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
(I⊗C)(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:
Accepted in Quantum 2020-02-15, click title to verify. Published under CC-BY 4.0. 2