Horn and Zhang, 2006, Theorem 1.12):

[TABLE]

where the matrix $M/P$ is called the Schur complement of $M$ with respect to $P$. For details concerning the degenerate case of non-invertible $P$, see Note (1). Using (6), for any QCM $V_{AB}$ as in (1), the inequality in (4) and the condition $\Delta_{B}\geq\mathrm{i}\Omega_{B}$ can be written together as

[TABLE]

Analogously, (5) is equivalent to

[TABLE]

Proof of Theorem 2.

We first establish necessity of (4) for $k$-extendibility of an arbitrary state $\rho_{AB}$. If $\rho_{AB}$ is $k$-extendible, then there exists a matrix $\widetilde{V}_{AB_{1}\ldots B_{k}}$ as in (2) that obeys the bona fide condition \widetilde{V}_{AB_{1}\ldots B_{k}}\geq\mathrm{i}\big{(}\Omega_{A}\oplus\Omega_{B_{1}\ldots B_{k}}\big{)}. Using (6), and noting that $V_{A}\geq\mathrm{i}\Omega_{A}$ holds because $\rho_{A}$ is a valid state, we arrive at the inequality \big{(}\widetilde{V}_{AB_{1}\ldots B_{k}}-\mathrm{i}\Omega_{A}\big{)}\big{/}\big{(}\widetilde{V}_{A}-\mathrm{i}\Omega_{A}\big{)}\geq\mathrm{i}\Omega_{B_{1}\ldots B_{k}}. Using (2), and letting $\ket{+}\coloneqq\frac{1}{\sqrt{k}}\sum_{j=1}^{k}\ket{j}\in\mathds{R}^{k}$, upon elementary manipulations this can be rephrased as

[TABLE]

Since the first factors of the above two addends are orthogonal to each other, positive semidefiniteness can be imposed separately on the second factors. Letting $\Delta_{B}\coloneqq V_{B}-Y$, we obtain (7), whose equivalence to (4) follows by applying (6). To deduce (5) from (4), simply substitute the complex conjugate bona fide condition $\Delta_{B}\geq-\mathrm{i}\Omega_{B}$ into (4).

By Theorem 1, the condition \widetilde{V}_{AB_{1}\ldots B_{k}}\geq\mathrm{i}\big{(}\Omega_{A}\oplus\Omega_{B_{1}\ldots B_{k}}\big{)} is also sufficient to ensure $k$-extendibility when $\rho_{AB}=\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is Gaussian. By the above reduction, this condition is equivalent to that in (4).

We now prove that when $n_{B}=1$, (5) implies the existence of a real $\Delta_{B}$ such that (7) is satisfied. By (Lami et al., 2018, Lemma 7), we know that (7) is satisfied for some real $\Delta_{B}$ if and only if

[TABLE]

meaning that both inequalities are satisfied. Using (6), we see that the condition with the $+$ reduces to $V_{AB}\geq\mathrm{i}\Omega_{AB}$, which is guaranteed to hold by hypothesis. That with the $-$ yields instead (8), which is in turn equivalent to (5). ∎

Although some of the above manipulations formally resemble those in Rajarama Bhat et al. (2017), the two arguments are conceptually different and lead to different conclusions Note (1): in fact, in Rajarama Bhat et al. (2017), the question of $k$-extendibility of Gaussian states is explicitly mentioned as an outstanding problem.

Recall that a bipartite state is separable if and only if it is $k$-extendible for all $k$ Fannes et al. (1988); Raggio and Werner (1989); Doherty et al. (2002, 2004) and that any $k$-extendible state is also $(k-1)$-extendible. Thus, taking the limit $k\to\infty$ of condition (4) shows that $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is separable if and only if there exists a $2n_{B}\times 2n_{B}$ matrix $\Delta_{B}\geq\mathrm{i}\Omega_{B}$ such that $V_{AB}\geq\mathrm{i}\Omega_{A}\oplus\Delta_{B}$. This reproduces the analytic condition for separability of Gaussian states found in (Lami et al., 2018, Theorem 5). In the same limit $k\to\infty$, it is also easy to verify that condition (5) reduces to the PPT criterion Peres (1996); Horodecki et al. (1996); Simon (2000); Werner and Wolf (2001); Lami et al. (2018).

It turns out that the necessary condition in (5) is no longer sufficient when $n_{B}>1$. This is demonstrated by the example of the $(2+2)$-mode bound entangled Gaussian state constructed in Werner and Wolf (2001), which obeys (5) for all $k$ (because it is PPT) yet it is not even $2$-extendible Note (1).

Theorem 2 also reveals an implication of $2$-extendibility for Gaussian steerability, i.e., Einstein–Podolsky–Rosen steerability via Gaussian measurements Wiseman et al. (2007); Kogias et al. (2015); Adesso and Simon (2016); Ji et al. (2015); Lami et al. (2016). The $k=2$ case of (5) shows that any Gaussian state that is 2-extendible on $B$ is necessarily $B\to A$ Gaussian unsteerable, and hence useless for one-sided-device-independent quantum key distribution. When $n_{B}=1$, this condition is also sufficient, i.e., $2$-extendibility is equivalent to $B\to A$ Gaussian unsteerability.

Extendibility of Gaussian channels.

We now apply Theorem 2 to study $k$-extendibility of single-sender single-receiver Gaussian quantum channels. A quantum channel $\mathcal{N}_{A\to B}$ is called $k$-extendible Pankowski et al. (2013); Kaur et al. (2018) if there exists another quantum channel $\widetilde{\mathcal{N}}_{A\to B_{1}\dotsb B_{k}}$ from the sender $A$ to $k$ receivers $B_{1},\dotsc,B_{k}$ such that the reduced channel from the sender to any one of the receivers is the same as the original channel $\mathcal{N}_{A\to B}$.

A Gaussian channel $\mathcal{N}_{A\to B}$ with $n$ input modes and $m$ output modes maps Gaussian states to Gaussian states and is uniquely characterized by a real $2m\times 2n$ matrix $X$, a real symmetric $2m\times 2m$ matrix $Y$, and a real vector $\delta\in\mathds{R}^{2m}$, such that $Y+\mathrm{i}\Omega\geq\mathrm{i}X\Omega X^{T}$ Serafini (2017). Its action can be described directly in terms of the mean vector $s$ and QCM $V$ of the input Gaussian state as follows: $s\mapsto Xs+\delta$, $V\mapsto XVX^{{\scriptscriptstyle\mathsf{T}}}+Y$. In what follows, we set $\delta=0$ without loss of generality.

To any channel $\mathcal{N}_{A\to B}$ we can associate its Choi–Jamiołkowski state $\rho_{AB}^{\mathcal{N}}(r)\coloneqq\mathcal{N}_{A^{\prime}\to B}\left(\ket{\psi_{r}}\!\!\bra{\psi_{r}}^{\otimes n}_{AA^{\prime}}\right)$, where for $r>0$ the two-mode squeezed vacuum is defined as $\ket{\psi_{r}}\coloneqq\mathrm{sech}(r)\sum_{j=0}^{\infty}\tanh(r)^{j}\ket{j,j}$ Holevo (2011). It can be seen that $\mathcal{N}_{A\to B}$ is $k$-extendible if and only if $\rho_{AB}^{\mathcal{N}}(r)$ is $k$-extendible on $B$ for some (and hence all) $r>0$ Note (1). The same conclusion follows from arguments in Giedke and Cirac (2002); Wolf et al. (2007); Niset et al. (2009). For any Gaussian channel $\mathcal{N}$, the state $\rho_{AB}^{\mathcal{N}}(r)$ is Gaussian. Hence, via Theorem 2, we deduce that a Gaussian channel is $k$-extendible if and only if there exists a $2m\times 2m$ real matrix $\Delta$ such that

[TABLE]

When $m=1$, this is equivalent to $Y+\mathrm{i}X\Omega X^{{\scriptscriptstyle\mathsf{T}}}+\left(1-2/k\right)\mathrm{i}\Omega\geq 0$. If also $n=1=m$, a simplified equivalent condition that incorporates also the complete positivity requirements is

[TABLE]

By applying (S55), we find necessary and sufficient conditions for the $k$-extendibility of all possible single-mode Gaussian channels, which play a prominent role in modelling optical quantum communication Holevo (2007); Eisert and Wolf (2007); Serafini (2017). By the results of Holevo (2007), the following characterization of $k$-extendibility for three fundamental single-mode Gaussian channels suffices to solve the problem for all single-mode Gaussian channels Note (1):

(i) The thermal channel of transmissivity $\eta\in(0,1)$ and environment thermal photon number $N_{B}\geq 0$ is defined by $X=\sqrt{\eta}\mathds{1}$ and $Y=(1-\eta)(2N_{B}+1)\mathds{1}$. It is $k$-extendible if and only if $\eta\leq\frac{N_{B}+1/k}{N_{B}+1}$. For the case $N_{B}=0$, corresponding to a pure-loss channel, this reduces to $\eta\leq 1/k$.

(ii) The amplifier channel of gain $G>1$ and environment thermal photon number $N_{B}\geq 0$ is defined by $X=\sqrt{G}\mathds{1}$ and $Y=(G-1)(2N_{B}+1)\mathds{1}$. This channel is $k$-extendible if and only if $N_{B}>0$ and $G\geq\frac{N_{B}+1-1/k}{N_{B}}$.

(iii) The additive noise channel with noise parameter $\xi>0$ is defined by $X=\mathds{1}$ and $Y=\xi\mathds{1}$. This channel is $k$-extendible if and only if $\xi\geq 2\left(1-1/k\right)$.

As expected, the above conditions reduce to their entanglement-breaking counterparts from Holevo (2008) for $k\to\infty$.

Distance between $k$-extendible and separable states.

A problem of central interest in quantum information theory is determining how close $k$-extendible states are to the set of separable states. In (Christandl et al., 2007, Theorem II.7’), it was found that a finite-dimensional $k$-extendible state is $4d^{2}/k$-close to the set of separable states in trace norm, where $d$ is the dimension of the extended system. Moreover, it was also shown (Christandl et al., 2007, Corollary III.9) that the error term in the approximation necessarily depends on $d$ at least linearly. One can instead obtain a $\ln d$ dependence by resorting to different norms Brandão et al. (2011).

Can similar estimates be provided in the Gaussian case? Results in this setting have been obtained in König and Wolf (2009) for fully symmetric systems of the form $B_{1}\ldots B_{k}$. Here we extend these de Finetti theorems to the case where the symmetry is relative to a fixed reference system $A$. We are interested in the distance of a given Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ to the set $\text{SEP}(A\!\!:\!\!B)$ of bipartite separable states on systems $A$ and $B$, as measured by either (i) the trace norm, yielding the quantity $\left\|\rho_{AB}^{\mathrm{\scriptscriptstyle G}}-\text{SEP}(A\!:\!B)\right\|_{1}\coloneqq\inf_{\sigma_{AB}\in\text{SEP}(A:B)}\left\|\rho_{AB}-\sigma_{AB}\right\|_{1}$, or (ii) the quantum Petz–Rényi relative entropy $D_{\alpha}(\rho\|\sigma)\coloneqq\frac{1}{\alpha-1}\ln\operatorname{Tr}[\rho^{\alpha}\sigma^{1-\alpha}]$ for $\alpha>0$ Petz (1986), which leads to the measure $E_{R,\alpha}(\rho^{\mathrm{\scriptscriptstyle G}}_{AB})\coloneqq\inf_{\sigma_{AB}\in\text{SEP}(A:B)}D_{\alpha}(\rho_{AB}\|\sigma_{AB})$. For $\alpha=1$ the Petz–Rényi relative entropy reduces to the Umegaki relative entropy Umegaki (1962), and we obtain the standard relative entropy of entanglement Vedral and Plenio (1998). We find the following:

Theorem 3**.**

Let $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ be a $k$-extendible Gaussian state of $n\coloneqq n_{A}+n_{B}$ modes. Then,

[TABLE]

where $\eta_{k,\alpha}=1$ if $\alpha\leq k+1$, and $\eta_{k,\alpha}=2$ otherwise.

The proof is in Note (1). Remarkably, the upper bounds in (12)–(13) hold universally for all Gaussian states, independently, e.g., of their mean photon number. This is in analogy with the main results of König and Wolf (2009), and constitutes a quantitative improvement over the finite-dimensional case, where—as we mentioned before—the bound has to depend on the underlying dimension. Furthermore, for two-mode states, the bounds in (12)–(13) can be shown to be tight up to a constant for all $k$ and all $\alpha\geq 1$. Namely, for all $k\geq 2$ there exists a $k$-extendible two-mode Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ such that $\left\|\rho_{AB}^{\mathrm{\scriptscriptstyle G}}-\text{SEP}(A\!:\!B)\right\|_{1}\geq\frac{1}{2k-1}$ and $E_{R,1}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}})\geq E_{D}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}})\geq\ln\frac{k}{k-1}-o(1)$ as $r\to\infty$, where $E_{D}$ denotes the distillable entanglement Note (1).

Entanglement of formation of Gaussian $k$-extendible states.

We now show that one can also obtain an upper bound on the entanglement of formation of Gaussian $k$-extendible states. This is a qualitative improvement over the finite-dimensional case, as a result of this kind has no analogue in that setting. We employ the recently developed theory of Rényi-2 Gaussian correlation quantifiers Adesso et al. (2012); Kogias et al. (2015); Lami et al. (2016, 2017), and especially the monogamy of the Gaussian Rényi-2 version of the entanglement of formation Lami et al. (2016), which stems in turn from the equality between this measure and the Gaussian Rényi-2 squashed entanglement Lami et al. (2017).

For a bipartite state $\rho_{AB}$ and for some $\alpha\geq 1$, the Rényi-$\alpha$ entanglement of formation $E_{F,\alpha}(\rho_{AB})$ is defined as the infimum of \sum_{i}p_{i}\,S_{\alpha}\big{(}\psi_{A}^{(i)}\big{)} over all pure-state decompositions $\sum_{i}p_{i}\psi_{AB}^{(i)}=\rho_{AB}$ of $\rho_{AB}$ Horodecki et al. (2009). Here, $S_{\alpha}(\sigma)\coloneqq\frac{1}{1-\alpha}\ln\operatorname{Tr}[\sigma^{\alpha}]$ is the Rényi-$\alpha$ entropy.

For a Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ with QCM $V_{AB}$, we can derive an upper bound on $E_{F,\alpha}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}})$ by restricting the decompositions to include pure Gaussian states only. This leads to the Gaussian Rényi-${\alpha}$ entanglement of formation, given by Wolf et al. (2004); Lami et al. (2017)

[TABLE]

where we denote by $S_{\alpha}(W)$ the Rényi-$\alpha$ entropy of a Gaussian state with QCM $W$, and “pure” QCMs are those that correspond to pure Gaussian states. While the typical choice $\alpha=1$ yields the standard entanglement of formation, Rényi-$2$ quantifiers arise naturally in the Gaussian setting, as they reproduce Shannon entropies of measurement outcomes Adesso et al. (2012); Lami et al. (2017). For $\alpha=2$, Eq. (14) becomes

[TABLE]

where for a positive definite matrix $V$ we set $M(V)\coloneqq S_{2}(V)=\frac{1}{2}\ln\det V$. We then find the following:

Theorem 4**.**

The Rényi-2 Gaussian entanglement of formation of a $k$-extendible Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ of $n_{A}+n_{B}$ modes with QCM $V_{AB}$ is bounded from above as $E^{\mathrm{\scriptscriptstyle G}}_{F,2}\left(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}\right)\leq\frac{M(V_{A})}{k}\,.$ Consequently, the standard entanglement of formation of $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ satisfies $E_{F,1}\left(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}\right)\leq E_{F,1}^{\mathrm{\scriptscriptstyle G}}\left(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}\right)\leq n_{A}\,\varphi\left(\frac{M(V_{A})}{n_{A}k}\right)$, where $\varphi(x)\coloneqq\frac{\mathrm{e}^{x}+1}{2}\ln\left(\frac{\mathrm{e}^{x}+1}{2}\right)-\frac{\mathrm{e}^{x}-1}{2}\ln\left(\frac{\mathrm{e}^{x}-1}{2}\right)$.

The function $M$ plays the role of some “effective dimension” in the bounds above. It is related to other quantities conventionally thought of as infinite-dimensional substitutes for the dimension, such as the mean photon number, defined for a state $\rho$ of $n$ modes as $\left<N\right>=\left<N\right>_{\rho}\coloneqq\operatorname{Tr}\left[\left(\sum_{j}a_{j}^{\dagger}a_{j}\right)\rho\right]$. When $\rho$ is zero-mean Gaussian and has QCM $V$, one has $\left<N\right>=\frac{1}{4}\left(\operatorname{Tr}V-2n\right)$. By applying the arithmetic–geometric mean inequality, one can show that $M(V)\leq n\ln\left(\frac{2\left<N\right>}{n}+1\right)$, which can be further relaxed to $M(V)\leq 2\left<N\right>$.

Summary & outlook.

We accomplished a comprehensive analysis of the $k$-extendibility of Gaussian quantum states. We determined that a Gaussian state is $k$-extendible if and only if it is Gaussian $k$-extendible, which allowed us to derive a simple semidefinite program that solves the problem completely in a computationally efficient way. When the extended system contains one mode only, we fully characterized the set of $k$-extendible Gaussian states by a simple analytic condition reminiscent of the PPT criterion. We demonstrated further applications to Gaussian state steerability, $k$-extendiblity of Gaussian channels, bounding the distance between $k$-extendible and separable states, and the Rényi entanglement of formation for Gaussian states. Our results also yield necessary criteria for $k$-extendibility of non-Gaussian states based on second moments. This work sheds novel light onto the fine structure of entanglement and its uses in continuous-variable systems.

It remains an intriguing open problem to find an analytic condition for $k$-extendibility of arbitrary Gaussian states. Another topic for future work is to explore applications of Theorem 2 to the non-asymptotic capacities of Gaussian channels, in light of recent work Kaur et al. (2018); Berta et al. (2018) exploiting $k$-extendibility to bound the performance of quantum processors.

Acknowledgements.

Acknowledgments.

LL and GA acknowledge financial support from the European Research Council under the Starting Grant GQCOP (Grant No. 637352). SK and MMW acknowledge support from the NSF under Grant No. 1714215. S.K. acknowledges support from the NSERC PGS-D.

I Background

For a continuous-variable system of $n$ modes, we let $r=(x_{1},p_{1},\dotsc,x_{n},p_{n})^{{\scriptscriptstyle\mathsf{T}}}$ denote the vector of position- and momentum- quadrature operators. These operators satisfy the canonical commutation relations $[x_{j},p_{k}]=\mathrm{i}\delta_{j,k}\mathds{1}$ for all $1\leq j,k\leq n$, which we can rewrite compactly as

[TABLE]

Given a quantum state represented by a density matrix $\rho$, we can construct the real vector $s\in\mathds{R}^{2n}$ defined by $s_{j}=\operatorname{Tr}[r_{j}\,\rho]$ for all $1\leq j\leq 2n$, called the mean vector of $\rho$, and the real symmetric $2n\times 2n$ matrix $V$ satisfying $V_{ij}=\operatorname{Tr}[\{r_{i}-s_{i},r_{j}-s_{j}\}\,\rho]$, called the quantum covariance matrix (QCM) of $\rho$, where $\{A,B\}\coloneqq AB+BA$ is the anticommutator. The QCM of every quantum state $\rho$ necessarily satisfies a Robertson–Schrödinger uncertainty principle of the form Simon et al. (1994)

[TABLE]

The QCM $V_{AB}$ of any bipartite state $\rho_{AB}$ has the block matrix form

[TABLE]

where $V_{A}$ and $V_{B}$ are the QCMs of the reduced states $\rho_{A}$ and $\rho_{B}$, respectively, and $X$ describes the correlations between the systems $A$ and $B$.

A state $\rho$ on $n$ modes is called a Gaussian state if it is either a thermal state of some quadratic Hamiltonian, i.e., if it can be written in the form $\rho=\frac{\mathrm{e}^{-\beta H(A,x)}}{\operatorname{Tr}[\mathrm{e}^{-\beta H(A,x)}]}$ for some $\beta>0$, $x\in\mathds{R}^{2n}$, and a real symmetric $2n\times 2n$ matrix $A$, where $H(A,x)=\frac{1}{2}r^{{\scriptscriptstyle\mathsf{T}}}Ar+r^{{\scriptscriptstyle\mathsf{T}}}x$ is quadratic in the canonical operators, or it is a limit of states of that form.

Gaussian states are uniquely described by their mean vector and quantum covariance matrix. Furthermore, any Gaussian state $\rho$ on $n$ modes with QCM $V$ can always be brought into a canonical form by means of a symplectic unitary $U_{S}$, which acts on the mode operators $r$ as $U_{\!S}^{\mathstrut}r\,U_{\!S}^{\dagger}=Sr$, where $S$ is a symplectic matrix, i.e., a matrix satisfying the defining relation $S\Omega S^{{\scriptscriptstyle\mathsf{T}}}=\Omega$. If the Williamson canonical form of $V$ is

[TABLE]

one has

[TABLE]

where the canonical form of one-mode Gaussian states is defined in the Fock basis by

[TABLE]

For $\lambda\in[0,1]$ we denote by $\mathcal{L}_{\lambda}$ the attenuator channel of parameter $\lambda$, defined by

[TABLE]

where $U_{\lambda}$ is the symplectic unitary that implements a beam splitter with transmissivity $\lambda$, and on two modes—whose annihilation operators we denote by $a,b$—takes the form

[TABLE]

where we use the function $\arccos:[0,1]\to[0,\pi/2]$, and the last identity is a rewriting of (Serafini, 2017, Eq. (5.116)). Upon tedious but straightforward algebraic manipulations, (S7) and (S8) together yield the following Kraus representation of the attenuator channel (Ivan et al., 2011, Eq. (4.5)):

[TABLE]

II Extensions of states of continuous-variable systems

Throughout this section we clarify some subtleties related to extensions in continuous-variable systems. We start by asking whether states with bounded energy are in some sense a closed set under $k$-extensions. The reason why this is important is because those states are naturally the most physically relevant.

Lemma 1**.**

Let $\rho_{AB}$ be a (not necessarily Gaussian) state with vanishing first moments and finite second moments, identified by a QCM $V_{AB}$ as in (1). Then any $k$-extension $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$ has (a) vanishing first moments and (b) finite second moments with a corresponding QCM of the form as in (2).

Proof.

We only show that the second moments must be finite, as the claims concerning the first moments are proved in an analogous fashion. To see this, it suffices to show that: (i) \operatorname{Tr}\big{[}r_{A,\alpha}^{2}\,\widetilde{\rho}_{AB_{1}\ldots B_{k}}\big{]}<\infty for all $1\leq\alpha\leq 2n_{A}$; and that (ii) \operatorname{Tr}\big{[}r_{B_{j},\,\beta}^{2}\,\widetilde{\rho}_{AB_{1}\ldots B_{k}}\big{]}<\infty for all $1\leq j\leq k$ and $1\leq\beta\leq 2n_{B}$. Here, $r_{B_{j},\,\beta}$ denotes the $\beta$-th component of the vector of canonical operators acting on $B_{j}$, and analogously for $r_{A,\alpha}$. For (ii) one writes

[TABLE]

The proof of (i) follows the same lines.

Since the second moments of $\,\widetilde{\rho}_{AB_{1}\ldots B_{k}}$ have been shown to be finite, we can now form the corresponding quantum covariance matrix $\widetilde{V}_{AB_{1}\dotsb B_{k}}$. The particular structure in (2) results from the requirements imposed on $k$-extensions. One such requirement is $\widetilde{\rho}_{AB_{j}}=\rho_{AB}$ for all $1\leq j\leq k$, which implies that the first row of $\widetilde{V}_{AB_{1}\dotsb B_{k}}$ must feature identical copies of the matrix $X$. Indeed, for all $1\leq j\leq k$,

[TABLE]

Symmetry with respect to the systems $B_{1},\dotsc,B_{k}$ implies that $\widetilde{\rho}_{B_{j_{1}}B_{j_{2}}}=\widetilde{\rho}_{B_{1}B_{2}}$ for all $1\leq j_{1},j_{2}\leq k$. Furthermore, all such two-party reduced states are invariant under swapping of the two subsystems. Therefore, the matrix $Y$ whose entries are given by

[TABLE]

is symmetric: $Y_{i,\ell}=Y_{\ell,i}$ for all $1\leq i,\ell\leq 2n_{B}$. The covariance matrix $\widetilde{V}_{AB_{1}\dotsb B_{k}}$ thus has the structure as in (2). ∎

Before we move on, we want to elaborate on the role of the symmetry requirement in Theorem 1. For the sake of this discussion, let us call a state $\rho_{AB}$ generally $k$-extendible on the $B$ system if there exists a quantum state $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$ on $A$ and $k$ copies of $B$ such that $\widetilde{\rho}_{AB_{i}}\equiv\rho_{AB}$ for all $i$, where $\widetilde{\rho}_{AB_{i}}$ refers to the reduced state over the systems $AB_{i}$, and for a fixed $i$ we identified $B_{i}\equiv B$. Unlike in the definition we adopt in the main text, here we are not requiring the whole state to be symmetric, only the reductions to always be the same. To stress the difference between the two cases, here and for the rest of this section we refer to standard $k$-extendible state as symmetrically $k$-extendible.

Clearly, if $\rho_{AB}=\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is Gaussian, we may want to look at Gaussian general $k$-extensions, in the same way as we did in the main text for symmetric $k$-extensions. Fortunately, it turns out that Theorem 1 implies that these concepts are equivalent.

Corollary 5**.**

Let $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ be a Gaussian state. Then the following are equivalent:

1. (i)

$\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$* is generally $k$-extendible;* 2. (ii)

$\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$* is symmetrically $k$-extendible;* 3. (iii)

$\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$* has a Gaussian general $k$-extension;* 4. (iv)

$\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$* has a Gaussian symmetric $k$-extension.*

Proof.

The general and well-known fact that $(i)\Leftrightarrow(ii)$ follows immediately by symmetrization arguments: given a general $k$-extension $\widetilde{\rho}_{AB_{1}\ldots B_{k}}$, we can construct the symmetric $k$-extension

[TABLE]

where $S_{k}$ denotes the symmetric group over $k$ elements, and $U_{B_{1}\ldots B_{k}}(\pi)$ is the unitary that permutes the $B$ systems according to $\pi\in S_{k}$. This shows that indeed $(i)\Leftrightarrow(ii)$. Theorem 1 guarantees that $(ii)\Leftrightarrow(iv)$. Also, the implications $(iv)\Rightarrow(iii)\Rightarrow(i)$ follow by definition. This completes the proof. ∎

Due to this equivalence result, the above definition of general $k$-extendibility does not add much to our discussion. We therefore do not consider it further, except very briefly in Remark 6 below.

III Gaussian extendibility: degenerate cases

In this section, we argue that the proof of Theorem 2 remains valid also in the degenerate cases where $V_{A}-\mathrm{i}\Omega_{A}$ is not invertible and thus some of the intermediate statements in the main text cease to hold as written. Namely, (7) and (8) do not seem to make sense when $V_{A}-\mathrm{i}\Omega_{A}$ does not possess an inverse. As it turns out, the right way to interpret these conditions is via a regularization procedure. For instance, (8) is said to hold if it holds for all $\epsilon>0$ when one performs the substitution $V_{A}\mapsto V_{A}+\epsilon\mathds{1}_{A}$.

A regularization procedure can be used to define the Schur complement with respect to a non-invertible block: given a positive semidefinite matrix $M$ as in (6), where $P$ is not necessarily invertible, we define $M/P$ as

[TABLE]

provided that such a limit exists. Observe that this happens if and only if $\mathrm{im}(Z)\perp\ker(P)$, where $\mathrm{im}$ denotes the image (or range), and orthogonality is between subspaces. When this is the case, one still has $M/P=Q-Z^{\dagger}P^{-1}Z$ provided that the inverse is taken on the support. This definition of generalized Schur complement fits well into the positive semidefiniteness condition in (6). In fact, note that

[TABLE]

Hence, one can still claim that the fundamental equivalence in (6) holds formally, provided that Schur complements are intended as generalized. It is understood that on the r.h.s. of (6) one still requires that $M/P$ actually exists. We now review the proof of Theorem 2 in light of these considerations.

Proof of Theorem 2 (complete version).

In what follows, we assume that $k\geq 2$. From Theorem 1, we know that a Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is $k$-extendible if and only if it has a Gaussian $k$-extension. If $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ has vanishing first moments, then the same is true for any extension, by Lemma 1. Hence, searching for a Gaussian $k$-extension amounts to asking whether there exists a legitimate QCM $\widetilde{V}_{AB_{1}\ldots B_{k}}$ of the form as in (2) that satisfies the bona fide condition

[TABLE]

In fact, it is not difficult to verify that Gaussian states with zero mean possess the same symmetries as their QCMs. Specifically, any Gaussian state with a QCM of the form as in (2) is both invariant under permutation of any two $B$ systems and such that its reduction to $AB_{1}$ coincides with the original state with QCM $V_{AB}$ as in (1).

We now rephrase (S20) using the suitably regularized conditions in (6), as given explicitly by (S19). Remember that $V_{A}\geq\mathrm{i}\Omega_{A}$ holds by hypothesis since the reduced state on $A$ is a legitimate density matrix. Making the dependence on the regularizing parameter $\epsilon>0$ explicit for clarity, we thus obtain that

[TABLE]

to be obeyed for all $\epsilon>0$. We can conveniently write the above inequality by making the identification $\bigoplus_{j=1}^{k}\mathds{R}^{2n_{B}}=\mathds{R}^{k}\otimes\mathds{R}^{2n_{B}}$ in terms of the underlying vector spaces. Introducing $\ket{+}\coloneqq\frac{1}{\sqrt{k}}\sum_{j=1}^{k}\ket{j}\in\mathds{R}^{k}$, the inequality (S22) becomes

[TABLE]

Now, since $\ket{+}\!\!\bra{+}$ and $\mathds{1}-\ket{+}\!\!\bra{+}$ are projectors onto orthogonal subspaces, the above relation is equivalent to

[TABLE]

Introducing the alternative parametrization $\Delta\coloneqq V_{B}-Y$, we can rephrase this as

[TABLE]

reproducing (7). Using again (S19) – this time backwards – we see that (S27) is equivalent to the existence of a real matrix $\Delta_{B}\geq\mathrm{i}\Omega_{B}$ such that (4) is obeyed.

As already mentioned in the main text, to see that (4) implies (5) one substitutes the (complex conjugate) bona fide condition $\Delta_{B}\geq-\mathrm{i}\Omega_{B}$ into (4). Observe that this is possible since $\Delta_{B}$ is real.

Now, the problem is to prove that (5) is also sufficient to guarantee the existence of a real $\Delta\geq\mathrm{i}\Omega_{B}$ that satisfies (4) when $n_{B}=1$. In order to do this, it suffices to prove that (5) is equivalent to (S27). To this end, we employ (Lami et al., 2018, Lemma 7), which states that given two $2\times 2$ Hermitian matrices $M,N$, there exists a real matrix $R$ such that $M\leq R\leq N$ if and only if both $M\leq N$ and $M^{*}\leq N$ hold true, with $M^{*}$ being the complex conjugate of $M$. Since all matrices in (S27) are $2\times 2$ when $n_{B}=1$, we can rephrase it as

[TABLE]

which upon elementary algebraic manipulations translates to the following two conditions:

[TABLE]

The first inequality follows from the bona fide condition $V_{AB}\geq\mathrm{i}\Omega_{AB}$ via an application of (S19), while the second is equivalent to (5) again via (S19).

Finally, the fact that (4) and (5) fail to be equivalent already for $n_{A}=n_{B}=k=2$ is demonstrated by the example of the Werner–Wolf state of Werner and Wolf (2001), as the discussion in the next section shows. ∎

Remark 6**.**

We take the chance here to draw a thorough comparison between our results and techniques and those of Rajarama Bhat et al. (2017). The starting point is the structure of the $k$-extended QCM in (2), to be compared with (Rajarama Bhat et al., 2017, Eq. (2.1)). We see that the fact that our matrix $Y$ (to be identified with their $\theta_{k}$) needs to be symmetric was not recognized in Rajarama Bhat et al. (2017) as descending directly from the required symmetry of the extended state $\rho_{AB_{1}\ldots B_{k}}$ under the exchange of any two $B$ systems. We have instead proved this explicitly (Lemma 1). Also, we have shown in Corollary 5 that a Gaussian state is generally $k$-extendible if and only if it has a Gaussian symmetric $k$-extension; this provides another way to show that if a general (possibly non-symmetric) $k$-extension of the state exists, then the corresponding matrix $Y$ that appears in its QCM as in (2) can always be taken to be symmetric.

Following the argument in Rajarama Bhat et al. (2017), one notes that the symmetry of $\theta_{k}$ is recovered via (Parthasarathy and Sengupta, 2015, Theorem 2.1) as a consequence of the bona fide condition when complete extendibility is assumed. At this precise point the reasoning in Rajarama Bhat et al. (2017) ceases to apply to $k$-extendible states with finite $k$, and holds instead only for completely extendible ones. Therefore, while the algebraic manipulations that lead to our (S25)–(S26) are identical to those in Rajarama Bhat et al. (2017), and both are indeed elementary applications of known properties of Schur complements, the conclusions that one is allowed to draw from them here are significantly more powerful than those obtained in Rajarama Bhat et al. (2017). For example, we are able to derive simple necessary and sufficient conditions for the $k$-extendibility of Gaussian states, solving an outstanding open problem that was explicitly stated as such in Rajarama Bhat et al. (2017).

Last but not least, in Rajarama Bhat et al. (2017) it does not seem to have been observed that the case $n_{B}=1$ can be solved analytically, yielding a necessary and sufficient condition that resembles PPT-ness. This special case is of paramount physical importance because of its applicability to the theory of quantum communication over single-sender single-receiver Gaussian channels.

Before we move on, we want to dwell on the possibility of deriving the separability condition for Gaussian states found in (Lami et al., 2018, Theorem 5) directly from Theorem 2. As argued in the main text, due to the fact that $k$-extendibility is a stronger property than $(k-1)$-extendibility, it suffices take the limit $k\to\infty$ of the condition in (4). Technically, the evaluation of this limit is made less obvious by the fact that the choice of $\Delta_{B}$ may depend on $k$. This is a priori a problem because the set of QCMs is not compact. However, it is not difficult to verify that any $\Delta_{B}$ satisfying (4) must automatically satisfy also $\Delta_{B}\leq V_{B}$. Since the set of QCMs with this property is compact, the sequence of matrices $\Delta_{B}$ admits a converging subsequence, and we can take the limit on that subsequence Rajarama Bhat et al. (2017). This shows that (Lami et al., 2018, Theorem 5) is in fact a corollary of our more general Theorem 2.

IV Extendibility of two-mode Gaussian states

We devote this section to the analytical characterization of $k$-extendibility in the simple case of two-mode Gaussian states, as can be deduced from our general Theorem 2. This is in complete analogy with the analytical characterization of separability given in Simon (2000). We start by recalling that the QCM of any two-mode Gaussian state can be brought into the normal ‘Simon’ form

[TABLE]

by means of local symplectic operations. Here, in compliance with the notation of (1), the first two rows and columns pertain to the first mode, and the remaining two to the second. As shown in Simon (2000), the bona fide condition $V_{AB}\geq\mathrm{i}\Omega_{AB}$ translates to a set of simple inequalities involving the local symplectic invariants $a,b,c_{+},c_{-}$:

[TABLE]

We can make the local symmmetry more explicit by rephrasing these conditions in terms of local symplectic invariants only. Referring to (1) for notation, we obtain that

[TABLE]

By appropriately modifying the second condition 222Let us sketch the derivation here for readability. First we use the Schur factorization theorem to write $\det V_{AB}=(\det V_{A})(\det(V_{B}-X^{\scriptscriptstyle\mathsf{T}}V_{A}^{-1}X))$. Then we apply the special determinant formula $\det(M-N)=\det M-\operatorname{Tr}[M\Omega N^{\scriptscriptstyle\mathsf{T}}\Omega^{\scriptscriptstyle\mathsf{T}}]+\det N$, valid for $2\times 2$ real matrices, to $M=V_{B}$ and $N=X^{\scriptscriptstyle\mathsf{T}}V_{A}^{-1}X$. Finally, we employ also the $2\times 2$ matrix inversion formula $(\det V_{A})V_{A}^{-1}=\Omega V_{A}\Omega^{\scriptscriptstyle\mathsf{T}}$. Formula (S33) is obtained by putting all together, remembering that $\Omega$ is skew-symmetric., one can also rewrite it as

[TABLE]

which reproduces (Simon, 2000, Eq. (17)), once one accounts for the extra factor $1/2$ that comes from the different conventions adopted to define QCMs 333Curiously, the rightmost inequality in (S33) – equivalently, the rightmost inequality in the second line of (S31) – does not appear in Simon (2000). It is however necessary and not implied by the others, as the example $a=b=1$, $c_{+}=1-\epsilon=-c_{-}$ shows..

We now write out the conditions for a two-mode Gaussian state with QCM as in (S30) to be $k$-extendible. In what follows, we usually assume that the given QCM already obeys the bona fide condition.

Proposition 7**.**

Let $V_{AB}\geq\mathrm{i}\Omega_{AB}$ be a two-mode QCM partitioned as in (1). Then it is $k$-extendible on $B$ if and only if

[TABLE]

which can be equivalently formulated as

[TABLE]

If the QCM is parametrised as in (S30), then (S34) and (S35) translate to

[TABLE]

Proof.

Without loss of generality, we can prove (S36). Applying the inequality (5) and permuting second and third rows and columns, we see that $k$-extendibility of $V_{AB}$ on $B$ is equivalent to the matrix inequality

[TABLE]

where the blocks $Q,P,D$ are all $2\times 2$. Since QCMs are positive definite, it automatically holds that $Q,P>0$. Hence, we can use the block positivity conditions in (6) (nondegenerate case) to rephrase this as

[TABLE]

and in turn as

[TABLE]

Here, we assumed that $D$ is invertible, i.e., that $k\neq 2$. We will see a posteriori that this assumption is anyway immaterial as far as the final result is concerned. Call $\lambda,\mu>0$ the two eigenvalues of the above positive definite matrix $M$. The conditions $\lambda,\mu\geq 1$ are equivalent to $1+\lambda\mu\geq\lambda+\mu\geq 2$. Note that

[TABLE]

We now argue that the condition $\lambda+\mu\geq 2$ is superfluous. In fact, if $c_{+}c_{-}<0$ it is already implied by the rightmost inequality in the second line of (S31). Otherwise, if $c_{+}c_{-}\geq 0$ it is well-known Simon (2000) that $V_{AB}$ corresponds to a separable state, hence it is $k$-extendible for all $k$, and $\lambda+\mu\geq 2$ must be satisfied. We are thus left with the condition $1+\lambda\mu\geq\lambda+\mu$, which is exactly (S36). For the special case $k=2$ this reduces to the simple inequality $(ab-c_{+}^{2})(ab-c_{-}^{2})\geq b^{2}$, which is elementarily seen to be equivalent to $M\geq\mathds{1}$, where $M$ is given by (S39). This shows that the working assumption $k\neq 2$ is in fact unnecessary, as claimed. Finally, by rewriting (S36) we obtain directly (S34), and from it also (S35). Observe that (S35) can be derived from (S34) by elementary linear algebra techniques, as explained in Simon (2000) and reviewed above. ∎

As a final remark, observe that in the limit $k\to\infty$ either of the inequalities (S34)–(S36) reduces to the separability condition in (Simon, 2000, Eq. (19)), as expected.

V The Werner–Wolf state is not 2-extendible

This subsection is devoted to the analysis of the Werner–Wolf bound entangled Gaussian state of Werner and Wolf (2001) from the point of view of $k$-extendibility. This bipartite Gaussian state of a system composed of $2+2$ modes is particularly interesting because it can be shown to be $2$-unextendible yet to obey (5) for all $k$. It thus demonstrates that condition (5) is no longer equivalent to (4) when the local subsystems consist of at least two modes each. We start by recalling that its QCM is given by (Werner and Wolf, 2001, Eq. (9))

[TABLE]

The above expression in understood to pertain to the following ordering of the four pairs of canonical operators: $x_{A,1},p_{A,1},x_{A,2},p_{A,2},x_{B,1},p_{B,1},x_{B,2},p_{B,2}$. We start with a simple result that connects the simplified condition in (5) with the positive partial transposition (PPT) criterion Peres (1996).

Lemma 2**.**

A QCM $V_{AB}$ obeys (5) if and only if the corresponding Gaussian state is PPT, i.e., if and only if $V_{AB}\geq(\mathrm{i}\Omega_{A})\oplus(-\mathrm{i}\Omega_{B})$.

Proof.

The matrices $V_{AB}-(\mathrm{i}\Omega_{A})\oplus\left(-\left(1-\frac{2}{k}\right)\mathrm{i}\Omega_{B}\right)$ are all positive semidefinite by hypothesis. Since positive semidefinite matrices form a closed set, the limit

[TABLE]

is also positive semidefinite. This is the same as saying that the Gaussian state with QCM $V_{AB}$ is PPT, as shown in (Simon, 2000, Eq. (10)). ∎

Thanks to Lemma 2 and leveraging the fact that the Werner–Wolf state is PPT by construction, we know that it obeys (5) for all $k$. However, we now proceed to show that it is not even two-extendible. Let us first establish some technical lemmata.

Lemma 3**.**

Let $\Delta\geq\mathrm{i}\Omega$ be a QCM. Then $\Delta\#(\Omega\Delta\Omega^{{\scriptscriptstyle\mathsf{T}}})\geq\mathds{1}$, where

[TABLE]

denotes the matrix geometric mean Pusz and Woronowicz (1975); Ando (1979).

Proof.

Since any QCM $\Delta$ is lower bounded by the QCM of some pure state, and the matrix geometric mean is monotonic in both entries, we can freely assume that $\Delta$ is the QCM of a pure state, i.e., that it is a symplectic matrix. This means that $\Delta\Omega\Delta=\Omega$ (remember that $\Delta=\Delta^{{\scriptscriptstyle\mathsf{T}}}$), which we can alternatively write as $\Omega\Delta\Omega^{{\scriptscriptstyle\mathsf{T}}}=\Delta^{-1}$. Then it is straightforward to see that $\Delta\#(\Omega\Delta\Omega^{{\scriptscriptstyle\mathsf{T}}})=\Delta\#\Delta^{-1}=\mathds{1}$, which completes the proof. ∎

The above result can be interpreted by noticing that the condition $\Delta\#(\Omega\Delta\Omega^{{\scriptscriptstyle\mathsf{T}}})\geq\mathds{1}$ amounts to saying that the Gaussian state with QCM $\Delta\#(\Omega\Delta\Omega^{{\scriptscriptstyle\mathsf{T}}})$ is a convex combination of coherent states, i.e., it is a classical state. Thus, even if $\Delta$ represents a highly squeezed state, taking the above geometric mean “averages out” all the squeezing.

Lemma 4**.**

For all $2n\times 2n$ QCMs $\Delta\geq\mathrm{i}\Omega$ and all vectors $\ket{v}\in\mathds{C}^{2n}$, we have that

[TABLE]

Proof.

The real-valued map $A\mapsto\bra{v}A\ket{v}$ is positive, i.e., it is nonnegative on positive semidefinite matrices. The claim follows from (Ando, 1979, Theorem 3) and Lemma 3. ∎

We are now ready to prove the main result of this section.

Proposition 8**.**

The Werner–Wolf Gaussian state with QCM given by (S42) is not $2$-extendible on the $B$ system.

Proof.

Since it can be readily verified that $\gamma_{A}>\mathrm{i}\Omega_{A}$ (more precisely, the symplectic spectrum of $\gamma_{A}$ is $\{\sqrt{2},\sqrt{2}\}$), we can rephrase (4) as (7) without the need to consider generalized Schur complements. Computing the r.h.s. of (7) for $V_{AB}=\gamma_{AB}$ and for $k=2$ yields

[TABLE]

We want to show that there does not exist a matrix $\Delta_{B}$ such that $\Delta_{B}\leq H_{B}$. It is straightforward to see that the normalized vector

[TABLE]

is an eigenvector of $H_{B}$ with corresponding eigenvalue $3-\sqrt{6}$. If there existed a real $\Delta_{B}$ such that $\mathrm{i}\Omega_{B}\leq\Delta_{B}\leq H_{B}$ then by Lemma 4 we would obtain

[TABLE]

which is a contradiction. Hence, $\gamma_{AB}$ does not satisfy (4) for any $\Delta_{B}\geq\mathrm{i}\Omega_{B}$, implying that the Werner–Wolf state is not $2$-extendible. ∎

VI Extendibility of Gaussian channels

We now provide further details of the $k$-extendibility of single-sender, single-receiver Gaussian channels. By such a $k$-extendible channel, as stated in the main text, we mean that it can be implemented as a broadcast channel from a single sender to $k$ receivers, such that the reduced channel from the sender to any one of the receivers is the same as the original channel. Recall that a Gaussian channel $\mathcal{N}$ with $n$ input and $m$ output modes is uniquely characterized by a pair of real matrices $X,Y$, where $X$ is $2m\times 2n$ and $Y$ is $2m\times 2m$, and a real vector $\delta\in\mathbb{R}^{2m}$, such that $Y+\mathrm{i}\Omega\geq\mathrm{i}X\Omega X^{\intercal}$. Since a Gaussian channel sends Gaussian states to Gaussian states, its action can be described directly at the level of the mean vector and covariance matrix:

[TABLE]

In what follows, we set $\delta=0$ without loss of generality.

Let $A,A^{\prime}$ be two isomorphic quantum systems, possibly infinite-dimensional. It is well-known that any pure state $\ket{\psi}_{AA^{\prime}}$ with invertible marginals defines a Choi–Jamiołkowski isomorphism between the set of quantum channels $\mathcal{N}_{A\to B}$ and the set of bipartite states $\rho_{AB}$ on $AB$ such that $\rho_{A}=\psi_{A}\coloneqq\operatorname{Tr}_{A^{\prime}}\ket{\psi}\!\!\bra{\psi}_{AA^{\prime}}$ (see Holevo (2011) or Holevo (2010)). Denoting the Schmidt decomposition of $\ket{\psi}_{AA^{\prime}}$ by

[TABLE]

the Choi–Jamiołkowski isomorphism is realized by defining (Holevo, 2010, Eq. (6))

[TABLE]

Conversely, every state $\rho_{AB}$ such that $\rho_{A}=\psi_{A}=\operatorname{Tr}_{A^{\prime}}\ket{\psi}\!\!\bra{\psi}_{AA^{\prime}}$ identifies a quantum channel $\mathcal{N}_{A\to B}$ via the formulae

[TABLE]

With this in mind, it is not difficult to realize that a channel $\mathcal{N}_{A\to B}$ is $k$-extendible if and only any (and hence all) of its Choi states $\rho_{AB}^{\mathcal{N}}$ is $k$-extendible on $B$. On the one hand, if $\widetilde{\mathcal{N}}_{A\to B_{1}\ldots B_{k}}$ is a $k$-extension of $\mathcal{N}_{A\to B}$, then clearly the corresponding Choi state $\rho_{AB_{1}\ldots B_{k}}^{\widetilde{\mathcal{N}}}$ is a $k$-extension of $\rho_{AB}^{\mathcal{N}}$. On the other hand, if $\sigma_{AB_{1}\ldots B_{k}}$ is a $k$-extension of $\rho_{AB}^{\mathcal{N}}$, due to the identity $\operatorname{Tr}_{B_{1}\ldots B_{k}}[\sigma_{AB_{1}\ldots B_{k}}]=\operatorname{Tr}_{B}\rho_{AB}^{\mathcal{N}}=\psi_{A}$ we see that $\sigma_{AB_{1}\ldots B_{k}}$ is the Choi state of a legitimate quantum channel $\widetilde{\mathcal{N}}_{A\to B_{1}\ldots B_{k}}$. By using (S52), it is not difficult to verify that this is indeed a $k$-extension of $\mathcal{N}_{A\to B}$.

Remark 9**.**

If a Gaussian channel is $k$-extendible, then $k$ instances of it can be implemented by the following procedure, related to the approach of Giedke and Cirac (2002); Wolf et al. (2007); Niset et al. (2009): (1) prepare a two-mode squeezed vacuum state, (2) send one share of it through the $k$-extension of the channel, (3) perform the usual bosonic continuous-variable teleportation protocol Braunstein and Kimble (1998) on the channel input state and the share of the entangled resource not sent into the channel, (4) broadcast the Bell measurement outcomes to all $k$ receivers, who then correct their local systems. In the limit as the squeezing strength goes to infinity, this simulation of the $k$-extension of the channel converges to the original one with respect to the strong topology.

The following result is a consequence of the above discussion:

Corollary 10**.**

A Gaussian channel $V\mapsto XVX^{\intercal}+Y$ from $n$ to $m$ modes is $k$-extendible in the sense of Pankowski et al. (2013); Kaur et al. (2018) if and only if there exists a $2m\times 2m$ real matrix $\Delta$ such that

[TABLE]

When $m=1$, this is equivalent to

[TABLE]

If also $n=1=m$, then a simplified equivalent condition that takes into accounts also the complete positivity requirement reads

[TABLE]

Proof.

Denote the channel under consideration by $\mathcal{N}$. Then, define its Choi state by $\rho_{AB}^{\mathcal{N}}\coloneqq(\text{id}_{A}\otimes\mathcal{N}_{A^{\prime}\to B})(\ket{\psi_{r}}\bra{\psi_{r}}^{\otimes n})$, where $\ket{\psi_{r}}$ is the two-mode squeezed vacuum given by

[TABLE]

The channel $\mathcal{N}$ is $k$-extendible (by definition) if and only if the state $\rho_{AB}^{\mathcal{N}}$ is $k$-extendible, which is true if and only if there exists $\Delta_{B}$ such that $\Delta_{B}\geq\mathrm{i}\Omega_{B}$ and such that the corresponding covariance matrix $V_{AB}^{\mathcal{N}}(r)$ satisfies (7), where

[TABLE]

Then,

[TABLE]

Identifying the block $X$ in (7) with the off-diagonal block $\sinh(2r)\Sigma_{n}X^{\intercal}$ of $V_{AB}^{\mathcal{N}}(r)$, and using $\Sigma_{n}\Omega_{n}\Sigma_{n}=-\Omega_{n}$, we get

[TABLE]

which leads to (S53).

In the case $m=1$, we apply the same calculations above to the condition in (8) in order to obtain (S54). Finally, (S55) is derived by applying the usual $2\times 2$ determinant formula $M\Omega M^{\scriptscriptstyle\mathsf{T}}=(\det M)\Omega$ to (S54). The same result can be obtained by resorting directly to Proposition 7. ∎

We now illustrate the $k$-extendibility conditions of single-mode Gaussian channels. A classification of all such channels into six different categories has been given in Holevo (2007) (see also Holevo (2012)). Here we can exploit this classification in order to determine $k$-extendibility of all single-mode Gaussian channels because $k$-extendibility of a single-sender, single-receiver channel is invariant under arbitrary input and output unitaries, and the procedure from Holevo (2007) exploits input and output Gaussian unitaries in order to arrive at the classification.

The categories of single-mode channels from the classification of Holevo (2007) are labeled as $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C$, and $D$. All channels in classes $A_{1}$, $A_{2}$, and $D$ are entanglement breaking, as proved in Holevo (2008). Thus, these channels are $k$-extendible for all $k\geq 2$. It thus remains to consider the channels in the classes $B_{1}$, $B_{2}$, and $C$. Channels in the class $B_{1}$ have $X=\mathds{1}$ and $Y=(\mathds{1}-\sigma_{Z})/2$. Applying the condition in (S55), we find, for all $k\geq 2$, that channels in this class are not $k$-extendible. This is consistent with the fact that their unconstrained quantum capacity is infinite Holevo (2007). The remaining channels are the most important for applications, as stressed in (Holevo and Giovannetti, 2012, Section 3.5) and (Holevo, 2012, Section 12.6.3). Channels in the class $B_{2}$ are called additive-noise channels, and channels in the class $C$ are either thermal channels or amplifier channels. By applying (S55), we find the necessary and sufficient conditions for their $k$-extendibility:

• •

The thermal channel of transmissivity $\eta\in(0,1)$ and environment thermal photon number $N_{S}\geq 0$, defined by $X=\sqrt{\eta}\mathds{1}$ and $Y=(1-\eta)(2N_{B}+1)\mathds{1}$, is $k$-extendible if and only if

[TABLE]

If the channel is a pure-loss channel with $N_{B}=0$, then we see that it is $k$-extendible if and only if $\eta\leq 1/k$.

• •

The amplifier channel of gain $G>1$ and environment thermal photon number $N_{B}\geq 0$, defined by $X=\sqrt{G}\mathds{1}$ and $Y=(G-1)(2N_{B}+1)\mathds{1}$, is $k$-extendible if and only if

[TABLE]

If $N_{B}=0$, as is the case for the pure-amplifier channel, then the channel is not $k$-extendible for all $k\geq 2$ and $G>1$.

• •

The additive noise channel defined by $X=\mathds{1}$ and $Y=\xi\mathds{1}$, with noise parameter $\xi>0$ is $k$-extendible for $k\geq 2$ if and only if

[TABLE]

As expected, these conditions for $k$-extendibility of the channels imply the entanglement-breaking conditions from Holevo (2008) in the limit $k\to\infty$. We also recover the conditions for two-extendibility (antidegradability) from (Caruso et al., 2006, Eq. (4.6)), for thermal and amplifier channels. See Fig. 1 for a plot of the parameter space of the single-mode Gaussian channels.

The multi-mode additive noise channel defined by $X=\mathds{1}$ and some $Y\geq 0$ is $k$-extendible ($k\geq 2$) if and only if $Y>0$ and $\nu_{\min}(Y)\geq 2\left(1-1/k\right)$, where $\nu_{\min}$ indicates the minimal symplectic eigenvalue.

VII Proof of Theorem 3

In this section, we prove Theorem 3, which is the statement that any Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ of $n=n_{A}+n_{B}$ modes that is $k$-extendible satisfies

[TABLE]

where

[TABLE]

are the distances of $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ to the set of separable states as measured by the trace norm and the Petz–Rényi relative entropy $D_{\alpha}(\rho\|\sigma)\coloneqq\frac{1}{\alpha-1}\ln\operatorname{Tr}[\rho^{\alpha}\sigma^{1-\alpha}]$, respectively, and

[TABLE]

The crucial fact that will allow us to derive all these results is the following.

Proposition 11**.**

Let $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ be a Gaussian state with QCM $V_{AB}$. If it is $k$-extendible on $B$, then the Gaussian state with QCM $\frac{k+1}{k-1}\,V_{AB}$ is separable.

Proof.

Thanks to Theorem 2, we know that if $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ is $k$-extendible on $B$, then (4) is satisfied for some $\Delta_{B}\geq\mathrm{i}\Omega_{B}$. Now, write the complex conjugate bona fide condition for $V_{AB}$ as $V_{AB}\geq(-\mathrm{i}\Omega_{A})\oplus(-\mathrm{i}\Omega_{B})$. Adding $\frac{1}{k}$ of this inequality to (4) yields

[TABLE]

i.e.,

[TABLE]

By (Lami et al., 2018, Theorem 5), this is equivalent to the separability of the Gaussian state with QCM $\frac{k+1}{k-1}\,V_{AB}$. ∎

Proof of Theorem 3.

We start by proving (S64). Drawing inspiration from the techniques in König and Wolf (2009), we construct an ansatz for the separable state $\sigma_{AB}$ to be inserted into (S66): namely, let $\sigma_{AB}=\sigma^{\mathrm{\scriptscriptstyle G}}_{AB}$ be the Gaussian state with the same (vanishing) first moments as $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ and with QCM $W_{AB}\coloneqq\frac{k+1}{k-1}V_{AB}$. By Proposition 11, we know that $\sigma_{AB}^{\mathrm{\scriptscriptstyle G}}$ is separable. The crucial property of $\sigma_{AB}^{\mathrm{\scriptscriptstyle G}}$, however, is that it commutes with $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$. In fact, since the two QCMs are proportional to each other, they can be brought into Williamson form by the same symplectic matrix. Comparing (S4) with (S5), we deduce that $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ and $\sigma_{AB}^{\mathrm{\scriptscriptstyle G}}$ are simultaneously diagonalizable, which is equivalent to them commuting. Naturally, the symplectic eigenvalues of $W_{AB}$ are $\mu_{j}=\lambda\nu_{j}$, where we set $\lambda\coloneqq\frac{k+1}{k-1}$ and $\nu_{j}$ are the symplectic eigenvalues of $V_{AB}$. We can then write:

[TABLE]

The above derivation can be justified as follows: step 1 descends from a telescopic inequality of the form

[TABLE]

The inequality in step 2 holds true because all symplectic eigenvalues $\nu_{j}$ must be at least $1$. As for step 3, we read from (König and Wolf, 2009, Eq. (10)) that

[TABLE]

which for $\mu=\lambda\nu$ and upon maximization over $\nu\geq 1$ yields König and Wolf (2009)

[TABLE]

This concludes the proof of (S64).

The inequality in (S65) can be proved with an analogous calculation. The relative entropy is even better behaved in this context, as it already factorizes over multiple copies; hence, there is no need for the above telescopic inequality. Namely, for a $k$-extendible Gaussian state $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ with QCM $V_{AB}$ we can denote as usual with $\sigma_{AB}^{\mathrm{\scriptscriptstyle G}}$ the Gaussian state with the same first moments and QCM $W_{AB}\coloneqq\frac{k+1}{k-1}V_{AB}$. Since $\sigma_{AB}^{\mathrm{\scriptscriptstyle G}}$ is separable by Proposition 11, we write

[TABLE]

where as above $\lambda=\frac{k+1}{k-1}$, and $\nu_{1},\ldots,\nu_{n}$ are the symplectic eigenvalues of $V_{AB}$. Now, observe that the Petz–Rényi relative entropy $D_{\alpha}(\rho\|\sigma)$ is a non-decreasing function of $\alpha\geq 0$ (Tomamichel, 2015, § 4.4). For commuting states $[\rho,\sigma]=0$, we have that $\lim_{\alpha\to\infty}D_{\alpha}(\rho\|\sigma)=\ln\left\|\sigma^{-1/2}\rho\sigma^{-1/2}\right\|_{\infty}=D_{\infty}(\rho\|\sigma)$, where the latter quantity is the max-relative entropy Datta (2009). Thus, we have to prove exactly two statements:

[TABLE]

In general, the Petz–Rényi relative entropies between Gaussian states can be computed thanks to the formulae found in Seshadreesan et al. (2018). In the present case, our task is made much easier by the fact that the states commute, and hence the quantum Petz–Rényi relative entropies reduce to their classical counterparts. Simple calculations using the expression (S6) reveal that

[TABLE]

We start by proving (S87). Let us define the function

[TABLE]

where $\lambda=\frac{k+1}{k-1}$. We will now show that $f_{k}$ is monotonically increasing in $\nu\geq 1$ for all $k>1$. In fact, computing its derivative, one obtains that

[TABLE]

where

[TABLE]

Now, we claim that $g(\lambda,\nu)>0$ for all $\lambda>1$ and $\nu\geq 1$. In fact, $g(1,\nu)\equiv 0$, and

[TABLE]

Here, the last inequality is a consequence of the elementary relation $\ln(1-x)+\frac{x(2-x)}{2(1-x)}>0$, valid for all $0<x<1$. This can in turn be proved by using for instance a power series expansion:

[TABLE]

We have thus shown that $g(\lambda,\nu)>0$ for all $\lambda>1$ and all $\nu\geq 1$. Going back to (S93), this implies that $f^{\prime}_{k}(\nu)>0$ for $\nu>1$ and hence that $f_{k}$ is monotonically increasing whenever $k>1$. Via (S89), this amounts to saying that D_{k+1}\left(\rho^{\mathrm{\scriptscriptstyle G}}(\nu)\,\big{\|}\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right) is decreasing in $\nu\geq 1$, in turn entailing that

[TABLE]

This proves (S87).

We now turn to the proof of (S88), which fortunately can be obtained much more straightforwardly from (S90). Noting that $\nu\mapsto\frac{\lambda\nu+1}{\nu+1}$ is monotonically increasing for $\nu\geq 1$ because $\lambda>1$, one finds that

[TABLE]

proving (S88). ∎

VIII Optimality of the bounds in Theorem 3

The purpose of this section is to prove that the bounds we established in Theorem 3 are in some sense optimal, at least in some regimes. We show the following.

Theorem 12**.**

Let $k\geq 2$ be fixed. Then there exists a two-mode $k$-extendible Gaussian state $\rho^{\!\mathrm{\scriptscriptstyle G}}$ such that

[TABLE]

Moreover, for all positive integers $m$ there is a family of $(m+m)$-mode bipartite $k$-extendible Gaussian states $\rho^{\mathrm{\scriptscriptstyle G}}_{k,m}(r)$ such that

[TABLE]

as $r\to\infty$, where $E_{D}$ denotes the distillable entanglement.

Remark 13**.**

The above Theorem 12 shows that (a) the bound in (12) is tight for $n=2$ and for all $k$ up to a universal multiplicative constant of $1/8$; (b) the bound in (13) is tight for all balanced systems ($n_{A}=n_{B}$), for all $k$, and for $\alpha\geq 1$, up to a universal multiplicative constant of either $1/2$ (if $\alpha\leq k+1$) or $1/4$ (if $\alpha>k+1$).

Proof of Theorem 12.

We start by proving (S105). Let $\ket{\psi_{r}}=\frac{1}{\cosh(r)}\sum_{j=0}^{\infty}\tanh(r)^{j}\ket{j,j}$ be the two-mode squeezed vacuum state. Consider a passive symplectic unitary $U=U_{B_{1}\ldots B_{k}}$ acting on $k$ modes $B_{1},\ldots,B_{k}$ so that $U^{\dagger}b_{1}U=\frac{b_{1}+\ldots+b_{k}}{\sqrt{k}}$, where $b_{j}$ is the creation operator associated with the $j$-th mode. Define

[TABLE]

Clearly, $\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)$ is $k$-extendible by construction. In fact, it is easy to verify that the state inside the partial trace at the right-hand side of (S107) is a symmetric extension of it. Another elementary property of the above state is that the passive unitary used to define it acts as an effective attenuator of parameter $\frac{1}{k}$, according with the definition in (S7). This means that

[TABLE]

To estimate $\left\|\,\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-\mathrm{SEP}\,\right\|_{1}$, we first observe that since for all traceless operators $X$ it holds that $\|X\|_{1}\geq 2\|X\|_{\infty}$, one can give the lower bound

[TABLE]

We now remember that for all bipartite pure states $\ket{\Psi}$ with maximal Schmidt coefficient $\lambda_{1}(\Psi)$ one has that $\bra{\Psi}\sigma\ket{\Psi}\leq\lambda_{1}(\Psi)$ for all separable states $\sigma$. Then,

[TABLE]

Choosing $\ket{\Psi}$ in the family of two-mode squeezed vacua, i.e., $\ket{\Psi}=\ket{\psi_{s}}$, one obtains

[TABLE]

To proceed further, we need to evaluate the matrix element $\bra{\psi_{s}}\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)\ket{\psi_{s}}$. This can be easily done by means of (S108) and the Kraus representation in (S9), which together yield

[TABLE]

Using the above expression one can verify that for all fixed $s$

[TABLE]

Putting all together, one obtains

[TABLE]

The above derivation can be justified as follows. Step 1 is obtained from (S113) by taking the supremum over $r$. In step 2 we exchanged the order of the suprema over $r$ and $s$. Step 3 comes from (S120). For step 4, we introduce the parameter $x=\sinh(s)^{2}$, so that

[TABLE]

where the last equality is a consequence of the fact that $f_{k}(x)\coloneqq\frac{x}{(1+x)((k-1)x+k)}$ achieves its maximum for $x=\sqrt{\frac{k}{k-1}}$. Finally, step 5 rests on the fact that $4k-2-\left(\sqrt{k}+\sqrt{k-1}\right)^{2}=\left(\sqrt{k}-\sqrt{k-1}\right)^{2}\geq 0$. Observe that for $k>1$ the combined supremum in $r,s$ is achieved at $(r,s)=(r_{0},s_{0})$, where $r_{0}$ and $s_{0}$ are the unique positive solutions of the equations

[TABLE]

Hence, the existence of a state $\rho^{\mathrm{\scriptscriptstyle G}}$ in the family $\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)$ with the property in (S105) follows.

To establish (S106), for a fixed $m$ we set $\rho_{k,m}^{\mathrm{\scriptscriptstyle G}}(r)\coloneqq\left(\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)\right)^{\otimes m}$, which is clearly an $(m+m)$-mode bipartite state on the system $AB$, where $A=A_{1}\ldots A_{m}$ represents the collection of all the $m$ subsystems each of which corresponds to the first mode in (S107). The distillable entanglement of $\rho_{k,m}^{\mathrm{\scriptscriptstyle G}}(r)$ (which is a well-known lower bound on the relative entropy of entanglement) can be estimated from below with the coherent information $I_{\mathrm{coh}}(B\,\rangle A)_{\rho}\coloneqq S(\rho_{A})-S(\rho_{AB})$ due to the hashing bound in Devetak and Winter (2005):

[TABLE]

Now, to compute the coherent information $I_{\mathrm{coh}}(B_{1}\rangle A_{1})$ of the state $\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)_{A_{1}B_{1}}$, we note that its reduced state on $A_{1}$ coincides with that of the two-mode squeezed vacuum $\ket{\psi_{r}}_{A_{1}B_{1}}$. Therefore, it is simply a one-mode Gaussian state with QCM $\cosh(2r)\mathds{1}_{2}$, whose entropy can be evaluated using, e.g., the $\alpha=1$ case of the following formula for the Rényi-$\alpha$ entropy $S_{\alpha}(V)$ of a Gaussian state with QCM $V$ (Adesso et al., 2014, Eq. (108)):

[TABLE]

Here, $\nu_{1},\ldots,\nu_{n}$ are the symplectic eigenvalues of $V$. We obtain that

[TABLE]

in the limit $r\to\infty$. Since the bipartite QCM of $\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)_{A_{1}B_{1}}$ has only one symplectic eigenvalue different from $1$, and this is equal to $\frac{1+(k-1)\cosh(2r)}{k}$, we also have that

[TABLE]

as $r\to\infty$. Putting all together, we see that

[TABLE]

Combining (S130) and (S134) yields the lower bound in (S106), concluding the proof. ∎

IX Proof of Theorem 4

This section is devoted to the proof of Theorem 4. We start by reminding the reader that the Rényi-2 Gaussian entanglement of formation, defined by the $\alpha=2$ case of (14), is given by

[TABLE]

where

[TABLE]

In what follows, we will consider the universal function $\varphi:\mathds{R}_{+}\to\mathds{R}$ given by

[TABLE]

Before delving into the proof of Theorem 4, let us establish a technical lemma that connects the Rényi-$2$ Gaussian entanglement of formation with its von Neumann version.

Lemma 5**.**

For all bipartite Gaussian states $\rho_{AB}^{\mathrm{\scriptscriptstyle G}}$ on $n_{A}+n_{B}$ modes, the entanglement of formation measured in natural units satisfies

[TABLE]

Proof.

Let $\delta\coloneqq E_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{AB}\right)$. By (S135), there exists a pure QCM $\gamma_{AB}\leq V_{AB}$ such that $M(V_{A})=\sum_{j=1}^{n_{A}}\ln\nu_{j}=\delta$. Using the readily verified concavity of $\varphi$, one observes that

[TABLE]

Recalling that

[TABLE]

for $\alpha=1$ we immediately obtain (S138). ∎

Proof of Theorem 4.

The Gaussian Rényi-2 entanglement of formation is known to be monogamous on Gaussian states (Lami et al., 2016, Corollary 7). Calling $\rho^{\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}}$ the $k$-extension of $\rho^{\mathrm{\scriptscriptstyle G}}_{AB}$, we then have

[TABLE]

i.e.,

[TABLE]

where the last inequality expresses the fact that $E_{F,\alpha}(\sigma_{AB})\leq S_{\alpha}(\sigma_{A})$ by concavity of the Rényi-$\alpha$ entropy; at the level of QCMs, this can also be thought of as a consequence of the fact that $M(\cdot)$ is a monotone function, and any $\gamma_{AB}$ in the set on the right-hand side of (S135) satisfies $\gamma_{A}\leq V_{A}$ and hence $M(\gamma_{A})\leq M(V_{A})$. Using (S138) and the fact that $\varphi$ is monotonically increasing we then obtain

[TABLE]

completing the proof. ∎