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Coexistence of unlimited bipartite and multipartite Einstein-Podolsky-Rosen entanglement

Gerardo Adesso1,2 *, L. Marie Ericsson2, Fabrizio Illuminati1

1Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, CNR-INFM Coherentia, and INFN Sezione di Napoli - Gruppo Collegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy.

2Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.

*To whom correspondence should be addressed. E-mail: gerardo.adesso@gmail.com

Quantum mechanics imposes ‘monogamy’ constraints on the sharing of entanglement. We show that, despite these limitations, entanglement can be simultaneously present in unlimited two-body and many-body forms in continuous variable systems. This is demonstrated in experimentally producible multimode states of light fields or atomic ensembles, which therefore enable infinitely more freedom in the distribution of information, as opposed to system of qubits where such entanglement structure is impossible. This is a central finding for the quantification, understanding and potential exploitation of shared quantum correlations, and may lead to novel realisations of robust multiparty communication networks and scalable quantum computation.

Introduction.

Quantum entanglement is the element distinguishing quantum from classical mechanics1, and a key resource for quantum communication and information processing2. Entanglement arises when the pure state of two or more subsystems of a compound quantum system cannot be factorised into pure local states of the subsystems. The subsystems thus share quantum correlations which can be stronger than any classical correlation3. Entanglement plays a fundamental role in the physics of many-

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body systems, in particular in critical phenomena like quantum phase transitions4, and in the description of the interactions between complex systems at the quantum scale. The investigation and understanding of the structure and distribution of quantum correlations encoded between individual constituents of a composite quantum system is therefore of utmost importance to characterise the collective quantum behaviours which determine the macroscopic properties of materials, such as superfluidity and superconductivity5, and magnetic response6.

A remarkable difference between classical correlations and quantum entanglement is that the latter cannot be freely shared. This basic limitation, standing on the same ground as the no-cloning theorem7, is addressed as ‘monogamy’ constraint on distributed entanglement8. In a multipartite compound system of two-state quantum objects (qubits), the basic logical units of quantum computation9, if two subsystems are maximally entangled, they cannot share any residual form of quantum correlations with the other remaining parties10. Analogous monogamy relations have been recently established for entanglement between canonical conjugate variables of systems with infinite-dimensional state spaces11,12, like harmonic oscillators, light modes and atomic ensembles, which are the elementary constituents in continuous-variable realisations of quantum information protocols13.

Here we show that a unique freedom is available for entanglement distribution in continuous variable systems, within the holding of the shareability constraints. We prove that Gaussian states of at least four modes exist, that can possess simultaneously arbitrarily large pairwise bipartite entanglement in two pairs of modes and arbitrarily large genuine multipartite entanglement among all modes without violating the monogamy inequality on entanglement sharing. These states asymptotically reach the form of two perfectly entangled Einstein-Podolsky-Rosen (EPR) pairs14 that can moreover be arbitrarily intercorrelated quantumly. The states exhibiting such unconstrained simultaneous distribution of quantum correlations are experimentally producible with standard optical means15, the achievable amount of entanglement being practically limited only by the attainable degree of squeezing. This novel property of entanglement, namely an unlimited promiscuous sharing structure, may play a leading role in the description of many-body systems, whose phase transitions are driven by strong correlations between the canonical variables of the quantum constituents, such as vibrational modes. On a more applicative ground, it serves as a prelude to implementations of quantum information processing in the continuous variable scenario that cannot be achieved with qubits, and that were thought a priori impossible.

Distributed entanglement.

Quantum information science2,9 aims at the treatment, manipulation and transfer of information in ways forbidden by classical physics. Both qubits and continuous variable systems have proven useful to this scope, relying on entanglement as a fundamental resource. While for two qubits the maximal unit of entanglement is

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encoded in the Bell singlet ( )12 ↑↓ − ↓↑ , continuous variable entanglement

manifests itself in the form of EPR correlations14. For the motional degrees of freedom of two particles, or the quadratures of a two-mode radiation field (where mode k=i,j is described by the canonical operators †ˆ ˆ,k ka a satisfying the bosonic commutation relation

†ˆ ˆ[ , ] 1k ka a = ), an unboundedly increasing degree of entanglement can be encoded in a two-mode squeezed state15 2 † † 2ˆ ˆ ˆ ˆexp[( e e ) / 2] 0 0i i

i j i j i jr a a r a aφ φ−− ⊗ (where r is a real

number, [0, 2 ]φ π∈ , and 0k denotes the vacuum state in the Fock space of mode k)

with increasing r. In the limit of infinite squeezing ( r →∞ ), the state approaches the ideal EPR state14, simultaneous eigenstate of total momentum and relative position of the two subsystems, which thus share infinite entanglement. The EPR state is unnormalisable and unphysical. Two-mode squeezed states, being arbitrarily good approximations of it with increasing squeezing, are key resources for practical implementations of continuous variable quantum information protocols involving two parties13, as experimentally demonstrated in quantum teleportation15. Mathematically, these states belong to the class of Gaussian states of multimode continuous variable systems. Gaussian states are ubiquitous in many-body physics, since they arise as ground and thermal states of harmonic lattice systems, and their structural and informational properties have been intensively studied in recent times17.

Leaving aside the obvious dimensional difference, no radically profound discrepancy had been discovered so far for what concerns entanglement properties of Gaussian states as opposed to states of qubits. When more than two parties are involved, in both instances a monogamy constraint on the entanglement sharing has been established10-12,18. In the general case of a state distributed among N parties (each owning a single qubit, or a single mode, respectively), the monogamy constraint takes the form of the Coffman-Kundu-Wootters inequality10

1 1 1|( ) |i i i N i j

N

S S S S S S Sj i

E E− +

≠

≥ ∑… … (1)

where the global system is multipartitioned in subsystems kS (k=1,…,N), each owned by a respective party, and E is a proper measure of bipartite entanglement, in particular monotonically decreasing under local operations and classical communication19. The left-hand side of inequality (1) quantifies the bipartite entanglement between a probe subsystem iS and the remaining subsystems taken as a whole. The right-hand side quantifies the total bipartite entanglement between iS and each one of the other subsystems j iS ≠ in the respective reduced states. The non-negative difference between these two entanglements, minimised over all choices of the probe subsystem, is referred to as the residual multipartite entanglement. It quantifies the purely quantum correlations that are not encoded in pairwise form, so it includes all manifestations of genuine K-partite entanglement, involving K subsystems at a time, with 2 K N< ≤ . In the simplest nontrvial instance of N = 3, the residual entanglement has the meaning of the genuine tripartite entanglement shared by the three subsystems10. Such a quantity

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has been proven to be a tripartite entanglement monotone both for three qubits20 and for three-mode Gaussian states11. The study of entanglement sharing and monogamy constraints thus offers a natural framework to interpret and quantify entanglement in multipartite quantum systems.

A first hint of a diversity between the structure of entanglement sharing in qubits and Gaussian states has been observed in the tripartite case. For three qubits, existence of maximum (unit) tripartite entanglement denies any bipartite entanglement in the reduced state of two qubits, as in the case of the Greenberger-Horne-Zeilinger state21

( )12GHZψ = ↑↑↑ + ↓↓↓ . Viceversa, existence of maximum (but strictly less than

unity) bipartite entanglement in all reduced two-qubit partitions leads to a zero residual tangle, as in the case of the W state20 ( )1

3Wψ = ↑↑↓ + ↑↓↑ + ↓↑↑ . On the

contrary, it has been recently established that there exist pure three-mode Gaussian states whose tripartite entanglement can become infinite, and at the same time the bipartite entanglement in any reduced two-mode state is non vanishing and increasing with increasing tripartite entanglement11. However, because of frustration effects due to the complete permutational invariance of the states22, the maximum achievable bipartite entanglement remains finite even in the limit of infinite squeezing. As a consequence, for instance, continuous variable teleportation23 using such reduced two-mode entangled resources can never reach perfect efficiency, as demonstrated experimentally24. The coexistence of maximal multipartite and non vanishing bipartite entanglement, which qualifies those three-mode Gaussian states as optimally robust carriers of continuous variable quantum information in the three-party setting25, has been referred to as the ‘promiscuity’ of entanglement sharing11: despite the fact that these states satisfy the monogamy inequality, bipartite and tripartite entanglement are mutually enhanced and are not competing, at variance with the case of states of three-qubit systems.

We report here that this result uncovered the tip of an iceberg. In states of continuous variable systems with more than three modes, entanglement can be distributed in an infinitely promiscuous way, with the simultaneous coexistence of arbitrarily large multipartite and bipartite entanglement in the global and in the pairwise reduced states, respectively. To illustrate the existence of this phenomenon, which sheds new light on the role of the fundamental laws of quantum mechanics in curtailing the distribution of information, we consider the simplest nontrivial instance of a class of four-mode Gaussian states, endowed with a partial symmetry under mode exchange.

Construction of four-mode Gaussian states and entanglement characterisation.

Gaussian states of N modes are completely described in phase space by the 2 2N N× real symmetric covariance matrix σ of the second moments 1

2ˆ ˆ ˆ ˆ

i j j iX X X X+

of the canonical bosonic operators † †ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ,k k k k N k k k kX q a a X p a a+≡ = + ≡ = − ( 1k N= … ).

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In this representation the unitary two-mode squeezing operator amounts to a symplectic

matrix13 , ( ) r r r ri j

r r r r

c s c sS r

s c s c−⎛ ⎞ ⎛ ⎞

= ⊕⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ (where cosh , sinhr rc r s r= = , and the

unimportant phase factor φ has been absorbed into the definition of the operators), which acts by congruence on the covariance matrix, TS Sσ σ .

We first consider an uncorrelated state of four modes, each one initially in the vacuum of the respective Fock space, whose corresponding covariance matrix is the identity. We apply a two-mode squeezing transformation with squeezing s to modes 2 and 3, then a two-mode squeezing transformations with squeezing a to modes 1 and 2, and another two-mode squeezing transformation with the same squeezing a to modes 3 and 4. The last two operations serve the purpose of distributing the original pairwise bipartite entanglement shared by modes 2 and 3, among all modes. The output results in a pure four-mode Gaussian state (see Figure 1A) with covariance matrix γ given by

3,4 1,2 2,3 2,3 1,2 3,4( ) ( ) ( ) ( ) ( ) ( )T T TS a S a S s S s S a S aγ = (2)

This state is experimentally realisable by quantum optical means13, where two-mode squeezing transformations occur in parametric down-conversions25. We also note that it is invariant under the double exchange of modes 1 4↔ and 2 3↔ , as , ,i j j iS S= and operations on disjoint pairs of modes commute.

In a pure four-mode Gaussian state and in its reductions, a necessary and sufficient condition for the existence of bipartite entanglement is the negativity of the partially transposed covariance matrix, obtained by reversing time in the subspace of any chosen single subsystem26,27. This inseparability condition is readily verified for the state in equation (2) yielding that, for all nonzero values of the squeezings s and a, the stateγ is entangled with respect to any global bipartition of the modes. The state is thus said to be fully inseparable2, that is it contains genuine four-partite entanglement.

Following previous studies on continuous variable entanglement sharing, to quantify bipartite entanglement we adopt the contangle11 τ , an entanglement monotone under Gaussian local operations and classical communication, defined for pure states as the squared logarithmic negativity29 (which quantifies how much the partially transposed state fails to be positive) and extended to mixed states via Gaussian convex roof29,30, that is as the minimum of the average pure-state entanglement over all decompositions of the mixed state in ensembles of pure Gaussian states. If |i jσ is the covariance matrix of a (generally mixed) bipartite Gaussian state where subsystem i comprises one mode only, then the contangle τ can be computed as11

2 2| | |( ) ( ) [ ], [ ] arcsinh [ 1],opt

i j i j i jg m g x xτ σ τ σ≡ = = − (3)

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where |opti jσ corresponds to a pure Gaussian state, and

| |( ) det detopt opt opti j i j i jm m σ σ σ≡ = = , with ( )

opti jσ the reduced covariance matrix of

subsystem i (j), obtained tracing over the degrees of freedom of subsystem j (i). The covariance matrix |

opti jσ denotes the pure bipartite Gaussian state which minimises

|( )pi jm σ among all pure-state covariance matrices |

pi jσ such that | |

pi j i jσ σ≤ . If |i jσ is a

pure state, then | |opti j i jσ σ= , while for a mixed Gaussian state equation (3) is

mathematically equivalent to constructing the Gaussian convex roof. For a separable state, |( ) 1opt

i jm σ = and the entanglement vanishes. The contangle τ is completely equivalent to the Gaussian entanglement of formation30, which quantifies the cost of creating a given mixed, entangled Gaussian state out of an ensemble of pure, entangled Gaussian states.

Structure of bipartite entanglement.

In the four-mode state with covariance matrix γ , we can compute the bipartite contangle in closed form31 for all pairwise reduced (mixed) states of two modes i and j, described by a covariance matrix |i jγ . By applying the criterion of the positive partial transpose27, we find that the two-mode states indexed by the partitions 1|3, 2|4, and 1|4, are in a separable state, that is, those subsystems share no pairwise bipartite entanglement. For the remaining two-mode states we find 1|2 3|4 cosh 2m m a= = , while

2|3m is equal to 2 2 2

2 2 2

1 2cosh (2 )cosh 3cosh(2 ) 4sinh sinh(2 )4[cosh e sinh ]s

a s s a sa a

− + + −+

if

1/ 2arcsinh[tanh ]a s< , and to 1 (implying separability) otherwise. Accordingly, one can compute the pure-state entanglements between one probe mode and the remaining three modes. One finds 2 2

1|(234) 4|(123) cosh cosh(2 )sinhm m a s a= = + and 2 2

2|(134) 3|(124) sinh cosh(2 )coshm m a s a= = + .

We can then verify whether the fundamental monogamy inequality (1) is satisfied on the four-mode state γ distributed among the four parties (each one owning a single mode). In fact, the problem reduces to proving that

2 2 2 2 21|(234) 1|2 2|(134) 1|2 2|3min{ [ ] [ ], [ ] [ ] [ ]}g m g m g m g m g m− − − is nonnegative. It is found that the

first quantity always achieves the minimum and we thus have

{ }2 2 2 2 21|(234) 1|2( ) ( ) ( ) arcsinh [cosh cosh(2 )sinh ] 1 4 0res a s a aτ γ τ γ τ γ≡ − = + − − > (4)

because cosh(2 ) 1s > for 0s > . The entanglement in the global state is therefore distributed according to the laws of quantum mechanics, in such a way that the residual

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contangle resτ quantifies the multipartite correlations that cannot be stored in bipartite form. Before analysing the residual multipartite entanglement, let us perform a closer scrutiny of the structure of the bipartite entanglement in the various partitions.

The contangle in the mixed two-mode states 1|2γ and 3|4γ is 24a , i.e. unrespective of the value of s. This quantity is exactly equal to the degree of entanglement in a pure two-mode squeezed state , ,( ) ( )T

i j i jS a S a of modes i and j generated with the same squeezing a. In fact, the two-mode mixed state 1|2γ (and, equivalently, 3|4γ ) serves as an optimal resource for continuous variable teleportation of coherent states (14,21), with a fidelity increasing with a and reaching unity (perfect transfer) in the limit of infinite squeezing. Thus, concerning the amount of pairwise bipartite entanglement, in the regime of very high a the four-mode state γ approaches the state of two EPR-like pairs (1,2 and 3,4). What is now crucial, is that there is entanglement between the two pairs. Namely, we find that the four-mode bipartite contangle (12)|(34)( )τ γ is equal to 24s , the

original entanglement in the two-mode squeezed state 2,3 2,3( ) ( )TS s S s , because the additional two-mode squeezings acting on the pair of modes 1,2 and 3,4 amount to local unitary operations with respect to the considered bipartition, which preserve entanglement by definition. Even more remarkably, thus, the entanglement between the two EPR-like pairs (each of them sharing arbitrarily increasing entanglement with increasing a) is a function only of the independent squeezing parameter s and can increase arbitrarily as well. This peculiar distribution of bipartite entanglement within the allowed limits set by the monogamy constraints (see Figure 1B) is a first remarkable signature of an unmatched freedom of entanglement sharing in Gaussian states as opposed to states of qubits, where a similar situation is impossible. Namely, if in a pure state of four qubits the first two share unit entanglement and the same holds for the last two, the global state is necessarily a product state of the two singlets: no entanglement between the two pairs is allowed by the fundamental monogamy constraint.

Residual entanglement and genuine multipartite quantum correlations.

Equation (4) entails that some multipartite entanglement among the single modes is present as well in the global state of the compound system. Those quantum correlations can be either tripartite involving three of the four modes, and/or genuinely four-partite among all of them. Concerning the tripartite entanglement, we first observe that in the tripartitions 1|2|4 and 1|3|4 the tripartite entanglement is zero, as mode 4 is not entangled with the block of modes 1,2, and mode 1 is not entangled with the block of modes 3,4 (the corresponding three-mode states are then said to be biseparable2). The only tripartite entanglement present, if any, is equal in content (due to the symmetry of the stateγ ) for the tripartitions 1|2|3 and 2|3|4, and can be quantified by the residual entanglement determined by the corresponding three-mode monogamy inequality. It is

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immediate to prove that such genuine tripartite entanglement 1|2|3 2|3|4( ) ( )res resτ γ τ γ= is

bounded from above by the quantity 1|2|3( )boundτ γ (see Figure 2),

2 2 2 21|2|3 1|2|3 1|(23) 1|2 3|(12) 2|3( ) ( ) min{ [( ) ] [ ], [( ) ] [ ]}res bound bound boundg m g m g m g mτ γ τ γ≤ ≡ − − , (5)

with 2 2

2 21 sech tanh

3|(12) 1 sech tanhbound a s

a sm +

−= , 2 2

1|(23) 3|(12)cosh sinhbound boundm a m a= + .

The upper bound 1|2|3( )boundτ γ is always nonnegative (as a consequence of monogamy), is decreasing with increasing squeezing a, and vanishes in the limit a →∞ , as shown pictorially in Figure 2. Therefore, in the regime of increasingly high a, eventually approaching infinity, any form of tripartite entanglement among any three modes in the state γ is negligible (exactly vanishing in the limit). As a crucial consequence, the residual entanglement ( )resτ γ determined by equation (4) is all stored in four-partite quantum correlations and quantifies the genuine four-partite entanglement. Finally, it is straightforward to see that ( )resτ γ is an increasing function of a for any value of s (see Figure 3), and it diverges in the limit a →∞ .

This proves that the class of pure four-mode Gaussian states with covariance matrix γ given by equation (2) exhibits genuine four-partite entanglement which grows unboundedly with increasing squeezing a and, simultaneously, possesses pairwise bipartite entanglement in the mixed two-mode reduced states of modes 1,2 and 3,4, that increases unboundedly as well with with increasing a. Moreover, as previously shown, the two pairs can themselves be arbitrarily entangled with each other with increasing squeezing s (see Figure 1B). By constructing a simple and experimentally realisable example we have thus demonstrated that unlimited entanglement in simultaneous multipartite and bipartite form is unconditionally possible in continuous variable systems associated to infinite-dimensional Hilbert spaces, with no violation of the fundamental monogamy constraint that retains its general validity in quantum mechanics.

Practical consequences: an eye to the future.

Two-mode squeezing transformations are basic tools in the domain of quantum optics15,26, and the amount of producible squeezing in real experiments is constantly improving32. Only practical, no a priori limitations need to be overcome to increase s and a to the point of engineering excellent approximations of such a structure of simultaneous, unlimited bipartite and multipartite entanglement in multimode states of light and atoms. We remark that the existence of states with analogous entanglement properties is strictly forbidden for systems of qubits, since it would lead to evident contradictions with the basic laws of quantum mechanics as embodied by the monogamy constraints10. The same laws, in continuous variable systems, allow instead

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for an unlimited promiscuity of entanglement sharing: when the quantum correlations arise among degrees of freedom which span an infinite-dimensional space of states, an accordingly infinite freedom is tolerated for quantum information to be dolen out.

We conclude with an outlook on the practical impact of this result. The peculiar structure of distributed entanglement renders the four-mode Gaussian states of equation (2), and in general the multimode states obtained extending the construction presented here, ideal resources for novel implementations of quantum communication with continuous variables, for instance in the context of teleportation of entanglement. In the case of qubits, quantum teleportation33 of two-qubit states is performed either via resources constituted by tensor products of two Bell singlets34, or (with a modified protocol) via a genuinely four-partite entangled state35; in the latter case, however, only partially known input states of two qubits can be faithfully teleported. As a consequence of the full promiscuity, instead, the continuous variable counterpart represented by teleportation of unknown two-mode states, is perfectly and faithfully realised for any input via the four-mode resources of equation (2) with a →∞ , either exploiting the bipartite entanglement in the pairs of modes 1,2 and 3,4, or exploiting the genuine four-partite entanglement among all modes. This suggests that a novel, hybrid teleportation scheme may be devised, where both forms of entanglement, coexisting to an unlimited extent, can be exploited simultaneously to enhance the quantum coherence of the state transfer and to protect the transmission from losses and eavesdropping. Such a robust quantum teleportation scheme may be at the heart of a future distributed quantum communication network between distant stations based on light pulses and atomic memories36, and a key element for continuous-variable implementations37 of one-way quantum computation38.

Conclusion.

We have only begun to fully grasp the significance and the outcome of an unlimitedly promiscuous entanglement sharing, that is the simultaneous presence of diverging bipartite entanglement shared by pairs of parties, and diverging genuine multipartite entanglement distributed among all parties in a continuous variable system. The quantum correlations which can arise in systems of harmonic oscillators, whose ground states are Gaussian states13,17, are not only infinitely stronger than those encodable in spin chains (physical realisations of qubit systems), but also infinitely more shareable, yet in accordance with the fundamental monogamy constraints on distributed entanglement8,10-12,18. This is likely to elevate entanglement, specifically EPR entanglement14, to the stage of an even more, infinitely more useful resource. The translation of this proof of principle into full-power practical implementations for what concerns processing and transmission of distributed information, awaits further investigation.

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Acknowledgements We acknowledge inspiring discussions with Ivette Fuentes-Schuller. G.A. and F.I. acknowledge financial support from CNR-INFM, INFN, and MIUR. M.E. is supported by The Leverhulme Trust. The authors declare no competing financial interests.

Correspondence and requests for materials should be addressed to G.A. ( gerardo.adesso@gmail.com )

13

a

a

a

s

b

a

s

a

a

a

a

s

b

a

s

a

Figure 1. Construction of four-mode Gaussian states and the structure of bipartite entanglement. a Starting with four initially uncorrelated modes of light, all residing in the respective vacuum states (yellow beams), one first applies a two-mode squeezed transformation (light blue box), with squeezing s, between the central modes 2 and 3, and then two additional two-mode squeezing transformations (light pink boxes), each one with equal squeezing a, acting on the pair of modes 1,2 and 3,4 respectively. The resulting state is endowed with a peculiar, yet insightful bipartite entanglement structure, pictorially depicted in b. The block of modes 1,2 shares with the block of modes 3,4 all the entanglement created originally between modes 2 and 3, which is an increasing function of s (blue springs). Moreover, modes 1 and 2 internally share an entanglement arbitrarily increasing as a function of a, and the same holds for modes 3 and 4 (pink springs). For a approaching infinity, each of the two pairs of modes 1,2 and 3,4 reproduces the entanglement content of an ideal EPR state, while being the same pairs arbitrarily entangled with each other according to value of s. Moreover, the four-mode Gaussian state constructed as in a, encodes simultaneously a genuine four-partite entanglement that increases unboundedly with increasing squeezing a (See text).

14

0 1 2 3 4 5

a0

12

34

5

s 0

1

2

3

tboundHg1»2»3L

1 2 3 4 5

Figure 2. Tripartite entanglement. Upper bound 1|2|3( )boundτ γ , equation (5), on the tripartite entanglement between modes 1, 2 and 3 (and equivalently 2, 3, and 4) of the four-mode Gaussian state defined in equation (2), plotted as a function of the squeezing parameters s and a. To quantify exactly the tripartite entanglement determined by the three-mode monogamy inequality (1), it is necessary to compute the three-mode bipartite contangle between one mode and the block of the two other modes. This requires solving the optimisation problem of equation (3). However, focusing on the tripartition 1|2|3, the bipartite contangle |( )( )i jkτ γ (with i,j,k a permutation of 1,2,3) is

bounded from above by the corresponding bipartite contangle |( )( )pi jkτ σ in any pure,

three-mode Gaussian state with covariance matrix |( ) |( )p

i jk i jkσ γ≤ . The state

1,2 2,3 2,3 1,2( ) ( ) ( ) ( )p T TS a S t S t S aσ = of the three modes 1, 2 and 3, with 2 2

2 21 sech tanh1

2 1 sech tanharccosh[ ]a s

a st +

−= , satisfies this condition and from equation (3) we have

2|( ) |( )( ) [( ) ]bound

i jk i jkg mτ γ ≤ . This leads to equation (5), where the quantity 2 2 2

2|(13) 1|2 2|3[( ) ] [ ] [ ]boundg m g m g m− − with 2 22|(13) 3|(12)sinh coshbound boundm a m a= + is not included in

the minimisation, being always larger than the other terms. The plotted upper bound on the tripartite entanglement among modes 1, 2, 3 (and equivalently 2, 3, 4) asymptotically vanishes for a going to infinity, while any other form of tripartite entanglement among any three modes is always zero.

15

0 1 2 3 4 5

a01

23

45

s 0

100

200

tresHg L

1 2 3 4 5

Figure 3. Residual four-partite entanglement. Plot of the residual multipartite entanglement ( )resτ γ [see equation (4)], which in the regime of large squeezing a is completely distributed in the form of genuine four-partite quantum correlations. The four-partite entanglement is monotonically increasing with increasing squeezing a, and diverges as a approaches infinity. The multimode Gaussian state γ constructed with an arbitrarily large degree of squeezing a, consequently, exhibits a coexistence of unlimited multipartite and pairwise bipartite entanglement in the form of EPR correlations. In systems of many qubits, and even in Gaussian states of continuous variable systems with a number of modes smaller than four, such an unlimited and unconstrained promiscuous distribution of entanglement is strictly forbidden.