Optical implementation and entanglement distribution in
Gaussian valence bond states
Gerardo Adesso
Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom;
Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185
Roma, Italy;
Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno, Via S. Allende,
84081 Baronissi (SA), Italy
Marie Ericsson
Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
PACS numbers: 42.50.Dv, 03.67.Mn, 03.67.Hk, 03.65.Ud.
Abstract. We study Gaussian valence bond states of continuous variable systems, obtained
as the outputs of projection operations from an ancillary space of M infinitely entangled
bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. The
entanglement distribution in Gaussian valence bond states can be controlled by varying the
input amount of entanglement engineered in a (2M + 1)-mode Gaussian state known as
the building block, which is isomorphic to the projector applied at a given site. We show
how this mechanism can be interpreted in terms of multiple entanglement swapping from
the chain of ancillary bonds, through the building blocks. We provide optical schemes to
produce bisymmetric three-mode Gaussian building blocks (which correspond to a single
bond, M = 1), and study the entanglement structure in the output Gaussian valence bond
states. The usefulness of such states for quantum communication protocols with continuous
variables, like telecloning and teleportation networks, is finally discussed.
http://arxiv.org/abs/0704.1580v1
Optical implementation and entanglement distribution in Gaussian valence bond states 2
1. Introduction
Quantum information aims at the treatment and transport of information in ways forbidden by
classical physics. For this goal, continuous variables (CV) of atoms and light have emerged
as a powerful tool [1]. In this context, entanglement is an essential resource. Recently, the
valence bond formalism, originally developed for spin systems [2], has been generalized to
the CV scenario [3, 4] for the special class of Gaussian states, which play a central role in
theoretical and practical CV quantum information and communication [5].
In this work we analyze feasible implementations of Gaussian valence bond states
(GVBS) for quantum communication between many users in a CV setting, as enabled by
their peculiar structure of distributed entanglement [4]. After recalling the necessary notation
(Sec. 2) and the construction of Gaussian valence bond states [3] (Sec. 3), we discuss
the characterization of entanglement and its distribution in such states as regulated by the
entanglement properties of simpler states involved in the valence bond construction [4]
(Sec. 4). We then focus on the realization of GVBS by means of quantum optics, provide
a scheme for their state engineering (Sec. 5), and discuss the applications of such resources in
the context of CV telecloning [6, 7] on multimode harmonic rings (Sec. 6).
2. Continuous variable systems and Gaussian states
A CV system [1, 5] is described by a Hilbert space H =
i=1 Hi resulting from the tensor
product of infinite dimensional Fock spaces Hi’s. Let ai and a†i be the annihilation and
creation operators acting on Hi (ladder operators), and q̂i = (ai + a†i ) and p̂i = (ai − a
i )/i
be the related quadrature phase operators. Let R̂ = (x̂1, p̂1, . . . , q̂N , p̂N ) denote the vector of
the operators q̂i and p̂i. The canonical commutation relations for the R̂i can be expressed in
terms of the symplectic form Ω as
[R̂i, R̂j ] = 2iΩij ,
with Ω ≡ ω⊕N , ω ≡
The state of a CV system can be equivalently described by quasi-probability distributions
defined on the 2N -dimensional space associated to the quadratic form Ω, known as quantum
phase space. In the phase space picture, the tensor productH =
i Hi of the Hilbert spaces
Hi’s of the N modes results in the direct sum Λ =
i Λi of the phase spaces Λi’s.
States with Gaussian quasi-probability distributions are referred to as Gaussian states.
Such states are at the heart of information processing in CV systems [1, 5] and are the subject
of our analysis. By definition, a Gaussian state is completely characterized by the first and
second statistical moments of the field operators, which will be denoted, respectively, by the
vector of first moments R̄ ≡
〈R̂1〉, 〈R̂2〉, . . . , 〈R̂2N−1〉, 〈R̂2N 〉
and the covariance matrix
(CM) γ of elements
γij ≡
〈R̂iR̂j + R̂jR̂i〉 − 〈R̂i〉〈R̂j〉 . (1)
Coherent states, resulting from the application of displacement operatorsDY = e
iY TΩR̂
(Y ∈ R2n) to the vacuum state, are Gaussian states with CM γ = 1 and first statistical
moments R̄ = Y . First moments can be arbitrarily adjusted by local unitary operations
(displacements), which cannot affect any property related to entropy or entanglement. They
Optical implementation and entanglement distribution in Gaussian valence bond states 3
can thus be assumed zero without any loss of generality. A N -mode Gaussian state will be
completely characterized by its real, symmetric, 2N × 2N CM γ.
The canonical commutation relations and the positivity of the density matrix ρ of a
Gaussian state imply the bona fide condition
γ + iΩ ≥ 0 , (2)
as a necessary and sufficient constraint the matrix γ has to fulfill to be a CM corresponding
to a physical state [8, 9]. Note that the previous condition is necessary for the CM of any
(generally non Gaussian) state, as it generalizes to many modes the Robertson-Schrödinger
uncertainty relation [10].
A major role in the theoretical and experimental manipulation of Gaussian states is
played by unitary operations which preserve the Gaussian character of the states on which
they act. Such operations are all those generated by terms of the first and second order in the
field operators. As a consequence of the Stone-Von Neumann theorem, any such operation
at the Hilbert space level corresponds, in phase space, to a symplectic transformation, i.e. to
a linear transformation S which preserves the symplectic form Ω, so that Ω = STΩS, i.e. it
preserves the commutators between the different operators. Symplectic transformations on a
2N -dimensional phase space form the (real) symplectic group, denoted by Sp(2N,R). Such
transformations act linearly on first moments and “by congruence” on the CM (i.e. so that
γ 7→ SγST ). One has DetS = 1, ∀S ∈ Sp(2N,R). A crucial symplectic operation is
the one achieving the normal mode decomposition. Due to Williamson theorem [11], any
N -mode Gaussian state can be symplectically diagonalized in phase space, so that its CM
is brought in the form ν, such that SγST = ν, with ν = diag {ν1, ν1, . . . νN , νN}. The
set {νi} of the positive-defined eigenvalues of |iΩγ| constitutes the symplectic spectrum of
γ and its elements, the so-called symplectic eigenvalues, must fulfill the conditions νi ≥ 1,
following from the uncertainty principle Eq. (2) and ensuring positivity of the density matrix
ρ corresponding to γ.
Ideal beam splitters, phase shifters and squeezers are described by symplectic
transformations. In particular, a phase-free two-mode squeezing transformation, which
corresponds to squeezing the first mode (say i) in one quadrature (say momentum, p̂i) and
the second mode (say j) in the orthogonal quadrature (say position, q̂j) with the same degree
of squeezing r, can be represented in phase space by the symplectic transformation
Sij(r) = diag{exp r, exp−r, exp−r, exp r} . (3)
These trasformations occur for instance in parametric down conversions [12]. Another
important example of symplectic operation is the ideal (phase-free) beam splitter, which acts
on a pair of modes i and j as [13]
B̂ij(θ) :
âi 7→ âi cos θ + âj sin θ
âj 7→ âi sin θ − âj cos θ
and corresponds to a rotation in phase space of the form
Bij(θ) =
cos(θ) 0 sin(θ) 0
0 cos(θ) 0 sin(θ)
sin(θ) 0 − cos(θ) 0
0 sin(θ) 0 − cos(θ)
. (4)
The transmittivity τ of the beam splitter is given by τ = cos2(θ) so that a 50:50 beam splitter
(τ = 1/2) amounts to a phase-space rotation of π/4.
Optical implementation and entanglement distribution in Gaussian valence bond states 4
The combined application of a two-mode squeezing and a 50:50 beam splitter realizes
the entangling twin-beam transformation [14]
Tij(r) = Bij(π/4) · Sij(r) , (5)
which, if applied to two uncorrelated vacuum modes i and j (whose initial CM is the identity
matrix), results in the production of a pure two-mode squeezed Gaussian state with CM
σi,j(r) = Tij(r)T
ij (r) given by
σi,j(r) =
cosh(2r) 0 sinh(2r) 0
0 cosh(2r) 0 − sinh(2r)
sinh(2r) 0 cosh(2r) 0
0 − sinh(2r) 0 cosh(2r)
. (6)
The CV entanglement in the state σi,j(r) increases unboundedly as a function of r, and
in the limit r → ∞ Eq. (6) approaches the (unnormalizable) Einstein-Podolski-Rosen (EPR)
state [15], simultaneous eigenstate of relative position and total momentum of the two modes
i and j. Concerning entanglement in general, the “positivity of partial transposition” (PPT)
criterion states that a Gaussian CM γ is separable (with respect to a 1×N bipartition) if and
only if the partially transposed CM γ̃ satisfies the uncertainty principle Eq. (2) [9, 16]. In
phase space, partial transposition amounts to a mirror reflection of one quadrature associated
to the single-mode partition. If {ν̃i} is the symplectic spectrum of the partially transposed
CM γ̃, then a (N + 1)-mode Gaussian state with CM γ is separable if and only if ν̃i ≥ 1 ∀ i.
A proper measure of CV entanglement is the logarithmic negativityEN [17], which is readily
computed in terms of the symplectic spectrum ν̃i of γ̃ as
EN = −
i: ν̃i<1
log ν̃i . (7)
Such an entanglement monotone [18] quantifies the extent to which the PPT condition ν̃i ≥ 1
is violated. For 1 × N Gaussian states, only the smallest symplectic eigenvalue ν̃− of the
partially transposed CM can be smaller than one [10], thus simplifying the expression of EN :
then the PPT criterion simply yields that γ is entangled as soon as ν̃− < 1, and infinite
entanglement (accompanied by infinite energy in the state) is reached for ν̃− → 0+.
For 1× 1 Gaussian states γi,j symmetric under mode permutations, the entanglement of
formation EF is computable as well via the formula [19]
EF (γi,j) = max{0, f(ν̃
− )} , (8)
f(x) =
(1 + x)2
(1 + x)2
− (1− x)
(1− x)2
Being a monotonically decreasing function of the smallest symplectic eigenvalue ν̃i,j− of the
partial transpose γ̃i,j of γi,j , the entanglement of formation is completely equivalent to the
logarithmic negativity in this case. For a two-mode state, ν̃i,j can be computed from the
symplectic invariants of the state [20], and experimentally estimated with measures of global
and local purities [21] (the purity µ = Tr ρ2 of a Gaussian state ρ with CM γ is equal to
µ = (Det γ)−1/2).
3. Gaussian valence bond states
Let us review the basic definitions and notations for GVBS, as adopted in Ref. [4]. The
so-called matrix product Gaussian states introduced in Ref. [3] are N -mode states obtained
Optical implementation and entanglement distribution in Gaussian valence bond states 5Optical implementation and entanglement distribution in Gaussian valence bond states
i i+1
Figure 1. Gaussian valence bond states. Γin is the state of N EPR bonds and γ is the three-
mode building block. After the EPR measurements (depicted as curly brackets), the chain
of modes γ
collapses into a Gaussian valence bond state with global state Γout. See also
Ref. [4].
by taking a fixed number, M , of infinitely entangled ancillary bonds (EPR pairs) shared
by adjacent sites, and applying an arbitrary 2M → 1 Gaussian operation on each site
i = 1, . . . , N . Such a construction, more properly definable as a “valence bond” picture
for Gaussian states, can be better understood by resorting to the Jamiolkowski isomorphism
between quantum operations and quantum states [22]. In this framework, one starts with a
chain ofN Gaussian states of 2M +1 modes (the building blocks). The global Gaussian state
of the chain is described by a CM Γ =
i=1 γ
[i]. As the interest in GVBS lies mainly in
their connections with ground states of Hamiltonians invariant under translation [3], we can
focus on pure (Detγ [i] = 1), translationally invariant (γ[i] ≡ γ ∀i) GVBS. Moreover, in this
work we consider single-bonded GVBS, i.e. withM = 1. This is also physically motivated in
view of experimental implementations of GVBS, as more than one EPR bond would result in a
building block with five or more correlated modes, which appears technologically demanding.
Under the considered prescriptions, the building block γ is a pure Gaussian state of three
modes. As we aim to construct a translationally invariant state, it is convenient to consider a
γ whose first two modes, which will be combined with two identical halves of consecutive
EPR bonds (see Fig. 3), have the same reduced CM. This yields a pure, three-mode Gaussian
building block with the property of being bisymmetric [23], that is with a CM invariant under
permutation of the first two modes. This choice of the building block is further justified
by the fact that, among all pure three-mode Gaussian states, bisymmetric states maximize
the genuine tripartite entanglement [24]: no entanglement is thus wasted in the projection
process. The 6 × 6 CM γ of the building block can be written as follows in terms of 2 × 2
submatrices,
γs εss εsx
εTss γs εsx
εTsx ε
sx γx
. (9)
The 4×4 CM of the first two modes (each of them having reduced CM γs) will be denoted by
γss, and will be regarded as the input port of the building block. On the other hand, the CM
γx of mode 3 will play the role of the output port. The intermodal correlations are encoded
in the off-diagonal ε matrices. Without loss of generality, we can assume γ to be, up to local
unitary operations, in the standard form [24] with
γs = diag{s, s} , γx = diag{x, x} , (10)
Optical implementation and entanglement distribution in Gaussian valence bond states 6
εss = diag{t+, t−} , εsx = diag{u+, u−} ;
x2 − 1±
16s4 − 8(x2 + 1)s2 + (x2 − 1)2
x2 − 1
(x − 2s)2 − 1±
(x+ 2s)2 − 1
The valence bond construction works as follows (see Fig. 3). The global CM Γ =
i=1 γ acts as the projector from the state Γ
in of the N ancillary EPR pairs, to the final
N -mode GVBS Γout. This is realized by collapsing the state Γin, transposed in phase space,
with the ‘input port’ Γss =
i γss of Γ, so that the ‘output port’ Γx =
i γx turns into
the desired Γout. Here collapsing means that, at each site, the two two-mode states, each
constituted by one mode (1 or 2) of γss and one half of the EPR bond between site i and
its neighbor (i − 1 or i + 1, respectively), undergo an “EPR measurement” i.e. are projected
onto the infinitely entangled EPR state [22, 3]. An EPR pair between modes i and j can be
described, see Eq. (6), as a two-mode squeezed state σi,j(r) in the limit of infinite squeezing
(r → ∞). The input state is then Γin = limr→∞
i σi,i+1(r), where we have set periodic
boundary conditions so that N + 1 = 1 in labeling the sites. The projection corresponds
mathematically to taking a Schur complement (see Refs. [4, 3, 22] for details), yielding an
output pure GVBS of N modes on a ring with a CM
out = Γx − ΓTsx(Γss + θΓ
θ)−1Γsx , (11)
where Γsx =
i γsx, and θ =
i diag{1, −1, 1, −1} represents transposition in phase
space (q̂i → q̂i, p̂i → −p̂i).
Within the building block picture, the valence bond construction can be in toto
understood as a multiple CV entanglement swapping [25], as shown in Fig. 3: the GVBS
is created as the entanglement in the bonds is swapped to the chain of output modes via
teleportation [26] through the input port of the building blocks. It is thus clear that at a given
initialization of the output port (i.e. at fixed x), changing the properties of the input port
(i.e. varying s), which corresponds to implementing different Gaussian projections from the
ancillary space to the physical one, will affect the structure and entanglement properties of
the target GVBS. This link is explored in the following section.
4. Entanglement distribution
In Ref. [4] the quantum correlations of GVBS of the form Eq. (11) have been studied,
and related to the entanglement properties of the building block γ. Let us first recall the
characterization of entanglement in the latter. As a consequence of the uncertainty principle
Eq. (2), the CM Eq. (9) of the building block describes a physical state if [24]
x ≥ 1 , s ≥ smin ≡
. (12)
Let us keep the output parameter x fixed. Straightforward applications of the PPT separability
conditions, and consequent calculations of the logarithmic negativity Eq. (7), reveal that the
entanglement in the CM γss of the first two modes (input port) is monotonically increasing
as a function of s, ranging from the case s = smin when γss is separable to the limit s → ∞
when the block γss is infinitely entangled. Accordingly, the entanglement between each of
the first two modes γs of γ and the third one γx decreases with s. One can also show that the
genuine tripartite entanglement in the building block increases with the difference s − smin
[24]. The entanglement properties of the building block are summarized in Fig. 4.
Optical implementation and entanglement distribution in Gaussian valence bond states 7
Figure 2. How a Gaussian valence bond state is created via continuous-variable entanglement
swapping. At each step, Alice attempts to teleport her mode 0 (half of an EPR bond, depicted
in yellow) to Bob, exploiting as an entangled resource two of the three modes of the building
block (denoted at each step by 1 and 2). The curly bracket denotes homodyne detection, which
together with classical communication and conditional displacement at Bob’s side achieves
teleportation. The state will be approximately recovered in mode 2, owned by Bob. Since
mode 0, at each step, is entangled with the respective half of an EPR bond, the process swaps
entanglement from the ancillary chain of the EPR bonds to the modes in the building block.
The picture has to be followed column-wise. For ease of clarity, we depict the process as
constituted by two sequences: in the first sequence [frames (1) to (4)] modes 1 and 2 are the
two input modes of the building block (depicted in blue); in the second sequence [frames (5)
to (8)] modes 1 and 2 are respectively an input and an output mode of the building block.
As a result of the multiple entanglement swapping [frame (9)] the chain of the output modes
(depicted in red), initially in a product state, is transformed into a translationally invariant
Gaussian valence bond state, possessing in general multipartite entanglement among all the
modes (depicted in magenta).
The main question addressed in Ref. [4] is how the initial entanglement in the building
block γ redistributes in the Gaussian MPS Γout. The answer is that the more entanglement
one prepares in the input port γss, the longer the range of pairwise quantum correlations in
the output GVBS is, as pictorially shown in Fig. 4.
In more detail, let us consider first a building block γ with s = smin = (x + 1)/2. In
this case, a separability analysis shows that, for an arbitrary numberN of modes in the GVBS
chain, the target state Γout exhibits bipartite entanglement only between nearest neighbor
modes, for any value of x > 1 (for x = 1 we trivially obtain a product state). In fact, each
reduced two-mode block γouti,j is separable for |i− j| > 1.
With increasing s in the choice of the building block, one finds that in the target GVBS
the correlations start to extend smoothly to distant modes. A series of thresholds sk can be
Optical implementation and entanglement distribution in Gaussian valence bond states 8
Figure 3. Entanglement properties of the three-mode building block γ, Eq. (9), of the
Gaussian valence bond construction, as functions of the standard form covariances x and
d ≡ s − smin. (a) Bipartite entanglement, as quantified by the logarithmic negativity,
between the first two input-port modes 1 and 2; (b) Bipartite entanglement, as quantified by
the logarithmic negativity, between each of the first two modes and the output-port mode 3;
(c) Genuine tripartite entanglement, as quantified by the residual Gaussian contangle [27, 24],
among all the three modes.
found such that for s > sk, two given modes i and j with |i − j| ≤ k are entangled. While
trivially s1(x) = smin for anyN (notice that nearest neighbors are entangled also for s = s1),
the entanglement boundaries for k > 1 are in general different functions of x, depending
on the number of modes. We observe however a certain regularity in the process: sk(x,N)
always increases with the integer k. Very remarkably, this means that the maximum range
of bipartite entanglement between two modes, or equivalently the maximum distribution of
multipartite entanglement, in a GVBS on a translationally invariant ring, is monotonically
related to the amount of entanglement in the reduced two-mode input port of the building
block [4]. Moreover, no complete transfer of entanglement to more distant modes occurs:
closer sites remain still entangled even when correlations between farther pairs arise.
The most interesting feature is perhaps obtained when infinite entanglement is fed in
the input port (s → ∞): in this limit, the output GVBS turns out to be a fully symmetric,
permutation-invariant, N -mode Gaussian state. This means that each individual mode is
equally entangled with any other, no matter how distant they are [4]. These states, being thus
Optical implementation and entanglement distribution in Gaussian valence bond states 9
Figure 4. Pictorial representation of the entanglement between a probe (green) mode and its
neighbor (magenta) modes on an harmonic ring with an underlying valence bond structure.
As soon as the parameter s (encoding entanglement in the input port of the valence bond
building block) is increased, pairwise entanglement between the probe mode and its farther
and farther neighbors gradually appears in the corresponding output Gaussian valence bond
states. By translational invariance, each mode exhibits the same entanglement structure with
its respective neighbors. In the limit s → ∞, every single mode becomes equally entangled
with every other single mode on the ring, independently of their relative distance: the Gaussian
valence bond state is in this case fully symmetric.
built by a symmetric distribution of infinite pairwise entanglement among multiple modes,
achieve maximum genuine multiparty entanglement among all Gaussian states (at a given
energy) while keeping the strongest possible bipartite one in any pair, a property known as
monogamous but promiscuous entanglement sharing [27].
Keeping Fig. 3 in mind, we can conclude that having the two input modes initially
entangled in the building blocks, increases the efficiency of the entanglement-swapping
mechanism, inducing correlations between distant modes on the GVBS chain, which enable to
store and distribute joint information. In the asymptotic limit of an infinitely entangled input
port of the building block, the entanglement range in the target GVBS states is engineered to
be maximum, and communication between any two modes, independently of their distance, is
enabled nonclassically. In the next sections, we investigate the possibility of producing GVBS
with linear optics, and discuss with a specific example the usefulness of such resource states
for multiparty CV quantum communication protocols such as telecloning [6] and teleportation
networks [13].
5. Optical implementation of Gaussian valence bond states
The power of describing the production of GVBS in terms of physical states, the building
blocks, rather than in terms of arbitrary non-unitary Gaussian maps, lies not only in the
immediacy of the analytical treatment. From a practical point of view, the recipe of Fig. 3 can
be directly implemented to produce GVBS experimentally in the domain of quantum optics.
We first note that the EPR measurements are realized by the standard toolbox of a beamsplitter
plus homodyne detection [22], as demonstrated in several CV teleportation experiments [28].
The next ingredient to produce aN -mode GVBS is constituted byN copies of the three-
mode building block γ. We provide here an easy scheme (see also Refs. [6, 29]) to realize
bisymmetric three-mode Gaussian states of the form Eq. (9). As shown in Fig. 5(a), one
can start from three vacuum modes and first apply a twin-beam operation to modes 1 and 3,
characterized by a squeezing r13, then apply another twin-beam operation to modes 1 and 2,
Optical implementation and entanglement distribution in Gaussian valence bond states 10Optical implementation and entanglement distribution in Gaussian valence bond states
50:50
γγγγ0000
γγγγ0000
γγγγ0000
Figure 5. Optical production of bisymmetric three-mode Gaussian states, used as buildingFigure 5. Optical production of bisymmetric three-mode Gaussian states, used as building
blocks for the valence bond construction. (a) Three initial vacuum modes are entangled
through two sequential twin-beam boxes, the first (parametrized by a squeezing degree r13)
acting on modes 1 and 3, and the second (parametrized by a squeezing degree r12) acting on
the transformed mode 1 and mode 2. The output is a pure three-mode Gaussian state whose
covariance matrix is equivalent, up to local unitary operations, to the standard form given in
Eq. (9). (b) Detail of the entangling twin-beam transformation. One input mode is squeezed
in a quadrature, say momentum, of a degree r (this transformation is denoted by stretching
arrows→| |←); the other input mode is squeezed in the orthogonal quadrature, say position, of
the same amount (this anti-squeezing transformation is denoted by the corresponding rotated
symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter. If the input
modes are both in the vacuum state, the output is a pure two-mode squeezed Gaussian state,
with entanglement proportional to the degree of squeezing r.
parametrized by r12. The symplectic operation describing the twin-beam transformation (two-
mode squeezing plus balanced beam splitter) is given by Eq. (5) and pictorially represented
in Fig. 5(b). The output of this optical network is a pure, bisymmetric, three-mode Gaussian
state with a CM γB = T12(r12)T13(r13)T
13(r13)T
12(r12) of the form Eq. (9), with
γs = diag
e−2r12
e4r12 cosh (2r13) + 1
e−2r12
cosh (2r13) + e
γx = diag {cosh (2r13) , cosh (2r13)} ,
εss = diag
e−2r12
e4r12 cosh (2r13)− 1
e−2r12
cosh (2r13) − e4r12
εsx = diag
2er12 cosh (r13) sinh (r13) , −
2e−r12 cosh (r13) sinh (r13)
By means of local symplectic operations (unitary on the Hilbert space), like additional single-
mode squeezings, the CM γB can be brought in the standard form of Eq. (10), from which
Optical implementation and entanglement distribution in Gaussian valence bond states 11
one has
r13 = arccos
x+ 1√
, r12 = arccos
−x3 + 2x2 + 4s2x− x
For a given r13 (i.e. at fixed x), the quantity r12 is a monotonic function of the standard-form
covariance s, so this squeezing parameter which enters in the production of the building block
(see Fig. 5) directly regulates the entanglement distribution in the target GVBS, as discussed
in Sec. 4.
The only unfeasible part of the scheme seems constituted by the ancillary EPR pairs.
But are infinitely entangled bonds truly necessary? In Ref. [4] the possibility is considered of
using a Γin given by the direct sum of two-mode squeezed states of Eq. (6), but with finite
r. Repeating the previous analysis to investigate the entanglement properties of the resulting
GVBS with finitely entangled bonds, it is found that, at fixed (x, s), the entanglement in the
various partitions is degraded as r decreases, as somehow expected. Crucially, this does not
affect the connection between input entanglement and output correlation length. Numerical
investigations show that, while the thresholds sk for the onset of entanglement between distant
pairs are quantitatively modified – a bigger s is required at a given x to compensate the less
entangled bonds – the overall structure stays untouched. This ensures that the possibility of
engineering the entanglement structure in GVBS via the properties of the building block is
robust against imperfect resources, definitely meaning that the presented scheme is feasible.
Alternatively, one could from the beginning observe that the triples consisting of two
projective measurements and one EPR pair can be replaced by a single projection onto the
EPR state, applied at each site i between the input mode 2 of the building block and the
consecutive input mode 1 of the building block of site i+1 [3]. The output of all the homodyne
measurements will conditionally realize the target GVBS.
6. Telecloning with Gaussian valence bond resources
The protocol of CV quantum telecloning [6] amongN parties is defined as a process in which
one of them (Alice) owns an unknown coherent state, and wants to distribute her state to
all the other N − 1 remote parties. The telecloning is achieved by a succession of standard
two-party CV teleportations [26] between the sender Alice and each of the N − 1 remote
receivers, exploiting each time the corresponding reduced two-mode state shared as resource
by the selected pair of parties. The 1 → 2 CV telecloning of unknown coherent states has
been recently demonstrated experimentally [7].
The no-cloning theorem [30] yields that the N − 1 remote clones can resemble the
original input state only to a certain extent. The fidelity, which quantifies the success of a
teleportation experiment, is defined as F ≡ 〈ψin|ρout|ψin〉, where “in” and “out” denote the
input and the output state. F reaches unity only for a perfect state transfer, ρout = |ψin〉〈ψin|.
Without using entanglement, by purely classical communication, an average fidelity of
Fcl = 1/2 is the best that can be achieved if the alphabet of input states includes all coherent
states with even weight [31]. The sufficient fidelity criterion states that, if teleportation
is performed with F > Fcl, then the two parties exploited an entangled state [31]. The
converse is generally false, i.e. some entangled resources may yield lower-than-classical
fidelities. In Ref. [32] it has been shown, however, that if the fidelity is optimized over all
possible local unitary operations performed on the shared Gaussian resource (which preserve
entanglement by definition), then it becomes equivalent, both qualitatively and quantitatively,
to the entanglement in the resource.
Optical implementation and entanglement distribution in Gaussian valence bond states 12
Let us also recall that the fidelity of CV two-user teleportation [26] of arbitrary single-
mode Gaussian states with CM γin (equal to the identity for coherent states) exploiting two-
mode Gaussian resources with CM γab =
γa εab
εTab γb
, can be computed [33] as
F = 2√
, Σ ≡ 2γin + ξγaξ + γb + ξεab + εTabξ , (14)
with ξ = diag{−1 , 1}. We can now consider the general setting of 1 → N − 1 telecloning,
where N parties share a N -mode GVBS as an entangled resource, and one of them plays the
role of Alice (the sender) distributing imperfect copies of unknown coherent states to all the
N − 1 receivers. For any N , the fidelity can be easily computed from the reduced two-mode
CMs via Eq. (14) and will depend, for translationally invariant states, on the relative distance
between the two considered modes.
In this work we focus on a practical example of a GVBS on a translationally invariant
harmonic ring, with N = 6 modes. As shown in the previous section, these states can be
produced with the current optical technology. They are completely characterized, up to local
unitary operations, by a 12 × 12 CM analytically obtained from Eq. (11) by considering the
building block in standard form Eq. (9), whose elements are algebraic functions of s and x
here omitted for brevity (as no particular insight is gained from their explicit expressions).
First of all we can construct the reduced CMs γouti,i+k of two modes with distance k, and
evaluate for each k the respective symplectic eigenvalue ν̃i,i+k− of the corresponding partial
transpose. The entanglement condition s > sk will correspond to the inequality ν̃
i,i+k
− < 1.
With this conditions one finds that s2(x) is the only acceptable solution to the equation: 72s
12(x2+1)s6+(−34x4+28x2−34)s4+(x6−5x4−5x2+1)s2+(x2−1)2(x4−6x2+1) = 0,
while for the next-next-nearest neighbors threshold one has simply s3(x) = x. This enables us
to classify the entanglement distribution and, more specifically, to observe the interaction scale
in the GVBS Γout: as discussed in Sec. 4 and explicitly shown in Ref. [4], by increasing initial
entanglement in γss one can gradually switch on pairwise quantum correlations between more
and more distant sites.
Accordingly, it is now interesting to test whether this entanglement is useful to achieve
nonclassical telecloning towards distant receivers. In this specific instance, Alice will send
two identical (approximate) clones to her nearest neighbors, two other identical clones (with
in principle different fidelity than the previous case) to her next-nearest neighbors, and one
final clone to the most distant site. The fidelities for the three transmissions can be computed
from Eq. (14) and are plotted in Fig. 6(a). For s = smin, obviously, only the two nearest
neighbor clones can be teleported with nonclassical fidelity, as the reduced states of more
distant pairs are separable. With increasing s also the state transfer to more distant sites is
enabled with nonclassical efficiency, but not in the whole region of the space of parameters s
and x in which the corresponding two-mode resources are entangled.
As mentioned before, one can optimize the telecloning fidelity considering resources
prepared in a different way but whose CM can be brought by local unitary operations (single-
mode symplectic transformations) in the standard form of Eq. (11). For GVBS resources,
this local-unitary freedom can be transferred to the preparation of the building block. A
more general γ locally equivalent to the standard form given in Eq. (10), can be realized by
complementing the presented state engineering scheme for the three-mode building block as in
Eq. (13) [see Fig. 5(a)], with additional single-mode rotations and squeezing transformations
aimed at increasing the output fidelity in the target GVBS states, while keeping both the
entanglement in the building block and consequently the entanglement in the final GVBS
unchanged by definition.
Optical implementation and entanglement distribution in Gaussian valence bond states 13
Figure 6. 1 → 5 quantum telecloning of unknown coherent states exploiting a six-mode
translationally invariant Gaussian valence bond state as a shared resource. Alice owns mode
i. Fidelities F for distributing clones to modes j such as k = |i − j| are plotted for k = 1
[(a),(d)]; k = 2 [(b),(e)]; and k = 3 [(c),(f)], as functions of the local invariants s and
x of the building block. In the first row [(a)–(c)] the fidelities are achieved exploiting the
non-optimized Gaussian valence bond resource in standard form. In the second row [(d)–
(f)] fidelities optimized over local unitary operations on the resource are displayed, which
are equivalent to the entanglement in the corresponding reduced two-mode states (see, as a
comparison, Fig. 3 in Ref. [4]). Only nonclassical values of the fidelities (F > 0.5) are
shown.
The optimal telecloning fidelity, obtained in this way exploiting the results of Ref. [32],
is plotted in Fig. 6(b) for the three teleportations between modes i and j with k = |i − j| =
1, 2, 3. In this case, one immediately recovers a non-classical fidelity as soon as the
separability condition s ≤ sk is violated in the corresponding resources. Moreover, the
optimal telecloning fidelity at a given k is itself a quantitative measure of the entanglement in
the reduced two-mode resource, being equal to [32]
Foptk = 1/(1 + ν̃
i,i+k
− ) , (15)
where ν̃i,i+k− is the smallest symplectic eigenvalue of the partially transposed CM in
the corresponding bipartition. The optimal fidelity is thus completely equivalent to the
entanglement of formation Eq. (8) and to the logarithmic negativity Eq. (7).
In the limit s → ∞, as discussed in Sec. 4, the GVBS become fully permutation-
invariant for anyN . Consequently, the (optimized and non-optimized) telecloning fidelity for
distributing coherent states is equal for any pair of sender-receiver parties. These resources are
thus useful for 1 → N−1 symmetric telecloning. However, due to the monogamy constraints
on distribution of CV entanglement [27], this two-party fidelity will decrease with increasing
N , vanishing in the limit N → ∞ where the resources become completely separable. In this
Optical implementation and entanglement distribution in Gaussian valence bond states 14
respect, it is worth pointing out that the fully symmetric GVBS resources are more useful
for teleportation networks [13, 34], where N − 2 parties first perform local measurements
(momentum detections) on their single-mode portion of the entangled resource to concentrate
as much entanglement as possible onto the two-mode state of Alice and Bob, who can
accomplish non-classical teleportation (after the outcomes of the N − 2 measurements are
classically communicated to Bob). In this case, the optimal fidelity of N -user teleportation
network is an estimator of multipartite entanglement in the shared N -mode resource [32],
which is indeed a GVBS obtained from an infinitely entangled building block.
7. Conclusion
The valence bond picture is a valuable framework to study the structure of correlations in
quantum states of harmonic lattices. In fact, the motivation for such a formalism is quite
different from the finite-dimensional case, where valence bond/matrix product states are
useful to efficiently approximate ground states of N -body systems – generally described by
a number of parameters exponential in N – with polynomial resources [2]. In continuous
variable systems, the key feature of GVBS lies in the understanding of their entanglement
distribution as governed by the properties of simpler structures [4]. This has also experimental
implications giving a robust recipe to engineer correlations in many-body Gaussian states
from feasible operations on the building blocks. We have provided a simple scheme to produce
bisymmetric three-mode building blocks with linear optics, and discussed the subsequent
implementation of the valence bond construction. We have also investigated the usefulness
of such GVBS as resources for nonclassical communication, like telecloning of unknown
coherent states to distant receivers on a harmonic ring. It would be interesting to employ the
valence bond picture to describe quantum computation with continuous-variable cluster states
[35], and to devise efficient protocols for its optical implementation.
Acknowledgments
This work is supported by MIUR (Italy) and by the European Union through the
Integrated Project RESQ (IST-2001-37559), QAP (IST-3-015848), SCALA (CT-015714), and
SECOQC.
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http://arxiv.org/abs/quant-ph/0703277
	Introduction
	Continuous variable systems and Gaussian states
	Gaussian valence bond states
	Entanglement distribution
	Optical implementation of Gaussian valence bond states
	Telecloning with Gaussian valence bond resources
	Conclusion

  We study Gaussian valence bond states of continuous variable systems,
obtained as the outputs of projection operations from an ancillary space of M
infinitely entangled bonds connecting neighboring sites, applied at each of $N$
sites of an harmonic chain. The entanglement distribution in Gaussian valence
bond states can be controlled by varying the input amount of entanglement
engineered in a (2M+1)-mode Gaussian state known as the building block, which
is isomorphic to the projector applied at a given site. We show how this
mechanism can be interpreted in terms of multiple entanglement swapping from
the chain of ancillary bonds, through the building blocks. We provide optical
schemes to produce bisymmetric three-mode Gaussian building blocks (which
correspond to a single bond, M=1), and study the entanglement structure in the
output Gaussian valence bond states. The usefulness of such states for quantum
communication protocols with continuous variables, like telecloning and
teleportation networks, is finally discussed.

Introduction
Quantum information aims at the treatment and transport of information in ways forbidden by
classical physics. For this goal, continuous variables (CV) of atoms and light have emerged
as a powerful tool [1]. In this context, entanglement is an essential resource. Recently, the
valence bond formalism, originally developed for spin systems [2], has been generalized to
the CV scenario [3, 4] for the special class of Gaussian states, which play a central role in
theoretical and practical CV quantum information and communication [5].
In this work we analyze feasible implementations of Gaussian valence bond states
(GVBS) for quantum communication between many users in a CV setting, as enabled by
their peculiar structure of distributed entanglement [4]. After recalling the necessary notation
(Sec. 2) and the construction of Gaussian valence bond states [3] (Sec. 3), we discuss
the characterization of entanglement and its distribution in such states as regulated by the
entanglement properties of simpler states involved in the valence bond construction [4]
(Sec. 4). We then focus on the realization of GVBS by means of quantum optics, provide
a scheme for their state engineering (Sec. 5), and discuss the applications of such resources in
the context of CV telecloning [6, 7] on multimode harmonic rings (Sec. 6).
2. Continuous variable systems and Gaussian states
A CV system [1, 5] is described by a Hilbert space H =
i=1 Hi resulting from the tensor
product of infinite dimensional Fock spaces Hi’s. Let ai and a†i be the annihilation and
creation operators acting on Hi (ladder operators), and q̂i = (ai + a†i ) and p̂i = (ai − a
i )/i
be the related quadrature phase operators. Let R̂ = (x̂1, p̂1, . . . , q̂N , p̂N ) denote the vector of
the operators q̂i and p̂i. The canonical commutation relations for the R̂i can be expressed in
terms of the symplectic form Ω as
[R̂i, R̂j ] = 2iΩij ,
with Ω ≡ ω⊕N , ω ≡
The state of a CV system can be equivalently described by quasi-probability distributions
defined on the 2N -dimensional space associated to the quadratic form Ω, known as quantum
phase space. In the phase space picture, the tensor productH =
i Hi of the Hilbert spaces
Hi’s of the N modes results in the direct sum Λ =
i Λi of the phase spaces Λi’s.
States with Gaussian quasi-probability distributions are referred to as Gaussian states.
Such states are at the heart of information processing in CV systems [1, 5] and are the subject
of our analysis. By definition, a Gaussian state is completely characterized by the first and
second statistical moments of the field operators, which will be denoted, respectively, by the
vector of first moments R̄ ≡
〈R̂1〉, 〈R̂2〉, . . . , 〈R̂2N−1〉, 〈R̂2N 〉
and the covariance matrix
(CM) γ of elements
γij ≡
〈R̂iR̂j + R̂jR̂i〉 − 〈R̂i〉〈R̂j〉 . (1)
Coherent states, resulting from the application of displacement operatorsDY = e
iY TΩR̂
(Y ∈ R2n) to the vacuum state, are Gaussian states with CM γ = 1 and first statistical
moments R̄ = Y . First moments can be arbitrarily adjusted by local unitary operations
(displacements), which cannot affect any property related to entropy or entanglement. They
Optical implementation and entanglement distribution in Gaussian valence bond states 3
can thus be assumed zero without any loss of generality. A N -mode Gaussian state will be
completely characterized by its real, symmetric, 2N × 2N CM γ.
The canonical commutation relations and the positivity of the density matrix ρ of a
Gaussian state imply the bona fide condition
γ + iΩ ≥ 0 , (2)
as a necessary and sufficient constraint the matrix γ has to fulfill to be a CM corresponding
to a physical state [8, 9]. Note that the previous condition is necessary for the CM of any
(generally non Gaussian) state, as it generalizes to many modes the Robertson-Schrödinger
uncertainty relation [10].
A major role in the theoretical and experimental manipulation of Gaussian states is
played by unitary operations which preserve the Gaussian character of the states on which
they act. Such operations are all those generated by terms of the first and second order in the
field operators. As a consequence of the Stone-Von Neumann theorem, any such operation
at the Hilbert space level corresponds, in phase space, to a symplectic transformation, i.e. to
a linear transformation S which preserves the symplectic form Ω, so that Ω = STΩS, i.e. it
preserves the commutators between the different operators. Symplectic transformations on a
2N -dimensional phase space form the (real) symplectic group, denoted by Sp(2N,R). Such
transformations act linearly on first moments and “by congruence” on the CM (i.e. so that
γ 7→ SγST ). One has DetS = 1, ∀S ∈ Sp(2N,R). A crucial symplectic operation is
the one achieving the normal mode decomposition. Due to Williamson theorem [11], any
N -mode Gaussian state can be symplectically diagonalized in phase space, so that its CM
is brought in the form ν, such that SγST = ν, with ν = diag {ν1, ν1, . . . νN , νN}. The
set {νi} of the positive-defined eigenvalues of |iΩγ| constitutes the symplectic spectrum of
γ and its elements, the so-called symplectic eigenvalues, must fulfill the conditions νi ≥ 1,
following from the uncertainty principle Eq. (2) and ensuring positivity of the density matrix
ρ corresponding to γ.
Ideal beam splitters, phase shifters and squeezers are described by symplectic
transformations. In particular, a phase-free two-mode squeezing transformation, which
corresponds to squeezing the first mode (say i) in one quadrature (say momentum, p̂i) and
the second mode (say j) in the orthogonal quadrature (say position, q̂j) with the same degree
of squeezing r, can be represented in phase space by the symplectic transformation
Sij(r) = diag{exp r, exp−r, exp−r, exp r} . (3)
These trasformations occur for instance in parametric down conversions [12]. Another
important example of symplectic operation is the ideal (phase-free) beam splitter, which acts
on a pair of modes i and j as [13]
B̂ij(θ) :
âi 7→ âi cos θ + âj sin θ
âj 7→ âi sin θ − âj cos θ
and corresponds to a rotation in phase space of the form
Bij(θ) =
cos(θ) 0 sin(θ) 0
0 cos(θ) 0 sin(θ)
sin(θ) 0 − cos(θ) 0
0 sin(θ) 0 − cos(θ)
. (4)
The transmittivity τ of the beam splitter is given by τ = cos2(θ) so that a 50:50 beam splitter
(τ = 1/2) amounts to a phase-space rotation of π/4.
Optical implementation and entanglement distribution in Gaussian valence bond states 4
The combined application of a two-mode squeezing and a 50:50 beam splitter realizes
the entangling twin-beam transformation [14]
Tij(r) = Bij(π/4) · Sij(r) , (5)
which, if applied to two uncorrelated vacuum modes i and j (whose initial CM is the identity
matrix), results in the production of a pure two-mode squeezed Gaussian state with CM
σi,j(r) = Tij(r)T
ij (r) given by
σi,j(r) =
cosh(2r) 0 sinh(2r) 0
0 cosh(2r) 0 − sinh(2r)
sinh(2r) 0 cosh(2r) 0
0 − sinh(2r) 0 cosh(2r)
. (6)
The CV entanglement in the state σi,j(r) increases unboundedly as a function of r, and
in the limit r → ∞ Eq. (6) approaches the (unnormalizable) Einstein-Podolski-Rosen (EPR)
state [15], simultaneous eigenstate of relative position and total momentum of the two modes
i and j. Concerning entanglement in general, the “positivity of partial transposition” (PPT)
criterion states that a Gaussian CM γ is separable (with respect to a 1×N bipartition) if and
only if the partially transposed CM γ̃ satisfies the uncertainty principle Eq. (2) [9, 16]. In
phase space, partial transposition amounts to a mirror reflection of one quadrature associated
to the single-mode partition. If {ν̃i} is the symplectic spectrum of the partially transposed
CM γ̃, then a (N + 1)-mode Gaussian state with CM γ is separable if and only if ν̃i ≥ 1 ∀ i.
A proper measure of CV entanglement is the logarithmic negativityEN [17], which is readily
computed in terms of the symplectic spectrum ν̃i of γ̃ as
EN = −
i: ν̃i<1
log ν̃i . (7)
Such an entanglement monotone [18] quantifies the extent to which the PPT condition ν̃i ≥ 1
is violated. For 1 × N Gaussian states, only the smallest symplectic eigenvalue ν̃− of the
partially transposed CM can be smaller than one [10], thus simplifying the expression of EN :
then the PPT criterion simply yields that γ is entangled as soon as ν̃− < 1, and infinite
entanglement (accompanied by infinite energy in the state) is reached for ν̃− → 0+.
For 1× 1 Gaussian states γi,j symmetric under mode permutations, the entanglement of
formation EF is computable as well via the formula [19]
EF (γi,j) = max{0, f(ν̃
− )} , (8)
f(x) =
(1 + x)2
(1 + x)2
− (1− x)
(1− x)2
Being a monotonically decreasing function of the smallest symplectic eigenvalue ν̃i,j− of the
partial transpose γ̃i,j of γi,j , the entanglement of formation is completely equivalent to the
logarithmic negativity in this case. For a two-mode state, ν̃i,j can be computed from the
symplectic invariants of the state [20], and experimentally estimated with measures of global
and local purities [21] (the purity µ = Tr ρ2 of a Gaussian state ρ with CM γ is equal to
µ = (Det γ)−1/2).
3. Gaussian valence bond states
Let us review the basic definitions and notations for GVBS, as adopted in Ref. [4]. The
so-called matrix product Gaussian states introduced in Ref. [3] are N -mode states obtained
Optical implementation and entanglement distribution in Gaussian valence bond states 5Optical implementation and entanglement distribution in Gaussian valence bond states
i i+1
Figure 1. Gaussian valence bond states. Γin is the state of N EPR bonds and γ is the three-
mode building block. After the EPR measurements (depicted as curly brackets), the chain
of modes γ
collapses into a Gaussian valence bond state with global state Γout. See also
Ref. [4].
by taking a fixed number, M , of infinitely entangled ancillary bonds (EPR pairs) shared
by adjacent sites, and applying an arbitrary 2M → 1 Gaussian operation on each site
i = 1, . . . , N . Such a construction, more properly definable as a “valence bond” picture
for Gaussian states, can be better understood by resorting to the Jamiolkowski isomorphism
between quantum operations and quantum states [22]. In this framework, one starts with a
chain ofN Gaussian states of 2M +1 modes (the building blocks). The global Gaussian state
of the chain is described by a CM Γ =
i=1 γ
[i]. As the interest in GVBS lies mainly in
their connections with ground states of Hamiltonians invariant under translation [3], we can
focus on pure (Detγ [i] = 1), translationally invariant (γ[i] ≡ γ ∀i) GVBS. Moreover, in this
work we consider single-bonded GVBS, i.e. withM = 1. This is also physically motivated in
view of experimental implementations of GVBS, as more than one EPR bond would result in a
building block with five or more correlated modes, which appears technologically demanding.
Under the considered prescriptions, the building block γ is a pure Gaussian state of three
modes. As we aim to construct a translationally invariant state, it is convenient to consider a
γ whose first two modes, which will be combined with two identical halves of consecutive
EPR bonds (see Fig. 3), have the same reduced CM. This yields a pure, three-mode Gaussian
building block with the property of being bisymmetric [23], that is with a CM invariant under
permutation of the first two modes. This choice of the building block is further justified
by the fact that, among all pure three-mode Gaussian states, bisymmetric states maximize
the genuine tripartite entanglement [24]: no entanglement is thus wasted in the projection
process. The 6 × 6 CM γ of the building block can be written as follows in terms of 2 × 2
submatrices,
γs εss εsx
εTss γs εsx
εTsx ε
sx γx
. (9)
The 4×4 CM of the first two modes (each of them having reduced CM γs) will be denoted by
γss, and will be regarded as the input port of the building block. On the other hand, the CM
γx of mode 3 will play the role of the output port. The intermodal correlations are encoded
in the off-diagonal ε matrices. Without loss of generality, we can assume γ to be, up to local
unitary operations, in the standard form [24] with
γs = diag{s, s} , γx = diag{x, x} , (10)
Optical implementation and entanglement distribution in Gaussian valence bond states 6
εss = diag{t+, t−} , εsx = diag{u+, u−} ;
x2 − 1±
16s4 − 8(x2 + 1)s2 + (x2 − 1)2
x2 − 1
(x − 2s)2 − 1±
(x+ 2s)2 − 1
The valence bond construction works as follows (see Fig. 3). The global CM Γ =
i=1 γ acts as the projector from the state Γ
in of the N ancillary EPR pairs, to the final
N -mode GVBS Γout. This is realized by collapsing the state Γin, transposed in phase space,
with the ‘input port’ Γss =
i γss of Γ, so that the ‘output port’ Γx =
i γx turns into
the desired Γout. Here collapsing means that, at each site, the two two-mode states, each
constituted by one mode (1 or 2) of γss and one half of the EPR bond between site i and
its neighbor (i − 1 or i + 1, respectively), undergo an “EPR measurement” i.e. are projected
onto the infinitely entangled EPR state [22, 3]. An EPR pair between modes i and j can be
described, see Eq. (6), as a two-mode squeezed state σi,j(r) in the limit of infinite squeezing
(r → ∞). The input state is then Γin = limr→∞
i σi,i+1(r), where we have set periodic
boundary conditions so that N + 1 = 1 in labeling the sites. The projection corresponds
mathematically to taking a Schur complement (see Refs. [4, 3, 22] for details), yielding an
output pure GVBS of N modes on a ring with a CM
out = Γx − ΓTsx(Γss + θΓ
θ)−1Γsx , (11)
where Γsx =
i γsx, and θ =
i diag{1, −1, 1, −1} represents transposition in phase
space (q̂i → q̂i, p̂i → −p̂i).
Within the building block picture, the valence bond construction can be in toto
understood as a multiple CV entanglement swapping [25], as shown in Fig. 3: the GVBS
is created as the entanglement in the bonds is swapped to the chain of output modes via
teleportation [26] through the input port of the building blocks. It is thus clear that at a given
initialization of the output port (i.e. at fixed x), changing the properties of the input port
(i.e. varying s), which corresponds to implementing different Gaussian projections from the
ancillary space to the physical one, will affect the structure and entanglement properties of
the target GVBS. This link is explored in the following section.
4. Entanglement distribution
In Ref. [4] the quantum correlations of GVBS of the form Eq. (11) have been studied,
and related to the entanglement properties of the building block γ. Let us first recall the
characterization of entanglement in the latter. As a consequence of the uncertainty principle
Eq. (2), the CM Eq. (9) of the building block describes a physical state if [24]
x ≥ 1 , s ≥ smin ≡
. (12)
Let us keep the output parameter x fixed. Straightforward applications of the PPT separability
conditions, and consequent calculations of the logarithmic negativity Eq. (7), reveal that the
entanglement in the CM γss of the first two modes (input port) is monotonically increasing
as a function of s, ranging from the case s = smin when γss is separable to the limit s → ∞
when the block γss is infinitely entangled. Accordingly, the entanglement between each of
the first two modes γs of γ and the third one γx decreases with s. One can also show that the
genuine tripartite entanglement in the building block increases with the difference s − smin
[24]. The entanglement properties of the building block are summarized in Fig. 4.
Optical implementation and entanglement distribution in Gaussian valence bond states 7
Figure 2. How a Gaussian valence bond state is created via continuous-variable entanglement
swapping. At each step, Alice attempts to teleport her mode 0 (half of an EPR bond, depicted
in yellow) to Bob, exploiting as an entangled resource two of the three modes of the building
block (denoted at each step by 1 and 2). The curly bracket denotes homodyne detection, which
together with classical communication and conditional displacement at Bob’s side achieves
teleportation. The state will be approximately recovered in mode 2, owned by Bob. Since
mode 0, at each step, is entangled with the respective half of an EPR bond, the process swaps
entanglement from the ancillary chain of the EPR bonds to the modes in the building block.
The picture has to be followed column-wise. For ease of clarity, we depict the process as
constituted by two sequences: in the first sequence [frames (1) to (4)] modes 1 and 2 are the
two input modes of the building block (depicted in blue); in the second sequence [frames (5)
to (8)] modes 1 and 2 are respectively an input and an output mode of the building block.
As a result of the multiple entanglement swapping [frame (9)] the chain of the output modes
(depicted in red), initially in a product state, is transformed into a translationally invariant
Gaussian valence bond state, possessing in general multipartite entanglement among all the
modes (depicted in magenta).
The main question addressed in Ref. [4] is how the initial entanglement in the building
block γ redistributes in the Gaussian MPS Γout. The answer is that the more entanglement
one prepares in the input port γss, the longer the range of pairwise quantum correlations in
the output GVBS is, as pictorially shown in Fig. 4.
In more detail, let us consider first a building block γ with s = smin = (x + 1)/2. In
this case, a separability analysis shows that, for an arbitrary numberN of modes in the GVBS
chain, the target state Γout exhibits bipartite entanglement only between nearest neighbor
modes, for any value of x > 1 (for x = 1 we trivially obtain a product state). In fact, each
reduced two-mode block γouti,j is separable for |i− j| > 1.
With increasing s in the choice of the building block, one finds that in the target GVBS
the correlations start to extend smoothly to distant modes. A series of thresholds sk can be
Optical implementation and entanglement distribution in Gaussian valence bond states 8
Figure 3. Entanglement properties of the three-mode building block γ, Eq. (9), of the
Gaussian valence bond construction, as functions of the standard form covariances x and
d ≡ s − smin. (a) Bipartite entanglement, as quantified by the logarithmic negativity,
between the first two input-port modes 1 and 2; (b) Bipartite entanglement, as quantified by
the logarithmic negativity, between each of the first two modes and the output-port mode 3;
(c) Genuine tripartite entanglement, as quantified by the residual Gaussian contangle [27, 24],
among all the three modes.
found such that for s > sk, two given modes i and j with |i − j| ≤ k are entangled. While
trivially s1(x) = smin for anyN (notice that nearest neighbors are entangled also for s = s1),
the entanglement boundaries for k > 1 are in general different functions of x, depending
on the number of modes. We observe however a certain regularity in the process: sk(x,N)
always increases with the integer k. Very remarkably, this means that the maximum range
of bipartite entanglement between two modes, or equivalently the maximum distribution of
multipartite entanglement, in a GVBS on a translationally invariant ring, is monotonically
related to the amount of entanglement in the reduced two-mode input port of the building
block [4]. Moreover, no complete transfer of entanglement to more distant modes occurs:
closer sites remain still entangled even when correlations between farther pairs arise.
The most interesting feature is perhaps obtained when infinite entanglement is fed in
the input port (s → ∞): in this limit, the output GVBS turns out to be a fully symmetric,
permutation-invariant, N -mode Gaussian state. This means that each individual mode is
equally entangled with any other, no matter how distant they are [4]. These states, being thus
Optical implementation and entanglement distribution in Gaussian valence bond states 9
Figure 4. Pictorial representation of the entanglement between a probe (green) mode and its
neighbor (magenta) modes on an harmonic ring with an underlying valence bond structure.
As soon as the parameter s (encoding entanglement in the input port of the valence bond
building block) is increased, pairwise entanglement between the probe mode and its farther
and farther neighbors gradually appears in the corresponding output Gaussian valence bond
states. By translational invariance, each mode exhibits the same entanglement structure with
its respective neighbors. In the limit s → ∞, every single mode becomes equally entangled
with every other single mode on the ring, independently of their relative distance: the Gaussian
valence bond state is in this case fully symmetric.
built by a symmetric distribution of infinite pairwise entanglement among multiple modes,
achieve maximum genuine multiparty entanglement among all Gaussian states (at a given
energy) while keeping the strongest possible bipartite one in any pair, a property known as
monogamous but promiscuous entanglement sharing [27].
Keeping Fig. 3 in mind, we can conclude that having the two input modes initially
entangled in the building blocks, increases the efficiency of the entanglement-swapping
mechanism, inducing correlations between distant modes on the GVBS chain, which enable to
store and distribute joint information. In the asymptotic limit of an infinitely entangled input
port of the building block, the entanglement range in the target GVBS states is engineered to
be maximum, and communication between any two modes, independently of their distance, is
enabled nonclassically. In the next sections, we investigate the possibility of producing GVBS
with linear optics, and discuss with a specific example the usefulness of such resource states
for multiparty CV quantum communication protocols such as telecloning [6] and teleportation
networks [13].
5. Optical implementation of Gaussian valence bond states
The power of describing the production of GVBS in terms of physical states, the building
blocks, rather than in terms of arbitrary non-unitary Gaussian maps, lies not only in the
immediacy of the analytical treatment. From a practical point of view, the recipe of Fig. 3 can
be directly implemented to produce GVBS experimentally in the domain of quantum optics.
We first note that the EPR measurements are realized by the standard toolbox of a beamsplitter
plus homodyne detection [22], as demonstrated in several CV teleportation experiments [28].
The next ingredient to produce aN -mode GVBS is constituted byN copies of the three-
mode building block γ. We provide here an easy scheme (see also Refs. [6, 29]) to realize
bisymmetric three-mode Gaussian states of the form Eq. (9). As shown in Fig. 5(a), one
can start from three vacuum modes and first apply a twin-beam operation to modes 1 and 3,
characterized by a squeezing r13, then apply another twin-beam operation to modes 1 and 2,
Optical implementation and entanglement distribution in Gaussian valence bond states 10Optical implementation and entanglement distribution in Gaussian valence bond states
50:50
γγγγ0000
γγγγ0000
γγγγ0000
Figure 5. Optical production of bisymmetric three-mode Gaussian states, used as buildingFigure 5. Optical production of bisymmetric three-mode Gaussian states, used as building
blocks for the valence bond construction. (a) Three initial vacuum modes are entangled
through two sequential twin-beam boxes, the first (parametrized by a squeezing degree r13)
acting on modes 1 and 3, and the second (parametrized by a squeezing degree r12) acting on
the transformed mode 1 and mode 2. The output is a pure three-mode Gaussian state whose
covariance matrix is equivalent, up to local unitary operations, to the standard form given in
Eq. (9). (b) Detail of the entangling twin-beam transformation. One input mode is squeezed
in a quadrature, say momentum, of a degree r (this transformation is denoted by stretching
arrows→| |←); the other input mode is squeezed in the orthogonal quadrature, say position, of
the same amount (this anti-squeezing transformation is denoted by the corresponding rotated
symbol). Then the two squeezed modes are combined at a 50:50 beam-splitter. If the input
modes are both in the vacuum state, the output is a pure two-mode squeezed Gaussian state,
with entanglement proportional to the degree of squeezing r.
parametrized by r12. The symplectic operation describing the twin-beam transformation (two-
mode squeezing plus balanced beam splitter) is given by Eq. (5) and pictorially represented
in Fig. 5(b). The output of this optical network is a pure, bisymmetric, three-mode Gaussian
state with a CM γB = T12(r12)T13(r13)T
13(r13)T
12(r12) of the form Eq. (9), with
γs = diag
e−2r12
e4r12 cosh (2r13) + 1
e−2r12
cosh (2r13) + e
γx = diag {cosh (2r13) , cosh (2r13)} ,
εss = diag
e−2r12
e4r12 cosh (2r13)− 1
e−2r12
cosh (2r13) − e4r12
εsx = diag
2er12 cosh (r13) sinh (r13) , −
2e−r12 cosh (r13) sinh (r13)
By means of local symplectic operations (unitary on the Hilbert space), like additional single-
mode squeezings, the CM γB can be brought in the standard form of Eq. (10), from which
Optical implementation and entanglement distribution in Gaussian valence bond states 11
one has
r13 = arccos
x+ 1√
, r12 = arccos
−x3 + 2x2 + 4s2x− x
For a given r13 (i.e. at fixed x), the quantity r12 is a monotonic function of the standard-form
covariance s, so this squeezing parameter which enters in the production of the building block
(see Fig. 5) directly regulates the entanglement distribution in the target GVBS, as discussed
in Sec. 4.
The only unfeasible part of the scheme seems constituted by the ancillary EPR pairs.
But are infinitely entangled bonds truly necessary? In Ref. [4] the possibility is considered of
using a Γin given by the direct sum of two-mode squeezed states of Eq. (6), but with finite
r. Repeating the previous analysis to investigate the entanglement properties of the resulting
GVBS with finitely entangled bonds, it is found that, at fixed (x, s), the entanglement in the
various partitions is degraded as r decreases, as somehow expected. Crucially, this does not
affect the connection between input entanglement and output correlation length. Numerical
investigations show that, while the thresholds sk for the onset of entanglement between distant
pairs are quantitatively modified – a bigger s is required at a given x to compensate the less
entangled bonds – the overall structure stays untouched. This ensures that the possibility of
engineering the entanglement structure in GVBS via the properties of the building block is
robust against imperfect resources, definitely meaning that the presented scheme is feasible.
Alternatively, one could from the beginning observe that the triples consisting of two
projective measurements and one EPR pair can be replaced by a single projection onto the
EPR state, applied at each site i between the input mode 2 of the building block and the
consecutive input mode 1 of the building block of site i+1 [3]. The output of all the homodyne
measurements will conditionally realize the target GVBS.
6. Telecloning with Gaussian valence bond resources
The protocol of CV quantum telecloning [6] amongN parties is defined as a process in which
one of them (Alice) owns an unknown coherent state, and wants to distribute her state to
all the other N − 1 remote parties. The telecloning is achieved by a succession of standard
two-party CV teleportations [26] between the sender Alice and each of the N − 1 remote
receivers, exploiting each time the corresponding reduced two-mode state shared as resource
by the selected pair of parties. The 1 → 2 CV telecloning of unknown coherent states has
been recently demonstrated experimentally [7].
The no-cloning theorem [30] yields that the N − 1 remote clones can resemble the
original input state only to a certain extent. The fidelity, which quantifies the success of a
teleportation experiment, is defined as F ≡ 〈ψin|ρout|ψin〉, where “in” and “out” denote the
input and the output state. F reaches unity only for a perfect state transfer, ρout = |ψin〉〈ψin|.
Without using entanglement, by purely classical communication, an average fidelity of
Fcl = 1/2 is the best that can be achieved if the alphabet of input states includes all coherent
states with even weight [31]. The sufficient fidelity criterion states that, if teleportation
is performed with F > Fcl, then the two parties exploited an entangled state [31]. The
converse is generally false, i.e. some entangled resources may yield lower-than-classical
fidelities. In Ref. [32] it has been shown, however, that if the fidelity is optimized over all
possible local unitary operations performed on the shared Gaussian resource (which preserve
entanglement by definition), then it becomes equivalent, both qualitatively and quantitatively,
to the entanglement in the resource.
Optical implementation and entanglement distribution in Gaussian valence bond states 12
Let us also recall that the fidelity of CV two-user teleportation [26] of arbitrary single-
mode Gaussian states with CM γin (equal to the identity for coherent states) exploiting two-
mode Gaussian resources with CM γab =
γa εab
εTab γb
, can be computed [33] as
F = 2√
, Σ ≡ 2γin + ξγaξ + γb + ξεab + εTabξ , (14)
with ξ = diag{−1 , 1}. We can now consider the general setting of 1 → N − 1 telecloning,
where N parties share a N -mode GVBS as an entangled resource, and one of them plays the
role of Alice (the sender) distributing imperfect copies of unknown coherent states to all the
N − 1 receivers. For any N , the fidelity can be easily computed from the reduced two-mode
CMs via Eq. (14) and will depend, for translationally invariant states, on the relative distance
between the two considered modes.
In this work we focus on a practical example of a GVBS on a translationally invariant
harmonic ring, with N = 6 modes. As shown in the previous section, these states can be
produced with the current optical technology. They are completely characterized, up to local
unitary operations, by a 12 × 12 CM analytically obtained from Eq. (11) by considering the
building block in standard form Eq. (9), whose elements are algebraic functions of s and x
here omitted for brevity (as no particular insight is gained from their explicit expressions).
First of all we can construct the reduced CMs γouti,i+k of two modes with distance k, and
evaluate for each k the respective symplectic eigenvalue ν̃i,i+k− of the corresponding partial
transpose. The entanglement condition s > sk will correspond to the inequality ν̃
i,i+k
− < 1.
With this conditions one finds that s2(x) is the only acceptable solution to the equation: 72s
12(x2+1)s6+(−34x4+28x2−34)s4+(x6−5x4−5x2+1)s2+(x2−1)2(x4−6x2+1) = 0,
while for the next-next-nearest neighbors threshold one has simply s3(x) = x. This enables us
to classify the entanglement distribution and, more specifically, to observe the interaction scale
in the GVBS Γout: as discussed in Sec. 4 and explicitly shown in Ref. [4], by increasing initial
entanglement in γss one can gradually switch on pairwise quantum correlations between more
and more distant sites.
Accordingly, it is now interesting to test whether this entanglement is useful to achieve
nonclassical telecloning towards distant receivers. In this specific instance, Alice will send
two identical (approximate) clones to her nearest neighbors, two other identical clones (with
in principle different fidelity than the previous case) to her next-nearest neighbors, and one
final clone to the most distant site. The fidelities for the three transmissions can be computed
from Eq. (14) and are plotted in Fig. 6(a). For s = smin, obviously, only the two nearest
neighbor clones can be teleported with nonclassical fidelity, as the reduced states of more
distant pairs are separable. With increasing s also the state transfer to more distant sites is
enabled with nonclassical efficiency, but not in the whole region of the space of parameters s
and x in which the corresponding two-mode resources are entangled.
As mentioned before, one can optimize the telecloning fidelity considering resources
prepared in a different way but whose CM can be brought by local unitary operations (single-
mode symplectic transformations) in the standard form of Eq. (11). For GVBS resources,
this local-unitary freedom can be transferred to the preparation of the building block. A
more general γ locally equivalent to the standard form given in Eq. (10), can be realized by
complementing the presented state engineering scheme for the three-mode building block as in
Eq. (13) [see Fig. 5(a)], with additional single-mode rotations and squeezing transformations
aimed at increasing the output fidelity in the target GVBS states, while keeping both the
entanglement in the building block and consequently the entanglement in the final GVBS
unchanged by definition.
Optical implementation and entanglement distribution in Gaussian valence bond states 13
Figure 6. 1 → 5 quantum telecloning of unknown coherent states exploiting a six-mode
translationally invariant Gaussian valence bond state as a shared resource. Alice owns mode
i. Fidelities F for distributing clones to modes j such as k = |i − j| are plotted for k = 1
[(a),(d)]; k = 2 [(b),(e)]; and k = 3 [(c),(f)], as functions of the local invariants s and
x of the building block. In the first row [(a)–(c)] the fidelities are achieved exploiting the
non-optimized Gaussian valence bond resource in standard form. In the second row [(d)–
(f)] fidelities optimized over local unitary operations on the resource are displayed, which
are equivalent to the entanglement in the corresponding reduced two-mode states (see, as a
comparison, Fig. 3 in Ref. [4]). Only nonclassical values of the fidelities (F > 0.5) are
shown.
The optimal telecloning fidelity, obtained in this way exploiting the results of Ref. [32],
is plotted in Fig. 6(b) for the three teleportations between modes i and j with k = |i − j| =
1, 2, 3. In this case, one immediately recovers a non-classical fidelity as soon as the
separability condition s ≤ sk is violated in the corresponding resources. Moreover, the
optimal telecloning fidelity at a given k is itself a quantitative measure of the entanglement in
the reduced two-mode resource, being equal to [32]
Foptk = 1/(1 + ν̃
i,i+k
− ) , (15)
where ν̃i,i+k− is the smallest symplectic eigenvalue of the partially transposed CM in
the corresponding bipartition. The optimal fidelity is thus completely equivalent to the
entanglement of formation Eq. (8) and to the logarithmic negativity Eq. (7).
In the limit s → ∞, as discussed in Sec. 4, the GVBS become fully permutation-
invariant for anyN . Consequently, the (optimized and non-optimized) telecloning fidelity for
distributing coherent states is equal for any pair of sender-receiver parties. These resources are
thus useful for 1 → N−1 symmetric telecloning. However, due to the monogamy constraints
on distribution of CV entanglement [27], this two-party fidelity will decrease with increasing
N , vanishing in the limit N → ∞ where the resources become completely separable. In this
Optical implementation and entanglement distribution in Gaussian valence bond states 14
respect, it is worth pointing out that the fully symmetric GVBS resources are more useful
for teleportation networks [13, 34], where N − 2 parties first perform local measurements
(momentum detections) on their single-mode portion of the entangled resource to concentrate
as much entanglement as possible onto the two-mode state of Alice and Bob, who can
accomplish non-classical teleportation (after the outcomes of the N − 2 measurements are
classically communicated to Bob). In this case, the optimal fidelity of N -user teleportation
network is an estimator of multipartite entanglement in the shared N -mode resource [32],
which is indeed a GVBS obtained from an infinitely entangled building block.
7. Conclusion
The valence bond picture is a valuable framework to study the structure of correlations in
quantum states of harmonic lattices. In fact, the motivation for such a formalism is quite
different from the finite-dimensional case, where valence bond/matrix product states are
useful to efficiently approximate ground states of N -body systems – generally described by
a number of parameters exponential in N – with polynomial resources [2]. In continuous
variable systems, the key feature of GVBS lies in the understanding of their entanglement
distribution as governed by the properties of simpler structures [4]. This has also experimental
implications giving a robust recipe to engineer correlations in many-body Gaussian states
from feasible operations on the building blocks. We have provided a simple scheme to produce
bisymmetric three-mode building blocks with linear optics, and discussed the subsequent
implementation of the valence bond construction. We have also investigated the usefulness
of such GVBS as resources for nonclassical communication, like telecloning of unknown
coherent states to distant receivers on a harmonic ring. It would be interesting to employ the
valence bond picture to describe quantum computation with continuous-variable cluster states
[35], and to devise efficient protocols for its optical implementation.
Acknowledgments
This work is supported by MIUR (Italy) and by the European Union through the
Integrated Project RESQ (IST-2001-37559), QAP (IST-3-015848), SCALA (CT-015714), and
SECOQC.
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http://arxiv.org/abs/quant-ph/0703277
	Introduction
	Continuous variable systems and Gaussian states
	Gaussian valence bond states
	Entanglement distribution
	Optical implementation of Gaussian valence bond states
	Telecloning with Gaussian valence bond resources
	Conclusion
