Bound entangled states fit for robust experimental
verification
Gael Sent
´
ıs
1,2
, Johannes N. Greiner
3
, Jiangwei Shang
4,1
, Jens Siewert
5,6
, and Matthias Kleinmann
1,2
1
Naturwissenschaftlich-Technische Fakult
¨
at, Universit
¨
at Siegen, 57068 Siegen, Germany
2
Departamento de F
´
ısica Te
´
orica e Historia de la Ciencia, Universidad del Pa
´
ıs Vasco UPV/EHU, E-48080 Bilbao, Spain
3
3rd Institute of Physics, University of Stuttgart and Institute for Quantum Science and Technology, IQST, Pfaffenwaldring
57, D-70569 Stuttgart, Germany
4
Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, Beijing 100081, China
5
Departamento de Qu
´
ımica F
´
ısica, Universidad del Pa
´
ıs Vasco UPV/EHU, E-48080 Bilbao, Spain
6
IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain
December 17, 2018
Preparing and certifying bound entan-
gled states in the laboratory is an intrinsi-
cally hard task, due to both the fact that
they typically form narrow regions in state
space, and that a certificate requires a to-
mographic reconstruction of the density
matrix. Indeed, the previous experiments
that have reported the preparation of a
bound entangled state relied on such tomo-
graphic reconstruction techniques. How-
ever, the reliability of these results cru-
cially depends on the extra assumption of
an unbiased reconstruction. We propose
an alternative method for certifying the
bound entangled character of a quantum
state that leads to a rigorous claim within
a desired statistical significance, while by-
passing a full reconstruction of the state.
The method is comprised by a search for
bound entangled states that are robust for
experimental verification, and a hypothe-
sis test tailored for the detection of bound
entanglement that is naturally equipped
with a measure of statistical significance.
We apply our method to families of states
of 3 ×3 and 4 ×4 systems, and find that the
experimental certification of bound entan-
gled states is well within reach.
Gael Sent
´
ıs: gael.sentis@uni-siegen.de
Jiangwei Shang: jiangwei.shang@bit.edu.cn
Jens Siewert: jens.siewert@ehu.eus
Matthias Kleinmann: matthias.kleinmann@uni-siegen.de
1 Introduction
To experimentally prepare, characterize and con-
trol entangled quantum states is an essential item
in the development of quantum-enhanced tech-
nologies, but it also serves the indispensable pur-
pose of testing the predictions of entanglement
theory in the laboratory. Among the most in-
triguing features of this theory stands the exis-
tence of bound entanglement [1], a form of en-
tanglement that cannot be distilled into singlet
states by any protocol that uses only local oper-
ations and classical communication. Originally
considered as useless for quantum information
processing, bound entangled states were later
established as a valid resource in the contexts
of quantum key distribution [2], entanglement
activation [3, 4], metrology [5, 6], steering [7],
and nonlocality [8], and their nondistillability has
been linked to irreversibility in thermodynam-
ics [9, 10].
Complementing these theoretical achieve-
ments, substantial efforts have been devoted to
experimentally producing and verifying bound
entanglement. The first experimental report
on the preparation of a bound entangled state
was presented in [11], although the result was
disputed [12] and subsequently amended [13].
The state prepared was the four-qubit Smolin
state [14], thus it showcases a multipartite in-
stance of bound entanglement, which is fun-
damentally distinct from the bipartite case:
when multiple parties are present, entanglement
can still potentially be distilled if two of the
Accepted in Quantum 2018-12-13, click title to verify 1
arXiv:1804.07562v3 [quant-ph] 14 Dec 2018
parties work together. Further experimental
works on multipartite bound entanglement in-
clude Refs. [15–20]. Examples of experimental
bipartite setups are more scarce. In Ref. [21] bi-
partite bound entanglement was produced using
four-mode continuous-variable Gaussian states,
and Ref. [22] focuses on a family of two-qutrit
states.
Since the property of nondistillability is exper-
imentally inaccessible in a direct manner, a nat-
ural route to verify a state as bound entangled
is to do a full tomographic reconstruction of the
density matrix from the experimental data [23]
and apply the only known computable criterion
on it [1]: if an entangled state has positive par-
tial transpose (PPT), then it is nondistillable
and therefore bound entangled
1
. However, it has
recently been pointed out that widely used re-
construction methods like maximum likelihood
and least squares [24, 25] inevitably suffer from
bias [26, 27], caused by imposing a positivity con-
straint over compatible density matrices. In some
cases, the bias can be large enough to drastically
change the entanglement properties of the esti-
mator with respect to the true state. In addition
to this state of affairs, the variance of the estima-
tor is usually calculated by bootstrapping [28],
which only accounts for statistical fluctuations,
and can result in a smaller variance than the ac-
tual bias of the estimator [27]. In contrast, a di-
rect reconstruction of the state by linear inversion
produces an unbiased estimator, but at the cost
of admitting unphysical density matrices. Then,
the PPT criterion simply looses all meaning.
All the experimental works cited above sup-
port their claims on some combination of max-
imum likelihood or least squares reconstruction
and bootstrapping. There exist more informative
methods to derive errors from tomographic data,
such as credibility [29] and confidence [30, 31] re-
gions, and also the alternative of using linear in-
version in addition to a sufficiently large number
of measurements that guarantees physical esti-
mates [32]. Should these methods be applied to
the detection of a bound entangled state, more
robust results may be generated, although they
might come at the expense of being computation-
ally expensive or even intractable [33]. However,
1
It is still an open question whether the PPT criterion
completely characterizes bound entanglement, namely
whether all nondistillable states are PPT.
regardless of the reconstruction method of choice,
the problem of experimentally testing bound en-
tanglement is intrinsically challenging. This is
so because bound entangled regions of the state
space are typically very small in volume [34]. Fur-
thermore, at least for the known cases in low-
dimensional systems, bound entangled states are
close to both the sets of separable states and
distillable entangled states. This translates into
the requirement of a highly precise experimental
setup, and the deepening of the potential pitfalls
of biased tomographic reconstructions.
In this paper, we set ourselves to improve on
the above situation by devising methods that en-
able robust experimental certification of bound
entanglement. Instead of advocating for a par-
ticular tomographic method for detecting bound
entanglement or considering the preparation of a
specific state, we address the more generic ques-
tion: Which are the best candidate states for
an experimental verification of bound entangle-
ment? In other words, for bipartite systems, we
aim at finding states that have the largest ball
of bound entangled states around them [35]. To
this end, we construct simple lower bounds that
the radius r of such a ball (in Hilbert–Schmidt
distance) has to obey for a given state, and for-
mulate an optimization problem that maximizes
r over parametrized families of states. Having a
value for the maximum radius, r
∗
, allows us to as-
sert the robustness of the target state at the cen-
ter of the ball, and in turn gives us an idea of the
required number of preparations of the state in a
potential experiment. We proceed by designing
a χ
2
hypothesis test directly applicable over un-
processed tomographic data that provides a cer-
tificate for bound entanglement within some sta-
tistical significance. We show that our proposed
method, combining the search of a robust can-
didate state with a statistical analysis through a
hypothesis test, makes rigorous bound entangle-
ment verification not only experimentally feasible
with current technology, but also computation-
ally cheap.
The paper is structured as follows. First we
derive the constraints for the existence of a ball
of bound entangled states of radius r around a
generic bipartite state of dimension d. Then, we
apply our method to two families of states of di-
mensions 3×3 and 4×4, known to contain bound
entanglement, and find robust candidates for its
Accepted in Quantum 2018-12-13, click title to verify 2
experimental verification. We proceed to devise a
hypothesis test for bound entanglement, and test
the robustness of the selected candidate states
in terms of the necessary number of samples to
achieve a statistically significant certification un-
der realistic experimental conditions.
2 A bound entangled ball
Verifying the bound entangled character of a bi-
partite state ρ requires, on the one hand, show-
ing that it is entangled, and on the other hand
proving that the entanglement of ρ is nondis-
tillable. Non-distillability is usually verified via
the (sufficient) PPT condition, which we denote
by Γ(ρ) ≥ 0. Throughout this paper, we iden-
tify bound entangled states with PPT entangled
states. As for verifying that ρ is entangled, there
exist several inequivalent criteria. We choose the
violation of the computable cross norm or realign-
ment criterion (CCNR) [36], since it is simple
and is generally tight enough to detect bound en-
tanglement. The CCNR criterion dictates that,
for some local orthonormal basis [with respect
to the Hilbert–Schmidt inner product (A, B) =
tr(A
†
B)] of the Hermitian matrices {g
k
}, ρ is en-
tangled if the matrix R(ρ)
k,l
= tr(ρ g
k
⊗g
l
) obeys
kR(ρ)k
1
> 1, where kY k
1
= tr
√
Y
†
Y is the trace
norm of Y . The points in the state space that vio-
late CCNR, fulfill PPT, and correspond to physi-
cal states (that is, satisfy the positivity condition
ρ ≥ 0), thus define a volume of bound entangled
density matrices.
Given a bound entangled state ρ that obeys
these conditions, we inquire how far we can move
away from it while remaining in the bound en-
tangled region. We construct a new state τ =
ρ + rX, where r ≥ 0, and X is a traceless Hermi-
tian matrix with bounded Hilbert–Schmidt norm
kXk
2
=
√
tr X
†
X ≤ 1. The set of all such matri-
ces forms a Hilbert–Schmidt ball that we denote
by B
0
. We then define the set of all states τ as
B(ρ, r) := ρ + rB
0
. Note that kρ −τ k
2
≤ r. We
can bound the minimum eigenvalue of τ as
λ
min
(τ) ≥ λ
min
(ρ) − rkXk
∞
≥ λ
min
(ρ) − r
q
(d − 1)/d ,
(1)
where kXk
∞
= λ
max
(X) is the uniform norm
of X. The first inequality holds since, in the
extreme case, the eigenvector associated to the
minimal eigenvalue of X is aligned with the cor-
responding one for ρ. For the second inequality
we have used that
max {kXk
∞
| X ∈ B
0
} =
q
(d − 1)/d (2)
(we provide a proof of this equation in Ap-
pendix A).
Similarly, since the partial transpose Γ(τ) does
not change the Hilbert–Schmidt ball, Γ(B) = B,
we have
λ
min
[Γ(τ)] ≥ λ
min
[Γ(ρ)] − r
q
(d − 1)/d . (3)
We can bound the value of the CCNR criterion
over τ in a similar fashion. We have
kR(τ)k
1
≥ kR(ρ)k
1
− rkR(X)k
1
≥ kR(ρ)k
1
− r
√
d ,
(4)
where we first applied the triangle inequality, and
for the last inequality we used that
max {kR(X)k
1
| X ∈ B
0
} =
√
d (5)
(we refer to Appendix A for a proof).
Our goal is to find a state ρ such that, if we de-
part from it by a distance r in any direction, the
resulting τ still fulfils PPT and violates CCNR,
that is, λ
min
[Γ(τ)] ≥ 0 and kR(τ)k
1
> 1. Then,
using Eqs. (3) and (4), we search for a state ρ
which admits the largest r under the constraints
λ
min
[Γ(ρ)] ≥ r
q
(d − 1)/d , (6a)
kR(ρ)k
1
> 1 + r
√
d . (6b)
For any admissible r, all states in B are bound
entangled. Note that, while the optimization is
naturally performed over physical target states ρ,
the resulting ball B can well be partly outside of
the state space, cf. Fig. 1, as the positivity of all
states inside B is not imposed as a constraint.
3 Symmetric families
The method described in the previous section is
completely general, but for the optimization to
actually become feasible one has to restrict the
free parameters in ρ. We consider two symmetric
families of bipartite states which are character-
ized by few parameters, and nevertheless contain
fairly large regions of bound entanglement.
Accepted in Quantum 2018-12-13, click title to verify 3
C
C
N
R
P
P
T
p
h
y
s
i
c
a
l
s
t
a
t
e
s
BE
Figure 1: Schematic picture of the state space, where the
boundaries of the PPT, CCNR and physical sets enclose
a bound entangled region (BE). A region of points τ =
ρ + rX, where X is a traceless Hermitian matrix with
kXk
2
≤ 1, is depicted in violet. The parameter r is the
radius of a Hilbert–Schmidt ball around a target state ρ
such that all physical states τ are bound entangled.
The first case that we consider is a family of
two-qutrit states of the form
ρ = a |φ
3
ihφ
3
| + b
2
X
k=0
|k, k ⊕ 1ihk, k ⊕1|
+ c
2
X
k=0
|k, k ⊕ 2ihk, k ⊕2| , (7)
where |φ
3
i =
P
2
k=0
|kki/
√
3, the symbol ⊕ de-
notes addition modulo 3, and a, b, c are real pa-
rameters. A similar three-parameter family of
two qutrits, known to contain bound entangle-
ment and arise from symmetry conditions, was
analyzed in Refs. [37–42], and considered in the
experiment reported in Ref. [22]. We search for
the optimal values of {a, b, c} that parametrize
an optimal target state ρ
∗
. This state will ad-
mit any r ≤ r
∗
, where r
∗
is the maximum radius
compatible with the constraints (6a) and (6b).
An exact solution for this optimization can be
found (see Appendix B.1 for details). We obtain
the optimal parameters
a ≈ 0.21289 , b ≈ 0.04834 , and c ≈ 0.21403 , (8)
yielding a maximal radius r
∗
≈ 0.02345. The
resulting ρ
∗
is a rank-7 state.
Since the ball B may contain unphysical states,
it is in principle possible that our estimate for the
maximal r
∗
is not tight. To explore this, we move
away from ρ
∗
by a distance r
∗
in suitable direc-
tions and evaluate the position of the resulting
state with respect to the boundaries of the PPT
and the CCNR sets. We provide the details of
this analysis in Appendix B.1. As a result, we
obtain that our estimate of r
∗
is indeed tight
with respect to the PPT boundary, but we ob-
serve that it slightly underestimates the distance
with respect to the CCNR boundary.
As our second case, we consider the two-
ququart states that are Bloch-diagonal, i.e., of
the form
ρ =
X
k
x
k
g
k
⊗ g
k
, (9)
where g
k
= (σ
µ
⊗ σ
ν
)/2, µ, ν = 0, 1, 2, 3, with
σ
0
= , σ
1
, σ
2
, σ
3
the Pauli matrices, and the
index k enumerates pairs of indices {µ, ν} in
lexicographic order. Bound entangled Bloch-
diagonal states have already been described in
Ref. [43]. The optimization over this family of
states, despite having more free parameters, is
much simpler than in the two-qutrit case dis-
cussed above. The reason is that the problem can
be reshaped as a linear program over the coeffi-
cients x
k
, and thus it can also be solved exactly
(see Appendix B.2 for details). A vertex enumer-
ation of the corresponding feasibility polytope is
possible, and leads us to a set of 4224 optimal
states achieving a maximal radius r
∗
≈ 0.0214.
One example of such optimal state, ρ
∗
, is given
by coefficients
x
1
=
1
4
,
x
α
≈ −0.0557066 , α ∈ {2, 3, 4, 5, 6, 9, 12, 14, 16},
x
β
≈ 0.0142664 , β ∈ {7, 11, 13},
x
γ
≈ 0.0971467 , γ ∈ {8, 10, 15}.
(10)
where x
1
is fixed by normalization. The state ρ
∗
has rank 10, which is the minimal rank among
bound entangled states achieving r
∗
that are of
the form (9) and are detectable by CCNR. As a
byproduct of our analysis, we also obtain that the
overall minimal rank for such bound entangled
states is 9, albeit with a fairly smaller radius.
4 Statistical analysis
In this section we put our method for finding op-
timal target states to work in a practical scenario.
That is, for an experiment aiming at the certifi-
cation of bound entanglement, we design a statis-
Accepted in Quantum 2018-12-13, click title to verify 4
tical analysis of the experimental data that cru-
cially hinges on knowing the radius of the bound
entangled ball around the target state. The idea
is to judge whether the data was obtained from a
bound entangled state by considering the mem-
bership of the preparation to the bound entan-
gled ball. In order to endow the certification with
a measure of statistical significance, we design a
hypothesis test for this membership problem.
Our null hypothesis, H
0
, is that the prepared
state is outside the bound entangled ball B(ρ
0
, r
0
)
of radius r
0
centered at the target state ρ
0
. We
make the assumption that an instance of experi-
mental data, ~x, is drawn from a multivariate nor-
mal distribution N(
~
ξ, Σ) with offset
~
ξ and covari-
ance matrix Σ. This is a good approximation for
realistic scenarios. Here the vector notation is
used over variables that belong to the same space
as the experimental data, that is, e.g., the space
of frequencies of measurement outcomes. When
H
0
holds true, then the offset
~
ξ is the expected
value of the data when a state ρ
exp
is prepared
such that kρ
0
− ρ
exp
k
2
≥ r
0
. The covariance ma-
trix Σ is determined by the particular experimen-
tal procedure used.
The goal is to design an appropriate hypothe-
sis test for H
1
of the form
ˆ
t(~x) ≤ t, where t is a
threshold parameter, and the function
ˆ
t is called
a test statistic. If the hypothesis test is true, the
data ~x is accepted as fulfilling the hypothesis H
1
,
that is, the state is bound entangled. The quan-
tity through which we assess the significance of
the hypothesis test is the worst-case probability
of the test accepting H
1
while H
0
is true. This is
formally written as
p(t, r
0
) = sup
ρ
n
P[
ˆ
t ≤ t] | kρ
0
− ρk
2
≥ r
0
o
, (11)
where P[
ˆ
t ≤ t] is the probability for the hypoth-
esis test to accept data sampled from N(
~
ξ, Σ),
~
ξ
depends to ρ, and kρ
0
− ρk
2
≥ r
0
is the assertion
that H
0
holds true. For given data ~x, the proba-
bility p[
ˆ
t(~x), r
0
] is the p-value of this hypothesis
test.
Now, let us define
ˆ
t: ~x 7→ kΣ
−1/2
[T (ρ
0
) −~x]k
2
, (12)
where T is a map that takes a density matrix to
the expected data (e.g., to a vector of probabil-
ities), thus it is determined by the experimental
procedure. Hence,
ˆ
t(~x) gives some notion of dis-
tance between the standardized versions of the
experimental data and the expected data of a
perfect measurement performed over the target
state. In Appendix C we show that, if
ˆ
t is of the
form in Eq. (12), the probability (11) is naturally
upper-bounded by
p(t, r
0
) ≤ q
m
(t
2
, r
2
1
) , (13)
where s 7→ q
m
(s, u) is the cumulative distribution
function of the noncentral χ
2
-distribution with m
degrees of freedom and noncentrality parameter
u, and r
1
is the equivalent distance in the exper-
imental data space of the Hilbert–Schmidt dis-
tance r
0
, or, more precisely, r
1
:= inf
∆6∈B
ˆ
t[T (∆)],
where ∆ is any Hermitian matrix with unit trace.
Naturally, the value of r
1
scales with r
0
and
strongly depends on the experimental procedure,
that is, on T and Σ.
In the following, we show how to evaluate
the hypothesis test for the two-qutrit state from
Eq. (8) as an example of a target state ρ
0
, and as-
sess the necessary experimental requirements for
a desired level of significance. For this, we need to
make some assumptions about the experimental
procedure. We associate the measurement out-
comes in the experiment to semidefinite-positive
Hermitian operators E
k
. We consider a com-
plete set of such operators, that is, the real linear
span of {E
k
} is the set of all Hermitian opera-
tors. The probability of obtaining the outcome
corresponding to E
k
when measuring the state ρ
is given by p
k
= tr(E
k
ρ), and we assume that
T (ρ)
k
≡ p
k
. The connection between the prob-
abilities p
k
and the data gathered in the exper-
iment crucially depends on how the experiment
is performed. A straightforward theoretical as-
sociation can be established if one assumes that
measurement outcomes correspond to indepen-
dent Poissonian trials with parameters nT (ρ)
k
,
renormalized by n, where n is the mean total
number of events per measurement setting. That
is, if we obtain n
k
events for the outcome k, we
use as data x
k
= n
k
/n. Note, however, that in
a realistic experiment T (ρ) and n are not known
exactly. A reasonable alternative is to use the
total number of observed events
P
k
n
k
as an es-
timate of n, but then the connection between p
k
and x
k
is more involved. For our examples below
we use this latter approach (a detailed discussion
is presented in Appendix C).
In order to get specific predictions, we assume
that mutually unbiased bases are used as local
Accepted in Quantum 2018-12-13, click title to verify 5
measurements (refer to Appendix C for an ex-
plicit construction) and that the experimental
state ρ
exp
has 5% white noise over the target,
that is, ρ
exp
= 0.95ρ
0
+ 0.05 /9. This amount
of noise still results in a state within B, since
kρ
0
− ρ
exp
k
2
≈ 0.6r
0
, where r
0
≈ 0.02345 is
the optimal radius associated to ρ
0
(cf. Sec-
tion 3). Then, one obtains r
2
1
≈ 0.0664
2
n (see
Appendix C for details on how to compute this
value). We now set a critical t
0
below which
the p-value will be larger than some acceptable
threshold p
0
, so that q
m
(t
2
0
, r
2
1
) = p
0
. To obtain
t
0
, one inverts the equation q
m
(t
2
, r
2
1
) = p to get
t
2
(p), called the quantile function. Then, t
0
:=
t(p
0
). Once t
0
is fixed, we compute the prob-
ability p
fail
that the test fails to certify bound
entanglement over data obtained by measuring
the prepared state ρ
exp
, i.e., that
ˆ
t(
~
ξ) > t
0
given
~
ξ = T (ρ
exp
). We can write this probability as
p
fail
= 1 − q
m
t
2
0
, r
2
2
, (14)
where r
2
2
:=
ˆ
t(
~
ξ)
2
≈ 0.0416
2
n. Note that, while
r
1
is used to determine t
0
considering a worst case
scenario for a false positive, r
2
is a distance be-
tween standardized probabilities given the partic-
ular preparation ρ
exp
. Therefore, r
1
> r
2
means
that the test
ˆ
t has a chance to single out the
particular experimental preparation as bound en-
tangled from the worst-case state outside B, and
hence p
fail
will decrease with the number of sam-
ples n, which is the case of interest. In Fig. 2 we
plot p
fail
as a function of n, for various levels of
significance p
0
expressed as multiples of the stan-
dard deviation kσ, so that p
0
= 1 − erf(k/
√
2).
Similarly, we carry out the same analysis for
the two-ququart target state specified in Eq. (10).
In this case we construct our measurement set-
tings by regarding each ququart as two qubits
and performing local Pauli measurements on
them. With r
2
1
≈ 0.0856
2
n, r
2
2
≈ 0.0469
2
n, and
an admixture of 2.5% white noise over the tar-
get ρ
0
, we obtain the results shown in Fig. 2. In
both cases, for the selected target states of two
qutrits and two ququarts given by Eqs. (8) and
(10), our analysis shows that their experimental
certification as bound entangled states under re-
alistic assumptions is within reach, with around
70000 samples per measurement setting to reach
a 3σ level of significance. Note that this num-
ber of samples justifies a posteriori our analy-
sis assuming that experimental data ~x is normal-
distributed.
5 Discussion
We have developed a comprehensive method for
the experimental characterization of bipartite
bound entanglement, from the selection of ro-
bust target states to the statistical analysis of the
data. Previous experimental works were based
on preparing a bound entangled state and infer-
ring its properties from those of the reconstructed
density matrix via maximum likelihood or least
squares. Unfortunately, such techniques have
been shown to produce unreliable results, par-
ticularly for entanglement certification [26, 27],
which casts a shadow of doubt over past exper-
imental demonstrations of bound entanglement.
Instead of using a (necessarily) biased reconstruc-
tion of the density matrix and assuming that it
shares the same properties that the true prepared
state, we show that it is feasible to perform a hy-
pothesis test over the unprocessed experimental
tomographic data to test for membership of the
preparation to a subset of the state space that
is guaranteed to only contain bound entangled
states. The hypothesis test is naturally equipped
with a measure of statistical significance, which is
easy to compute. The subset of bound entangled
states is specified as a ball of radius r, and the
design of the hypothesis test directly depends on
this parameter. We have shown, through explicit
examples of families of bipartite qutrit states and
bipartite ququart states, how the certification of
bound entanglement with high statistical signif-
icance is well within current experimental capa-
bilities by using our method.
While our statistical analysis of the data in the
form of a hypothesis test is a standalone tech-
nique applicable to the preparation and measure-
ment of any state, we would like to stress that
the value of the radius r has an important ef-
fect in its detection power, hence it is worth aim-
ing at target states with maximal r. To show
this, we run our analysis for the two-qutrit state
prepared in Ref. [22], assuming a noiseless ex-
perimental preparation. This state has a ball
of bound entangled states around it of radius
r ≈ 0.01182. This value is computed as the maxi-
mal r that satisfies Eqs. (6a) and (6b). We obtain
that the required number of samples for achiev-
Accepted in Quantum 2018-12-13, click title to verify 6
1
10
–1
10
–2
10
–3
10
–4
10
–5
10
–6
10
–7
10 20 30 40 50 60 70 80 90 100
significance 2σ
significance 3σ
s
i
g
n
i
fi
c
a
n
c
e
4
σ
s
i
g
n
i
fi
c
a
n
c
e
5
σ
significance 1σ
samples per setting n (in thousands)
p
fail
(a)
1
10
–1
10
–2
10
–3
10
–4
10
–5
10
–6
10
–7
10 20 30 40 50 60 70 80 90 100
significance 2σ
significance 3σ
significance 4σ
s
i
g
n
i
fi
c
a
n
c
e
5
σ
significance 1σ
samples per setting n (in thousands)
p
fail
(b)
Figure 2: Probability p
fail
to obtain data which does not confirm bound entanglement at a level of significance
corresponding to at least kσ standard deviations. The experiments consist of locally measuring (a) mutually unbiased
bases over preparations of the two-qutrit state in Eq. (8) with 5% white noise, and (b) local Pauli measurements
over the two-ququart state in Eq. (10) with 2.5% white noise.
ing a 3σ level of significance with a failure prob-
ability p
fail
≈ 10
−3
is roughly twice as much as
for the two-qutrit state in Eq. (8) (which admits
a radius r ≈ 0.02345) when assuming a noisy
preparation. As a comparison, optimizing over
the one-parameter Horodecki family [1]—which
is part of the family in Eq. (7)—yields a radius
r ≈ 0.01681, for the parameters a ≈ 0.28571,
b ≈ 0.07931, and c ≈ 0.15879.
As a concluding remark, some comments on
the generality of our result are in order. If one
has prior knowledge of the expected amount and
type of experimental noise, this could be incor-
porated into the statistical analysis by assuming
the noisy state as the target. As a result, one
should expect a smaller value of the test statistic
ˆ
t(~x) for a given set of data ~x and hence a smaller
p-value. However, our calculations for several
examples indicate that this advantage does not
compensate in general the drawback of having
a smaller bound entangled ball. A refinement of
our method could also be achieved by taking into
account the form of the covariance matrix Σ into
the optimization over target states, as we did in-
corporate it into the design of the hypothesis test.
This would generally yield an ellipsoid around
the target, instead of a ball, potentially captur-
ing a larger volume of bound entangled states and
thus leading to a stronger test. The possible dis-
advantage is that the bounds (6a) and (6b) will
likely be much more complicated (if computable
at all), hence the optimization step will be much
harder to carry out. We leave the question open
of whether such refinement provides a significant
reduction in the experimental requirements.
Acknowledgments
We acknowledge Otfried G¨uhne, G´eza T´oth,
Beatrix Hiesmayr, Wolfgang L¨offler, and Dag-
mar Bruß for useful discussions. This project
was funded by the ERC Starting Grant No.
258647/GEDENTQOPT, the Basque Govern-
ment grant No. IT986-16, the Spanish
MINECO/FEDER/UE grant FIS2015-67161-P,
the UPV/EHU program UFI 11/55, the ERC
Consolidator Grant 683107/TempoQ, the DFG
(No. KL 2726/2-1), and the Beijing Institute of
Technology Research Fund Program for Young
Scholars.
Accepted in Quantum 2018-12-13, click title to verify 7
A Proofs of Eqs. (2) and (5)
Given a target state ρ, we construct the displaced state τ = ρ + rX, where r ≥ 0, and X is a bounded
traceless Hermitian matrix fulfilling kXk
2
≤ 1. We claim that the minimal eigenvalue of τ can be
lower bounded as [cf. Eq. (1)]
λ
min
(τ) ≥ λ
min
(ρ) − rkXk
∞
≥ λ
min
(ρ) − r
q
(d − 1)/d , (15)
where for the second inequality we have used that [cf. Eq. (2)]
max {kXk
∞
| X ∈ B
0
} =
q
(d − 1)/d , (16)
and B
0
= {X|kXk
2
≤ 1, tr X = 0, X = X
†
}. This holds from the following reasoning. Clearly, the
maximum is attained for X = diag(x, y
1
, . . . , y
d−1
) with some vector ~y fulfilling k~yk
∞
≤ x = −~y · ~e
and x
2
+ ~y
2
≤ 1. Here, e
k
= 1 for k = 1, . . . , d − 1. Note that the choice of ~y that allows x to be
largest is the uniform vector ~y = −x~e/~e
2
≡ ~z, since it minimizes ~y
2
. The maximization now reduces
to
max {x | x
2
+ x
2
/~e
2
≤ 1 } ,
which immediately yields the assertion due to ~e
2
= d − 1.
In a similar fashion, we then obtain a lower bound of the 1-norm of the realigned state τ as [cf.
Eq. (4)]
kR(τ)k
1
≥ kR(ρ)k
1
− rkR(X)k
1
≥ kR(ρ)k
1
− r
√
d , (17)
where we use that [cf. Eq. (5)]
max {kR(X)k
1
| X ∈ B
0
} =
√
d . (18)
This immediately follows from the facts that the 1-norm is bounded by the 2-norm as kAk
1
≤
p
rank(A)kAk
2
, via e.g. Cauchy-Schwarz inequality, and that kR(X)k
2
= kXk
2
≤ 1. Hence
√
d
is an upper bound. It remains to show that this bound is attainable. For arbitrary d, we see that
tr X = 0 ⇔ [R(X)]
1,1
= 0 (choosing the local basis such that g
1
= ). Then, R(X) is a constant
anti-diagonal matrix.
B Optimally detectable bound entangled states
B.1 A state of two-qutrits
The first case that we consider is a family of two-qutrit states of the form
ρ = a |φ
3
ihφ
3
| + b
2
X
k=0
|k, k ⊕ 1ihk, k ⊕1|
+ c
2
X
k=0
|k, k ⊕ 2ihk, k ⊕2| , (19)
where |φ
3
i =
P
2
k=0
|kki/
√
3, the symbol ⊕ denotes addition modulo 3, and a, b, c are real and non-
negative. We denote the dimension of the total Hilbert space by d = 9. There are three distinct
eigenvalues for ρ,
λ(ρ) = {a, b, c}, (20)
which satisfy the constraint
a + 3(b + c) = 1 . (21)
Accepted in Quantum 2018-12-13, click title to verify 8
Figure 3: The density matrix of our optimal two-qutrit state ρ
∗
in the computational basis. The heights of the bars
correspond to the entries of the density matrix. Note that all entries of ρ
∗
are real-valued.
Then, the three distinct eigenvalues of its partial transpose Γ(ρ) are given by
λ(Γ(ρ)) =
a
3
,
1
6
1 − a ±
q
4a
2
+ 9(b −c)
2
. (22)
Now, the trace-norm of the realigned matrix R(ρ) is given by
kR(ρ)k
1
=
1
3
+ 2a + 2
r
3(b
2
+ c
2
+ bc) −(b + c) +
1
9
. (23)
Plugging in Eqs. (22) and (23) in the bounds (6a) and (6b) one obtains two inequalities that, together
with Eqs. (21) and λ(ρ) ≥ 0, form the set of constraints that a valid target state should obey. The
maximization over r such that these constraints hold can be solved exactly, although the analytical
form of the optimal parameters is rather involved. Here we give the approximate values a ≈ 0.21289,
b ≈ 0.04834, and c ≈ 0.21403 that yield an optimal state ρ
∗
, which admits any r ≤ r
∗
≈ 0.02345.
Note that the optimization is invariant under the interchange b ↔ c, therefore r
∗
is also not affected
by it. A visual representation of the density matrix of ρ
∗
is shown in Fig. 3.
As argued in the main text, since we do not require in our optimization that the full ball B is
contained in the state space, an analysis of the tightness of our estimate for r
∗
is in order. To do
so, we move away from ρ
∗
by a distance r
∗
in suitable directions and evaluate the position of the
resulting state with respect to the boundaries of the PPT and the CCNR sets. We first move towards
the PPT boundary. To this end, we take a normalized eigenvector |ηi from the eigenspace of Γ(ρ
∗
)
with minimal eigenvalue and find, for the matrix after the displacement ˜ρ = ρ
∗
−r
∗
Π[Γ
−1
(|ηihη|)], the
values
λ
min
[Γ(˜ρ)] ≈ 0 , kR(˜ρ)k
1
≈ 1.094 , and λ
min
(˜ρ) ≈ −0.005 , (24)
where we wrote Π[X] for [X − tr(X) /9]/ξ for some appropriate ξ, so that kΠ[X]k
2
= 1. Hence, ˜ρ is
effectively sitting on the PPT boundary, it is still detected as bound entangled by the CCNR criterion,
and it is slightly outside the space of physical states. This shows that our estimate for r
∗
is tight with
respect to PPT.
Similarly, we now consider a displacement towards the CCNR boundary. With a singular value
decomposition UDV
†
= R(ρ
∗
), we let ¯ρ = ρ
∗
− r
∗
Π[R
−1
(UV
†
)] and find the values
λ
min
[Γ(¯ρ)] ≈ 0.03 , kR(¯ρ)k
1
≈ 1.004 , and λ
min
(¯ρ) ≈ 0.01 . (25)
Therefore, it is possible that our state ρ
∗
is not yet optimal for obtaining the largest value r
∗
, since
we are still not touching the CCNR boundary.
Accepted in Quantum 2018-12-13, click title to verify 9
B.2 States of two-ququarts
We consider two-ququart states that are Bloch-diagonal, i.e., of the form
ρ =
X
k
x
k
g
k
⊗ g
k
, (26)
where g
k
= (σ
µ
⊗ σ
ν
)/2, µ, ν = 0, 1, 2, 3, with σ
0
= , σ
1
, σ
2
, σ
3
the Pauli matrices, and the index k
enumerates pairs of indices {µ, ν} in lexicographic order. Since tr(ρ) = 1, we get x
1
= 1/4.
By making the assumption that |x
k
| = s for k = 2, . . . , 16, the optimization can be easily carried
out analytically. We will see later that the maximum radius r
∗
for the general states (26) is indeed
achieved under this assumption. The trace norm of the realignment is
kR(ρ)k
1
=
X
k
|x
k
| =
1
4
+ 15s . (27)
Computing the minimal eigenvalue of Γ(ρ) is not so straightforward, as the signs of the coefficients
in Eq. (26) ought to be taken into account. We consider all possible sign combinations, and we see
that the minimal eigenvalue of Γ(ρ) is always of the form λ
min
[Γ(ρ)] =
1
16
[1 − 4(2k − 1)s], where
k = 1, 2, . . . , 8. Further, we check that k = 2 is the only combination with which the constraints
ρ ≥ 0, Γ(ρ) ≥ 0, and kR(ρ)k
1
> 1 are satisfied, so we have
λ
min
[Γ(ρ)] =
1
16
−
3
4
s . (28)
Let us denote
r
a
:=
q
d/(d − 1) λ
min
[Γ(ρ)] =
4
√
15
1
16
−
3
4
s
,
r
b
:= (kR(ρ)k
1
− 1)/
√
d =
1
4
15s −
3
4
.
Since r
a
and r
b
are affine functions of the parameter s, the solution to the equation r
a
= r
b
will single
out a unique s
∗
, and the radius r
∗
associated to it will be maximal. These are
s
∗
=
1
12
4 + 3
√
15
4 + 5
√
15
≈ 0.05571 ,
r
∗
=
1
4
15s
∗
−
3
4
≈ 0.0214 . (30)
As argued before, the parameter s
∗
is not enough to characterize a state of the form (26), as sign
combinations of the coefficients x
k
matter. An example of a state that achieves r
∗
is given by the
parameters x
1
= 1/4, x
k
= s
∗
for k ∈ S = {2, 3, 4, 5, 6, 7, 9, 10, 12, 14}, and x
k
= −s
∗
for k ∈ S
c
\ {1}.
Let us now tackle the optimization over the family of states in Eq. (26) without any extra assumption,
for which we will resort to an exact but computer-aided method. First, note that the search for the
optimal state can be concisely expressed as
maximize r
subject to Mx +
1
4
≥ 0
MDx +
1
4
≥
√
15r
X
k
|x
k
| −
3
4
≥ 4r
(31)
where x = {x
k
}
16
k=2
is a 15-dimensional real vector, r is a real nonnegative number, D is a diagonal
matrix of signs such that the map x 7→ Dx effectively induces the map ρ 7→ Γ(ρ), M is a 16 × 15
Accepted in Quantum 2018-12-13, click title to verify 10
Figure 4: The density matrix of the optimal two-ququart state with parameters given in Eq. (32) in the computational
basis. The heights of the bars correspond to the entries of the density matrix. Note that these are all real-valued.
matrix of signs such that
1
4
Mx +
1
16
is a vector of eigenvalues of ρ, and recall that x
1
= 1/4 is fixed
by normalization. Under these considerations it becomes apparent that the constraints in Eq. (31)
correspond to semidefinite-positiveness, PPT, and CCNR, respectively.
In order to solve Eq. (31), we linearize the CCNR constraint by removing the absolute values and
considering all the 2
15
possible sign combinations in the sum. Moreover, we incorporate the constraint
r ≥ 0 and regard r as an extra free parameter, instead of as an objective function. Then, each sign
combination chosen in the CCNR constraint results in a linear constraint that, along with positivity,
PPT, and r ≥ 0, will define a polytope of feasible parameters and, since all constraints are linear in
r, its maximal value will occur for one of the vertices. Of course, each single linear program of this
kind is in principle more restrictive than Eq. (31), but it is granted that the solution we are looking
for will correspond to a vertex of the feasibility polytope of one of them. We enumerate the vertices
of all the 2
15
polytopes using the polyhedral computation library cddlib
2
. We then take the union
of all vertices, eliminate redundant and nonextremal ones, and end up with a set of 4224 vertices
with a maximal radius r
∗
≈ 0.0214, which coincides with the value analytically obtained above. The
advantage of this method is that we obtain a much larger set of optimal states with varying rank.
Having optimal states with smaller rank can significantly simplify their experimental preparation.
In contrast, note that any state for which the assumption made in the analytical calculation above,
|x
k
| = s for all k = 2, . . . , 16, holds, is full-rank. The minimal rank for states of the form (26) achieving
a maximal radius r
∗
is 10. An example of such a rank-10 optimal state is given by parameters
x
1
=
1
4
,
x
α
≈ −0.0557066 , α ∈ {2, 3, 4, 5, 6, 9, 12, 14, 16},
x
β
≈ 0.0142664 , β ∈ {7, 11, 13},
x
γ
≈ 0.0971467 , γ ∈ {8, 10, 15}.
(32)
In Fig. 4 we depict the resulting density matrix as a bar plot.
The vertex enumeration of all feasibility polytopes contained in Eq. (31) allows us to go beyond
the set of optimal states and characterize states with even smaller ranks, naturally at the expense of
sacrificing some volume of bound entangled states around them. With this in mind, we see that the
absolute minimal rank for a Bloch-diagonal two-ququart bound entangled state is 9, with associated
2
cddlib version 0.94h, https://www.inf.ethz.ch/personal/fukudak/cdd_home/
Accepted in Quantum 2018-12-13, click title to verify 11
radius r
9
≈ 0.0128. An example of such state is given by parameters
x
1
=
1
4
,
x
α
≈ 0.0999184 , α ∈ {2, 5},
x
β
≈ 0.0750408 , β ∈ {3, 4, 6, 10, 14},
x
γ
≈ 0.0501632 , γ ∈ {7, 9},
x
δ
≈ −0.0252856 , δ ∈ {8, 13},
x
= −x
γ
, ∈ {11, 15, 16},
x
12
= −x
δ
.
(33)
C A hypothesis test for bound entanglement
In this appendix we give a proof of Eq. (13) and show how to compute the parameters r
1
, r
2
, and m,
needed to reproduce Fig. 2. Let us start with the proof.
The χ
2
-distribution arises naturally in a hypothesis test where the hypotheses are defined in terms
of bounds on Euclidean distances in a vector space of normal-distributed random variables. To see
this, we consider a generic situation where we draw a sample ~x from an m-variate normal distribution
N(
~
ξ, Σ) with offset
~
ξ and covariance matrix Σ. The offset shall be of the form
~
ξ = Sλ + ~η with a
matrix S and a constant vector ~η. We denote by p(t, r) the maximal probability that the hypothesis
test
ˆ
t(~x) ≤ t holds true under the hypothesis kµ − λk
2
≥ r for some fixed µ, where ~x is sampled from
N(
~
ξ, Σ), and
ˆ
t is a test statistic. Then, for a given sample ~x, the probability p[
ˆ
t(~x), r] is a p-value for
the hypothesis test
ˆ
t(~x) ≤ t. By choosing
ˆ
t: ~x 7→ kΣ
−1/2
(Sµ + ~η − ~x)k
2
, (34)
the following lemma holds:
Lemma 1. We have
p(t, r) = q
m
[t
2
, r
2
λ
min
(S
T
Σ
−1
S)] , (35)
where s 7→ q
m
(s, u) is the cumulative distribution function of the noncentral χ
2
-distribution with m
degrees of freedom and noncentrality parameter u. Here, λ
min
(X) denotes the smallest eigenvalue of
the symmetric matrix X.
Proof. By our assumptions, we have
p(t, r) = sup
λ
{P[
ˆ
t ≤ t] | kµ − λk
2
≥ r } , (36)
where
ˆ
t takes ~x distributed according to the normal distribution N(
~
ξ, Σ). Then, ~y = Σ
−1/2
(~x −
~
ξ) is
normal-distributed with offset
~
0 and covariance matrix , and we find
p(t, r) = sup
υ
{P[k~y − Σ
−1/2
Sυk
2
≤ t : ~y ∼ N (
~
0, )] | kυk
2
≥ r }
≡ sup
υ
{q
m
(t
2
, kΣ
−1/2
Sυk
2
2
) | kυk
2
≥ r } ,
(37)
where we used the definition of q
m
(s, u) in the second step. In order to see that this yields Eq. (35),
it is enough to note that q
m
(s, u) is a monotonously decreasing function in u [44].
The cumulative distribution function of the noncentral χ
2
- distribution is given in terms of Marcum’s
Q-function as q
m
(s, u) = 1 − Q
m/2
(
√
u,
√
s). In particular,
q
m
(s, u) = e
−u/2
∞
X
l=0
(u/2)
l
l!
˜γ(l +
m
2
,
s
2
) , (38)
Accepted in Quantum 2018-12-13, click title to verify 12
where ˜γ is the regularized lower incomplete gamma function, so that s 7→ q
m
(s, 0) is the cumulative
distribution function of the central χ
2
-distribution
3
.
Lemma 1 is presented in a generic form. At variance with the p-value defined in the main text
[cf. Eq. (11)], note that the supremum in Eq. (36) is taken over a larger space, as λ is not affected
by any positivity constraint. Hence, Eq. (35) is an upper bound of Eq. (11). In the context of the
quantum experiment at hand, µ and λ are Hermitian operators, and we make the identifications
T (ρ
0
) ≡ S(µ) + ~η and T (ρ
exp
) ≡ S(λ) + ~η, where ρ
0
is the target state with associated radius r
0
, and
ρ
exp
is the actual state prepared in the experiment. Then, we find that r
2
0
λ
min
(S
T
Σ
−1
S) ≡ r
2
1
[cf. Eq.
(13)].
We now assume that the data is of the form x
`
k
= n
`
k
/
P
j
n
`
j
, where n
`
k
is the number of events for
outcome k and measurement setting `. Then, the elements of the vector T (X) are tr(E
`
k
X), given the
effects {E
`
k
} with
P
k
E
`
k
= , i.e., T takes quantum states to probabilities, and S is the restriction of
T to the traceless operators. This determines ~η = T [ / tr( )]. Note that one effect per measurement
setting shall not be included in S, since its associated outcome will not be independent and this will
lead to r
1
= 0. Note also that introducing the restricted map S is necessary in order to have a
nonsingular covariance matrix Σ with a well defined inverse. With this in mind, one can determine
the number of degrees of freedom m for the examples of qutrits and ququarts described in the main
text. In the case of qutrits, measuring locally a complete set of mutually unbiased bases
4
(MUBs) in
principle produces data of dimension 144 (we have 4 settings per party with 3 outcomes per setting),
but eliminating dependent outcomes reduces this dimension to m = 128. For the sake of completeness,
we give an explicit construction of the MUBs {M
`
}
3
`=0
, where each row in M
`
corresponds to a basis
vector. Then, M
0
is the identity matrix,
M
1
=
1
√
3
1 1 1
1 ω
2
ω
1 ω ω
2
, M
2
=
1
√
3
1 1 ω
1 ω
2
ω
2
1 ω 1
, M
3
=
1
√
3
1 1 ω
2
1 ω
2
1
1 ω ω
, (39)
and ω = exp(2πi/3). Likewise, for the ququart case we assume that the measurement settings for
each party are built as pairs of local Pauli measurements {σ
µ
, σ
ν
}, with µ, ν = 1, 2, 3, hence we have
9 settings per party with 4 outcomes per setting, which leads to a reduced dimension of m = 1215.
For the moment, let us assume that the measurement outcomes k for each setting ` follow a Poisson
distribution with parameters N
`
T (ρ)
`
k
, so that data is now of the form x
`
k
= n
`
k
/N
`
, where N
`
is
the mean total value of events for measurement setting `. Then, the covariance matrix is a simple
diagonal matrix with entries T (ρ)
`
k
/N
`
. In this situation, also the elimination of dependent outcomes
is not necessary since Σ is invertible. This procedure is not optimal. If N
`
is large enough, we can
well approximate N
`
by
P
k
n
`
k
, at the cost of Σ becoming singular. However, after the dimensional
reduction introduced above, the reduced covariance matrix is again invertible. The advantage of this
approach is that we reduced the dimension m without discarding data, and hence the overall statistical
performance is increased. This is due to the fact that r
1
and r
2
in this approach are the same as in the
previous one. Note that an analysis using multinomial statistics yields the same reduced covariance
matrix and hence the same parameters m, r
1
, and r
2
. For our examples in the main text we use the
latter approach and assume N
`
≡ n for all settings `. The parameters used for computing p
fail
in
Fig. 2 are computed to be r
2
1
≈ 0.0664
2
n and r
2
2
≈ 0.0416
2
n for the qutrit example, and r
2
1
≈ 0.0856
2
n
and r
2
2
≈ 0.0469
2
n in the ququart case.
3
From Eq. (38) it is also easy to see that f
k
: u 7→ q
2k
(s, 2u) is a decreasing function in u, since ∂
u
f
k
(u) = f
k+1
(u) −
f
k
(u) and ˜γ(a, z) − ˜γ(a + 1, z) = ∂
z
˜γ(a + 1, z) = e
−z
z
a
/Γ(a + 1) ≥ 0.
4
Two orthonormal bases {|e
i
i}
d
i=1
and {|f
i
i}
d
i=1
are called mutually unbiased if it holds that |he
j
|f
k
i|
2
= 1/d ∀j, k =
1, . . . , d.
Accepted in Quantum 2018-12-13, click title to verify 13
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