the variables X
A
, X
B
and Y
A
, Y
B
will refer to those
local measurements applied to the respective sides.
We only consider projective measurements, but be-
side this we impose no further restrictions on the in-
dividual measurements. So the only property that
measurements like X
A
and X
B
have to share is the
common label ’X’, besides this, they could be non-
commuting or even defined on Hilbert spaces with
different dimensions.
The main result of this work is stated in Prop.1 in
Sec. 3. In that section, we also collect some remarks
on possible and impossible generalizations and the
construction of entanglement witnesses. The proof of
Prop.1 is placed at the end of this paper, as it relies
on two basic theorems stated in Sec.4 and Sec.5.
Thm.1, in Sec.4, clarifies and expands the known
connection between the logarithm of (p, q)-norms and
entropic uncertainty relations. As a special case of
this theorem we obtain Lem.1 which states the nat-
ural generalization of the well known Maassen and
Uffink bound [21] to weighted uncertainty relations.
Thm.2, in Sec.5, states that (p, q)-norms, in a certain
parameter range, are multiplicative, which at the end
leads to the desired statement on the additivity of
uncertainty relations.
Before stating the main result, we collect, in Sec.1,
some general observations on the behavior of uncer-
tainty relations for product measurements with re-
spect to different classes of correlated states. Fur-
thermore, in Sec. 2, we will motivate and explain the
explicit form of linear uncertainty relations used in
this work.
1 Uncertainty in bipartitions
All uncertainty relations considered is this paper
are state-independent. In practice, finding a state-
independent relation leads to the problem of jointly
minimizing a tuple of given uncertainty measures,
here the Shannon entropy of X
AB
and Y
AB
, over all
states. This minimum, or a lower bound on it, then
gives the aforementioned trade-off, which then allows
to formulate statements like: "whenever the uncer-
tainty of X
AB
is small, the uncertainty of Y
AB
has to
be bigger than some state-independent constant" .
Considering the measured state, ρ
AB
, it is natural
to distinguish between the three classes: uncorrelated,
classically correlated and non-classical correlated. In
regard of the uncertainty in a corresponding global
measurement, states in these classes share some com-
mon features:
If the measured state is uncorrelated, i.e a prod-
uct state ρ
AB
= ρ
A
⊗ ρ
B
, the outcomes of the lo-
cal measurements are uncorrelated as well. Hence,
the uncertainty of a global measurement is completely
determined by the uncertainty of the local measure-
ments on the respective local states ρ
A
and ρ
B
. More-
over, in our case, the additivity of the Shannon en-
tropy, tells us that the uncertainty of a global mea-
surement is simply the sum of the uncertainty of the
local ones. In the same way any trade-off on the global
uncertainties can be deduced from local ones.
If the measured state is classically correlated,
i.e a convex combination of product states [22], ad-
ditivity of local uncertainties does not longer hold.
More generally, whenever we consider a concave un-
certainty measure [23], like the Shannon entropy, the
global uncertainty of a single global measurement is
smaller than the sum of the local uncertainties. Intu-
itively this makes sense because a correlation allows
to deduce information on the potential measurement
outcomes of one side given a particular measurement
outcome on the other. However, a linear uncertainty
relation for a pair of global measurements is not af-
fected by this, i.e a trade-off will again be saturated
by product states. This is because the uncertainty re-
lation between two measurements, restricted to some
convex set of states, will always be attained on an
extreme point of this set.
However, if measurements are applied to an entan-
gled state, more precisely to a state which shows
EPR-steering [24–26] with respect to the measure-
ments X
AB
and Y
AB
, it is in general not clear how
a trade-off between global uncertainties relates to the
corresponding trade-off between local ones. Just have
in mind that steering implies the absence of any lo-
cal state model, which is usually proven by showing
that any such model would violate a local uncertainty
relation.
In principle one would expect to obtain smaller un-
certainty bounds by also considering entangled states,
and there are many entanglement witnesses known
based on this idea (see also Rem. 3 in the following
section).
2 Linear uncertainty relations
We note that there are many uncertainty measures,
most prominently variances [8, 10]. Variance, and
similar constructed measures [17, 27], describe the
deviation from a mean value, which clearly demands
to assign a metric structure to the set of measure-
ment outcomes. From a physicist’s perspective this
makes sense in many situations [11] but can also cause
strange behaviours in situations where this metric
structure has to be imposed artificially [28]. However,
from the perspective of information theory, this seems
to be an unnecessary dependency. Especially when
uncertainties with respect to multipartitions are con-
sidered, it is not clear at all how such a metric should
be constructed. Hence, it can be dropped and a quan-
tity that only depends on probability distributions of
measurement outcomes has to be used. We will use
the Shannon entropy. It fulfills the above require-
ment, does not change when the labeling of the mea-
surement outcomes are permuted, and has a clear op-
Accepted in Quantum 2018-03-20, click title to verify 2