Extension of the Alberti-Ulhmann criterion beyond qubit
dichotomies
Michele Dall’Arno
1,2
, Francesco Buscemi
3
, and Valerio Scarani
1,4
1
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore
2
Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050,
Japan
3
Graduate Scho ol of Informatics, Nagoya University, Chikusa-ku, 464-8601 Nagoya, Japan
4
Department of Physics, National University of Singapore, 2 Science Drive 3, 117542, Singapore
The Alberti-Ulhmann criterion states
that any given qubit dichotomy can be
transformed into any other given qubit
dichotomy by a quantum channel if and
only if the testing region of the former di-
chotomy includes the testing region of the
latter dichotomy. Here, we generalize the
Alberti-Ulhmann criterion to the case of
arbitrary number of qubit or qutrit states.
We also derive an analogous result for the
case of qubit or qutrit measurements with
arbitrary number of elements. We demon-
strate the possibility of applying our crite-
rion in a semi-device independent way.
1 Introduction
When quantum states are looked at as resources,
it is natural to study which states can be trans-
formed into which others by means of an al-
lowed set of operations. This question has been
rephrased in many ways: entanglement process-
ing, thermal operations... In this paper, we
consider generalizations of the following task:
given a pair of quantum states (ρ
0
, ρ
1
), called
a dichotomy, determine which other dichotomies
(σ
0
, σ
1
) can be obtained from it by application
of a completely positive trace preserving (CPTP)
map. The simplicity of the problem is only ap-
parent: very few results are known about this
problem. Before reviewing them, and stating our
contribution, let us take a detour to consider the
analogous task in classical statistics.
A classical dichotomy is a pair of probabil-
Michele Dall’Arno: michele.dallarno@aoni.waseda.jp
Francesco Buscemi: buscemi@i.nagoya-u.ac.jp
Valerio Scarani: physv@nus.edu.sg
ity distributions (p
0
, p
1
). It appears naturally in
the simplest formulation of hypothesis testing, in
which there are two inputs (the null and the al-
ternative hypotheses) and two outputs (accept or
reject). In this case, any test is represented by a
point in the dichotomy’s hypothesis testing region,
deﬁned as the region {(p
0
, p
1
)} R
2
where p
0
is
the probability of correctly accepting the null hy-
pothesis and p
1
is the probability of wrongly ac-
cepting the null hypothesis with the given test [1].
Tests can be then designed, for instance, to max-
imize p
0
while keeping p
1
under a certain thresh-
old. In particular, the wider the testing region,
the more “testable”, that is, the more “distinguish-
able” the pair of hypotheses is.
That the testing region is all that matters when
dealing with pairs of hypotheses is made partic-
ularly clear by the celebrated Blackwell’s theo-
rem for dichotomies [2]: given two dichotomies
(p
0
, p
1
) and (q
0
, q
1
), possibly on diﬀerent sample
spaces, there exists a stochastic transformation
that transforms p
0
into q
0
and p
1
into q
1
simul-
taneously (“statistical suﬃciency”) if and only if
the testing region for (p
0
, p
1
) contains the testing
region for (q
0
, q
1
). In other words, the former di-
chotomy can be deterministically processed into
(or, can deterministically simulate) the latter. In
the special case in which p
1
= q
1
= u, the uniform
distribution, the ordering induced by comparing
the testing region coincides with the ubiquitous
majorization ordering: indeed, the Lorenz curve
corresponding to a probability distribution p is
nothing but the boundary of the testing region
corresponding to the dichotomy (p, u) [2, 4, 3, 1].
Such a compact characterization is not known
in the quantum case that concerns us [8, 9, 5,
6, 7]: quantum statistical suﬃciency is in gen-
eral expressed in terms of an inﬁnite number of
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arXiv:1910.04294v2 [quant-ph] 18 Feb 2020
conditions [10, 11, 12] that are, therefore, very
diﬃcult to check in practice [13]. Some results,
based on [14], are known when the conversion
is relaxed to be approximate [15, 16], but the
problem remains hard in general. A notable ex-
ception is the case in which both quantum di-
chotomies, (ρ
0
, ρ
1
) and (σ
0
, σ
1
), only comprise
two-dimensional (i.e., qubit) states. Then, as a
consequence of a well-known result by Alberti and
Uhlmann, there exists a CPTP map transforming
(ρ
0
, ρ
1
) into (σ
0
, σ
1
) if and only if the testing re-
gion of the former contains the testing region of
the latter [17]. This is the perfect analog of Black-
well’s theorem; but counterexamples are known
as soon as (ρ
0
, ρ
1
) is a qutrit dichotomy [5].
In this paper, building upon previous works of
some of the present authors [18, 19, 20, 21, 22],
we derive the following results. First, we show
that any family of n qubit states which can all
become simultaneously real under a single uni-
tary transformation can be transformed into any
other family of n qubit (or, under some condi-
tions, qutrit) states by a CPTP map if the testing
region of the former includes the testing region of
the latter (the Alberti-Uhlmann case is recovered
for n = 2, since any pair of qubit states can be
made simultaneously real). Second, we show that
an analogous result holds for qubit or qutrit mea-
surements with n elements which can all become
simultaneously real under a single unitary trans-
formation. Our results follow as a natural conse-
quence of the Woronowicz decomposition [23] of
linear maps, once families of states and measure-
ments are regarded as linear transformations. We
demonstrate the possibility of witnessing statisti-
cal suﬃciency in a semi-device independent way,
that is, without any assumption on the devices
except their Hilbert space dimension.
The paper is structured as follows. In Section 2
we present our main results. We ﬁrst introduce
our extensions of the Alberti-Ulhmann criterion,
ﬁrst for families of states (in Section 2.1) and then
for measurements (in Section 2.2). We then dis-
cuss semi-device independent applications, ﬁrst
for families of states (in Section 2.3), and then
for measurements (in Section 2.4). In Section 3,
we provide technical proofs for our results. In
particular, Sections 3.1 and 3.2 prove the results
of Sections 2.1 and 2.2, respectively. We conclude
by summarizing our results in Section 4.
2 Main results
We will make use of standard deﬁnitions in quan-
tum information theory [24]. A quantum state is
represented by a density matrix, that is, a posi-
tive semi-deﬁnite operator ρ such that Tr[ρ] = 1.
A quantum measurement is represented by a pos-
itive operator-valued measure, that is, a family
{π
a
} of positive semi-deﬁnite operators that sat-
isfy the completeness condition
P
a
π
a
= , where
denotes the identity operator.
A channel is represented by a completely posi-
tive trace preserving map, that is, a map C such
that for any state ρ one has Tr[C(ρ)] = Tr[ρ] and
(IC)(ρ) 0. In the Heisenberg picture, a chan-
nel is represented by a completely positive unit
preserving map, that is, a map C such that for
any π 0 one has C( ) = and (I C)(π) 0.
2.1 Simulability of families of states
We say that a family {ρ
x
} of m states simulates
another (possibly diﬀerent dimensional) family
{σ
x
} of m states, in formula
{σ
x
} {ρ
x
}, (1)
if and only if there exists a channel C such that
σ
x
= C(ρ
x
) for any x.
If condition (1) is veriﬁed, it immediately fol-
lows that
R
{σ
x
}
R
{ρ
x
}
, (2)
where R({ρ
x
}) denotes the testing region of fam-
ily {ρ
x
}, deﬁned as the set of all vectors whose
x-th entry is the probability Tr[ρ
x
π] for any mea-
surement element π, in formula
R
{ρ
x
}
:
=
(
q
0 π s.t. q
x
= Tr [ρ
x
π] x
)
.
In other words, for any measurement element τ
there exists a measurement element π such that
Tr[ρ
x
π] = Tr[σ
x
τ] for any x.
Here, we derive conditions under which the re-
verse implication is also true, that is Eq. (2) im-
plies Eq. (1):
Theorem 1. For any family {σ
x
} of qubit states
and any real family {ρ
x
} of qubit states (that is,
states that have only real entries in some basis),
the following are equivalent:
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{σ
x
} {ρ
x
}.
R({σ
x
}) R({ρ
x
}).
If {ρ
x
} contains the identity operator in its lin-
ear span, the statement holds even if {σ
x
} is a
family of qutrit states.
The proof is given in Section 3.1.
Notice that the assumption that the family
{ρ
x
} of states is real cannot be relaxed. As a
counterexample, take {σ
x
} and {ρ
x
} to be sym-
metric informationally complete (or tetrahedral)
families of states with ρ
0
= σ
1
and ρ
1
= σ
0
, while
ρ
k
= σ
k
for k = 2, 3. A family {ρ
x
} of four pure
qubit states is tetrahedral if and only if Tr[ρ
x
ρ
z
]
is constant for any x 6= z. It immediately follows
that there exists a transposition map T (with re-
spect to some basis) such that σ
x
= T (ρ
x
) for
any x. Due to the informational completeness of
{σ
x
} and {ρ
x
}, map T is the only map such that
this is the case. However, map T is not a channel
as it is not completely positive.
2.2 Simulability of measurements
We say that an n-outcome measurement {π
a
}
simulates another (possibly, diﬀerent dimen-
sional) n-outcome measurement {τ
a
}, in formula
{τ
a
} {π
a
}, (3)
if and only if there exists a channel C such that
τ
a
= C
(π
a
) for any a, where C
denotes channel
C in the Heisenberg picture.
If condition (3) is veriﬁed, it follows immedi-
ately that
R
{τ
a
}
R
{π
a
}
, (4)
where the range R({π
a
}) of measurement {π
a
}
is deﬁned as the set of all probability distribu-
tions Tr[ρπ
a
] on the outcomes a for any state ρ,
in formula
R
{π
a
}
:
=
(
p
ρ 0, Tr ρ = 1, s.t. p
a
= Tr [ρπ
a
] a
)
.
In other words, for any state σ there exists a state
ρ such that Tr[ρπ
a
] = Tr[στ
a
] for any a.
Similarly to what we did before, we derive con-
ditions under which the reverse implication is also
true, that is Eq. (4) implies Eq. (3):
Theorem 2. For any qubit or qutrit measure-
ment {τ
a
} and any real qubit measurement {π
a
}
(that is, one whose elements are all real in some
basis), the following are equivalent:
{τ
a
} {π
a
}.
R({τ
a
}) R({π
a
}).
The proof is given in Section 3.2. As before, the
assumption that measurement {π
a
} is real cannot
be relaxed.
2.3 Semi-device independent simulability of
families of states
Suppose that a black box preparator with m but-
tons is given, and let us denote with ρ
x
the un-
known state prepared upon the pressure of button
x. Consider the setup where a black box tester
with n buttons is connected to the black box
preparator, and let us denote with {π
0|y
, π
1|y
:
=
π
0|y
} the test performed upon the pressure of
button y. One has
x [ 0, m 1]
ρ
x
π
a|y
a [ 0, 1]
y [0, n 1]
(5)
For each y, by running the experiment asymptot-
ically many times one collects the vectors q
y
and
u q
y
(u denotes the vector with unit entries)
whose x-th entry are the probabilities Tr[ρ
x
π
0|y
]
and Tr[ρ
x
π
1|y
], respectively, that is
q
y
x
:
= Tr
h
ρ
x
π
0|y
i
.
We call semi-device independent simulability
the problem of characterizing the class of all fam-
ilies of states that can be simulated by the black
box {ρ
x
}, for which simulability can be certiﬁed
based on distributions {q
y
} and {uq
y
} without
any characterisation of the tests {π
a|y
}, under an
assumption on the Hilbert space dimension.
Here, we will address the semi-device indepen-
dent simulability problem under the promise that
{ρ
x
} is a family of qubit states. In this case,
the testing region [19, 20] is the convex hull of
the isolated points 0 and u with a (possibly de-
generate) ellipsoid centered in u/2. Conversely,
for any (possibly degenerate) ellipsoid centered
in u/2 and contained in the hypercube [0, 1]
m
,
its convex hull with 0 and u is the testing region
of a qubit family of states. In general, such a
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testing region identiﬁes the family of states up to
unitaries and anti-unitaries.
We will further make the restriction that the
black box {ρ
x
} has m = 2 buttons, that is,
{ρ
x
} is a dichotomy. Notice that any qubit di-
chotomy is necessarily real. Hence, in the dis-
cussion above the (possibly degenerate) ellipsoid
becomes a (possibly degenerate) ellipse. Addi-
tionally, since two anti-unitarily related qubit di-
chotomies are also unitarily-related, a qubit di-
chotomy is identiﬁed by its range up to unitaries
only.
Due to Theorem 1, we have the following result:
Corollary 1. If the convex hull of points 0 and
u with any given ellipse centered in u/2 is a sub-
set of conv(0, u, {q
y
}, {u q
y
}) it is the testing
region of some qubit dichotomy that can be sim-
ulated by {ρ
x
}.
Notice that, on the one hand, the hypothesis
of Corollary 1 represents only a suﬃcient con-
dition for a qubit dichotomy to be simulable by
{ρ
x
}. On the other hand, for any other qubit di-
chotomy that can be simulated by {ρ
x
} (if any),
simulability cannot be certiﬁed in a semi-device
independent way unless further data is collected.
As an application, consider the case when one
of the states prepared by the dichotomy (say ρ
1
)
is the thermal state at inﬁnite temperature, that
is ρ
1
= /2 (for this example we are assuming
more than just the Hilbert space dimension, al-
though no knowledge of ρ
0
is assumed). Consider
the problem of ﬁnding the dichotomy with max-
imal free energy among those that can be sim-
ulated by {ρ
0
, /2} through a Gibbs-preserving
channel (in this case, a unit-preserving channel).
In this case, it immediately follows that the free
energy is monotone in the area of the range. This
can be seen as follows. First, notice that the free
energy in this case is equal to the neg-entropy
S(ρ
0
), since the free energy is equal to the rela-
tive entropy S(ρ
0
|| /2) and by deﬁnition one has
S(ρ
0
|| /2) = S(ρ
0
). In turn, S(ρ
0
) = h(λ
±
),
where h(·) denotes the binary entropy and λ
±
the
eigenvalues of ρ
0
. By setting λ
±
= 1/2 ± a, by
explicit computation it immediately follows that
the volume of the range of {ρ
0
, /2} is propor-
tional to a, hence the statement is proved.
Suppose one test is performed on the black box
dichotomy and the following probability vectors
are observed:
q
0
=
1
2
1
1
!
, u q
0
=
1
2
1 +
1
!
. (6)
for some value of parameter 0 1. The
situation is illustrated in Fig. 1. We assume that
0
1
1
u
u/2q
0
q
0
Tr[ρ
0
π]
Tr[ρ
1
π]
Figure 1: Probability vectors q
0
and u q
0
as given
by Eq. (6) lie at the vertices of a a line segment of
length and centered in u/2. The maximally committal
testing region for qubit dichotomy {σ
0
, /2} enclosed in
conv(0, u, q
0
, u q
0
) is given by conv(0, u, q
0
, u q
0
)
itself.
the black box implements a qubit dichotomy, a
justiﬁed assumption since the probability vector
in Eq. (6) belongs, for example, to the range of
any qubit dichotomy {φ, /2}, for any pure state
φ. It is easy to derive the maximum volume range
enclosed in conv(0, u, q
0
, u q
0
), and to verify
using Ref. [18] that it correspond to the range
of any -depolarized dichotomy {D
(φ), /2}, for
any pure state φ.
2.4 Semi-device independent simulability of
measurements
Suppose that a black box measurement with n
outcomes is given, and let us denote with π
a
the
unknown measurement element corresponding to
outcome a. Consider the setup where a black box
preparator with m buttons is connected to the
black box measurement, and let us denote with
ρ
x
the unknown state prepared upon the pressure
of button x. One has
x [ 0, m 1]
ρ
x
π
a
a [ 0, n 1]
(7)
For each x, by running the experiment asymp-
totically many times one collects the probability
distribution p
x
of outcome a, that is
[p
x
]
a
:
= Tr [ρ
x
π
a
] .
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We call semi-device independent simulability
the problem of characterizing the class of all mea-
surements that can be simulated by the black box
{π
a
}, for which simulability can be certiﬁed based
on distributions {p
x
} without any characterisa-
tion of the states {ρ
x
}, under an assumption on
the Hilbert space dimension.
Here, we will address the semi-device indepen-
dent simulability problem under the promise that
{π
y
} is a qubit measurement. In this case, the
range [18, 20] is a (possibly degenerate) ellip-
soid. Conversely, any (possibly degenerate) el-
lipsoid subset of the probability simplex is the
range of a qubit measurement. In general, such a
range identiﬁes the measurement up to unitaries
and anti-unitaries.
We will further make the restriction that the
black box {π
a
} has n = 3 outcomes. Notice
that any three-outcome qubit measurement is
necessarily real due to the completeness condi-
tion. Hence, in the discussion above the (pos-
sibly degenerate) ellipsoid becomes a (possibly
degenerate) ellipse. Additionally, since three-
outcome anti-unitarily related qubit measure-
ments are also unitarily related, a three-outcome
measurement is identiﬁed by its range up to uni-
taries only.
Due to Theorem 2, we have the following result:
Corollary 2. Any ellipse subset of conv({p
x
}) is
the range of some qubit three-outcome measure-
ment that can be simulated by {π
a
}.
Notice that, on the one hand, the hypothesis
of Corollary 2 represents only a suﬃcient con-
dition for a qubit three-outcome measurement to
be simulable by {π
a
}. On the other hand, for any
other qubit three-outcome measurement that can
be simulated by {π
a
} (if any), simulability can-
not be certiﬁed in a semi-device independent way
unless further data is collected.
As an application, consider the problem of ﬁnd-
ing, among the measurements that can be simu-
lated by the black box {π
a
}, the one with max-
imal simulation power, quantiﬁed according to
Theorem 2 by the volume of its range. Suppose
m states are fed into the black-box measurement
and the following distributions are observed:
p
x
=
2 2 cos θ
x
2 + cosθ
x
3 sin θ
x
2 + cosθ
x
+
3 sin θ
x
, (8)
where θ
x
:
= 2πx/m and x [0, m 1]. This
situation is depicted in Fig. 2. We assume
(1, 0, 0)
(0, 1, 0)(0, 0, 1) p
0
p
1
p
2
(1, 0, 0)
(0, 1, 0)(0, 0, 1) p
0
p
1
p
2
p
3
p
4
p
5
Figure 2: In both left and right ﬁgures, the outer dashed
triangle represents the simplex of three-outcome prob-
ability distributions. The distributions {p
x
} given by
Eq. (8) lie on the vertices of regular polygons (m = 3
and m = 6 in left and right ﬁgures, respectively). The
maximum volume ellipsoid enclosed in conv({p
x
}) is the
inner circle, which is the range of a -depolarized trine
qubit measurement ( = 1/2 and =
3/2 in left and
right ﬁgures, respectively).
that the black box implements a qubit measure-
ment, a justiﬁed assumption since such distri-
butions belong to the range of, for example, a
trine qubit measurement, that is, a measurement
whose elements lie on the vertices of a regu-
lar triangle in the Bloch sphere representation.
It is easy to derive the maximum volume el-
lipse [26, 27, 28, 29] enclosed in conv({p
x
}), and
to verify using Ref. [18] that it corresponds to the
range of any [cos(π/m)]-depolarized trine mea-
surement.
3 Proofs of Theorems 1 and 2
The formalism of quantum information theory,
used to present our results in Section 2, is not
the most eﬃcient to prove such statements. Here,
we introduce a more eﬃcient formalism [30], that
provides the additional beneﬁt of holding for any
bilinear physical theory, not just quantum theory.
Each system is associated with a dimension ,
and states and measurement elements are rep-
resented by vectors in R

. Let us denote with
S

R

and E

R

the set of all states
and the set of all measurement elements, respec-
tively. The probability that measurement ele-
ment e E

clicks upon the input of state s S

is given by the inner product s·e. Let u
n
R
n
de-
note the vector with unit entries and let us choose
a basis in which u

E

is the measurement el-
ement that has unit probability of click over any
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state. For example, for quantum systems  is the
squared Hilbert-space dimension, states and mea-
surement elements can be represented by (gener-
alized) Pauli vectors, their inner product reduces
to the Born rule, and u

corresponds to the iden-
tity operator.
A family of n states {s
k
} can be conveniently
represented by arranging the states as the rows
of an n ×  matrix S. This way, the correspond-
ing linear map S
:
R

R
n
maps any eﬀect e
into the vector whose k-th entry is the probability
s
k
·e. It immediately follows that, for any family
S of states, one has Su

= u
n
. Analogously, an n
outcome measurement can be conveniently repre-
sented by arranging its elements {e
k
} as the rows
of an n× matrix M. This way, the corresponding
linear map MR

R
n
maps any state s into the
probability distribution whose k-th entry is the
probability e
k
·s. It immediately follows that, for
any measurement M , one has M
T
u
n
= u

.
Finally, we discuss maps from states to states
and from eﬀects to eﬀects.
Deﬁnition 1 (State morphism). A linear map
C
:
R

0
R

1
is a state morphism if and only if
CS
0
S
1
.
Deﬁnition 2 (Statistical morphism). A linear
map C
:
R

0
R

1
is a statistical morphism if
and only if CE
0
E
1
and Cu

0
= u

1
.
In standard quantum theory, a state morphism
is a positive (not necessarily completely positive)
trace-preserving (PTP) map, that is, a transfor-
mation of states in the Schrödinger picture. Anal-
ogously, a statistical morphism is a positive (not
necessarily completely positive) unit-preserving
(PUP) map, that is, a transformation of measure-
ment elements in the Heisenberg picture. For our
proofs, we do not need an analogous of complete
positivity for arbitrary bilinear theories.
3.1 Simulability of families of states
Here we prove Theorem 1, that we report here for
the reader’s convenience.
Theorem 1. For any family {σ
x
} of qubit states
and any real family {ρ
x
} of qubit states, the fol-
lowing are equivalent:
{σ
x
} {ρ
x
}.
R({σ
x
}) R({ρ
x
}).
If {ρ
x
} contains the identity operator in its lin-
ear span, the statement holds even if {σ
x
} is a
family of qutrit states.
By adopting the formalism of bilinear theories,
we denote with S
0
and S
1
the families of states
corresponding to {σ
x
} and {ρ
x
}, respectively. To
prove the theorem, we need to distinguish two
cases. First, let us consider the case when the
linear span of {ρ
x
} contains the identity operator
, that is, S
+
1
S
1
u

1
= u

1
, where (·)
+
denotes
the Moore-Penrose pseudoinverse. In the follow-
ing Lemma we show that, under the hypothesis
S
+
1
S
1
E
1
E
1
, range inclusion between two fam-
ilies of states is equivalent to the existence of a
statistical morphism between them.
Lemma 1. For any families of states S
0
:
R

0
R
n
and S
1
:
R

1
R
n
such that S
+
1
S
1
E
1
E
1
and S
+
1
S
1
u

1
= u

1
, the following are equivalent:
1. S
0
E
0
S
1
E
1
,
2. there exists a statistical morphism C
:
R

0
R

1
such that S
0
= S
1
C.
Proof. Implication 1 2 is trivial.
Implication 1 2 can be shown as follows.
Let
C
:
= S
+
1
S
0
. (9)
Let us ﬁrst show that map C is a statistical
morphism. One has
CE
0
= S
+
1
S
0
E
0
S
+
1
S
1
E
1
E
1
,
where the equality follows from Eq. (9) and the
inclusions follow from the hypothesis S
0
E
0
S
1
E
1
and S
+
1
S
1
E
1
E
1
, respectively. Moreover,
Cu

0
= S
+
1
S
0
u

0
= S
+
1
u
n
,
where the equalities follow from Eq. (9) and from
the hypothesis S
0
u

0
= u
n
, respectively. Since by
hypothesis S
1
u

1
= u
n
, one also has S
+
1
S
1
u

1
=
S
+
1
u
n
, and hence
Cu

0
= S
+
1
S
1
u

1
= u

1
,
where the second inequality follows by hypothe-
sis. Hence map C is a statistical morphism.
Let us now show that S
0
= S
1
C. By multiply-
ing Eq. (9) from the left by S
1
one has
S
1
C = S
1
S
+
1
S
0
.
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Since span E
0
= R

0
and span E
1
= R

1
, from
S
0
E
0
S
1
E
1
one has rng S
0
rng S
1
. Since
S
1
S
+
1
is the projector on rng S
1
, one has S
0
=
S
1
C.
It is easy to see that the hypothesis S
+
1
S
1
E
1
E
1
in Lemma 1 is satisﬁed for any family S
1
of states if E
1
is a 
1
–dimensional (hyper)–cone
with (
1
1)–dimensional (hyper)–spherical sec-
tion with axis along u

1
, as is the case for the
qubit system, where
1
= 4. In this case, by ad-
ditionally assuming that the family {ρ
x
} of states
is real, that is S
1
T = S
1
where T denotes the
transposition map with respect to some basis, the
following Lemma completes the ﬁrst part of the
proof of Theorem 1.
Lemma 2. For any qubit or qutrit family S
0
:
R

0
R
n
of states, with
0
= 4 or 
0
= 9, and
any qubit family S
1
:
R
4
R
n
of states such that
S
+
1
S
1
u

1
= u

1
and S
1
T = S
1
, the following are
equivalent:
1. S
0
E
0
S
1
E
1
,
2. there exists CPTP map C
:
R

0
R
4
such
that S
0
= S
1
C.
Proof. Implication 1 2 is trivial.
Implication 1 2 can be shown as follows.
Due to Lemma 1, there exists a statistical mor-
phism C
0
:
R

0
R
4
, hence a PUP map, such
that S
0
= S
1
C
0
. Let us prove that there exists a
CPUP map C
:
R

0
R
4
such that S
0
= S
1
C.
For any PUP map C
0
:
R

0
R
4
, Woronow-
icz [23] proved that there exist 0 p 1 and
CPUP maps C
0
:
R

0
R
4
and C
1
:
R

0
R
4
such that
C
0
= pC
0
+ (1 p) T C
1
. (10)
One has
S
1
C
0
=S
1
pC
0
+ (1 p) T C
1
=S
1
pC
0
+ (1 p) C
1
,
where the equalities follow from Eq. (10) and
from the hypothesis S
1
T = S
1
, respectively.
Since the convex combination of CPUP maps is
CPUP, map C
:
= pC
0
+ (1 p) C
1
is CPUP.
Second, let us consider the case when the linear
span of {ρ
x
} does not contain the identity opera-
tor . Since by linearity it immediately follows
that any linear dependency in the states {ρ
x
}
must be present also in the states {σ
x
}, as for-
mally show in Lemma 3, the remaining part of the
proof directly follows from the Alberti-Ulhmann
criterion, as formally shown in Lemma 4.
Lemma 3. For any families S
0
:
R

0
R
n
and
S
1
:
R

1
R
n
of states such that S
0
E
0
S
1
E
1
,
if for some k there exists {λ
i
R} such that
s
k
1
=
X
i6=k
λ
i
s
i
1
,
then one has
s
k
0
=
X
i6=k
λ
i
s
i
0
.
Proof. By hypothesis, for any e
0
E
0
there ex-
ists e
1
E
1
such that
s
k
0
· e
0
= s
k
1
· e
1
.
Hence, for any set {e
j
0
} E
0
one has
s
k
0
· e
j
0
= s
k
1
· e
j
1
=
X
i6=k
λ
i
s
i
1
· e
j
1
=
X
i6=k
λ
i
s
i
0
· e
j
0
.
Since span E
0
= R

0
, it is possible to take set
{e
j
0
E
0
} a spanning set. Hence the thesis.
Lemma 4. For any qubit families S
0
:
R
4
R
n
and S
1
:
R
4
R
n
of states such that S
+
1
S
1
u

1
6=
u

1
and S
1
T = S
1
where T denotes the transposi-
tion map with respect to some basis, the following
are equivalent:
1. S
0
E
0
S
1
E
1
,
2. there exists CPTP map C
:
R
4
R
4
such
that S
0
= S
1
C.
Proof. Implication 1 2 is trivial.
Implication 1 2 can be shown as follows.
By the hypothesis S
0
E
0
S
1
E
1
, one has that
for any e
0
E
0
there exists a e
1
E
1
such that
s
k
0
· e
0
= s
k
1
e
1
for any k. Since in particular this
holds for k = 0, 1, by denoting with S
0
0
and S
0
1
the families of states whose rows are (s
0
0
, s
1
0
) and
(s
0
1
, s
1
1
), respectively, one has
S
0
0
E
0
S
0
1
E
1
.
Hence, due to a result [7] by Buscemi and Gour,
in turn based on a result [17] by Alberti and Ulh-
mann, there exists a CPTP map C
:
R
4
R
4
such that S
0
0
= S
0
1
C.
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Due to the hypotheses S
+
1
S
1
u

1
6= u

1
and
S
1
T = S
1
, for any k there exists λ
k
R such
that
s
k
1
= λ
k
s
0
1
+
1 λ
k
s
1
1
,
that is, state s
k
1
is a convex combination of states
s
0
1
and s
1
1
. Hence, due to Lemma 3, one also has
s
k
0
= λ
k
s
0
0
+
1 λ
k
s
1
0
,
that is, state s
k
0
is a convex combination of states
s
0
0
and s
1
0
. Hence, by linearity, S
0
= S
1
C.
This concludes the proof of Theorem 1.
3.2 Simulability of measurements
Here we prove Theorem 2, that we report here for
the reader’s convenience.
Theorem 2. For any qubit or qutrit measure-
ment {τ
a
} and any real qubit measurement {π
a
},
the following are equivalent:
{τ
a
} {π
a
}.
R({τ
a
}) R({π
a
}).
By adopting the formalism of bilinear theories,
we denote with M
0
and M
1
the measurements
corresponding to {τ
a
} and {π
a
}, respectively. In
contrast to Theorem 1, for whose proof we needed
to distinguish two cases based on whether the
linear span of {ρ
x
} contained the identity op-
erator or not, due to completeness for any
measurement {π
a
} one has
P
a
π
a
= , that is,
M
+
1
M
1
u

1
= u

1
, where (·)
+
denotes the Moore-
Penrose pseudoinverse. Hence, the proof pro-
ceeds along the lines of Lemmas 1 and 2, that
are in this case replaced by Lemmas 5 and 6, re-
spectively. In the following Lemma we show that,
under the hypothesis M
+
1
M
1
S
1
S
1
, range inclu-
sion between two measurements is equivalent to
the existence of a state morphism between them.
Lemma 5. For any measurements M
0
:
R

0
R
n
and M
1
:
R

1
R
n
such that M
+
1
M
1
S
1
S
1
,
the following are equivalent:
1. M
0
S
0
M
1
S
1
,
2. there exists a state morphism C
:
R

0
R

1
such that M
0
= M
1
C.
Proof. Implication 1 2 is trivial.
Implication 1 2 can be shown as follows.
Let
C
:
= M
+
1
M
0
. (11)
Let us ﬁrst show that map C is a state mor-
phism. One has
CS
0
= M
+
1
M
0
S
0
M
+
1
M
1
S
1
S
1
,
where the equality follows from Eq. (11) and the
inclusions follow from the hypothesis M
0
S
0
M
1
S
1
and M
+
1
M
1
S
1
S
1
, respectively. Hence
map C is a state morphism.
Let us now show that M
0
= M
1
C. By multi-
plying Eq. (11) from the left by M
1
one has
M
1
C = M
1
M
+
1
M
0
.
Since span S
0
= R

0
and span S
1
= R

1
, from
M
0
S
0
M
1
S
1
one has rng M
0
rng M
1
. Since
M
1
M
+
1
is the projector on rng M
1
, one has M
0
=
M
1
C.
It is easy to see that the hypothesis M
+
1
M
1
S
1
S
1
in Lemma 5 is satisﬁed for any measurement
M
1
if and only if S
1
is a ( 1)–dimensional
(hyper)–sphere with center along u

1
, as is the
case for the qubit system, where 
1
= 4.
To see this, notice that the (hyper)–sphere is
the only body for which there exists a point (the
center) such that any line through the point is
orthogonal to the surface of the body. Hence,
the (hyper)–sphere is also the only body for
which the projection of the body on any sub-
space containing such a point is a subset of the
body. Finally, notice that by multiplying con-
dition M
T
1
u
n
= u

1
on the left by M
+
1
M
1
and
using the elementary property of pseudoinverse
that M
+
1
M
1
M
T
1
= M
T
1
, one immediately has
M
+
1
M
1
u

1
= u

1
,
that is, M
+
1
M
1
is the projector on a subspace that
contains the center of the (hyper)–sphere S
1
.
In this case, by additionally assuming that the
measurement {π
a
} is real, that is M
1
T = M
1
where T denotes the transposition map with re-
spect to some basis, the following Lemma com-
pletes the proof of Theorem 2.
Lemma 6. For any qubit or qutrit measurement
M
0
:
R

0
R
n
, with
0
= 4 or 
0
= 9, and
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any qubit measurement M
1
:
R
4
R
n
such that
M
1
T = M
1
where T denotes the transposition
map with respect to some basis, the following are
equivalent:
1. M
0
S
0
M
1
S
1
,
2. there exists CPTP map C
:
R

0
R
4
such
that M
0
= M
1
C.
Proof. Implication 1 2 is trivial.
Implication 1 2 can be shown as follows.
Due to Lemma 5, there exists a state morphism
C
0
:
R

0
R
4
, hence a PTP map, such that
M
0
= M
1
C
0
. Let us prove that there exists a
CPTP map C
:
R

0
R
4
such that M
0
= M
1
C.
For any PTP map C
0
:
R

0
R
4
, there ex-
ists [23] 0 p 1 and CPTP maps C
0
:
R

0
R
4
and C
1
:
R
`
0
R
4
such that
C
0
= pC
0
+ (1 p) T C
1
. (12)
One has
M
1
C
0
=M
1
pC
0
+ (1 p) T C
1
=M
1
pC
0
+ (1 p) C
1
,
where the equalities follow from Eq. (12) and
from the hypothesis M
1
T = M
1
, respectively.
Since the convex combination of CPTP maps is
CPTP, map C
:
= pC
0
+ (1 p) C
1
is CPTP.
This concludes the proof of Theorem 2.
4 Conclusion
In this work we addressed the problem of quan-
tum simulability, that is, the existence of a quan-
tum channel transforming a given device into an-
other. We considered the cases of families of n
qubit or qutrit states and of qubit or qutrit mea-
surements with n elements, thus extending the
Alberti-Ulhmann criterion for qubit dichotomies.
Based on these results, we demonstrated the pos-
sibility of certifying the simulability in a semi-
device independent way, that is, without any as-
sumption of the devices except their Hilbert space
dimension.
Acknowledgement
M.D. is grateful to A. Bisio, A. Jenčová, and K.
Matsumoto for insightful discussions. This work
is supported by the MEXT Quantum Leap Flag-
ship Program (MEXT Q-LEAP) Grant No. JP-
MXS0118067285; the National Research Foun-
dation and the Ministry of Education, Singa-
pore, under the Research Centres of Excellence
programme; the Japan Society for the Promo-
tion of Science (JSPS) KAKENHI, Grant No.
19H04066; the program for FRIAS-Nagoya IAR
Joint Project Group.
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