A measure of the non-Gaussian character of a quantum state
Marco G. Genoni,1 Matteo G. A. Paris,1, 2, ∗ and Konrad Banaszek3
1Dipartimento di Fisica dell’Università di Milano, I-20133, Milano, Italia.
2Institute for Scientific Interchange, I-10133 Torino, Italia
3Institute of Physics, Nicolaus Copernicus University, PL-87-100 Toruń, Poland
(Dated: November 3, 2018)
We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce
a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a
reference Gaussian state. We analyze in details the properties of the proposed measure and exploit it to evaluate
the non-Gaussianity of some relevant single- and multi-mode quantum states. The evolution of non-Gaussianity
is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by
photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows
to define a corresponding measure for the non-Gaussian character of a quantum operation.
PACS numbers: 03.67.-a, 03.65.Bz, 42.50.Dv
I. INTRODUCTION
Gaussian states play a crucial role in quantum information
processing with continuous variables. This is especially true
for quantum optical implementations since radiation at ther-
mal equilibrium, including the vacuum state, is itself a Gaus-
sian state and most of the Hamiltonians achievable within the
current technology are at most bilinear in the field operators,
i.e. preserve the Gaussian character [1, 2, 3]. As a matter
of fact, using single-mode and entangled Gaussian states, lin-
ear optical circuits and Gaussian operations, like homodyne
detection, several quantum information protocols have been
implemented, including teleportation, dense coding and quan-
tum cloning [4].
On the other hand quantum information protocols required
for long distance communication, as for example entangle-
ment distillation and entanglement swapping, rely on non-
Gaussian operations. In addition, it has been demonstrated
that teleportation [5, 6, 7] and cloning [8] of quantum states
may be improved by using non-Gaussian states and non-
Gaussian operations. Indeed, de-Gaussification protocols for
single-mode and two-mode states have been proposed [5, 6, 7]
and realized [9]. It should be also noticed that any strongly
superadditive function is minimized, at fixed covariance ma-
trix, by Gaussian states. This is crucial to prove extremality
of Gaussian states and Gaussian operations [10, 11] for what
concerns various quantities as channel capacities [12], multi-
partite entanglement measures [13] and distillable secret key
in quantum key distribution protocols. Since in most cases
these quantities can be computed only for Gaussian states, a
non-Gaussianity measure may serve as a guideline to quan-
tify them for the class of non-Gaussian states. Overall, non-
Gaussianity is revealing itself as a resource for continuous
variable quantum information, and thus we urge a measure
able to quantify the non-Gaussian character of a quantum
state.
In this paper we introduce a novel quantity, the non-
∗Electronic address: matteo.paris@fisica.unimi.it
Gaussianity δ[̺] of a quantum state, which quantifies how
much a state fails to be Gaussian. Our measure, which is based
on the Hilbert-Schmidt distance between the state itself and a
reference Gaussian state, can be easily computed for any state,
either single-mode or multi-mode.
The paper is structured as follows. In the next Section we
introduce notation and review the basic properties of Gaussian
states. Then, in Section III we introduce the formal definition
of δ[̺] and study its properties in details. In Section IV we
evaluate non-Gaussianity of relevant quantum states whereas
in Section V we analyze the evolution of non-Gaussianity for
known Gaussification and de-Gaussification maps. Section VI
closes the paper with some concluding remarks.
II. GAUSSIAN STATES
For concreteness, we will use here the quantum optical ter-
minology of modes carrying photons, but our theory applies
to general bosonic systems. Let us consider a system of n
modes described by mode operators ak, k = 1 . . . n, satis-
fying the commutation relations [ak, a
j ] = δkj . A quantum
state ̺ of the n modes is fully described by its characteristic
function [14]
χ[̺](λ) = Tr[̺D(λ)]
where D(λ) =
k=1Dk(λk) is the n-mode displacement
operator, with λ = (λ1, . . . , λn)
T , λk ∈ C, and where
Dk(λk) = exp{λka†k − λ
is the single-mode displacement operator. The canonical op-
erators are given by:
(ak + a
(ak − a†k)
with commutation relations given by [qj , pk] = iδjk. Upon in-
troducing the real vector R = (q1, p1, . . . , qn, pn)
T , the com-
http://arxiv.org/abs/0704.0639v4
mailto:matteo.paris@fisica.unimi.it
mutation relations rewrite as
[Rk, Rj ] = iΩkj
where Ωkj are the elements of the symplectic matrix Ω =
k=1 σ2, σ2 being the y-Pauli matrix. The covariance ma-
trix σ ≡ σ[̺] and the vector of mean values X ≡ X[̺] of a
quantum state ̺ are defined as
Xj = 〈Rj〉
σkj =
〈{Rk, Rj}〉 − 〈Rj〉〈Rk〉
where {A,B} = AB+BA denotes the anti-commutator, and
〈O〉 = Tr[̺ O] is the expectation value of the operator O.
A quantum state ̺G is referred to as a Gaussian state if its
characteristic function has the Gaussian form
χ[̺G](Λ) = exp
σΛ+XTΩΛ
where Λ is the real vector Λ =
(Reλ1, Imλ1, . . . ,Reλn, Imλn)
T . Of course, once the
covariance matrix and the vector of mean values are given, a
Gaussian state is fully determined. For a single-mode system
the most general Gaussian state can be written as
̺G = D(α)S(ζ)ν(nt)S
†(ζ)D†(α),
D(α) being the displacement operator, S(ζ) =
exp[ 1
ζ(a†)2 − 1
ζ∗a2] the squeezing operator, α, ζ ∈ C, and
ν(nt) = (1 + nt)
−1[nt/(1 + nt)]
a†a a thermal state with nt
average number of photons.
III. A MEASURE OF THE NON-GAUSSIAN CHARACTER
OF A QUANTUM STATE
In order to quantify the non-Gaussian character of a quan-
tum state ̺ we use a quantity based on the distance between
̺ and a reference Gaussian state τ , which itself depends on ̺.
Specifically, we define the non-Gaussianity δ[̺] of the state ̺
δ[̺] =
D2HS [̺, τ ]
where DHS [̺, τ ] denotes the Hilbert-Schmidt distance be-
tween ̺ and τ
D2HS [̺, τ ] =
Tr[(̺− τ)2] = µ[̺] + µ[τ ] − 2κ[̺, τ ]
, (3)
with µ[̺] = Tr[̺2] and κ[̺, τ ] = Tr[̺τ ] denoting the purity of
̺ and the overlap between ̺ and τ respectively. The Gaussian
reference τ is the Gaussian state such that
X[̺] = X[τ ]
σ[̺] = σ[τ ]
i.e. τ is the Gaussian state with the same covariance matrix σ
and the same vector X of the state ̺.
The relevant properties of δ[̺], which confirm that it repre-
sents a good measure of the non-Gaussian character of ̺, are
summarized by the following Lemmas:
Lemma 1: δ[̺] = 0 iff ̺ is a Gaussian state.
Proof: If δ[̺] = 0 then ̺ = τ and thus it is a Gaussian state.
If ̺ is a Gaussian state, then it is uniquely identified by its first
and second moments and thus the reference Gaussian state τ
is given by τ = ̺, which, in turn, leads to DHS [̺, τ ] = 0 and
thus to δ[̺] = 0.
Lemma 2: If U is a unitary map corresponding to a symplec-
tic transformation in the phase space, i.e. if U = exp{−iH}
with hermitianH that is at most bilinear in the field operators,
then δ[U̺U †] = δ[̺]. This property ensures that displace-
ment and squeezing operations do not change the Gaussian
character of a quantum state.
Proof: Let us consider ̺′ = U̺U †. Then the covariance ma-
trix transforms as σ[̺′] = Σσ[̺]ΣT , Σ being the symplectic
transformation associated to U . At the same time the vector of
mean values simply translates to X ′ = X +X0, where X0
is the displacement generated by U . Since any Gaussian state
is fully characterized by its first and second moments, then the
reference state must necessarily transform as τ ′ = UτU †, i.e.
with the same unitary transformation U . Since the Hilbert-
Schmidt distance and the purity of a quantum state are invari-
ant under unitary transformations the lemma is proved.
Lemma 3: δ[̺] is proportional to the squared L2(Cn) dis-
tance between the characteristic functions of ̺ and of the ref-
erence Gaussian state τ . In formula:
δ[̺] ∝
d2nλ [χ[̺](λ)− χ[τ ](λ)]2 . (4)
Since the notion of Gaussianity of a quantum state is de-
fined through the shape of its characteristic function, and since
the characteristic function of a quantum state belongs to the
L2(Cn) space [14], we address L2(C) distance to as a good
indicator for the non Gaussian character of ̺.
Proof: Since characteristic functions of self-adjoint operators
are even functions of λ and by means of the identity
Tr[O1O2] =
χ[O1](λ)χ[O2](−λ) ,
we obtain
D2HS [̺, τ ] =
[χ[̺](λ)− χ[τ ](λ)]2 .
Lemma 4: Consider a bipartite state ̺ = ̺A ⊗ ̺G. If ̺G is a
Gaussian state then δ[̺] = δ[̺A].
Proof: we have
µ[̺] = µ[̺A]µ[̺G]
µ[τ ] = µ[τA]µ[τG]
κ[̺, τ ] = κ[̺A, τA]κ[̺G, ̺G] .
Therefore, since κ[̺G, ̺G] = µ[̺G] we arrive at
δ[̺] =
µ[̺A]µ[̺G] + µ[τA]µ[̺G]− 2κ[̺A, τA]κ[̺G, ̺G]
2µ[̺A]µ[̺G]
= δ[̺A] (5)
The four properties illustrated by the above lemmas are the
natural properties required for a good measure of the non-
Gaussian character of a quantum state. Notice that by using
the trace distanceDT [̺, τ ] =
Tr|̺−τ | instead of the Hilbert-
Schmidt distance we would lose Lemmas 3 and 4, and that the
invariance expressed by Lemma 4 holds thanks to the renor-
malization of the Hilbert-Schmidt distance through the purity
µ[̺]. We stress the fact that our measure of non-Gaussianity
is a computable one: It may be evaluated for any quantum
state of n modes by the calculation of the first two moments
of the state, followed by the evaluation of the overlap with the
corresponding Gaussian state.
Notice that δ[̺] is not additive (nor multiplicative) with re-
spect to the tensor product. If we consider a (separable) multi-
partite quantum state in the product form ̺ = ⊗nk=1̺k, the
non-Gaussianity is given by
δ[̺] =
k=1 µ[̺k] +
k=1 µ[τk]− 2
k=1 κ[̺k, τk]
k=1 µ[̺k]
where τk is the Gaussian state with the same moments of ̺k.
In fact, since the state ̺ is factorisable, we have that the cor-
responding Gaussian τ is a factorisable state too.
IV. NON-GAUSSIANITY OF RELEVANT QUANTUM
STATES
Let us now exploit the definition (2) to evaluate the non-
Gaussianity of some relevant quantum states. At first we con-
sider Fock number states |p〉 of a single mode as well as mul-
timode factorisable states |p〉⊗n made of n copies of a num-
ber state. The reference Gaussian states are a thermal state
τp = ν(p) with average photon number p and a factorisable
thermal state τN = [ν(p)]
⊗n with average photon number p
in each mode [15]. Non-Gaussianity may be analytically eval-
uated, leading to
δ[|p〉〈p|] = 1
2p+ 1
δ[(|p〉〈p|)⊗n] = 1
2p+ 1
In the multimode case of |p〉⊗n, we seek for the number of
copies that maximizes the non-Gaussianity. In Fig. 1 we
show both δp ≡ δ[|p〉〈p|] and δ̄p = maxn δ[(|p〉〈p|)⊗n] as a
function of p. As it is apparent from the plot non-Gaussianity
of Fock states |p〉 increases monotonically with the number
of photon p with the limiting value δp = 1/2 obtained for
p → ∞. Upon considering multi-mode copies of Fock states
we obtain larger value of non-Gaussianity: δ̄p is a decreasing
function of p, approaching δ̄ = 1/2 from above. The value
1 5 10 15 20 25 30
FIG. 1: (Top): Non-Gaussianity of single mode Fock states (gray)
|p〉 and of multi-mode Fock states |p〉⊗n (black) as a function of p.
Non-Gaussianity for multi-mode states has been maximized over the
number of copies n. (Bottom): Non-Gaussianity, as a function of the
parameter φ, for the two-mode superpositions |Φ〉〉 (dashed gray),
|Ψ〉〉 (solid gray), and for the single-mode superposition of coherent
states |ψS〉 for α = 0.5 (solid black) and α = 5 (dashed black).
of δ̄p corresponds to n = 3 for p < 26 and to n = 2 for
27 ≤ p . 250.
Another example is the superposition of coherent states
|ψS〉 = N−1/2 (cosφ|α〉 + sinφ| − α〉) (7)
with normalization N = 1 + sin(2φ) exp{−2α2} which for
φ = ±π/4 reduces to the so-called Schrödinger cat states,
and whose reference Gaussian state is a displaced squeezed
thermal state τS = D(C)S(r)ν(N)S
†(r)D†(C), where the
real parameters C, r, and N are analytical functions of φ and
α. Finally we evaluate the non-Gaussianity of the two-mode
Bell-like superpositions of Fock states
|Φ〉〉 = cosφ|0, 0〉+ sinφ|1, 1〉
|Ψ〉〉 = cosφ|0, 1〉+ sinφ|1, 0〉,
which for φ = ±π/4 reduces to the Bell states |Φ±〉
and |Ψ±〉. The corresponding reference Gaussian states
are respectively a two mode squeezed thermal state τΦ =
S2(ξ)[ν(N) ⊗ ν(N)]S†2(ξ), where S2(ξ) = exp(ξa
ξ∗ab) denotes the two-mode squeezing operator, and τΨ =
R(θ)[ν(N1)⊗ν(N2)]R†(θ), namely the correlated two-mode
state obtained by mixing a single-mode thermal state with
the vacuum at a beam splitter of transmissivity cos2 θ, i.e.
R(θ) = exp[iθ(a
1a2+a
2a1)]. All the parameters involved in
these reference Gaussian states are analytical functions of the
superposition parameter φ. Non-Gaussianities are thus evalu-
ated by means of (2) and are reported in Fig. 1 as a function
of the parameter φ. As it is apparent from the plot, the non-
Gaussianity of single-mode states does not surpass the value
δ = 1/2, and this fact is confirmed by other examples not
reported here.
As concern the cat-like states, we notice that for small val-
ues of α the non-Gaussianity of the superposition |ψS〉 shows
a different behavior for positive and negative values of the pa-
rameter φ: for φ > 0 and α = 0.5 we have almost zero δ,
while higher values are achieved for φ < 0. For higher val-
ues of α (α = 5 in Fig. 1), non-Gaussianity becomes an even
function of φ. This different behavior can be understood by
looking at the Wigner functions of even and odd Schrödinger
cat states for different values of α: for small values of α the
even cat’s Wigner function is similar to a Gaussian function,
while the odd cat’s Wigner function shows a non-Gaussian
hole in the origin of the phase space; increasing the value of α
the Wigner functions of the two kind of states become similar
and deviate from a Gaussian function.
We have also done a numerical analysis of non-Gaussianity
of single-mode quantum states represented by finite superpo-
sition of Fock states
n,k=0
̺nk|n〉〈k| . (8)
To this aim we generate randomly quantum states in a finite
dimensional subspaces, dim(H) ≡ d+ 1 ≤ 21, following the
algorithm proposed by Zyczkowski et al [16, 17], i.e. by gen-
erating a random diagonal state (i.e. a point on the simplex)
and a random unitary matrix according to the Haar measure.
In Fig. 2 we report the distribution of non-Gaussianity δ[̺d],
as evaluated for 105 random quantum states, for three different
value of the maximum number of photons d. As it is apparent
from the plots the distribution of δ[̺d] becomes Gaussian-like
for increasing d. In the fourth panel of Fig. 2 we thus re-
port the mean values and variances of the the distributions as
a function of the maximum number of photons d. The mean
value increases with the dimension whereas the variance is a
monotonically decreasing function of d.
Also for finite superpositions simulations we did not ob-
serve non-Gaussianity higher than 1/2. Therefore, although
we have no proof, we conjecture that δ = 1/2 is a limiting
value for the non-Gaussianity of a single-mode state. Higher
values are achievable for two-mode or multi-mode quantum
states (e.g. δ = 2/3 for the Bell states |Ψ±〉〉).
V. GAUSSIFICATION AND DE-GAUSSIFICATION
PROCESSES
We have also studied the evolution of non-Gaussianity
of quantum states undergoing either Gaussification or de-
Gaussification processes. First we have considered the Gaus-
sification of Fock states due do the interaction of the system
FIG. 2: Distribution of non-Gaussianity δ[̺d] as evaluated for 10
random quantum states, for three different value of the maximum
number of photons d. Top: d = 2 (left), d = 10 (right); Bottom:
d = 20 (left). (Bottom-right): Mean values and variances of the non-
Gaussianities evaluated for 105 random quantum states, as a function
of the maximum number of photons d.
with a bath of oscillators at zero temperature. This is per-
haps the simplest example of a Gaussification protocol. In
fact the interaction drives asymptotically any quantum state to
the vacuum state of the harmonic system, which, in turn, is a
Gaussian state. The evolution of the system is governed by the
Lindblad Master equation ˙̺ = γ
L[a]̺, where ˙̺ denotes time
derivative, γ is the damping factor and the Lindblad superop-
erator acts as follows L[a]̺ = 2a†̺a − a†a̺ − ̺a†a. Upon
writing η = e−γt the solution of the Master equation can be
written as
̺(η) =
Vm ̺ V
m (9)
Vm = [(1− η)m/m!]
2 am η
(a†a−m) ,
where ̺ is the initial state. In particular for the system ini-
tially prepared in a Fock state ̺p = |p〉〈p|, we obtain, after
evolution, the mixed state
̺p(η) =
Vm̺pV
αl,p(η)|l〉〈l| (10)
with αl,p(η) =
(1−η)p−lηl. The reference Gaussian state
corresponding to ̺p(η) is a thermal state τp(η) = ν(pη) with
average photon number pη. Non-Gaussianity of ̺p(η) can be
evaluated analytically, we have
δpη ≡ δ[̺p(η)]
2(1− η)2m 2F1
−m,−m, 1; η2
(η−1)2
(1− η)2m 2F1
−m,−m, 1; η
(η − 1)2
+ (1 + 2mη)−1 − 2(1 + (m− 1)η)
(1 +mη)m+1
2F1(a, b, c;x) being a hypergeometric function. We show the
behavior of δpη in Fig. 3 as a function of 1 − η for different
values of p. As it is apparent from the plot δpη is a monotoni-
cally decreasing function of 1 − η as well as a monotonically
increasing function of p. That is, at fixed time t the higher
is the initial photon number p, the larger is the resulting non-
Gaussianity.
0.2 0.4 0.6 0.8 1
1 - Η
0.2 0.4 0.6 0.8 1
FIG. 3: (Left): Non-Gaussianity of Fock states |p〉 undergoing Gaus-
sification by loss mechanism due to the interaction with a bath of os-
cillators at zero temperature. We show δηp as a function of 1 − η
for different values of p: from bottom to top p = 1, 10, 100, 1000.
(Right): Non-Gaussianity of ̺IPS as a function of T for r = 0.5 and
for different values of ǫ = 0.2, 0.4, 0.6, 0.8 (from bottom to top).
δIPS results to be a monotonous increasing function of T , while ǫ
only slightly changes the non-Gaussian character of the state.
Let us now consider the de-Gaussification protocol ob-
tained by the process of photon subtraction. Inconclusive Pho-
ton Subtraction (IPS) has been introduced for single-mode and
two-mode states in [6, 7, 18] and experimentally realized in
[9]. In the IPS protocol an input state ̺(in) is mixed with
the vacuum at a beam splitter (BS) with transmissivity T and
then, on/off photodetection with quantum efficiency ǫ is per-
formed on the reflected beam. The process can be thus charac-
terized by two parameters: the transmissivity T and the detec-
tor efficiency ǫ. Since the detector can only discriminate the
presence from the absence of light, this measurement is in-
conclusive, namely it does not resolve the number of detected
photons. When the detector clicks, an unknown number of
photons is subtracted from the initial state and we obtain the
conditional IPS state ̺IPS . The conditional map induced by
the measurement is non-Gaussian [7], and the output state is
de-Gaussified. Upon applying the IPS protocol to the (Gaus-
sian) single-mode squeezed vacuum S(r)|0〉 (r ∈ R), where
S(r) is the real squeezing operation we obtain [18] the con-
ditional state ̺IPS , whose characteristic function χ[̺IPS ](λ)
is a sum of two Gaussian functions and therefore is no longer
Gaussian. The corresponding Gaussian reference state is a
squeezed thermal state τIPS = S(ξIPS)ν(NIPS)S
†(ξIPS)
where the parameters ξIPS andNIPS are analytic functions of
r, T and ǫ. Non-Gaussianity δIPS = δIPS(T, ǫ, r) has been
evaluated, and in Fig. 3 (right) we report δIPS for r = 0.5
as a function of the transmittivity T for different values of
the quantum efficiency ǫ. As it is apparent from the plot the
IPS protocol indeed de-Gaussifies the input state, i.e. nonzero
values of the non-Gaussianity are obtained. We found that
δIPS is an increasing function of the transmissivity T which
is the relevant parameter, while the quantum efficiency ǫ only
slightly affects the non-Gaussian character of the output state.
The highest value of non-Gaussianity is achieved in the limit
of unit transmissivity and unit quantum efficiency
T,η→1
δIPS = δ[|1〉〈1|] = δ[S(r)|1〉〈1|S†(r)],
where the last equality is derived from Lemma 2. This result
is in agreement with the fact that a squeezed vacuum state
undergoing the IPS protocol is driven towards the target state
S(r)|1〉 in the limit of T, ǫ → 1 [18]. Finally, we notice that
for T, ǫ 6= 1 and for r → ∞ the non-Gaussianity vanishes. In
turn, this corresponds to the fact that one of the coefficients
of the two Gaussians of χ[̺IPS ](λ) vanishes, i.e. the output
state is again a Gaussian one.
VI. CONCLUSION AND OUTLOOKS
Having at disposal a good measure of non-Gaussianity for
quantum state allows us to define a measure of the non-
Gaussian character of a quantum operation. Let us denote
by G the whole set of Gaussian states. A convenient defi-
nition for the non-Gaussianity of a map E reads as follows
δ[E ] = max̺∈G δ[E(̺)], where E(̺) denotes the quantum
state obtained after the evolution imposed by the map. Indeed,
for a Gaussian map Eg , which transforms any input Gaussian
state into a Gaussian state, we have δ[Eg] = 0. Work along
this line is in progress and results will be reported elsewhere.
In conclusion, we have proposed a measure of the non-
Gaussian character of a CV quantum state. We have shown
that our measure satisfies the natural properties expected from
a good measure of non-Gaussianity, and have evaluated the
non-Gaussianity of some relevant states, in particular of states
undergoing Gaussification and de-Gaussification protocols.
Using our measure an analogue non-Gaussianity measure for
quantum operations may be introduced.
Acknowledgments
This work has been supported by MIUR project
PRIN2005024254-002, the EC Integrated Project QAP (Con-
tract No. 015848) and Polish MNiSW grant 1 P03B 011 29.
APPENDIX A: GAUSSIAN REFERENCE WITH
UNCONSTRAINED MEAN VALUE
As we have seen from the above examples δ[̺] of Eq. (2)
represents a good measure of the non-Gaussian character of a
quantum state. A question arises on whether different choices
for the reference Gaussian state τ may lead to alternative,
valid, definitions. As for example (for single-mode states) we
may define
δ′[̺] = min
D2HS [̺, τ ]/µ[̺], (A1)
where τ = D(C)S(ξ)ν(N)S†(ξ)D†(C) is a Gaussian state
with the same covariance matrix of ̺ and unconstrained vec-
tor of mean values X = (ReC, ImC) used to minimize the
Hilbert-Schmidt distance. Here we report few examples of
the comparison between the results already obtained using (2)
with that coming from (A1). As we will see either the two
definitions coincide or δ′ and δ are monotone functions of
each other. Since the definition (2) corresponds to an easily
computable measure we conclude that it represents the most
convenient choice.
Let us first consider the Fock state ̺ = |p〉〈p|. According
to (A1), the reference Gaussian state is given by a displaced
thermal states τ ′ = D(C)νpD
†(C). The overlap between ̺
and τ ′ is given by
κ[|p〉〈p|, τ ′] = 1
1 + p
1 + p
1 + p
p(1 + p)
The maximum of (A2) is achieved forC = 0, which coincides
with the assumptions C = Tr[a|p〉〈p|].
Let us consider the quantum state (10) obtained as the so-
lution of the loss Master Equation for an initial Fock state
|p〉〈p|. The unconstrained Gaussian reference is again a dis-
placed thermal state τ ′ = D(C)νpηD
†(C), and the overlap is
given by
κ[̺p(η), τ
′] = Tr[τ̺p(η)] =
(1 + η(p− 1))p
(1 + pη)p+1
η|C|2
(1 + pη)(η(1 − p)− 1)
Again, since the overlap is maximum for C = Tr[a̺p(η)] =
0, both definitions give the same results for the non-
Gaussianity.
Let us now consider the Schrödinger cat-like states of (7).
The reference Gaussian state is a displaced squeezed thermal
state, with squeezing and thermal photons as calculated be-
fore. The optimization over the free parameterC may be done
numerically. In Fig. 4 we show the non-Gaussianitiy, both as
resulting from (A1) and by choosing C = Tr[a̺S ] as in (2),
as a function of ǫ. The two curves are almost the same, with
no qualitative differences.
[1] A. Ferraro, S. Olivares and M. G. A. Paris, Gaussian States in
Quantum Information, (Bibliopolis, Napoli, 2005)
[2] J. Eisert, M. B. Plenio, Int. J. Quant. Inf. 1, 479 (2003)
[3] F. Dell’Anno et al., Phys. Rep. 428, 53 (2006).
[4] S. L. Braunstein, P. van Loock, Rev. Mod. Phys 77, 513 (2005).
[5] T. Opatrny et al., Phys.Rev. A 61, 032302 (2000).
[6] P. T. Cochrane et al., Phys Rev. A 65, 062306 (2002).
[7] S. Olivares et al., Phys. Rev. A 67, 032314 (2003); S. Olivares,
M. G. A. Paris, Las. Phys. 16, 1533 (2006).
[8] N. J. Cerf et al., Phys. Rev. Lett. 95, 070501 (2005).
[9] J. Wenger et al., Phys. Rev. Lett. 92, 153601 (2004); A. Our-
joumtsev et al., Science 312, 83 (2006).
[10] M. M. Wolf et al., Phys. Rev. Lett. 96, 080502 (2006).
[11] M. M. Wolf et al, Phys. Rev. Lett. 98, 130501 (2007). 0
[12] A. S. Holevo, R. F. Werner, Phys. Rev. A 63, 032312 (2001).
[13] L. M. Duan at al, Phys. Rev. Lett. 84, 4002 (2000); R. F. Werner,
M. M. Wolf, Phys. Rev. Lett. 86, 3658 (2001).
[14] K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
[15] P. Marian, T. Marian, Phys. Rev. A 47, 4474 (1993).
[16] K. Zyczkowski and M. Kus, J. Phys. A 27, 4235 (1994).
[17] K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein,
Phys. Rev. A 58, 883 (1994).
[18] S. Olivares and M. G. A. Paris, J. Opt. B, 7, S392 (2005).
FIG. 4: Non-Gaussianity of a Schrödinger cat-like state as a func-
tion of the superposition parameter φ, with either C obtained by nu-
merical minimization (solid) or with C = Tr[a̺] (dotted). (Left):
α = 0.5; (Right): α = 5.

  We address the issue of quantifying the non-Gaussian character of a bosonic
quantum state and introduce a non-Gaussianity measure based on the
Hilbert-Schmidt distance between the state under examination and a reference
Gaussian state. We analyze in details the properties of the proposed measure
and exploit it to evaluate the non-Gaussianity of some relevant single- and
multi-mode quantum states. The evolution of non-Gaussianity is also analyzed
for quantum states undergoing the processes of Gaussification by loss and
de-Gaussification by photon-subtraction. The suggested measure is easily
computable for any state of a bosonic system and allows to define a
corresponding measure for the non-Gaussian character of a quantum operation.

A measure of the non-Gaussian character of a quantum state
Marco G. Genoni,1 Matteo G. A. Paris,1, 2, ∗ and Konrad Banaszek3
1Dipartimento di Fisica dell’Università di Milano, I-20133, Milano, Italia.
2Institute for Scientific Interchange, I-10133 Torino, Italia
3Institute of Physics, Nicolaus Copernicus University, PL-87-100 Toruń, Poland
(Dated: November 3, 2018)
We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce
a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a
reference Gaussian state. We analyze in details the properties of the proposed measure and exploit it to evaluate
the non-Gaussianity of some relevant single- and multi-mode quantum states. The evolution of non-Gaussianity
is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by
photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows
to define a corresponding measure for the non-Gaussian character of a quantum operation.
PACS numbers: 03.67.-a, 03.65.Bz, 42.50.Dv
I. INTRODUCTION
Gaussian states play a crucial role in quantum information
processing with continuous variables. This is especially true
for quantum optical implementations since radiation at ther-
mal equilibrium, including the vacuum state, is itself a Gaus-
sian state and most of the Hamiltonians achievable within the
current technology are at most bilinear in the field operators,
i.e. preserve the Gaussian character [1, 2, 3]. As a matter
of fact, using single-mode and entangled Gaussian states, lin-
ear optical circuits and Gaussian operations, like homodyne
detection, several quantum information protocols have been
implemented, including teleportation, dense coding and quan-
tum cloning [4].
On the other hand quantum information protocols required
for long distance communication, as for example entangle-
ment distillation and entanglement swapping, rely on non-
Gaussian operations. In addition, it has been demonstrated
that teleportation [5, 6, 7] and cloning [8] of quantum states
may be improved by using non-Gaussian states and non-
Gaussian operations. Indeed, de-Gaussification protocols for
single-mode and two-mode states have been proposed [5, 6, 7]
and realized [9]. It should be also noticed that any strongly
superadditive function is minimized, at fixed covariance ma-
trix, by Gaussian states. This is crucial to prove extremality
of Gaussian states and Gaussian operations [10, 11] for what
concerns various quantities as channel capacities [12], multi-
partite entanglement measures [13] and distillable secret key
in quantum key distribution protocols. Since in most cases
these quantities can be computed only for Gaussian states, a
non-Gaussianity measure may serve as a guideline to quan-
tify them for the class of non-Gaussian states. Overall, non-
Gaussianity is revealing itself as a resource for continuous
variable quantum information, and thus we urge a measure
able to quantify the non-Gaussian character of a quantum
state.
In this paper we introduce a novel quantity, the non-
∗Electronic address: matteo.paris@fisica.unimi.it
Gaussianity δ[̺] of a quantum state, which quantifies how
much a state fails to be Gaussian. Our measure, which is based
on the Hilbert-Schmidt distance between the state itself and a
reference Gaussian state, can be easily computed for any state,
either single-mode or multi-mode.
The paper is structured as follows. In the next Section we
introduce notation and review the basic properties of Gaussian
states. Then, in Section III we introduce the formal definition
of δ[̺] and study its properties in details. In Section IV we
evaluate non-Gaussianity of relevant quantum states whereas
in Section V we analyze the evolution of non-Gaussianity for
known Gaussification and de-Gaussification maps. Section VI
closes the paper with some concluding remarks.
II. GAUSSIAN STATES
For concreteness, we will use here the quantum optical ter-
minology of modes carrying photons, but our theory applies
to general bosonic systems. Let us consider a system of n
modes described by mode operators ak, k = 1 . . . n, satis-
fying the commutation relations [ak, a
j ] = δkj . A quantum
state ̺ of the n modes is fully described by its characteristic
function [14]
χ[̺](λ) = Tr[̺D(λ)]
where D(λ) =
k=1Dk(λk) is the n-mode displacement
operator, with λ = (λ1, . . . , λn)
T , λk ∈ C, and where
Dk(λk) = exp{λka†k − λ
is the single-mode displacement operator. The canonical op-
erators are given by:
(ak + a
(ak − a†k)
with commutation relations given by [qj , pk] = iδjk. Upon in-
troducing the real vector R = (q1, p1, . . . , qn, pn)
T , the com-
http://arxiv.org/abs/0704.0639v4
mailto:matteo.paris@fisica.unimi.it
mutation relations rewrite as
[Rk, Rj ] = iΩkj
where Ωkj are the elements of the symplectic matrix Ω =
k=1 σ2, σ2 being the y-Pauli matrix. The covariance ma-
trix σ ≡ σ[̺] and the vector of mean values X ≡ X[̺] of a
quantum state ̺ are defined as
Xj = 〈Rj〉
σkj =
〈{Rk, Rj}〉 − 〈Rj〉〈Rk〉
where {A,B} = AB+BA denotes the anti-commutator, and
〈O〉 = Tr[̺ O] is the expectation value of the operator O.
A quantum state ̺G is referred to as a Gaussian state if its
characteristic function has the Gaussian form
χ[̺G](Λ) = exp
σΛ+XTΩΛ
where Λ is the real vector Λ =
(Reλ1, Imλ1, . . . ,Reλn, Imλn)
T . Of course, once the
covariance matrix and the vector of mean values are given, a
Gaussian state is fully determined. For a single-mode system
the most general Gaussian state can be written as
̺G = D(α)S(ζ)ν(nt)S
†(ζ)D†(α),
D(α) being the displacement operator, S(ζ) =
exp[ 1
ζ(a†)2 − 1
ζ∗a2] the squeezing operator, α, ζ ∈ C, and
ν(nt) = (1 + nt)
−1[nt/(1 + nt)]
a†a a thermal state with nt
average number of photons.
III. A MEASURE OF THE NON-GAUSSIAN CHARACTER
OF A QUANTUM STATE
In order to quantify the non-Gaussian character of a quan-
tum state ̺ we use a quantity based on the distance between
̺ and a reference Gaussian state τ , which itself depends on ̺.
Specifically, we define the non-Gaussianity δ[̺] of the state ̺
δ[̺] =
D2HS [̺, τ ]
where DHS [̺, τ ] denotes the Hilbert-Schmidt distance be-
tween ̺ and τ
D2HS [̺, τ ] =
Tr[(̺− τ)2] = µ[̺] + µ[τ ] − 2κ[̺, τ ]
, (3)
with µ[̺] = Tr[̺2] and κ[̺, τ ] = Tr[̺τ ] denoting the purity of
̺ and the overlap between ̺ and τ respectively. The Gaussian
reference τ is the Gaussian state such that
X[̺] = X[τ ]
σ[̺] = σ[τ ]
i.e. τ is the Gaussian state with the same covariance matrix σ
and the same vector X of the state ̺.
The relevant properties of δ[̺], which confirm that it repre-
sents a good measure of the non-Gaussian character of ̺, are
summarized by the following Lemmas:
Lemma 1: δ[̺] = 0 iff ̺ is a Gaussian state.
Proof: If δ[̺] = 0 then ̺ = τ and thus it is a Gaussian state.
If ̺ is a Gaussian state, then it is uniquely identified by its first
and second moments and thus the reference Gaussian state τ
is given by τ = ̺, which, in turn, leads to DHS [̺, τ ] = 0 and
thus to δ[̺] = 0.
Lemma 2: If U is a unitary map corresponding to a symplec-
tic transformation in the phase space, i.e. if U = exp{−iH}
with hermitianH that is at most bilinear in the field operators,
then δ[U̺U †] = δ[̺]. This property ensures that displace-
ment and squeezing operations do not change the Gaussian
character of a quantum state.
Proof: Let us consider ̺′ = U̺U †. Then the covariance ma-
trix transforms as σ[̺′] = Σσ[̺]ΣT , Σ being the symplectic
transformation associated to U . At the same time the vector of
mean values simply translates to X ′ = X +X0, where X0
is the displacement generated by U . Since any Gaussian state
is fully characterized by its first and second moments, then the
reference state must necessarily transform as τ ′ = UτU †, i.e.
with the same unitary transformation U . Since the Hilbert-
Schmidt distance and the purity of a quantum state are invari-
ant under unitary transformations the lemma is proved.
Lemma 3: δ[̺] is proportional to the squared L2(Cn) dis-
tance between the characteristic functions of ̺ and of the ref-
erence Gaussian state τ . In formula:
δ[̺] ∝
d2nλ [χ[̺](λ)− χ[τ ](λ)]2 . (4)
Since the notion of Gaussianity of a quantum state is de-
fined through the shape of its characteristic function, and since
the characteristic function of a quantum state belongs to the
L2(Cn) space [14], we address L2(C) distance to as a good
indicator for the non Gaussian character of ̺.
Proof: Since characteristic functions of self-adjoint operators
are even functions of λ and by means of the identity
Tr[O1O2] =
χ[O1](λ)χ[O2](−λ) ,
we obtain
D2HS [̺, τ ] =
[χ[̺](λ)− χ[τ ](λ)]2 .
Lemma 4: Consider a bipartite state ̺ = ̺A ⊗ ̺G. If ̺G is a
Gaussian state then δ[̺] = δ[̺A].
Proof: we have
µ[̺] = µ[̺A]µ[̺G]
µ[τ ] = µ[τA]µ[τG]
κ[̺, τ ] = κ[̺A, τA]κ[̺G, ̺G] .
Therefore, since κ[̺G, ̺G] = µ[̺G] we arrive at
δ[̺] =
µ[̺A]µ[̺G] + µ[τA]µ[̺G]− 2κ[̺A, τA]κ[̺G, ̺G]
2µ[̺A]µ[̺G]
= δ[̺A] (5)
The four properties illustrated by the above lemmas are the
natural properties required for a good measure of the non-
Gaussian character of a quantum state. Notice that by using
the trace distanceDT [̺, τ ] =
Tr|̺−τ | instead of the Hilbert-
Schmidt distance we would lose Lemmas 3 and 4, and that the
invariance expressed by Lemma 4 holds thanks to the renor-
malization of the Hilbert-Schmidt distance through the purity
µ[̺]. We stress the fact that our measure of non-Gaussianity
is a computable one: It may be evaluated for any quantum
state of n modes by the calculation of the first two moments
of the state, followed by the evaluation of the overlap with the
corresponding Gaussian state.
Notice that δ[̺] is not additive (nor multiplicative) with re-
spect to the tensor product. If we consider a (separable) multi-
partite quantum state in the product form ̺ = ⊗nk=1̺k, the
non-Gaussianity is given by
δ[̺] =
k=1 µ[̺k] +
k=1 µ[τk]− 2
k=1 κ[̺k, τk]
k=1 µ[̺k]
where τk is the Gaussian state with the same moments of ̺k.
In fact, since the state ̺ is factorisable, we have that the cor-
responding Gaussian τ is a factorisable state too.
IV. NON-GAUSSIANITY OF RELEVANT QUANTUM
STATES
Let us now exploit the definition (2) to evaluate the non-
Gaussianity of some relevant quantum states. At first we con-
sider Fock number states |p〉 of a single mode as well as mul-
timode factorisable states |p〉⊗n made of n copies of a num-
ber state. The reference Gaussian states are a thermal state
τp = ν(p) with average photon number p and a factorisable
thermal state τN = [ν(p)]
⊗n with average photon number p
in each mode [15]. Non-Gaussianity may be analytically eval-
uated, leading to
δ[|p〉〈p|] = 1
2p+ 1
δ[(|p〉〈p|)⊗n] = 1
2p+ 1
In the multimode case of |p〉⊗n, we seek for the number of
copies that maximizes the non-Gaussianity. In Fig. 1 we
show both δp ≡ δ[|p〉〈p|] and δ̄p = maxn δ[(|p〉〈p|)⊗n] as a
function of p. As it is apparent from the plot non-Gaussianity
of Fock states |p〉 increases monotonically with the number
of photon p with the limiting value δp = 1/2 obtained for
p → ∞. Upon considering multi-mode copies of Fock states
we obtain larger value of non-Gaussianity: δ̄p is a decreasing
function of p, approaching δ̄ = 1/2 from above. The value
1 5 10 15 20 25 30
FIG. 1: (Top): Non-Gaussianity of single mode Fock states (gray)
|p〉 and of multi-mode Fock states |p〉⊗n (black) as a function of p.
Non-Gaussianity for multi-mode states has been maximized over the
number of copies n. (Bottom): Non-Gaussianity, as a function of the
parameter φ, for the two-mode superpositions |Φ〉〉 (dashed gray),
|Ψ〉〉 (solid gray), and for the single-mode superposition of coherent
states |ψS〉 for α = 0.5 (solid black) and α = 5 (dashed black).
of δ̄p corresponds to n = 3 for p < 26 and to n = 2 for
27 ≤ p . 250.
Another example is the superposition of coherent states
|ψS〉 = N−1/2 (cosφ|α〉 + sinφ| − α〉) (7)
with normalization N = 1 + sin(2φ) exp{−2α2} which for
φ = ±π/4 reduces to the so-called Schrödinger cat states,
and whose reference Gaussian state is a displaced squeezed
thermal state τS = D(C)S(r)ν(N)S
†(r)D†(C), where the
real parameters C, r, and N are analytical functions of φ and
α. Finally we evaluate the non-Gaussianity of the two-mode
Bell-like superpositions of Fock states
|Φ〉〉 = cosφ|0, 0〉+ sinφ|1, 1〉
|Ψ〉〉 = cosφ|0, 1〉+ sinφ|1, 0〉,
which for φ = ±π/4 reduces to the Bell states |Φ±〉
and |Ψ±〉. The corresponding reference Gaussian states
are respectively a two mode squeezed thermal state τΦ =
S2(ξ)[ν(N) ⊗ ν(N)]S†2(ξ), where S2(ξ) = exp(ξa
ξ∗ab) denotes the two-mode squeezing operator, and τΨ =
R(θ)[ν(N1)⊗ν(N2)]R†(θ), namely the correlated two-mode
state obtained by mixing a single-mode thermal state with
the vacuum at a beam splitter of transmissivity cos2 θ, i.e.
R(θ) = exp[iθ(a
1a2+a
2a1)]. All the parameters involved in
these reference Gaussian states are analytical functions of the
superposition parameter φ. Non-Gaussianities are thus evalu-
ated by means of (2) and are reported in Fig. 1 as a function
of the parameter φ. As it is apparent from the plot, the non-
Gaussianity of single-mode states does not surpass the value
δ = 1/2, and this fact is confirmed by other examples not
reported here.
As concern the cat-like states, we notice that for small val-
ues of α the non-Gaussianity of the superposition |ψS〉 shows
a different behavior for positive and negative values of the pa-
rameter φ: for φ > 0 and α = 0.5 we have almost zero δ,
while higher values are achieved for φ < 0. For higher val-
ues of α (α = 5 in Fig. 1), non-Gaussianity becomes an even
function of φ. This different behavior can be understood by
looking at the Wigner functions of even and odd Schrödinger
cat states for different values of α: for small values of α the
even cat’s Wigner function is similar to a Gaussian function,
while the odd cat’s Wigner function shows a non-Gaussian
hole in the origin of the phase space; increasing the value of α
the Wigner functions of the two kind of states become similar
and deviate from a Gaussian function.
We have also done a numerical analysis of non-Gaussianity
of single-mode quantum states represented by finite superpo-
sition of Fock states
n,k=0
̺nk|n〉〈k| . (8)
To this aim we generate randomly quantum states in a finite
dimensional subspaces, dim(H) ≡ d+ 1 ≤ 21, following the
algorithm proposed by Zyczkowski et al [16, 17], i.e. by gen-
erating a random diagonal state (i.e. a point on the simplex)
and a random unitary matrix according to the Haar measure.
In Fig. 2 we report the distribution of non-Gaussianity δ[̺d],
as evaluated for 105 random quantum states, for three different
value of the maximum number of photons d. As it is apparent
from the plots the distribution of δ[̺d] becomes Gaussian-like
for increasing d. In the fourth panel of Fig. 2 we thus re-
port the mean values and variances of the the distributions as
a function of the maximum number of photons d. The mean
value increases with the dimension whereas the variance is a
monotonically decreasing function of d.
Also for finite superpositions simulations we did not ob-
serve non-Gaussianity higher than 1/2. Therefore, although
we have no proof, we conjecture that δ = 1/2 is a limiting
value for the non-Gaussianity of a single-mode state. Higher
values are achievable for two-mode or multi-mode quantum
states (e.g. δ = 2/3 for the Bell states |Ψ±〉〉).
V. GAUSSIFICATION AND DE-GAUSSIFICATION
PROCESSES
We have also studied the evolution of non-Gaussianity
of quantum states undergoing either Gaussification or de-
Gaussification processes. First we have considered the Gaus-
sification of Fock states due do the interaction of the system
FIG. 2: Distribution of non-Gaussianity δ[̺d] as evaluated for 10
random quantum states, for three different value of the maximum
number of photons d. Top: d = 2 (left), d = 10 (right); Bottom:
d = 20 (left). (Bottom-right): Mean values and variances of the non-
Gaussianities evaluated for 105 random quantum states, as a function
of the maximum number of photons d.
with a bath of oscillators at zero temperature. This is per-
haps the simplest example of a Gaussification protocol. In
fact the interaction drives asymptotically any quantum state to
the vacuum state of the harmonic system, which, in turn, is a
Gaussian state. The evolution of the system is governed by the
Lindblad Master equation ˙̺ = γ
L[a]̺, where ˙̺ denotes time
derivative, γ is the damping factor and the Lindblad superop-
erator acts as follows L[a]̺ = 2a†̺a − a†a̺ − ̺a†a. Upon
writing η = e−γt the solution of the Master equation can be
written as
̺(η) =
Vm ̺ V
m (9)
Vm = [(1− η)m/m!]
2 am η
(a†a−m) ,
where ̺ is the initial state. In particular for the system ini-
tially prepared in a Fock state ̺p = |p〉〈p|, we obtain, after
evolution, the mixed state
̺p(η) =
Vm̺pV
αl,p(η)|l〉〈l| (10)
with αl,p(η) =
(1−η)p−lηl. The reference Gaussian state
corresponding to ̺p(η) is a thermal state τp(η) = ν(pη) with
average photon number pη. Non-Gaussianity of ̺p(η) can be
evaluated analytically, we have
δpη ≡ δ[̺p(η)]
2(1− η)2m 2F1
−m,−m, 1; η2
(η−1)2
(1− η)2m 2F1
−m,−m, 1; η
(η − 1)2
+ (1 + 2mη)−1 − 2(1 + (m− 1)η)
(1 +mη)m+1
2F1(a, b, c;x) being a hypergeometric function. We show the
behavior of δpη in Fig. 3 as a function of 1 − η for different
values of p. As it is apparent from the plot δpη is a monotoni-
cally decreasing function of 1 − η as well as a monotonically
increasing function of p. That is, at fixed time t the higher
is the initial photon number p, the larger is the resulting non-
Gaussianity.
0.2 0.4 0.6 0.8 1
1 - Η
0.2 0.4 0.6 0.8 1
FIG. 3: (Left): Non-Gaussianity of Fock states |p〉 undergoing Gaus-
sification by loss mechanism due to the interaction with a bath of os-
cillators at zero temperature. We show δηp as a function of 1 − η
for different values of p: from bottom to top p = 1, 10, 100, 1000.
(Right): Non-Gaussianity of ̺IPS as a function of T for r = 0.5 and
for different values of ǫ = 0.2, 0.4, 0.6, 0.8 (from bottom to top).
δIPS results to be a monotonous increasing function of T , while ǫ
only slightly changes the non-Gaussian character of the state.
Let us now consider the de-Gaussification protocol ob-
tained by the process of photon subtraction. Inconclusive Pho-
ton Subtraction (IPS) has been introduced for single-mode and
two-mode states in [6, 7, 18] and experimentally realized in
[9]. In the IPS protocol an input state ̺(in) is mixed with
the vacuum at a beam splitter (BS) with transmissivity T and
then, on/off photodetection with quantum efficiency ǫ is per-
formed on the reflected beam. The process can be thus charac-
terized by two parameters: the transmissivity T and the detec-
tor efficiency ǫ. Since the detector can only discriminate the
presence from the absence of light, this measurement is in-
conclusive, namely it does not resolve the number of detected
photons. When the detector clicks, an unknown number of
photons is subtracted from the initial state and we obtain the
conditional IPS state ̺IPS . The conditional map induced by
the measurement is non-Gaussian [7], and the output state is
de-Gaussified. Upon applying the IPS protocol to the (Gaus-
sian) single-mode squeezed vacuum S(r)|0〉 (r ∈ R), where
S(r) is the real squeezing operation we obtain [18] the con-
ditional state ̺IPS , whose characteristic function χ[̺IPS ](λ)
is a sum of two Gaussian functions and therefore is no longer
Gaussian. The corresponding Gaussian reference state is a
squeezed thermal state τIPS = S(ξIPS)ν(NIPS)S
†(ξIPS)
where the parameters ξIPS andNIPS are analytic functions of
r, T and ǫ. Non-Gaussianity δIPS = δIPS(T, ǫ, r) has been
evaluated, and in Fig. 3 (right) we report δIPS for r = 0.5
as a function of the transmittivity T for different values of
the quantum efficiency ǫ. As it is apparent from the plot the
IPS protocol indeed de-Gaussifies the input state, i.e. nonzero
values of the non-Gaussianity are obtained. We found that
δIPS is an increasing function of the transmissivity T which
is the relevant parameter, while the quantum efficiency ǫ only
slightly affects the non-Gaussian character of the output state.
The highest value of non-Gaussianity is achieved in the limit
of unit transmissivity and unit quantum efficiency
T,η→1
δIPS = δ[|1〉〈1|] = δ[S(r)|1〉〈1|S†(r)],
where the last equality is derived from Lemma 2. This result
is in agreement with the fact that a squeezed vacuum state
undergoing the IPS protocol is driven towards the target state
S(r)|1〉 in the limit of T, ǫ → 1 [18]. Finally, we notice that
for T, ǫ 6= 1 and for r → ∞ the non-Gaussianity vanishes. In
turn, this corresponds to the fact that one of the coefficients
of the two Gaussians of χ[̺IPS ](λ) vanishes, i.e. the output
state is again a Gaussian one.
VI. CONCLUSION AND OUTLOOKS
Having at disposal a good measure of non-Gaussianity for
quantum state allows us to define a measure of the non-
Gaussian character of a quantum operation. Let us denote
by G the whole set of Gaussian states. A convenient defi-
nition for the non-Gaussianity of a map E reads as follows
δ[E ] = max̺∈G δ[E(̺)], where E(̺) denotes the quantum
state obtained after the evolution imposed by the map. Indeed,
for a Gaussian map Eg , which transforms any input Gaussian
state into a Gaussian state, we have δ[Eg] = 0. Work along
this line is in progress and results will be reported elsewhere.
In conclusion, we have proposed a measure of the non-
Gaussian character of a CV quantum state. We have shown
that our measure satisfies the natural properties expected from
a good measure of non-Gaussianity, and have evaluated the
non-Gaussianity of some relevant states, in particular of states
undergoing Gaussification and de-Gaussification protocols.
Using our measure an analogue non-Gaussianity measure for
quantum operations may be introduced.
Acknowledgments
This work has been supported by MIUR project
PRIN2005024254-002, the EC Integrated Project QAP (Con-
tract No. 015848) and Polish MNiSW grant 1 P03B 011 29.
APPENDIX A: GAUSSIAN REFERENCE WITH
UNCONSTRAINED MEAN VALUE
As we have seen from the above examples δ[̺] of Eq. (2)
represents a good measure of the non-Gaussian character of a
quantum state. A question arises on whether different choices
for the reference Gaussian state τ may lead to alternative,
valid, definitions. As for example (for single-mode states) we
may define
δ′[̺] = min
D2HS [̺, τ ]/µ[̺], (A1)
where τ = D(C)S(ξ)ν(N)S†(ξ)D†(C) is a Gaussian state
with the same covariance matrix of ̺ and unconstrained vec-
tor of mean values X = (ReC, ImC) used to minimize the
Hilbert-Schmidt distance. Here we report few examples of
the comparison between the results already obtained using (2)
with that coming from (A1). As we will see either the two
definitions coincide or δ′ and δ are monotone functions of
each other. Since the definition (2) corresponds to an easily
computable measure we conclude that it represents the most
convenient choice.
Let us first consider the Fock state ̺ = |p〉〈p|. According
to (A1), the reference Gaussian state is given by a displaced
thermal states τ ′ = D(C)νpD
†(C). The overlap between ̺
and τ ′ is given by
κ[|p〉〈p|, τ ′] = 1
1 + p
1 + p
1 + p
p(1 + p)
The maximum of (A2) is achieved forC = 0, which coincides
with the assumptions C = Tr[a|p〉〈p|].
Let us consider the quantum state (10) obtained as the so-
lution of the loss Master Equation for an initial Fock state
|p〉〈p|. The unconstrained Gaussian reference is again a dis-
placed thermal state τ ′ = D(C)νpηD
†(C), and the overlap is
given by
κ[̺p(η), τ
′] = Tr[τ̺p(η)] =
(1 + η(p− 1))p
(1 + pη)p+1
η|C|2
(1 + pη)(η(1 − p)− 1)
Again, since the overlap is maximum for C = Tr[a̺p(η)] =
0, both definitions give the same results for the non-
Gaussianity.
Let us now consider the Schrödinger cat-like states of (7).
The reference Gaussian state is a displaced squeezed thermal
state, with squeezing and thermal photons as calculated be-
fore. The optimization over the free parameterC may be done
numerically. In Fig. 4 we show the non-Gaussianitiy, both as
resulting from (A1) and by choosing C = Tr[a̺S ] as in (2),
as a function of ǫ. The two curves are almost the same, with
no qualitative differences.
[1] A. Ferraro, S. Olivares and M. G. A. Paris, Gaussian States in
Quantum Information, (Bibliopolis, Napoli, 2005)
[2] J. Eisert, M. B. Plenio, Int. J. Quant. Inf. 1, 479 (2003)
[3] F. Dell’Anno et al., Phys. Rep. 428, 53 (2006).
[4] S. L. Braunstein, P. van Loock, Rev. Mod. Phys 77, 513 (2005).
[5] T. Opatrny et al., Phys.Rev. A 61, 032302 (2000).
[6] P. T. Cochrane et al., Phys Rev. A 65, 062306 (2002).
[7] S. Olivares et al., Phys. Rev. A 67, 032314 (2003); S. Olivares,
M. G. A. Paris, Las. Phys. 16, 1533 (2006).
[8] N. J. Cerf et al., Phys. Rev. Lett. 95, 070501 (2005).
[9] J. Wenger et al., Phys. Rev. Lett. 92, 153601 (2004); A. Our-
joumtsev et al., Science 312, 83 (2006).
[10] M. M. Wolf et al., Phys. Rev. Lett. 96, 080502 (2006).
[11] M. M. Wolf et al, Phys. Rev. Lett. 98, 130501 (2007). 0
[12] A. S. Holevo, R. F. Werner, Phys. Rev. A 63, 032312 (2001).
[13] L. M. Duan at al, Phys. Rev. Lett. 84, 4002 (2000); R. F. Werner,
M. M. Wolf, Phys. Rev. Lett. 86, 3658 (2001).
[14] K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969).
[15] P. Marian, T. Marian, Phys. Rev. A 47, 4474 (1993).
[16] K. Zyczkowski and M. Kus, J. Phys. A 27, 4235 (1994).
[17] K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein,
Phys. Rev. A 58, 883 (1994).
[18] S. Olivares and M. G. A. Paris, J. Opt. B, 7, S392 (2005).
FIG. 4: Non-Gaussianity of a Schrödinger cat-like state as a func-
tion of the superposition parameter φ, with either C obtained by nu-
merical minimization (solid) or with C = Tr[a̺] (dotted). (Left):
α = 0.5; (Right): α = 5.
