Extendibility of bosonic Gaussian states

Extendibility of bosonic Gaussian states

Ludovico Lami ludovico.lami@gmail.com School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom    Sumeet Khatri skhatr5@lsu.edu Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, 70803, USA    Gerardo Adesso School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom    Mark M. Wilde mwilde@lsu.edu Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, 70803, USA Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana, 70803, USA
July 26, 2019
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 Rényi 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.

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 terhal2004; coffman2000; FLV88; RW89; DPS02; complete-extendibility; lancien2016, 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 (with respect to B) if there exists a quantum state \widetilde{\rho}_{AB_{1}\cdots B_{k}} on A and k copies B_{1},\ldots,B_{k} of B that is permutation-invariant with respect to the systems B_{i} and satisfies \operatorname{Tr}_{B_{2}\cdots B_{n}}\left[\,\widetilde{\rho}_{AB_{1}\cdots B_% {n}}\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 FLV88; RW89; DPS02; complete-extendibility. 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 deFinetti0; deFinetti1; 1-1/2-de-Finetti; deFinetti2; Koenig2009; deFinetti4; CJYZ16 and other studies of entanglement NOP09; BC12. Extendibility also arises in the contexts of security of quantum key distribution MCL06; MRDL09; KL17, capacities of quantum channels NH09; Kaur2018; BBFS18, Bell’s inequalities TDS03; KGM17, and other information-theoretic scenarios Lan16; VV-Markov. More broadly, the extendibility problem is a special case of the quantum marginal problem (see, e.g., Klyachko_2006; tyc; CLL13; Schilling14; Tyc2015), which has been referred to in quantum chemistry as the N-representability problem for several decades Tred57; Coul60; Cole63. The quantum marginal problem is known to be QMA-complete Liu06; LCV07. For fixed k, the extendibility problem can be formulated as a semidefinite program (SDP), making it efficient to determine k-extendibility of bipartite states of low-dimensional systems A and B DPS02; complete-extendibility. General analytic conditions for k-extendibility in finite-dimensional systems are known only for particular values of k and/or for special classes of states Ranade09; JV13; CJKLZ14; FLSC14; KGM17.

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 adesso14; BUCCO; revisited. However, more general questions about extendibility have been approached sparingly. The only work we are aware of is Bhat16, in which it was shown that a Gaussian state is separable if and only if it is Gaussian k-extendible for all k.

In this paper, we study and characterize the full hierarchy of extendibility of quantum Gaussian states. After showing that any quantum 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 that the extended system contains one mode only. This condition is reminiscent of the well-known positive partial transpose criterion PeresPPT; HorodeckiPPT; Simon00, and reduces to it when applied to all k. We then discuss several applications of this result, deriving along the way: (i) analytic conditions for k-extendibility for all single-mode bosonic 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 start by recalling the basic theory of quantum Gaussian states WANG20071; BUCCO; adesso14; weedbrook12. 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 simon94. 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, i.e., \operatorname{Tr}[r_{j}^{2}\,\rho_{AB}]<\infty for all 1\leq j\leq 2n. We can then construct the QCM, which has the form

V_{AB}=\begin{pmatrix}V_{A}&X\\ X^{{\scriptscriptstyle\mathsf{T}}}&V_{B}\end{pmatrix}. (1)

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

\widetilde{V}_{AB_{1}\ldots B_{k}}=\begin{pmatrix}V_{A}&X&X&\ldots&X\\ X^{{\scriptscriptstyle\mathsf{T}}}&V_{B}&Y&\ldots&Y\\ X^{{\scriptscriptstyle\mathsf{T}}}&Y&V_{B}&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&Y\\ X^{{\scriptscriptstyle\mathsf{T}}}&Y&\ldots&Y&V_{B}\end{pmatrix}, (2)

where Y is a symmetric matrix. A similar structure had already been identified in Bhat16; however, there the fact that Y needs to be symmetric seems not to have been observed. We will see that this is indeed instrumental to our main results. 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 systems B_{1j},\ldots,B_{mj} as U_{j}^{\dagger}b_{1j}U_{j}=\frac{b_{1j}+\ldots+b_{mj}}{\sqrt{m}}, where b_{\ell j} is the annihilation operator of the system B_{\ell j}. Consider the state

\omega^{(m)}_{A_{1}B_{11}\ldots B_{mk}}\coloneqq(U_{1}\otimes\ldots\otimes U_{% k})\left(\bigotimes_{\ell=1}^{m}\widetilde{\rho}_{A_{\ell}B_{\ell 1}\ldots B_{% \ell k}}\right)(U_{1}\otimes\ldots\otimes U_{k})^{\dagger}. (3)

By the quantum central limit theorem Cushen1971; Petz1992; Petz, 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}} denotes the “Gaussification” of \widetilde{\rho}_{AB_{1}\ldots B_{k}}, i.e., the Gaussian state with the same first and second moments, and we identify A_{1}\equiv A and B_{1j}\equiv B_{j} Note1.

We now show that \widetilde{\rho}^{\,\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}} is in fact a Gaussian k-extension of \rho_{AB}^{\mathrm{\scriptscriptstyle G}}. First of all, 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 the above result, when considering the existence of k-extensions of Gaussian states, we can confine the search to the same realm of Gaussian states. 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

V_{AB}\geq\mathrm{i}\Omega_{A}\oplus\big{(}\left(1-1/k\right)\Delta_{B}+(1/k)% \mathrm{i}\Omega_{B}\big{)}\,. (4)

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

V_{AB}\geq\mathrm{i}\Omega_{A}\oplus\big{(}-\left(1-2/k\right)\mathrm{i}\Omega% _{B}\big{)}\,. (5)

In the proof of Theorem 2, we employ the following well-known fact about the positive semidefiniteness of Hermitian block matrices (ZHANG1, Theorem 1.12):

\displaystyle M\!=\!\begin{pmatrix}P&Z\\ Z^{\dagger}&Q\end{pmatrix}\geq 0\ \Leftrightarrow\ P\geq 0,\ M/P\coloneqq Q-Z^% {\dagger}P^{-1}Z\geq 0, (6)

where the matrix M/P is called the Schur complement of M with respect to P. For details on how to treat the degenerate case of non-invertible P, see Note1. 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

\mathrm{i}\Omega_{B}\leq\Delta_{B}\leq\frac{k}{k-1}\left(V_{B}-X^{{% \scriptscriptstyle\mathsf{T}}}(V_{A}-\mathrm{i}\Omega_{A})^{-1}X\right)-\frac{% 1}{k-1}\,\mathrm{i}\Omega_{B}. (7)

Analogously, (5) can be seen to be equivalent to

V_{B}-X^{{\scriptscriptstyle\mathsf{T}}}(V_{A}-\mathrm{i}\Omega_{A})^{-1}X\geq% -\left(1-2/k\right)\mathrm{i}\Omega_{B}. (8)
Proof of Theorem 2.

We first establish necessity of the condition in (4) for k-extendibility of an arbitrary (not necessarily Gaussian) 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 (\mathds{1}_{k}-\ket{+}\!\!\bra{+})\otimes(V_{B}-Y-\mathrm{i}\Omega_{B})+\ket{% +}\!\!\bra{+}\otimes\left(V_{B}+(k-1)Y-kX^{{\scriptscriptstyle\mathsf{T}}}% \left(V_{A}-\mathrm{i}\Omega_{A}\right)^{-1}X-\mathrm{i}\Omega_{B}\right)\geq 0. 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. To proceed, we employ (revisited, Lemma 7), which guarantees that (7) is satisfied for some real \Delta_{B} if and only if

\frac{k}{k-1}\left(V_{B}-X^{{\scriptscriptstyle\mathsf{T}}}(V_{A}-\mathrm{i}% \Omega_{A})^{-1}X\right)-\frac{1}{k-1}\,\mathrm{i}\Omega_{B}\geq\pm\mathrm{i}% \Omega_{B}\,, (9)

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). ∎

Note that although some of the above manipulations resemble formally those in Bhat16, the two arguments are conceptually quite different and lead to different conclusions Note1: in fact, in Bhat16, the question of k-extendibility of Gaussian states is explicitly mentioned as an outstanding open problem.

Recall that a bipartite state is separable if and only if it is k-extendible for all k FLV88; RW89; DPS02; complete-extendibility 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} 222Technically, the evaluation of the limit k\to\infty of (4) 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 Bhat16.. This reproduces the analytic condition for separability of Gaussian states found in (revisited, Theorem 5). In the same limit k\to\infty, it is also easy to verify that condition (5) reduces to the positivity of the partial transpose (PPT) PeresPPT; HorodeckiPPT; Simon00; Werner01; revisited.

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 Werner01, which obeys (5) for all k (because it is PPT) yet it is not even 2-extendible Note1.

Theorem 2 also reveals an implication of 2-extendibility for Gaussian steerability, i.e., Einstein–Podolsky–Rosen steerability via Gaussian measurements wise; steerability; Simon16; Kor; Lami16. 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 Pankowski2013; Kaur2018 if there exists another quantum channel \widetilde{\mathcal{N}}_{A\to B_{1}\cdots B_{k}} from the sender A to k receivers B_{1},\ldots,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} BUCCO. The action of a Gaussian channel can be described directly in terms of the mean vector s and covariance matrix V of the the input Gaussian state 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.

The Choi–Jamiołkowski isomorphism in infinite dimension Holevo-CJ ensures that to any channel \mathcal{N}_{A\to B} we can associate in a reversible way a quantum 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}. 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 Note1. The same conclusion follows from arguments in nogo3; Wolf2007; NFC09. For any Gaussian channel \mathcal{N}, the state \rho_{AB}^{\mathcal{N}}(r) is Gaussian, which means that we can apply Theorem 2 to conclude that a Gaussian channel is k-extendible if and only if there exists a 2m\times 2m real matrix \Delta such that

\mathrm{i}\Omega\leq\Delta\leq\frac{k}{k-1}\left(Y+\mathrm{i}X\Omega X^{{% \scriptscriptstyle\mathsf{T}}}\right)-\frac{1}{k-1}\mathrm{i}\Omega. (10)

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 reads

\sqrt{\det Y}\geq 1-1/k+\left|\det X-1/k\right|. (11)

By applying (S42), 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 Holevo2007; EW07; BUCCO. By the results of Holevo2007, the following characterization of k-extendibility for three fundamental single-mode Gaussian channels suffices to solve the problem for all single-mode Gaussian channels Note1:

(i) The thermal channel of transmissivity \eta\in(0,1) and environment thermal photon number N_{B}\geq 0, which 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, which 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, which 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 Holevo2008 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 (1-1/2-de-Finetti, 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 (1-1/2-de-Finetti, 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 faithful.

Can similar estimates be provided in the Gaussian case? Results in this setting have been obtained in Koenig2009 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 PetzRenyi, 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 U62, and we obtain the standard relative entropy of entanglement VP98. We find

Theorem 3.

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

\displaystyle\left\|\rho_{AB}^{\mathrm{\scriptscriptstyle G}}-\operatorname{% SEP}(A\!:\!B)\right\|_{1} \displaystyle\leq\frac{2n}{k}, (12)
\displaystyle E_{R,\alpha}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}})\leq n\,% \ln\left(1+\frac{\eta_{k,\alpha}}{k-1}\right) \displaystyle\leq\frac{n\,\eta_{k,\alpha}}{k-1}, (13)

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

The proof is in Note1. Remarkably, note that 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 Koenig2009, and is in stark contrast to the finite-dimensional case, where—as we mentioned before—the bound has to depend in a nontrivial way 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\log\frac{k}{k-1}-o(1) as r\to\infty, where E_{D} denotes the distillable entanglement; see Note1 for details.

Entanglement of formation of Gaussian k-extendible states.

The fact that the bounds in Theorem 3 are independent of the mean photon number constitutes a quantitative improvement over finite-dimensional quantum de Finetti theorems. 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 AdessoSerafini; steerability; Lami16; LL-log-det, and especially the monogamy of the Gaussian Rényi-2 version of the entanglement of formation Lami16, which stems in turn from the equality between such a measure and the Gaussian Rényi-2 squashed entanglement LL-log-det.

In general, for a bipartite state \rho_{AB} and for some \alpha\geq 1, the Rényi-\alpha entanglement of formation is defined as Horodecki-review

E_{F,\alpha}(\rho_{AB})\coloneqq\inf\big{\{}\sum_{i}p_{i}\,S_{\alpha}\big{(}% \psi_{A}^{(i)}\big{)}:\ \rho_{AB}=\sum_{i}p_{i}\psi_{AB}^{(i)}\big{\}}\,, (14)

where \psi^{(i)}_{AB} are bipartite pure states, \psi^{(i)}_{A}=\operatorname{Tr}_{B}[\psi^{(i)}_{AB}] are the corresponding marginals, and S_{\alpha}(\sigma)\coloneqq\frac{1}{1-\alpha}\ln\operatorname{Tr}[\sigma^{% \alpha}] denotes the Rényi-\alpha entropy measured in natural units.

For a Gaussian state \rho_{AB}^{\mathrm{\scriptscriptstyle G}} with covariance matrix V_{AB}, we can derive an upper bound on E_{F,\alpha}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}) by restricting the decompositions in (14) to be comprised of pure Gaussian states only. This leads to the Gaussian Rényi-{\alpha} entanglement of formation, given by the simpler formula Wolf03

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{\}}, (15)

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. The typical choice is \alpha=1, in which case (14) becomes the standard entanglement of formation. However, Rényi-2 quantifiers arise naturally in the Gaussian setting, as they reproduce Shannon entropies of measurement outcomes AdessoSerafini; LL-log-det. In the case \alpha=2, (15) becomes E^{\mathrm{\scriptscriptstyle G}}_{F,2}\left(\rho_{AB}\right)=\min\left\{M(% \gamma_{A}):\text{$\gamma_{AB}$ pure QCM and $\gamma_{AB}\leq V_{AB}$}\right\}, 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 (von Neumann) 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).

Observe that 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 using 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 first determined that a Gaussian state is k-extendible if and only if it is Gaussian k-extendible. This insight allowed us to derive a simple semidefinite program that determines whether a Gaussian state is k-extendible in a computationally efficient way, thus resolving the problem completely. When the system being extended 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 capacities of Gaussian channels in the non-asymptotic setting, in light of recent work Kaur2018; BBFS18 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.

References

Supplemental Material

I Background

For a continuous-variable system of n modes, we let r=(x_{1},p_{1},\ldots,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

[r,r^{\scriptscriptstyle\mathsf{T}}]=\mathrm{i}\Omega=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}^{\oplus n}. (S1)

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 simon94

V\geq\mathrm{i}\Omega. (S2)

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

V_{AB}=\begin{pmatrix}V_{A}&X\\ X^{{\scriptscriptstyle\mathsf{T}}}&V_{B}\end{pmatrix}, (S3)

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}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

V=S\left(\bigoplus_{j=1}^{n}\nu_{j}\mathds{1}_{2}\right)S^{{\scriptscriptstyle% \mathsf{T}}}, (S4)

one has

\rho=U_{S}^{\dagger}\left(\bigotimes_{j=1}^{n}\rho^{\mathrm{\scriptscriptstyle G% }}(\nu_{j})\right)U_{S}\,, (S5)

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

\rho^{\mathrm{\scriptscriptstyle G}}(\nu)\coloneqq\frac{2}{\nu+1}\sum_{\ell=0}% ^{\infty}\left(\frac{\nu-1}{\nu+1}\right)^{\ell}\ket{\ell}\!\!\bra{\ell}. (S6)

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

\mathcal{L}_{\lambda}(\cdot)\coloneqq\operatorname{Tr}_{B}\left[U_{\lambda}((% \cdot)_{A}\otimes{|0\rangle\langle 0|}_{B})U_{\lambda}^{\dagger}\right], (S7)

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

U_{\lambda}\coloneqq e^{-\arccos\left(\!\sqrt{\lambda}\right)\,(ab^{\dagger}-a% ^{\dagger}b)}=e^{-\sqrt{\frac{1-\lambda}{\lambda}}ab^{\dagger}}\lambda^{\frac{% a^{\dagger}a-b^{\dagger}b}{2}}e^{\sqrt{\frac{1-\lambda}{\lambda}}a^{\dagger}b}, (S8)

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

\mathcal{L}_{\lambda}(\cdot)=\sum_{j=0}^{\infty}\frac{\left(\frac{1}{\lambda}-% 1\right)^{j}}{j!}\ a^{j}\lambda^{a^{\dagger}a/2}(\cdot)\lambda^{a^{\dagger}a/2% }(a^{\dagger})^{j}\,. (S9)

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

\operatorname{Tr}\left[r_{B_{j},\,\beta}^{2}\,\widetilde{\rho}_{AB_{1}\ldots B% _{k}}\right]=\operatorname{Tr}\left[r_{B_{j},\,\beta}^{2}\,\widetilde{\rho}_{% AB_{j}}\right]=\operatorname{Tr}\left[r_{B,\,\beta}^{2}\,\rho_{AB}^{\mathrm{% \scriptscriptstyle G}}\right]<\infty\,. (S10)

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}\cdots 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}\cdots B_{k}} must feature identical copies of the matrix X. Indeed, for all 1\leq j\leq k,

\displaystyle\left(\widetilde{V}_{AB_{1}\cdots B_{k}}\right)_{A,\alpha\,;\,B_{% j},\,\beta} \displaystyle=\operatorname{Tr}[(r_{A,\alpha}r_{B_{j},\,\beta}+r_{B_{j},\,% \beta}r_{A,\alpha})\,\widetilde{\rho}_{AB_{1}\cdots B_{k}}] (S11)
\displaystyle=\operatorname{Tr}[(r_{A,\alpha}r_{B_{j},\,\beta}+r_{B_{j},\,% \beta}r_{A,\alpha})\,\widetilde{\rho}_{AB_{j}}] (S12)
\displaystyle=\operatorname{Tr}[(r_{A,\alpha}r_{B_{j},\,\beta}+r_{B_{j},\,% \beta}r_{A,\alpha})\,\rho_{AB}] (S13)
\displaystyle=X_{\alpha,\,\beta}. (S14)

Symmetry with respect to the systems B_{1},\ldots,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

\displaystyle Y_{i,\ell} \displaystyle\coloneqq\operatorname{Tr}[(r_{B_{1},i}r_{B_{2},\ell}+r_{B_{2},% \ell}r_{B_{1},\ell})\widetilde{\rho}_{AB_{1}\cdots B_{k}}] (S15)
\displaystyle=\operatorname{Tr}[(r_{B_{1},i}r_{B_{2},\ell}+r_{B_{2},\ell}r_{B_% {1},\ell})\widetilde{\rho}_{B_{1}B_{2}}] (S16)

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

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 replaces 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

M/P\coloneqq\lim_{\epsilon\to 0^{+}}\left(Q-Z^{\dagger}(P+\epsilon\mathds{1})^% {-1}Z\right), (S17)

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

M=\begin{pmatrix}P&Z\\ Z^{\dagger}&Q\end{pmatrix}\geq 0\ \Leftrightarrow\ P\geq 0,\ M/(P+\epsilon% \mathds{1})\coloneqq Q-Z^{\dagger}(P+\epsilon\mathds{1})^{-1}Z\geq 0\quad% \forall\ \epsilon>0\,. (S18)

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 said (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 will 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

\widetilde{V}_{AB_{1}\ldots B_{k}}\geq\mathrm{i}\Omega_{AB_{1}\ldots B_{k}}=(% \mathrm{i}\Omega_{A})\oplus(\mathrm{i}\Omega_{B_{1}})\oplus\cdots\oplus(% \mathrm{i}\Omega_{B_{k}})\,. (S19)

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 (S19) using the suitably regularized conditions in (6), as given explicitly by (S18). 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

\displaystyle 0 \displaystyle\leq\Big{(}\widetilde{V}_{AB_{1}\ldots B_{k}}+\epsilon\mathds{1}_% {A}\oplus 0_{B_{1}\ldots B_{k}}-\mathrm{i}\Omega_{AB_{1}\ldots B_{k}}\Big{)}\,% \Big{/}\,\Big{(}V_{A}+\epsilon\mathds{1}_{A}-\mathrm{i}\Omega_{A}\Big{)} (S20)
\displaystyle=\begin{pmatrix}V_{B}-\mathrm{i}\Omega_{B}&Y&\ldots&Y\\ Y&V_{B}-\mathrm{i}\Omega_{B}&\ddots&\vdots\\ \vdots&\ddots&\ddots&Y\\ Y&\ldots&Y&V_{B}-\mathrm{i}\Omega_{B}\end{pmatrix}-\begin{pmatrix}X^{% \scriptscriptstyle\mathsf{T}}\\ X^{\scriptscriptstyle\mathsf{T}}\\ \vdots\\ X^{\scriptscriptstyle\mathsf{T}}\end{pmatrix}\left(V_{A}+\epsilon\mathds{1}_{A% }-\mathrm{i}\Omega_{A}\right)^{-1}\begin{pmatrix}X&X&\ldots&X\end{pmatrix}, (S21)

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 (S21) becomes

\displaystyle 0 \displaystyle\leq\mathds{1}_{k}\otimes(V_{B}-\mathrm{i}\Omega_{B}-Y)+k\ket{+}% \!\!\bra{+}\otimes Y-k\ket{+}\!\!\bra{+}\otimes\left(X^{\scriptscriptstyle% \mathsf{T}}\left(V_{A}+\epsilon\mathds{1}_{A}-\mathrm{i}\Omega_{A}\right)^{-1}% X\right) (S22)
\displaystyle=\left(\mathds{1}_{k}-\ket{+}\!\!\bra{+}\right)\otimes(V_{B}-% \mathrm{i}\Omega_{B}-Y)+\ket{+}\!\!\bra{+}\otimes\left(V_{B}+(k-1)Y-\mathrm{i}% \Omega_{B}-kX^{\scriptscriptstyle\mathsf{T}}\left(V_{A}+\epsilon\mathds{1}_{A}% -\mathrm{i}\Omega_{A}\right)^{-1}X\right) (S23)

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

\displaystyle V_{B}-\mathrm{i}\Omega_{B}-Y \displaystyle\geq 0\,, (S24)
\displaystyle V_{B}+(k-1)Y-\mathrm{i}\Omega_{B}-kX^{\scriptscriptstyle\mathsf{% T}}\left(V_{A}+\epsilon\mathds{1}_{A}-\mathrm{i}\Omega_{A}\right)^{-1}X \displaystyle\geq 0\,. (S25)

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

\mathrm{i}\Omega_{B}\leq\Delta_{B}\leq\frac{k}{k-1}\left(V_{B}-X^{% \scriptscriptstyle\mathsf{T}}\left(V_{A}+\epsilon\mathds{1}_{A}-\mathrm{i}% \Omega_{A}\right)^{-1}X\right)-\frac{1}{k-1}\,\mathrm{i}\Omega_{B}\qquad% \forall\ \epsilon>0\,, (S26)

reproducing (7). Using again (S18) – this time backwards – we see that (S26) 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-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 i\Omega_{B} that satisfies (4) when n_{B}=1. In order to do this, it suffices to prove that (5) is equivalent to (S26). To this end, we employ (revisited, 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 when n_{B}=1 all matrices in (S26) are 2\times 2, we can rephrase it as

\pm\mathrm{i}\Omega_{B}\leq\frac{k}{k-1}\left(V_{B}-X^{\scriptscriptstyle% \mathsf{T}}\left(V_{A}+\epsilon\mathds{1}_{A}-\mathrm{i}\Omega_{A}\right)^{-1}% X\right)-\frac{1}{k-1}\,\mathrm{i}\Omega_{B}\qquad\forall\ \epsilon>0\,, (S27)

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

\begin{aligned} \displaystyle V_{B}-X^{\scriptscriptstyle\mathsf{T}}\left(V_{A% }+\epsilon\mathds{1}_{A}-\mathrm{i}\Omega_{A}\right)^{-1}X&\displaystyle\geq% \mathrm{i}\Omega_{B}\,,\\ \displaystyle\frac{k}{k-1}\left(V_{B}-X^{\scriptscriptstyle\mathsf{T}}\left(V_% {A}+\epsilon\mathds{1}_{A}-\mathrm{i}\Omega_{A}\right)^{-1}X\right)&% \displaystyle\geq-\frac{k-2}{k-1}\,\mathrm{i}\Omega_{B}\,,\end{aligned}\qquad% \quad\forall\ \epsilon>0\,. (S28)

The first inequality follows from the bona fide condition V_{AB}\geq\mathrm{i}\Omega_{AB} via an application of (S18), while the second is equivalent to (5) again via (S18).

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 Werner01, as the discussion in the next section shows. ∎

Remark 5.

We take the chance here to draw a thorough comparison between our results and techniques and those of Bhat16. The starting point is the structure of the k-extended QCM in (2), to be compared with (Bhat16, 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 Bhat16 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). Following the argument in Bhat16, one notes that the symmetry of \theta_{k} is anyway recovered via (Parthasarathy2015, Theorem 2.1) as a consequence of the bona fide condition when complete extendibility is assumed. At this precise point the reasoning in Bhat16 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 (S24)–(S25) are identical to those in Bhat16, 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 Bhat16. For example, we were 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 Bhat16.

Last but not least, in Bhat16 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.

IV The Werner–Wolf state is not 2-extendible

This subsection is devoted to the analysis of the Werner–Wolf bound entangled Gaussian state of Werner01 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 (Werner01, Eq. (9))

\gamma_{AB}=\begin{pmatrix}2&0&0&0&1&0&0&0\\ 0&1&0&0&0&0&0&-1\\ 0&0&2&0&0&0&-1&0\\ 0&0&0&1&0&-1&0&0\\ 1&0&0&0&2&0&0&0\\ 0&0&0&-1&0&4&0&0\\ 0&0&-1&0&0&0&2&0\\ 0&-1&0&0&0&0&0&4\\ \end{pmatrix}. (S29)

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 PeresPPT.

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

\lim_{k\to\infty}\left\{V_{AB}-(\mathrm{i}\Omega_{A})\oplus\left(-\left(1-% \frac{2}{k}\right)\mathrm{i}\Omega_{B}\right)\right\}=V_{AB}-(\mathrm{i}\Omega% _{A})\oplus(-\mathrm{i}\Omega_{B}) (S30)

is also positive semidefinite. This is the same as saying that the Gaussian state with QCM V_{AB} is PPT, as shown in (Simon00, 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

A\#B\coloneqq A^{1/2}\left(A^{-1/2}BA^{-1/2}\right)^{1/2}A^{1/2} (S31)

denotes the matrix geometric mean geometric-mean; ando79.

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

\bra{v}\Delta\ket{v}\bra{v}\Omega\Delta\Omega^{{\scriptscriptstyle\mathsf{T}}}% \ket{v}\geq 1. (S32)
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 (ando79, Theorem 3) and Lemma 3. ∎

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

Proposition 6.

The Werner–Wolf Gaussian state with QCM given by (S29) is not 2-extendible on the B system.

Proof.

Since it can be readily verified that \gamma_{A}>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

H_{B}\coloneqq 2\left(\gamma_{B}-X_{\gamma}^{\scriptscriptstyle\mathsf{T}}(% \gamma_{A}-\mathrm{i}\Omega_{A})^{-1}X_{\gamma}\right)-\mathrm{i}\Omega_{B}=% \begin{pmatrix}2&-\mathrm{i}&0&2\mathrm{i}\\ \mathrm{i}&4&2\mathrm{i}&0\\ 0&-2\mathrm{i}&2&-\mathrm{i}\\ -2\mathrm{i}&0&\mathrm{i}&4\end{pmatrix}. (S33)

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

\ket{v}\coloneqq\frac{1}{\sqrt{2\left(6+\sqrt{6}\right)}}\begin{pmatrix}-\left% (1+\sqrt{6}\right)\mathrm{i}&-1&0&2\end{pmatrix}^{{\scriptscriptstyle\mathsf{T% }}} (S34)

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

1\leq\braket{v}{\Delta_{B}}{v}\braket{v}{\Omega_{B}\Delta_{B}\Omega_{B}^{{% \scriptscriptstyle\mathsf{T}}}}{v}\leq\braket{v}{H_{B}}{v}\braket{v}{\Omega_{B% }H_{B}\Omega_{B}^{{\scriptscriptstyle\mathsf{T}}}}{v}=\left(3-\sqrt{6}\right)% \frac{10}{\left(1+\sqrt{6}\right)^{2}}<0.463, (S35)

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. ∎

V 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:

\mathcal{N}:\ \left\{\begin{array}[c]{lcl}V&\longmapsto&XVX^{\intercal}+Y\,,\\ s&\longmapsto&Xs+\delta\,.\end{array}\right. (S36)

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}} Holevo-CJ. Denoting the Schmidt decomposition of \ket{\psi}_{AA^{\prime}} by

\ket{\psi}_{AA^{\prime}}=\sum_{i}\lambda_{i}^{1/2}\ket{e_{i}}_{A}\otimes\ket{f% _{i}}_{A^{\prime}}\,, (S37)

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

\rho_{AB}^{\mathcal{N}}\coloneqq\left(I_{A}\otimes\mathcal{N}\right)(\ket{\psi% }\!\!\bra{\psi}_{AA^{\prime}})\,. (S38)

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

\mathcal{N}(\ket{e_{i}}\!\!\bra{e_{j}}_{A})\coloneqq\frac{1}{\sqrt{\lambda_{i}% \lambda_{i}}}\operatorname{Tr}_{A}\left[\left(\ket{e_{j}}\!\!\bra{e_{i}}_{A}% \otimes\operatorname{id}_{B}\right)\rho_{AB}\right] (S39)

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 (S39), it is not difficult to verify that this is indeed a k-extension of \mathcal{N}_{A\to B}.

Corollary 7.

A Gaussian channel V\mapsto XVX^{\intercal}+Y from n to m modes is k-extendible in the sense of Pankowski2013; Kaur2018 if and only if there exists a 2m\times 2m real matrix \Delta such that

i\Omega\leq\Delta\leq\frac{k}{k-1}\left(Y+iX\Omega X^{\intercal}\right)-\frac{% 1}{k-1}\,i\Omega\,. (S40)

When m=1, this is equivalent to

Y+iX\Omega X^{\intercal}+\left(1-\frac{2}{k}\right)i\Omega\geq 0\,. (S41)

If also n=1=m, then a simplified equivalent condition reads

\sqrt{\det Y}\geq 1-\frac{1}{k}+\left|\det X-\frac{1}{k}\right|. (S42)
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

\ket{\psi_{r}}=\frac{1}{\cosh(r)}\sum_{j=0}^{\infty}\tanh(r)^{j}\ket{j,j}. (S43)

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 i\Omega_{B}\leq\Delta_{B} and such that the corresponding covariance matrix V_{AB}^{\mathcal{N}}(r) satisfies (7), where

V_{AB}^{\mathcal{N}}(r)=\begin{pmatrix}\cosh(2r)\operatorname{id}_{2n}&\sinh(2% r)\Sigma_{n}X^{\intercal}\\ \sinh(2r)X\Sigma_{n}&\cosh(2r)XX^{\intercal}+Y\end{pmatrix},\qquad\Sigma_{n}=% \begin{pmatrix}1&0\\ 0&-1\end{pmatrix}^{\oplus n}. (S44)

Then,

(V_{A}^{\mathcal{N}}(r)-i\Omega_{A})^{-1}=(\cosh(2r)\operatorname{id}_{2n}-i% \Omega_{n})^{-1}=\frac{1}{\sinh^{2}(2r)}(\cosh(2r)\operatorname{id}_{2n}+i% \Omega_{n}). (S45)

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

\displaystyle V_{B}^{\mathcal{N}}(r)-\sinh(2r)X\Sigma_{n}(\cosh(2r)% \operatorname{id}_{2n}-i\Omega_{n})^{-1}\Sigma_{n}X^{\intercal}\sinh(2r)
\displaystyle=\cosh(2r)XX^{\intercal}+Y-\sinh(2r)X\Sigma_{n}(\cosh(2r)% \operatorname{id}_{2n}-i\Omega_{n})^{-1}\Sigma_{n}X^{\intercal}\sinh(2r) (S46)
\displaystyle=Y+iX\Omega X^{\intercal}, (S47)

which leads to (S40).

In the case m=1, we apply the same calculations above to the condition in (8) in order to obtain (S41). ∎

We now determine necessary and sufficient conditions for the k-extendibility of all single-mode Gaussian channels. A classification of all such channels into six different categories has been given in Holevo2007 (see also H12). 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 Holevo2007 exploits input and output Gaussian unitaries in order to arrive at the classification.

The categories of single-mode channels from the classification of Holevo2007 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 Holevo2008. 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=I and Y=(I-\sigma_{Z})/2. Applying the condition in (S42), 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 Holevo2007. The remaining channels are the most important for applications, as stressed in (HG12, Section 3.5) and (H12, 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 (S42), 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}\operatorname{id} and Y=(1-\eta)(2N_{B}+1)\operatorname{id}, is k-extendible if and only if

    \eta\leq\frac{N_{B}+1/k}{N_{B}+1}. (S48)

    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}\operatorname{id} and Y=(G-1)(2N_{B}+1)\operatorname{id}, is k-extendible if and only if

    G\geq\frac{N_{B}+1-1/k}{N_{B}}. (S49)

    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=\operatorname{id} and Y=\xi\operatorname{id}, with noise parameter \xi>0 is k-extendible for k\geq 2 if and only if

    \xi\geq 2\left(1-1/k\right). (S50)

As expected, these conditions for k-extendibility of the channels imply the entanglement-breaking conditions from Holevo2008 in the limit k\to\infty. We also recover the conditions for two-extendibility (antidegradability) from (CGH2006, Eq. (4.6)), for thermal and amplifier channels.

The multi-mode additive noise channel defined by X=\operatorname{id} 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.

VI 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

\displaystyle\left\|\,\rho_{AB}^{\mathrm{\scriptscriptstyle G}}-\text{SEP}(A\!% :\!B)\,\right\|_{1} \displaystyle\leq\frac{2n}{k} (S51)
\displaystyle E_{R,\alpha}\left(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}% \right)\leq n\ln\left(1+\frac{\eta_{k,\alpha}}{k-1}\right) \displaystyle\leq\frac{n\,\eta_{k,\alpha}}{k-1}, (S52)

where

\displaystyle\left\|\,\rho_{AB}^{\mathrm{\scriptscriptstyle G}}-\text{SEP}(A\!% :\!B)\,\right\|_{1} \displaystyle\coloneqq\inf_{\sigma_{AB}\in\text{SEP}(A:B)}\left\|\,\rho_{AB}^{% \mathrm{\scriptscriptstyle G}}-\sigma_{AB}\,\right\|_{1} (S53)
\displaystyle E_{R,\alpha}\left(\rho_{AB}^{\mathrm{\scriptscriptstyle G}}\right) \displaystyle\coloneqq\inf_{\sigma_{AB}\in\text{SEP}(A:B)}D_{\alpha}\left(\rho% _{AB}^{\mathrm{\scriptscriptstyle G}}\,\big{\|}\,\sigma_{AB}\right) (S54)

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

\eta_{k,\alpha}=\left\{\begin{array}[]{ll}1&\text{if $\alpha\leq k+1$,}\\ 2&\text{otherwise.}\end{array}\right. (S55)

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

Proposition 8.

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 i\Omega_{B}. Now, write the complex conjugate bona fide condition for V_{AB} as V_{AB}\geq(-i\Omega_{A})\oplus(-i\Omega_{B}). Adding \frac{1}{k} of this inequality to (4) yields

\left(1+\frac{1}{k}\right)V_{AB}\geq\left(1-\frac{1}{k}\right)\left(\mathrm{i}% \Omega_{A}\oplus\Delta_{B}\right), (S56)

i.e.,

\frac{k+1}{k-1}\,V_{AB}\geq\mathrm{i}\Omega_{A}\oplus\Delta_{B}. (S57)

By (revisited, 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 (S51). Drawing inspiration from the techniques in Koenig2009, we construct an ansatz for the separable state \sigma_{AB} to be inserted into (S53): 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 8, 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:

\displaystyle\left\|\,\rho^{\mathrm{\scriptscriptstyle G}}_{AB}-\mathrm{SEP}\,% \right\|_{1} \displaystyle\leq\left\|\,\rho^{\mathrm{\scriptscriptstyle G}}_{AB}-\sigma_{AB% }^{\mathrm{\scriptscriptstyle G}}\,\right\|_{1} (S58)
\displaystyle=\left\|U_{S}^{\dagger}\left(\bigotimes\nolimits_{j}\rho^{\mathrm% {\scriptscriptstyle G}}(\nu_{j})-\bigotimes\nolimits_{j}\rho^{\mathrm{% \scriptscriptstyle G}}(\lambda\nu_{j})\right)U_{S}\right\|_{1} (S59)
\displaystyle=\left\|\bigotimes\nolimits_{j=1}^{n}\rho^{\mathrm{% \scriptscriptstyle G}}(\nu_{j})-\bigotimes\nolimits_{j=1}^{n}\rho^{\mathrm{% \scriptscriptstyle G}}(\lambda\nu_{j})\right\|_{1} (S60)
\displaystyle\overset{1}{\leq}\sum_{j=1}^{n}\left\|\,\rho^{\mathrm{% \scriptscriptstyle G}}(\nu_{j})-\rho^{\mathrm{\scriptscriptstyle G}}(\lambda% \nu_{j})\,\right\|_{1} (S61)
\displaystyle\overset{2}{\leq}n\sup_{\nu_{j}\geq 1}\left\|\,\rho^{\mathrm{% \scriptscriptstyle G}}(\nu_{j})-\rho^{\mathrm{\scriptscriptstyle G}}(\lambda% \nu_{j})\,\right\|_{1} (S62)
\displaystyle\overset{3}{=}2n\,\frac{\lambda-1}{\lambda+1} (S63)
\displaystyle=\frac{2n}{k}\,. (S64)

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

\displaystyle\left\|\,\rho_{1}\otimes\ldots\otimes\rho_{N}-\sigma_{1}\otimes% \ldots\otimes\sigma_{N}\,\right\|_{1}
\displaystyle=\left\|\,(\rho_{1}-\sigma_{1})\otimes\rho_{2}\otimes\ldots% \otimes\rho_{N}-\sigma_{1}\otimes\left(\rho_{2}\otimes\ldots\otimes\rho_{N}-% \sigma_{2}\otimes\ldots\otimes\sigma_{N}\right)\,\right\|_{1} (S65)
\displaystyle\leq\|\,\rho_{1}-\sigma_{1}\,\|_{1}+\left\|\,\rho_{2}\otimes% \ldots\otimes\rho_{N}-\sigma_{2}\otimes\ldots\otimes\sigma_{N}\,\right\|_{1} (S66)
\displaystyle\,\;\vdots (S67)
\displaystyle\leq\sum_{j}\|\,\rho_{j}-\sigma_{j}\,\|_{1}\,; (S68)

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 (Koenig2009, Eq. (10)) that

\left\|\,\rho^{\mathrm{\scriptscriptstyle G}}(\nu)-\rho^{\mathrm{% \scriptscriptstyle G}}(\mu)\,\right\|_{1}=\max_{l\in\mathds{N}}2\left\{\left(% \frac{\mu-1}{\mu+1}\right)^{l+1}-\left(\frac{\nu-1}{\nu+1}\right)^{l+1}\right\}, (S69)

which for \mu=\lambda\nu and upon maximization over \nu\geq 1 yields Koenig2009

\sup_{\nu\geq 1}\left\|\,\rho^{\mathrm{\scriptscriptstyle G}}(\nu)-\rho^{% \mathrm{\scriptscriptstyle G}}(\lambda\nu)\,\right\|_{1}=2\,\frac{\lambda-1}{% \lambda+1}=\frac{2}{k}\,. (S70)

This concludes the proof of (S51).

The inequality in (S52) 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 8, we write

\displaystyle E_{R,\alpha}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{AB}\right) \displaystyle\leq D_{\alpha}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{AB}\,% \|\,\sigma^{\mathrm{\scriptscriptstyle G}}_{AB}\right) (S71)
\displaystyle=\sum_{j=1}^{n}D_{\alpha}\left(\rho^{\mathrm{\scriptscriptstyle G% }}(\nu_{j})\,\|\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu_{j})\right) (S72)
\displaystyle\leq n\sup_{\nu\geq 1}D_{\alpha}\left(\rho^{\mathrm{% \scriptscriptstyle G}}(\nu)\,\|\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda% \nu)\right), (S73)

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, § 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 Datta2009. Thus, we have to prove exactly two statements:

\displaystyle\sup_{\nu\geq 1}D_{k+1}\left(\rho^{\mathrm{\scriptscriptstyle G}}% (\nu)\,\|\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right) \displaystyle=\ln\left(\frac{k}{k-1}\right), (S74)
\displaystyle\sup_{\nu\geq 1}D_{\infty}\left(\rho^{\mathrm{\scriptscriptstyle G% }}(\nu)\,\|\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right) \displaystyle=\ln\left(\frac{k+1}{k-1}\right). (S75)

In general, the Petz–Rényi relative entropies between Gaussian states can be computed thanks to the formulae found in LL-Renyi. 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

\displaystyle D_{k+1}\left(\rho^{\mathrm{\scriptscriptstyle G}}(\nu)\,\big{\|}% \,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right) \displaystyle=\frac{1}{k}\left(\ln 2-\ln\left((\nu+1)^{k+1}(\lambda\nu+1)^{-k}% -(\nu-1)^{k+1}(\lambda\nu-1)^{-k}\right)\right), (S76)
\displaystyle D_{\infty}\left(\rho^{\mathrm{\scriptscriptstyle G}}(\nu)\,\big{% \|}\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right) \displaystyle=\ln\left(\frac{\lambda\nu+1}{\nu+1}\right). (S77)

We start by proving (S74). Let us define the function

f_{k}(\nu)\coloneqq(\nu+1)^{k+1}(\lambda\nu+1)^{-k}-(\nu-1)^{k+1}(\lambda\nu-1% )^{-k}, (S78)

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

\displaystyle f^{\prime}_{k}(\nu) \displaystyle=\lambda(\nu^{2}-1)\frac{(\nu+1)^{k-1}}{(\lambda\nu+1)^{k+1}}% \left\{1-\left(\frac{\nu-1}{\nu+1}\right)^{k-1}\left(\frac{\lambda\nu+1}{% \lambda\nu-1}\right)^{k+1}\right\} (S79)
\displaystyle=\lambda(\nu^{2}-1)\frac{(\nu+1)^{k-1}}{(\lambda\nu+1)^{k+1}}% \left\{1-e^{-(k-1)g(\lambda,\nu)}\right\}, (S80)

where

g(\lambda,\nu)\coloneqq\ln\left(\frac{\nu+1}{\nu-1}\right)-\lambda\ln\left(% \frac{\lambda\nu+1}{\lambda\nu-1}\right). (S81)

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

\displaystyle\frac{\partial g(\lambda,\nu)}{\partial\lambda} \displaystyle=\frac{2\lambda\nu}{\lambda^{2}\nu^{2}-1}+\ln\left(\frac{\lambda% \nu-1}{\lambda\nu+1}\right) (S82)
\displaystyle=\frac{2\lambda\nu}{\lambda^{2}\nu^{2}-1}+\ln\left(1-\frac{2}{% \lambda\nu+1}\right) (S83)
\displaystyle>0\,. (S84)

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:

\displaystyle\ln(1-x) \displaystyle=-\sum_{r=1}^{\infty}\frac{x^{r}}{r} (S85)
\displaystyle>-x-\sum_{r=2}^{\infty}\frac{x^{r}}{2} (S86)
\displaystyle=-x-\frac{x^{2}}{2}\sum_{r=0}^{\infty}x^{r} (S87)
\displaystyle=-x-\frac{x^{2}}{2}\frac{1}{1-x} (S88)
\displaystyle=-\frac{x(2-x)}{2(1-x)}\,. (S89)

We have thus shown that g(\lambda,\nu)>0 for all \lambda>1 and all \nu\geq 1. Going back to (S80), this implies that f^{\prime}_{k}(\nu)>0 for \nu>1 and hence that f_{k} is monotonically increasing whenever k>1. Via (S76), 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

\sup_{\nu\geq 1}D_{k+1}\left(\rho^{\mathrm{\scriptscriptstyle G}}(\nu)\,\big{% \|}\,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right)=D_{k+1}\left(\rho% ^{\mathrm{\scriptscriptstyle G}}(1)\,\big{\|}\,\rho^{\mathrm{% \scriptscriptstyle G}}(\lambda)\right)=\ln\left(\frac{\lambda+1}{2}\right)=\ln% \left(\frac{k}{k-1}\right). (S90)

This proves (S74).

We now turn to the proof of (S75), which fortunately can be obtained much more straightforwardly from (S77). Noting that \nu\mapsto\frac{\lambda\nu+1}{\nu+1} is monotonically increasing for \nu\geq 1 because \lambda>1, one finds that

\sup_{\nu\geq 1}D_{\infty}\left(\rho^{\mathrm{\scriptscriptstyle G}}(\nu)\,\|% \,\rho^{\mathrm{\scriptscriptstyle G}}(\lambda\nu)\right)=\lim_{\nu\to\infty}% \ln\left(\frac{\lambda\nu+1}{\nu+1}\right)=\ln\lambda=\ln\left(\frac{k+1}{k-1}% \right), (S91)

proving (S75). ∎

VII 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 9.

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

\left\|\,\rho^{\!\mathrm{\scriptscriptstyle G}}-\mathrm{SEP}\,\right\|_{1}\geq 2% \left\|\,\rho^{\!\mathrm{\scriptscriptstyle G}}-\mathrm{SEP}\,\right\|_{\infty% }\geq\frac{1}{2k-1}. (S92)

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

E_{R,1}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{k,m}(r)\right)\geq E_{D}% \left(\rho^{\mathrm{\scriptscriptstyle G}}_{k,m}(r)\right)\geq m\log\frac{k}{k% -1}-o(1) (S93)

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

Remark 10.

The above Theorem 9 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 9.

We start by proving (S92). 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

\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)=\left(\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)\right)_{AB_{1}}\coloneqq\operatorname{Tr}_{B_{2}% \ldots B_{k}}\left[U_{B_{1}\ldots B_{k}}\left(\ket{\psi_{r}}\!\!\bra{\psi_{r}}% _{AB_{1}}\otimes\bigotimes\nolimits_{j=2}^{k}\ket{0}\!\!\bra{0}_{B_{j}}\right)% U_{B_{1}\ldots B_{k}}^{\dagger}\right] (S94)

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 (S94) 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

\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)=\left(\operatorname{id}\otimes% \mathcal{L}_{1/k}\right)\left(\psi_{r}\right). (S95)

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

\left\|\,\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-\mathrm{SEP}\,\right\|_{1% }\geq 2\left\|\,\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-\mathrm{SEP}\,% \right\|_{\infty}\,. (S96)

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,

\displaystyle\left\|\,\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-\mathrm{SEP}% \,\right\|_{\infty} \displaystyle=\inf_{\sigma\in\mathrm{SEP}}\left\|\,\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)-\sigma\,\right\|_{\infty} (S97)
\displaystyle=\inf_{\sigma\in\mathrm{SEP}}\sup_{\ket{\Psi}}\left|\braket{\Psi}% {\left(\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-\sigma\right)}{\Psi}\right| (S98)
\displaystyle\geq\sup_{\ket{\Psi}}\left(\braket{\Psi}{\,\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)\,}{\Psi}-\lambda_{1}(\Psi)\right) (S99)

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

\left\|\,\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-\mathrm{SEP}\,\right\|_{% \infty}\geq\sup_{s}\left(\braket{\psi_{s}}{\,\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)\,}{\psi_{s}}-\frac{1}{\cosh(s)^{2}}\right) (S100)

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 (S95) and the Kraus representation in (S9), which together yield

\displaystyle\braket{\psi_{s}}{\,\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)\,% }{\psi_{s}} \displaystyle=\braket{\psi_{s}}{\left(\operatorname{id}\otimes\mathcal{L}_{1/k% }\right)\left(\psi_{r}\right)}{\psi_{s}} (S101)
\displaystyle=\frac{1}{\cosh(r)^{2}\cosh(s)^{2}}\sum_{j,\ell=0}^{\infty}(\tanh% (r)\tanh(s))^{j+\ell}\braket{j}{\,\mathcal{L}_{1/k}(\ket{j}\!\!\bra{\ell})\,}{\ell} (S102)
\displaystyle=\frac{1}{\cosh(r)^{2}\cosh(s)^{2}}\sum_{j,\ell=0}^{\infty}(\tanh% (r)\tanh(s))^{j+\ell}\left(\frac{1}{k}\right)^{(j+\ell)/2} (S103)
\displaystyle=\frac{1}{\cosh(r)^{2}\cosh(s)^{2}}\left(\sum_{j=0}^{\infty}(% \tanh(r)\tanh(s))^{j}\left(\frac{1}{k}\right)^{j/2}\right)^{2} (S104)
\displaystyle=\frac{1}{\cosh(r)^{2}\cosh(s)^{2}}\frac{1}{\left(1-\tanh(r)\tanh% (s)k^{-1/2}\right)^{2}} (S105)
\displaystyle=\frac{k}{\left(\sqrt{k}\cosh(r)\cosh(s)-\sinh(r)\sinh(s)\right)^% {2}} (S106)

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

\sup_{r}\bra{\psi_{s}}\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)\ket{\psi_{s}% }=\frac{k}{k\cosh(s)^{2}-\sinh(s)^{2}} (S107)

Putting all together, one obtains

\displaystyle\sup_{r}\left\|\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)-% \mathrm{SEP}\right\|_{\infty} \displaystyle\overset{1}{\geq}\sup_{r,s}\left(\bra{\psi_{s}}\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)\ket{\psi_{s}}-\frac{1}{\cosh(s)^{2}}\right) (S108)
\displaystyle\overset{2}{=}\sup_{s}\left(\sup_{r}\bra{\psi_{s}}\rho_{k}^{% \mathrm{\scriptscriptstyle G}}(r)\ket{\psi_{s}}-\frac{1}{\cosh(s)^{2}}\right) (S109)
\displaystyle\overset{3}{=}\sup_{s}\left(\frac{k}{k\cosh(s)^{2}-\sinh(s)^{2}}-% \frac{1}{\cosh(s)^{2}}\right) (S110)
\displaystyle=\sup_{s}\frac{\tanh(s)^{2}}{k\cosh(s)^{2}-\sinh(s)^{2}} (S111)
\displaystyle\overset{4}{=}\frac{1}{\left(\sqrt{k}+\sqrt{k-1}\right)^{2}} (S112)
\displaystyle\overset{5}{\geq}\frac{1}{4k-2}. (S113)

The above derivation can be justified as follows. Step 1 is obtained from (S100) by taking the supremum over r. In step 2 we exchanged the order of the suprema over r and s. Step 3 comes from (S107). For step 4, we introduce the parameter x=\sinh(s)^{2}, so that

\sup_{s}\frac{\tanh(s)^{2}}{k\cosh(s)^{2}-\sinh(s)^{2}}=\sup_{x\geq 0}\frac{x}% {(1+x)((k-1)x+k)}=\frac{1}{\left(\sqrt{k}+\sqrt{k-1}\right)^{2}}, (S114)

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

\displaystyle\tanh(r_{0})^{2} \displaystyle=\frac{1}{\sqrt{k}\left(\sqrt{k}+\sqrt{k-1}\right)}, (S115)
\displaystyle\tanh(s_{0})^{2} \displaystyle=\frac{\sqrt{k}}{\sqrt{k}+\sqrt{k-1}}\,. (S116)

Hence, the existence of a state \rho^{\mathrm{\scriptscriptstyle G}} in the family \rho_{k}^{\mathrm{\scriptscriptstyle G}}(r) with the property in (S92) follows.

To establish (S93), 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 (S94). 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 devetak2005:

\displaystyle E_{R,1}\left(\rho_{k,m}^{\mathrm{\scriptscriptstyle G}}(r)\right) \displaystyle\geq E_{D}\left(\rho_{k,m}^{\mathrm{\scriptscriptstyle G}}(r)\right) (S117)
\displaystyle=E_{D}\left(\left(\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)% \right)^{\otimes m}\right)
\displaystyle\geq I_{\mathrm{coh}}(B\,\rangle A)_{\left(\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)\right)^{\otimes m}}
\displaystyle=m\,I_{\mathrm{coh}}(B_{1}\rangle A_{1})_{\rho_{k}^{\mathrm{% \scriptscriptstyle G}}(r)}\,.

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 (adesso14, Eq. (108)):

S_{\alpha}(V)\coloneqq\left\{\begin{array}[]{ll}-\frac{1}{\alpha-1}\,\sum_{j=1% }^{n}\log\frac{2^{\alpha}}{\vphantom{\widetilde{E}}\left(\nu_{j}+1\right)^{% \alpha}-\left(\nu_{j}-1\right)^{\alpha}},&\quad\text{if $\alpha>1$,}\\ \sum_{j=1}^{n}\left(\frac{\nu_{j}+1}{2}\,\log\frac{\nu_{j}+1}{2}-\frac{\nu_{j}% -1}{2}\,\log\frac{\nu_{j}-1}{2}\right),&\quad\text{if $\alpha=1$.}\end{array}\right. (S118)

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

S\left(\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)_{A_{1}}\right)=S_{1}(\cosh(% 2r)\mathds{1}_{2})=\log\frac{e\cosh(2r)}{2}+o(1) (S119)

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

S\left(\rho_{k}^{\mathrm{\scriptscriptstyle G}}(r)_{A_{1}B_{1}}\right)=\log% \left(\frac{e}{2}\,\frac{1+(k-1)\cosh(2r)}{k}\right)+o(1) (S120)

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

I_{\mathrm{coh}}(B_{1}\rangle A_{1})_{\rho_{k}^{\mathrm{\scriptscriptstyle G}}% (r)}=\log\frac{k\cosh(2r)}{1+(k-1)\cosh(2r)}+o(1)=\log\frac{k}{k-1}+o(1)\,. (S121)

Combining (S117) and (S121) yields the lower bound in (S93), concluding the proof. ∎

VIII 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 (15), is given by

E^{\text{G}}_{F,2}\left(\rho_{AB}\right)=\min\left\{M(\gamma_{A}):\ \text{$% \gamma_{AB}$ pure QCM and $\gamma_{AB}\leq V_{AB}$}\right\}, (S122)

where

M(V)\coloneqq S_{2}(V)=\sum_{j}\log\nu_{j}=\frac{1}{2}\log\det V\,. (S123)

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

\varphi(x)\coloneqq\frac{e^{x}+1}{2}\ln\left(\frac{e^{x}+1}{2}\right)-\frac{e^% {x}-1}{2}\ln\left(\frac{e^{x}-1}{2}\right). (S124)

Before delving into the proof of Theorem 4, let us establish a techical 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

E_{F}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}})\leq E_{F}^{\mathrm{% \scriptscriptstyle G}}(\rho_{AB}^{\mathrm{\scriptscriptstyle G}})\leq n_{A}\,% \varphi\left(\frac{E_{F,2}^{\mathrm{\scriptscriptstyle G}}(\rho_{AB}^{\mathrm{% \scriptscriptstyle G}})}{n_{A}}\right). (S125)
Proof.

Let \delta\coloneqq E_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(\rho^{\mathrm{% \scriptscriptstyle G}}_{AB}\right). By (S122), there exists a pure QCM \gamma_{AB}\leq V_{AB} such that M(V_{A})=\sum_{j=1}^{n_{A}}\log\nu_{j}=\delta. Using the readily verified concavity of \varphi, one observes that

\displaystyle S_{1}(V) \displaystyle=\sum_{j=1}^{n_{A}}\left(\frac{\nu_{j}+1}{2}\,\log\frac{\nu_{j}+1% }{2}-\frac{\nu_{j}-1}{2}\,\log\frac{\nu_{j}-1}{2}\right) (S126)
\displaystyle=\sum_{j=1}^{n_{A}}\varphi(\log\nu_{j}) (S127)
\displaystyle=n_{A}\sum_{j=1}^{n_{A}}\frac{1}{n_{A}}\,\varphi(\log\nu_{j}) (S128)
\displaystyle\leq n_{A}\,\varphi\left(\sum_{j=1}^{n_{A}}\frac{1}{n_{A}}\log\nu% _{j}\right) (S129)
\displaystyle=n_{A}\,\varphi\left(\frac{\delta}{n_{A}}\right). (S130)

Recalling that

\displaystyle E_{F,\alpha}(\rho_{AB}) \displaystyle=\inf\left\{\sum_{i}p_{i}\,S_{\alpha}\left(\psi_{A}^{(i)}\right):% \ \rho_{AB}=\sum_{i}p_{i}\psi_{AB}^{(i)}\right\}, (S131)
\displaystyle E^{\text{G}}_{F,\alpha}\left(\rho_{AB}\right) \displaystyle=\inf\left\{S_{\alpha}(\gamma_{A}):\ \text{$\gamma_{AB}$ pure QCM% and $\gamma_{AB}\leq V_{AB}$}\right\}, (S132)

for \alpha=1 we immediately obtain (S125). ∎

Proof of Theorem 4.

The Gaussian Rényi-2 entanglement of formation is known to be monogamous on Gaussian states (Lami16, Corollary 7). Calling \rho^{\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}} the k-extension of \rho^{\mathrm{\scriptscriptstyle G}}_{AB}, we then have

E_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(\rho^{\mathrm{\scriptscriptstyle G% }}_{AB_{1}\ldots B_{k}}\right)\geq\sum_{j=1}^{k}E_{F,2}^{\mathrm{% \scriptscriptstyle G}}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{AB_{j}}% \right)=kE_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(\rho^{\mathrm{% \scriptscriptstyle G}}_{AB}\right), (S133)

i.e.,

E_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(\rho^{\mathrm{\scriptscriptstyle G% }}_{AB}\right)\leq\frac{1}{k}\,E_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(% \rho^{\mathrm{\scriptscriptstyle G}}_{AB_{1}\ldots B_{k}}\right)\leq\frac{1}{k% }\,M(V_{A})\,, (S134)

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 (S122) satisfies \gamma_{A}\leq V_{A} and hence M(\gamma_{A})\leq M(V_{A}). Using (S125) and the fact that \varphi is monotonically increasing we then obtain

E_{F}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{AB}\right)\leq E_{F}^{\mathrm% {\scriptscriptstyle G}}\left(\rho^{\mathrm{\scriptscriptstyle G}}_{AB}\right)% \leq n_{A}\,\varphi\left(\frac{E_{F,2}^{\mathrm{\scriptscriptstyle G}}\left(% \rho^{\mathrm{\scriptscriptstyle G}}_{AB}\right)}{k}\right)\leq n_{A}\,\varphi% \left(\frac{M(V_{A})}{n_{A}k}\right), (S135)

completing the proof. ∎

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