# Additivity of entropic uncertainty relations

René Schwonnek  Institut für Theoretische Physik, Leibniz Universität
Hannover, Germany

We consider the uncertainty between two pairs of local projective measurements
performed on a multipartite system. We show that the optimal bound in any
linear uncertainty relation, formulated in terms of the Shannon entropy, is
additive. This directly implies, against naive intuition, that the minimal
entropic uncertainty can always be realized by fully separable states. Hence,
in contradiction to proposals by other authors, no entanglement witness can be
constructed solely by comparing the attainable uncertainties of entangled and
separable states. However, our result gives rise to a huge simplification for
computing global uncertainty bounds as they now can be deduced from local
ones.

Furthermore, we provide the natural generalization of the Maassen and Uffink
inequality for linear uncertainty relations with arbitrary positive
coefficients.

## Introduction

Uncertainty and entanglement are doubtless two of the most prominent and
drastic properties that set apart quantum physics from a classical view on the
world. Their interplay contains a rich structure, which is neither
sufficiently understood nor fully discovered. In this work, we reveal a new
aspect of this structure: the additivity of entropic uncertainty relations.

For product measurements in a multipartition, we show that the optimal bound
$c_{ABC\ldots}$ in a linear uncertainty relation satisfies

$\displaystyle c_{ABC\cdots}=c_{A}+c_{B}+c_{C}+\ldots\quad,$ (1)

where $c_{A},c_{B},c_{C},\ldots$ are bounds that only depend on local
measurements. This result implies that minimal uncertainty for product
measurements can always be realized by uncorrelated states. Hence, we have an
example for a task which is not improved by the use of entanglement.

We will quantify the uncertainty of a measurement by the Shannon entropy of
its outcome distribution. For this case, the corresponding linear uncertainty
bound $c_{ABC\ldots}$ gives the central estimate in many applications like:
entropic steering witnesses [1, 2, 3, 4], uncertainty relations with side-
information [5], some security proofs [6] and many more.

When speaking about uncertainty, we consider so called preparation uncertainty
relations [7, 8, 9, 10, 11, 12, 13, 14]. From an operational point of view, a
preparation uncertainty describes fundamental limitations, i.e. a tradeoff, on
the certainty of predicting outcomes of several measurements that are
performed on instances of the same state. This should not be confused [15]
with its operational counterpart named measurement uncertainty[16, 17, 18, 19,
20]. A measurement uncertainty relation describes the ability of producing a
measurement device which approximates several incompatible measurement devices
in one shot.

The calculations in this work focus on uncertainty relations in a bipartite
setting. However, all results can easily be generalized to a multipartite
setting by an iteration of statements on bipartitions. The basic measurement
setting, which we consider for bipartitions, is depicted in Fig. 1.

$\rho_{AB}$$X_{A}$$Y_{A}$$\\{1,0\\}$$1$$0$$X_{B}$$Y_{B}$$1$$0$AB$\lambda\,\mathbf{p}_{X_{AB}}$$\oplus$$\mu\,\mathbf{p}_{Y_{AB}}$
Figure 1: Basic setting of product measurements on a bipartition: pairs of
measurements $X_{A},X_{B}$ or $Y_{A},Y_{B}$ are applied to a joint state
$\rho_{AB}$ at the respective sides of a bipartition. One bit of information
is transmitted for communicating whether the $X$ or the $Y$ measurements are
performed. The weights $(\lambda,\mu)$ denote the probabilities corresponding
to this choice.

We consider a pair of measurements, $X_{AB}=X_{A}X_{B}$ and
$Y_{AB}=Y_{A}Y_{B}$, to which we will refer as the global measurements of
(tensor) product form. Each of those global measurements of product form is
implemented by applying local measurements at the respective sides of a
bipartition between parties denoted by $A$ and $B$. Hereby, 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 beside this we impose no further
restrictions on the individual 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 natural 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 uncertainty relations for product measurements
with respect to different classes of correlated states. Furthermore, 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 uncertainty of $X_{AB}$ is small, the
uncertainty of $Y_{AB}$ has to be bigger than some state-independent constant"
.

Considering the measured state, $\rho_{AB}$, 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 common features:

If the measured state is uncorrelated, i.e a product state
$\rho_{AB}=\rho_{A}\otimes\rho_{B}$, the outcomes of the local measurements
are uncorrelated as well. Hence, the uncertainty of a global measurement is
completely determined by the uncertainty of the local measurements on the
respective local states $\rho_{A}$ and $\rho_{B}$. Moreover, in our case, the
additivity of the Shannon entropy, tells us that the uncertainty of a global
measurement 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], additivity of local uncertainties does not longer hold.
More generally, whenever we consider a concave uncertainty measure [23], like
the Shannon entropy, the global uncertainty of a single global measurement is
smaller than the sum of the local uncertainties. Intuitively 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 affected by this, i.e a trade-off will again be saturated
by product states. This is because the uncertainty relation 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 entangled state, more precisely to
a state which shows EPR-steering [24, 25, 26] with respect to the measurements
$X_{AB}$ 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 local 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 uncertainty 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
[10, 8]. Variance, and similar constructed measures [27, 17], describe the
deviation from a mean value, which clearly demands to assign a metric
structure to the set of measurement 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 considered, it is not clear at all how such a metric
should be constructed. Hence, it can be dropped and a quantity that only
depends on probability distributions of measurement outcomes has to be used.
We will use the Shannon entropy. It fulfills the above requirement, does not
change when the labeling of the measurement outcomes are permuted, and has a
clear operational interpretation [29, 30]. Remarkably, Claude Shannon himself
used the term ’uncertainty’ as an intuitive paraphrase for the quantity today
known as ’entropy’ [29]. Historically, the decision to call the Shannon
entropy an ’entropy’ goes back to a suggestion John von Neumann gave to
Shannon, when he was visiting Weyl in 1940 (there are, at least, three
versions of this anecdote known [31], the most popular is [32]).

Because we are not interested in assigning values to measurement outcomes, a
measurement, say $X$, is sufficiently described by its POVM elements,
$\\{X_{i}\\}$. So, given a state $\rho$, the probability of obtaining the
$i$-th outcome is computed by $\operatorname{tr}(\rho X_{i})$. The respective
probability distribution of all outcomes is denoted by the vector
$\mathbf{p}^{x}_{\rho}$. Within this notation the Shannon entropy of a $X$
measurement is given by
$H(X|\rho):=-\sum_{i}\left(\mathbf{p}^{x}_{\rho}\right)_{i}\log\left(\mathbf{p}^{x}_{\rho}\right)_{i}$.
As we restrict ourselves to non-degenerate projective measurements, all
necessary information on a pair of measurements, $X$ and $Y$, is captured by a
unitary $U$ that links the measurement basis. We will use the convention to
write $U$ as transformation from the $\\{X_{i}\\}$ to the $\\{Y_{i}\\}$-basis,
i.e. we will take $U$ such that $Y_{i}=UX_{i}U^{\dagger}$ holds.

Our basic objects of interest are optimal, state-independent and linear
relations. This is, for fixed weights $\lambda,\mu\in\mathbb{R}^{+}$ we are
interested in the best constant $c(\lambda,\mu)$ for which the linear
inequality

$\displaystyle\lambda H(X|\rho)+\mu H(Y|\rho)\geq c(\lambda,\mu)$ (2)

holds on all states $\rho$.

Such a relation has two common interpretations: On one hand one can consider a
guessing game, see also [33]. On the other, a relation like (2) can be
interpreted geometrically as in Fig. 2.

Linear uncertainty: a guessing game
For the moment, consider a player, called Eve, who plays against an opponent,
called Alice. Dependent on a coin throw, in each round, Alice performs
measurement $X_{A}$ or $Y_{A}$ on a local quantum state. Thereby the weights
$\lambda$ and $\mu$ are the weights of the coin and the l.h.s. of (2)
describes the total uncertainty Eve has on Alice’s outcomes in each round. To
be more precise, up to a $(\lambda,\mu)$-dependent constant, the l.h.s of (2)
equals the Shannon entropy of the outcome distribution
$\lambda\mathbf{p}^{X_{A}}_{\rho}\oplus\mu\mathbf{p}^{Y_{A}}_{\rho}$.

Eve’s role in this game is to first choose a state $\rho$, observe the coin
throw, wait for the measurements to be performed by Alice, and then ask binary
questions to her opponent in order to get certainty on the outcomes. Thereby,
the Shannon entropy sum on the l.h.s of (2) (with logarithm to the base 2)
equals the expected amount questions Eve has to ask using an optimal strategy
based on a fixed $\rho$. Hence, the value $c(\lambda,\mu)$ denotes the minimal
amount of expected questions, attainable by choosing an optimal $\rho$.

For a bipartite setting, Fig. 1, a second player, say Bob, joins the game.
Here, Eve will play the above game against Alice and Bob, simultaneously.
Thereby, Alice and Bob share a common coin, and, therefore, apply measurements
with the same labels ($X_{AB}$ or $Y_{AB}$). The obvious question that arises
in this context is if Eve gets an advantage in this simultaneous game by using
an entangled state or not. Prop. 1 in the next section answers the above
question negatively, which is somehow unexpected as in principle the possible
usage of non-classical correlations enlarges Eve’s strategies. For example:
Eve could have used a maximally entangled state, adjusted such that all
measurements Alice and Bob perform are maximally correlated. In this case the
remaining uncertainty Eve has, would only be the uncertainty on the outcomes
of one of the parties. However, the marginals of a maximally entangled state
are maximally mixed. Hence, Eve still has a serious amount of uncertainty
($\log d)$, which turns out to be not small enough for beating a strategy
based on minimizing the uncertainty of the local measurements individually.
For the case of product-MUBs in prime square dimension [34], it turns out that
the minimal uncertainty realizable by a maximally entangled state actually
equals the optimal bound.

Linear uncertainty: the positive convex hull

Figure 2: Uncertainty set for measurements performed on a qubit. Any linear
uncertainty relation, (2), with weights $(\lambda,\mu)$, gives the description
of a tangent to the uncertainty set. All attainable pairs of entropies lie
above this tangent.

The second interpretation comes from considering the set of all attainable
uncertainty pairs, the so called uncertainty set

$\displaystyle\mathcal{U}=\left\\{\left(H(X|\rho),H(Y|\rho)\right)|\rho\textit{
is a quantum state}\right\\}.$ (3)

In principle this set contains all information on the uncertainty trade-off
between two measurements. More precisely, the white space in the lower-left
corner of a diagram like Fig. 2 indicates that both uncertainties cannot be
small simultaneously. In this context, a state-independent uncertainty gives a
quantitative description of this white space. Unfortunately, it turns out that
computing $\mathcal{U}$ can be very hard, because the whole state-space has to
be considered. Here a linear inequality, like (2), gives an outer
approximation of this set. More precisely, if $c(\lambda,\mu)$ is the optimal
constant in (2), this inequality describes a halfspace bounded from the lower-
left by a tangent on $\mathcal{U}$. This tangent has the slope $\mu/\lambda$.
The points on which this tangent touches the boundary of $\mathcal{U}$
corresponds to states which realize equality in (2). Those states are called
minimal-uncertainty states. Given all those tangents, i.e. $c(\lambda,\mu)$
for all positive $(\lambda,\mu)$, we can intersect all corresponding
halfspaces and get a convex set which we call the positive convex hull of
$\mathcal{U}$, denoted by $\overline{\mathcal{U}}$ in the following.
Geometrically, the positive convex hull can be constructed by taking the
convex hull of $\mathcal{U}$ and adding to it all points that have bigger
uncertainties then, at least, some point in $\mathcal{U}$.

If $\mathcal{U}$ is convex, like in the example above,
$\overline{\mathcal{U}}$ contains the full information on the relevant parts
of $\mathcal{U}$. If $\mathcal{U}$ is not convex, $\overline{\mathcal{U}}$
still gives a variety of state independent uncertainty relations, but there is
still place for finding improvements, see [34].

##  3 Additivity, implications and applications

We are now able to state our main result

######  Proposition 1 (Additivity of linear uncertainty relations).

Let $c_{A}(\lambda,\mu)$ and $c_{B}(\lambda,\mu)$ be state-independent lower
bounds on the linear entropic uncertainty for local measurements $X_{A},X_{B}$
and $Y_{A},Y_{B}$, with weights $(\lambda,\mu)$. This means we have that

$\displaystyle\lambda H(X_{A}|\rho_{A})+\mu H(Y_{A}|\rho_{A})\geq
c_{A}(\lambda,\mu)$ $\displaystyle\lambda H(X_{B}|\rho_{B})+\mu
H(Y_{B}|\rho_{B})\geq c_{B}(\lambda,\mu)$ (4)

holds on any state $\rho_{A}$ from $\mathcal{B}(\mathcal{H}_{A})$ and
$\rho_{B}$ from $\mathcal{B}(\mathcal{H}_{B})$. Let $X_{AB}$ and $Y_{AB}$ be
the joint global measurements that arise from locally performing $X_{A},X_{B}$
and $Y_{A},Y_{B}$ respectively. Then

$\displaystyle\lambda H(X_{AB}|\rho_{AB})+\mu H(Y_{AB}|\rho_{AB})\geq
c_{A}(\lambda,\mu)+c_{B}(\lambda,\mu)$ (5)

holds for all states $\rho_{AB}$ from
$\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$. Furthermore, if $c_{A}$
and $c_{B}$ are optimal bounds, then

$\displaystyle c_{AB}(\lambda,\mu):=c_{A}(\lambda,\mu)+c_{B}(\lambda,\mu)$ (6)

is the optimal bound in (5), i.e. linear entropic uncertainty relations are
additive.

The proof of this proposition is placed at the end of Sec. 5. We will proceed
this section by collecting some remarks related to the above proposition:

######  Remark 1 (Product states).

Assume that $c_{A}(\lambda,\mu)$ and $c_{B}(\lambda,\mu)$ are optimal
constants, and $\phi_{A}$ and $\phi_{B}$ are the states that saturate the
corresponding uncertainty relations (1). Then the product state
$\phi_{AB}:=\phi_{A}\otimes\phi_{B}$ saturates (5), due to the additivity of
the Shannon-entropy. However, this does not imply that all states that
saturate (1) have to be product states. Examples for this, involving MUBs of
product form, are provided in [34].

######  Remark 2 (Minkowski sums of uncertainty regions).

Prop. 1 shows how the uncertainty set $\mathcal{U}_{AB}$, of the product
measurement, relates to the uncertainty sets $\mathcal{U}_{A}$ and
$\mathcal{U}_{B}$ of corresponding local measurements: For the case of an
optimal $c_{AB}(\lambda,\mu)$, and fixed $(\lambda,\mu)$, equality in (5) can
always be realized by product states (see Rem. 1). In an uncertainty diagram,
like Fig. 3, those states correspond to points on the lower-left boundary of
an uncertainty set, and, in general, they produce the finite extreme points of
the positive convex hull of an uncertainty set.

Figure 3:  Uncertainty sets of local measurements can be combined by the
Minkowski sum: Uncertainty sets (green and yellow) for two pairs of local
measurements on Qubits and the uncertainty set of the corresponding global
measurements (blue).

For product states we have the additivity of the Shannon entropy, which gives

$\displaystyle\begin{pmatrix}H(X_{AB}|\phi_{A}\otimes\phi_{B})\\\
H(Y_{AB}|\phi_{A}\otimes\phi_{B})\end{pmatrix}=\begin{pmatrix}H(X_{A}|\phi_{A})\\\
H(Y_{A}|\phi_{A})\end{pmatrix}+\begin{pmatrix}H(X_{B}|\phi_{B})\\\
H(Y_{B}|\phi_{B})\end{pmatrix}$ (7)

This implies that we can get every extreme point of
$\overline{\mathcal{U}}_{AB}$ by taking the sum of two extreme points of
$\overline{\mathcal{U}}_{A}$ and $\overline{\mathcal{U}}_{B}$. Due to
convexity the same holds for all points in $\overline{\mathcal{U}}_{AB}$ and
we can get this set as Minkowski sum [35].

$\displaystyle\overline{\mathcal{U}}_{AB}=\overline{\mathcal{U}}_{AB}\boxplus\overline{\mathcal{U}}_{B}$
(8)

For convex uncertainty regions, arising from local measurements, this is
depicted in Fig. 3. For this example, it is also true that $\mathcal{U}_{AB}$
itself is given as Minkowski sum of local uncertainty sets. However, we have
to note, this behavior cannot be concluded from Prop. 1 alone.

######  Remark 3 (Relation to existing entanglement witnesses).

A well know method for constructing non-linear entanglement witnesses is based
on computing the minimal value of a functional, like the sum of uncertainties
[36, 37, 38], attainable on separable states. Given an unknown quantum state,
the value of this functional is measured. If the measured value undergoes the
limit set by separable states, the presence of entanglement is witnessed. For
uncertainty relations based on the sum of general Schur concave functionals
this method was proposed in [4], including Shannon entropy, i.e. the l.h.s. of
(5), as central example.

Our result Prop. 1 shows that this method will not work for Shannon entropies,
because there is no entangled state that undergoes the limit set by separable
states. We note that there is no mathematical contradiction between Prop. 1
and [4]. We only show that the set of examples for the method proposed in [4]
is empty.

For uncertainty relations in terms of Shannon, Tsallis and Renyi entropies a
similar procedure for constructing witnesses was proposed by [39, 37]. Here
explicit examples for states, that can be witnessed to be entangled, were
provided. Again, our proposition Prop. 1 is not in contradiction to this work
because in [39, 37] observables with a non-local degeneracy where considered.

Prop. 1 can easily be generalized to a multipartite setting, see Fig. 4 :

Figure 4:  Multiparite setting: Additivity of entropic uncertainty relations
also holds if a pair of global product measurements for many local parties is
considered.

######  Corollary 1 (Generalization to multipartite measurements).

Assume parties $A_{1}\ldots A_{n}$ that locally perform measurements,
$X_{A_{1}},\ldots,X_{A_{n}}$ or $Y_{A_{1}},\ldots,Y_{A_{n}}$, with weights
$\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{n})$. In analogy to (1), let
$c_{A_{1}}(\vec{\lambda}),\ldots,c_{A_{n}}(\vec{\lambda})$ denote optimal
local bounds and let $c_{A_{1}\cdots A_{n}}(\vec{\lambda})$ be the optimal
bound corresponding to product measurements $X_{A_{1}\ldots A_{n}}$ and
$Y_{A_{1}\ldots A_{n}}$. We have

$\displaystyle c_{A_{1}\ldots
A_{n}}(\vec{\lambda})=\sum_{i=1}^{n}c_{A_{i}}(\vec{\lambda})$ (9)

This follows by iterating (6).

######  Remark 4 (Generalization to three measurements).

The generalization of Prop. 1 to three measurements, say $X_{AB}$, $Y_{AB}$
and $Z_{AB}$, fails in general. The following counterexample was provided by
O. Gühne [40]: For both parties we consider local measurements deduced from
the three Pauli operators on a qubit and take all weights equal to one. In
short hand notation we write $X_{AB}=\sigma_{X}\otimes\sigma_{X}$,
$Y_{AB}=\sigma_{Y}\otimes\sigma_{Y}$, and
$Z_{AB}=\sigma_{Z}\otimes\sigma_{Z}$. In this case, the minimal local
uncertainty sum is attained on eigenstates of the Pauli operators. If such a
state is measured, the entropy for one of the measurements is zero and maximal
for the others. Hence, the local uncertainty sum is always bigger than $2$
[bit]. Therefore we have

$\displaystyle
H\left(\sigma_{X}\otimes\sigma_{X}|\phi_{A}\otimes\phi_{B}\right)+$
$\displaystyle
H\left(\sigma_{Y}\otimes\sigma_{Y}|\phi_{A}\otimes\phi_{B}\right)+$
$\displaystyle
H\left(\sigma_{Z}\otimes\sigma_{Z}|\phi_{A}\otimes\phi_{B}\right)\geq 4$ (10)

for all product states. In contrast to this a Bell state, say $\Psi^{-}$, will
give the entropy of $1$[bit], for all above measurements. Hence we have,

$\displaystyle H\left(\sigma_{X}\otimes\sigma_{X}|\Psi^{-}\right)+$
$\displaystyle H\left(\sigma_{Y}\otimes\sigma_{Y}|\Psi^{-}\right)+$
$\displaystyle H\left(\sigma_{Z}\otimes\sigma_{Z}|\Psi^{-}\right)=3\ngeq 4.$
(11)

##  4 Lower bounds from $(p,q)$-norms

The quite standard technique for analyzing a linear uncertainty relation is to
connect it to the $(p,q)$-norm (see (12) below) of the basis transformation
$U$. Thereby, the majority of previous works in this field is concentrating
only on handling the case of equal weights $\lambda=\mu=1$, which is connected
to the $(p,q)$-norm for the case $1/p+1/q=1$. However, for the purpose of this
work, i.e. for proving Prop. 1, we have to extend this connection to arbitrary
$(\lambda,\mu)$. We will do this by Thm. 1 on the next page.

A historically important example for the use of the connection between
$(p,q)$-norms and entropic uncertainties, is provided by Bialynicki-Birula and
Mycielski [41]. They used Beckner’s result [42], who computed the $(p,q)$-norm
of the Fourier-Transfromation, for proving the corresponding uncertainty
relation, between position and momentum, conjectured by Hirschmann [43]. Also
Maassen and Uffink [21] took this way for proving their famous relation. Our
result gives a direct generalization of this, meaning we will recover the
Maassen and Uffink relation at the end of this section as special case of
(50). Albeit, before stating our result, we will start this section by shortly
reviewing the previously known way for connecting $(p,q)$-norms with linear
uncertainty relation, see also [44, 45] for further details:

The $(p,q)$-norm, i.e the $l^{p}\rightarrow l^{q}$ operator norm, of a basis
transformation $U$ is given by

$\displaystyle\|U\|_{q,p}:=\sup_{\phi\in\mathcal{H}}\frac{\|U\phi\|_{q}}{\|\phi\|_{p}}.$
(12)

Here, the limit of $\|U\|_{q,p}$ for $(p,q)\rightarrow(2,2)$ goes to $1$.
However, when $p$ and $q$ are fixed on the curve $1/p+1/q=1$, the leading
order of $\|U\|_{q,p}$ around $(p,q)=(2,2)$ recovers the uncertainty relation
(2) in the case of equal weights $\lambda=\mu=1/2$, see [43, 41].

More precisely, taking the negative logarithm of (12) gives

$\displaystyle-\log\|U\|_{q,p}=\inf_{\phi\in\mathcal{H}}\;\log\|\phi\|_{p}-\log\|U\phi\|_{q}\;.$
(13)

Here, we can identify the squared modulus of the components of $\phi$ as
probabilities of the $X$ and $Y$ measurement outcomes

$\displaystyle|(\phi)_{i}|^{2}$
$\displaystyle=\bra{\phi}X_{i}\ket{\phi}=(\mathbf{p}_{\phi}^{X})_{i}$
$\displaystyle|(U\phi)_{i}|^{2}$
$\displaystyle=\bra{\phi}Y_{i}\ket{\phi}\;=(\mathbf{p}_{\phi}^{Y})_{i}$ (14)

and substitute

$\displaystyle\|\phi\|_{p}=\left(\|\mathbf{p}_{\phi}^{X}\|_{p/2}\right)^{2}\text{\;
and \;}\|U\phi\|_{q}=\left(\|\mathbf{p}_{\phi}^{Y}\|_{q/2}\right)^{2}\;.$ (15)

By this, (13) gives a linear relation in terms of the $\alpha$-Renyi entropy
[46],
$H_{\alpha}(\mathbf{p})=\frac{\alpha}{1-\alpha}\log(\|\mathbf{p}\|_{\alpha})$.
Here we get

$\displaystyle\inf_{\phi\in\mathcal{H}}\frac{2-p}{p}H_{p/2}(X|\phi)-\frac{2-q}{q}H_{q/2}(Y|\phi)=-\log\|U\|_{q,p}^{2}\;.$
(16)

If we evaluate this on the curve $1/p+1/q=1$, for $p\leq 2\leq q$, we can use

$\displaystyle\frac{2-p}{p}=\frac{1}{p}-\frac{1}{q}=\frac{q-2}{q},$ (17)

which can be employed to (16), in order to get

$\displaystyle\inf_{\phi\in\mathcal{H}}\;H_{p/2}(X|\phi)+H_{q/2}(Y|\phi)=\left(\frac{1}{q}-\frac{1}{p}\right)^{-1}\log\|U\|_{q,p}^{2}\;.$
(18)

Here, the limit $(p,q)\rightarrow(2,2)$, in the l.h.s of (18), gives the limit
from the Renyi to the Shannon entropy. This gives the l.h.s. of the
uncertainty relation (2) for $\lambda=\mu=1$. Hence, the functional dependence
of $\|U\|_{q,p}$ on $(p,q)$ in the limit $(p,q)\rightarrow(2,2)$ gives the
optimal bound $c(1,1)$, in (2). For the case of the
$\mathcal{L}^{2}(\mathbb{R})$-Fourier transformation the norm
$\|U_{\mathcal{F}}\|_{q,p}=\sqrt{p^{1/p}}/\sqrt{q^{1/q}}$ was computed by
Beckner [42], leading to $c(1,1)=\log(\pi\rm e)$. However, to the best of our
knowledge, computing $\|U\|_{q,p}$, for general $U$ and $(p,q)$, is an
outstanding problem, and presumably very hard [47, 48]. Albeit, for special
choices of $(p,q)$ this problem gets treatable, see [49] for a list of those.
The known cases include $p=q=2$, $p=\infty$ or $q=\infty$ such as $p=1$ or
$q=1$.

The central idea of Maassen’s and Uffink’s work [21] is to show that the easy
case of $(p=1,q=\infty)$, here we have $\|U\|_{1,\infty}=\max_{ij}|U_{ij}|$,
gives a lower bound on $c(1,1)$. More precisely, they show that, for $1\leq
p\leq 2$ and on the line $1/p+1/q=1$, the r.h.s. of (18) approaches $c(1,1)$
from below. Note that this is far from being obvious. Explicitly, for $p\leq
2\leq q$ we have $H_{q/2}(Y|\phi)\geq H(Y|\phi)$ and $H_{p/2}(X|\phi)\leq
H(X|\phi)$, so one term approaches the limit from above and the other
approaches the limit from below. Whereas Maassen and Uffink showed, using the
Riesz-Thorin interpolation [50, 51], that the $\inf_{\phi}$ of the sum of both
approaches the limit from below.

The following Theorem, Thm.1, extends the above to the case of arbitrary
$(\lambda,\mu)$. Notably, we have to take $(p,q)$ from curves with
$1/p+1/q\neq 1$, those are depicted in Fig. 5. In contrast to Maassen and
Uffink, the central inequality we use is the $\infty$-norm versions of the
Golden Thompson inequality (see [52, 53, 54] and the blog of T.Tao [55] for a
proof and related discussions).

######  Theorem 1.

Let $c(\lambda,\mu)$, with $\lambda,\mu\in\mathbb{R}_{+}$, be the optimal
constant in the linear weighted entropic uncertainty relation

$\displaystyle c(\lambda,\mu):=\inf_{\rho}\lambda H\left(X|\rho\right)+\mu
H\left(Y|\rho\right).$ (19)

Then:

(i) $c(\lambda,\mu)$ is bounded from below by
$-N\log\left(\omega_{N}(\lambda,\mu)\right)$

$\displaystyle\text{with
\quad}\omega_{N}(\lambda,\mu)=\sup_{\begin{subarray}{c}\mathbf{x}\in
B_{r}(\mathbb{C}^{d})\\\ \mathbf{y}\in
B_{s}(\mathbb{C}^{d})\end{subarray}}\left|\mathbf{x}^{\dagger}U\mathbf{y}\right|$
(20) $\displaystyle\text{and \quad}r=\frac{2N}{N+2\lambda}\quad
s=\frac{2N}{N+2\mu}$ (21)

where

$B_{r}(\Omega):=\left\\{\mathbf{x}\in\Omega|\;1\geq\|\mathbf{x}\|_{r}\right\\}$

denotes the unit $r$-norm Ball on $\Omega$.

(ii) For $\lambda,\mu\leq N/2$ we can write

$\displaystyle\omega_{N}(\lambda,\mu)=\sup_{\phi\in\mathbb{C}^{d}}\frac{\|U\phi\|_{r^{\prime}}}{\|\phi\|_{s}}=\sup_{\phi\in\mathbb{C}^{d}}\frac{\|U\phi\|_{s^{\prime}}}{\|\phi\|_{r}}$
(22) $\displaystyle\text{ with }r^{\prime}=\frac{2N}{N-2\lambda}\quad
s^{\prime}=\frac{2N}{N-2\mu}$ (23)

(iii) For $\mu,\lambda\in\mathbb{R}^{+}\backslash\\{0\\}$, we have

$\displaystyle
c(\lambda,\mu)=\lim_{N\rightarrow\infty}-N\log\left(\omega_{N}(\lambda,\mu)\right)$
(25)

$4$$3$$2$$0$$1$$2$$r^{\prime}$$s$$\lambda/\mu=1$$\lambda/\mu=0.1$$\mu/\lambda=0.1$
Figure 5: Evaluating $\|U\|_{r^{\prime},s}$ on the depicted curves gives a
lower bound for $c(\lambda,\mu)$, (see Thm. 1). Because $c(\lambda,\mu)$ is a
linear bound it is $1$-homogenious in $(\lambda,\mu)$. Hence all information
on the optimal bound $c(\lambda,\mu)$ can be recovered by knowing it for any
fixed ratio $\lambda/\mu$. The thick red curve corresponds to the case
$1/{r^{\prime}}+1/s=1$ which gives bounds $c(1,1)$ from below. For $s=1$ the
norm $\|U\|_{r^{\prime},s=1}$ can be computed analytically, this gives a
generalization of the Masssen and Uffink bound (see Lem. 1).

###### Proof.

The starting point of this proof is a modification of a technique, used by
Frank and Lieb in [56], for reproving the Maassen and Uffink bound (see also
the talk of Hans Maassen [44], for a finite dimensional version).

For probability distributions $\mathbf{p},\mathbf{q}\in
B_{1}(\mathbb{R}^{d}_{+})$ we define the operators

$\displaystyle A(\mathbf{p}):=-\sum X_{i}\log(p_{i})\text{ and
}B(\mathbf{q}):=-\sum Y_{i}\log(q_{i})$ (26)

such that we can rewrite the Shannon entropy as

$\displaystyle H\left(X|\rho\right)=\operatorname{tr}(\rho
A(p^{X}_{\rho}))\text{ and }H\left(Y|\rho\right)=\operatorname{tr}(\rho
B(p^{Y}_{\rho}))$ (27)

Based on this, we can further rewrite the Shannon entropy as an optimization
over a linear function in $\rho$ by using the positivity of the relative
entropy, i.e. we have $D(\mathbf{p}||\mathbf{q})=\sum p_{i}\log(p_{i})-\sum
p_{i}\log(q_{i})\geq 0$, which implies $-\sum p_{i}\log(q_{i})\geq
H(\mathbf{p})$. We obtain

$\displaystyle H\left(X|\rho\right)=\inf_{\mathbf{p}}\operatorname{tr}(\rho
A(\mathbf{p})),$ (28)

such as the respective statement for $H\left(Y|\rho\right)$ and
$B(\mathbf{q})$. If we employ this rewriting to $c(\lambda,\mu)$, we obtain
the minimal entropy sum as a minimization over a parametrized eigenvalue
problem, namely

$\displaystyle c(\lambda,\mu)$ $\displaystyle=\inf_{\rho}\lambda
H\left(X|\rho\right)+\mu H\left(Y|\rho\right)$
$\displaystyle=\inf_{\mathbf{p},\mathbf{q},\rho}\operatorname{tr}\left(\rho\left(\lambda
A(\mathbf{p})+\mu B(\mathbf{q})\right)\right)$ (29)

Now we will turn the minimization, over $\rho$, into a maximization by
applying the convex function $e^{-x/N}$, with $N\geq 1$, to the weighted sum
of $A$ and $B$. This will map the smallest eigenvalue of $\lambda A+\mu B$ to
the largest of $e^{-\frac{\lambda A(p)+\mu B(q)}{N}}$ and so on. In order to
get back the correct value of $c$ we will have to apply the inverse function,
$-N\log(x)$, afterwards. We get

$\displaystyle
c(\lambda,\mu)=-N\log\left(\sup_{\mathbf{p},\mathbf{q},\rho}\operatorname{tr}\left(\rho
e^{-\frac{\lambda A(p)+\mu B(q)}{N}}\right)\right).$ (30)

Due to the positivity of the operator exponential, i.e. $A$ and $B$ are
hermitian, the optimization over $\rho$ is equivalent to the Schatten-$\infty$
norm. We have

$\displaystyle
c(\lambda,\mu)=-N\log\left(\sup_{\mathbf{p},\mathbf{q}}\left\|e^{-\frac{\lambda
A(p)+\mu B(q)}{N}}\right\|_{\infty}\right)$ (31)

At this point we apply the Golden-Thompson inequality

$\displaystyle\|e^{S+T}\|_{p}\leq\|e^{S}e^{T}\|_{p}$ (32)

and expand the resulting exponentials, as well as the Schatten norm. We get

$\displaystyle c(\lambda,\mu)$
$\displaystyle\geq\text{-}N\log\left(\sup_{\mathbf{p},\mathbf{q}}\left\|e^{-\frac{\lambda
A(p)}{N}}e^{-\frac{\mu B(q)}{N}}\right\|_{\infty}\right)$ (33)
$\displaystyle=\text{-}N\log\left(\sup_{\mathbf{p},\mathbf{q}}\left\|\sum_{ij}X_{i}p_{i}^{\lambda/N}Y_{j}q_{j}^{\mu/N}\right\|_{\infty}\right)$
(34)
$\displaystyle=\text{-}N\log\left(\sup_{\begin{subarray}{c}\mathbf{p},\mathbf{q}\\\
\ket{x},\ket{y}\end{subarray}}\bigl{\langle}x\bigl{|}\sum_{ij}X_{i}p_{i}^{\lambda/N}Y_{j}q_{j}^{\mu/N}\bigr{|}y\bigr{\rangle}\right)$
(35)

Now we substitute $p_{i}^{\lambda/N}=:\chi_{i}$ and $q_{j}^{\mu/N}=:\xi_{j}$,
and expand $\ket{x}=\sum x_{i}\ket{e_{i}}$ and $\ket{y}=\sum
y_{j}\ket{f_{j}}$, with component vectors $\mathbf{x},\mathbf{y}\in
B_{2}(\mathbb{C}^{d})$. By this the r.h.s of (35) becomes

$\displaystyle-N\log\left(\sup_{\mathbf{\chi},\mathbf{x}}\sup_{\mathbf{\xi},\mathbf{y}}\left|\sum_{ij}\chi_{i}x_{i}\braket{e_{i}}{f_{j}}\xi_{j}y_{j}\right|\right).$
(36)

Here we can identify $\braket{e_{i}}{f_{j}}=U_{ij}$, i.e. the overlaps are the
components of $U$ when represented in the basis $X$. At this point, it is
straightforward to check that $\chi\in B_{N/\lambda}(\mathbb{R}^{d}_{+})$ and
$\xi\in B_{N/\mu}(\mathbb{R}^{d}_{+})$. Using the generalized Hölder
inequality we can fuse some of the maximizations above as follows: On one
hand, we have

$\displaystyle\left(\sum|\chi_{i}x_{i}|^{r}\right)^{\frac{1}{r}}$
$\displaystyle\leq\|\mathbf{x}\|_{2}\|\chi\|_{N/\lambda}\leq 1$ (37) and
$\displaystyle\left(\sum|\xi_{j}y_{j}|^{s}\right)^{\frac{1}{s}}$
$\displaystyle\leq\|\mathbf{y}\|_{2}\|\xi\|_{N/\mu}\leq 1$ for
$\displaystyle\frac{1}{r}=\frac{1}{2}+\frac{\lambda}{N}$
$\displaystyle\quad\text{and}\quad\frac{1}{s}=\frac{1}{2}+\frac{\mu}{N}\quad,$
(38)

which means that the vectors $\mathbf{v}$ and $\mathbf{w}$, with
$v_{i}=\chi_{i}x_{i}$ and $w_{j}=\xi_{j}y_{j}$, are in $B_{r}(\mathbb{C})$ and
$B_{s}(\mathbb{C})$ respectively.

On the other hand, the converse is also true, i.e. every $\mathbf{v}$ and
$\mathbf{w}$ from $B_{r}(\mathbb{C}^{d})$ and $B_{s}(\mathbb{C}^{d})$ can be
realized by suitable choices of $\mathbf{x},\chi$ and $\mathbf{y},\xi$. For
example, we can always set

$\displaystyle x_{i}=|v_{i}|^{r\lambda/N}\text{ and
}\chi_{i}=|v_{i}|^{2/r}e^{i\operatorname{arg}(v_{i})}$ (39)

for getting $\mathbf{x}$ and $\chi$ from $\mathbf{v}$, componentwise. For this
particular choice we can check that

$\displaystyle x_{i}\chi_{i}$
$\displaystyle=|v_{i}|^{r(\lambda/N+1/2)}e^{i\operatorname{arg}(v_{i})}$
$\displaystyle=|v_{i}|^{r/r}e^{i\operatorname{arg}(v_{i})}=v_{i}$ (40)

holds, such that we will get back $\mathbf{v}$. Furthermore $\mathbf{x}\in
B_{N/\lambda}(\mathbb{R}_{+}^{d})$ and $\chi\in B_{2}(\mathbb{C}^{d})$ follows
by writing out

$\displaystyle\sum_{i}x_{i}^{N/\lambda}=\sum_{i}v_{i}^{r}\leq 1\text{ and
}\sum_{i}\chi_{i}^{2}=\sum_{i}v_{i}^{r}\leq 1.$ (41)

If we use the above in (36), we can replace $\sup_{x,\chi}$ by
$\sup_{\mathbf{v}}$ and $\sup_{y,\xi}$ by $\sup_{\mathbf{w}}$, in order to get
the statement (i) with

$\displaystyle\omega_{N}:=\sup_{\begin{subarray}{c}\mathbf{v}\in
B_{r}(\mathbb{C}^{d})\\\ \mathbf{w}\in
B_{s}(\mathbb{C}^{d})\end{subarray}}\left|\mathbf{v}^{\dagger}U\mathbf{w}\right|\quad.$
(42)

For showing the statement (ii), we take $r^{\prime}$, with
$1=1/r+1/r^{\prime}$. If $\lambda\leq N/2$ holds we have $r^{\prime}\geq 0$
and we can use the tightness of the Hölder inequality to rewrite

$\displaystyle\sup_{\mathbf{v}\in
B_{r}}\left|\mathbf{v}^{\dagger}U\mathbf{w}\right|=\|U\mathbf{w}\|_{r^{\prime}}\;,$
(43)

i.e. the maximization over $B_{r}$ gives the dual norm of $r$. Substituting
$\mathbf{w}$ by $\phi=\mathbf{w}\|\phi\|_{s}$ then gives

$\displaystyle\omega_{N}=\sup_{\phi\in\mathbb{C}^{d}}\frac{\|U\phi\|_{r^{\prime}}}{\|\phi\|_{s}}$
(44)

Here the analogous rewriting applies with $s^{\prime}$ given by
$1=1/s+1/{s^{\prime}}$, if $\mu\leq\lambda/2$ holds.

For showing (iii), i.e.

$\displaystyle c=\lim_{N\rightarrow\infty}-N\log(\omega_{N})\quad,$ (45)

it suffices to expand all exponentials in (31) and (33) up to the first order
in $N$. On this order the Golden-Thomson inequality is a equality. ∎

######  Remark 5 (The Maassen and Uffink bound).

For the case of $N=2$ and $\lambda=\mu=1$, in Thm.1, we get $s=r=1$ and
$s^{\prime}=r^{\prime}=\infty$. Hence, we recover the Maassen-Uffink bound
[21]. Explicitly, we have

$\displaystyle\omega_{2}(1,1)=\sup_{\begin{subarray}{c}\mathbf{x}\in
B_{1}(\mathbb{C}^{d})\\\ \mathbf{y}\in
B_{1}(\mathbb{C}^{d})\end{subarray}}\left|\mathbf{x}^{\dagger}U\mathbf{y}\right|=\max_{ij}\left|U_{ij}\right|\;.$
(46)

Here we used that $\left|\mathbf{x}^{\dagger}U\mathbf{y}\right|$ is convex in
$\mathbf{x}$ and $\mathbf{y}$. Hence, $\sup_{\mathbf{x},\mathbf{y}}$ is
attained at the extreme points of $B_{1}(\mathbb{C}^{d})$. Up to a phase,
those extreme points have the form $(0,\cdots,0,1,0\cdots,0)$ , i.e. they have
their support only on a single site. So, choosing $\mathbf{x}$ and
$\mathbf{y}$, with support on the $i-th$ and $j-th$ site, will give
$\left|\mathbf{x}^{\dagger}U\mathbf{y}\right|=\left|U_{ij}\right|$.

######  Remark 6 (Renyi-Entropies).

Alternatively, the bound obtained in Thm. 1 can be expressed in terms of
Renyi-entropies: Using statement (i), (ii) and (iii) together directly gives

$\displaystyle c(\lambda,\mu)$ $\displaystyle\geq-N\log\|U\|_{r^{\prime},s}$
$\displaystyle=\inf_{\phi\in\mathcal{H}}N\log\|\phi\|_{r}-N\log\|U\phi\|_{s^{\prime}}.$
(47)

Here a straightforward computation shows

$\displaystyle\frac{2-r}{r}=\lambda/N\text{ and
}\frac{2-s^{\prime}}{s^{\prime}}=-\mu/N.$ (48)

So, when we proceed as in (13), substituting the Renyi entropy in (6) gives

$\displaystyle c(\lambda,\mu)\geq\inf_{\phi\in\mathcal{H}}\lambda
H_{r/2}(X|\phi)+\mu H_{s^{\prime}/2}(Y|\phi).$ (49)

######  Lemma 1 (Generalization of the Maassen and Uffink bound).

Let $\mathbf{u}_{i}$ denote the $i$-th column of the basis transformation $U$
that links the measurements $X$ and $Y$. Then, for $1\geq\lambda\geq\mu\geq 0$
and all states $\rho$ we have

$\displaystyle\lambda H(X|\rho)+\mu
H(Y|\rho)\geq-2\lambda\log\left(\sup_{i=1\cdots
d}\bigl{\|}\mathbf{u}_{i}\bigr{\|}_{t}\right).$ (50)

with

$\displaystyle t=\frac{2}{(1-\mu/\lambda)}$ (51)

Note that for the case $1\geq\mu\geq\lambda\geq 0$ the same holds, if $U$ is
replaced by $U^{\dagger}$, i.e. by the transformation between $Y$ and $X$.

###### Proof.

The linear uncertainty bound $c(\lambda,\mu)$ is homogeneous in
$(\lambda,\mu)$. Hence, we can consider

$\displaystyle c(\lambda,\mu)=\lambda c(1,\mu/\lambda)$ (52)

We will apply Thm. 1, with $N=2$, in order to get a lower bound. Here, we have
$s=\frac{2}{1+\mu/\lambda}$ and

$\displaystyle\omega_{2}(1,\mu/\lambda)=\sup_{\begin{subarray}{c}\mathbf{x}\in
B_{1}(\mathbb{C}^{d})\\\ \mathbf{y}\in
B_{s}(\mathbb{C}^{d})\end{subarray}}\left|\mathbf{x}^{\dagger}U\mathbf{y}\right|=\sup_{\begin{subarray}{c}i=1,\cdots,d\\\
\mathbf{y}\in
B_{s}(\mathbb{C}^{d})\end{subarray}}\left|\mathbf{u}_{i}\;\mathbf{y}\right|.$
(53)

Here the second equality stems from the same argumentation as in Rem. 5. The
$\sup$ over $B_{s}(\mathbb{C}^{d})$ on the most right of (53), gives the norm
dual to $s$, given by $t=\frac{2}{1-\mu/\lambda}$. All in all we have,

$\displaystyle c(1,\mu/\lambda)$
$\displaystyle\geq-2\log\left(\omega(1,\mu/\lambda)\right)$
$\displaystyle=-2\log\left(\sup_{i=1\cdots
d}\bigl{\|}\mathbf{u}_{i}\bigr{\|}_{t}\right)$ (54)

∎

######  Remark 7 (More than two observables).

As mentioned in Sec. 3, the proposition Prop. 1 does not generalize to three
measurements. A reasoning, or at least a hint, for this can be found by
carefully following the proof of Thm. 1. In principle, the ansatz in (4) can
be generalized to more than two measurements as well, and all following steps
work out in a similar way, up to (33). Here the Golden-Thompson inequality was
used. It is well known, that the direct generalization of this inequality to
three operators fails to hold. Hence, the technique of our proof cannot be
generalized for this case. We note that there is an ongoing work of exploring
more sophisticated generalizations of this inequality [57, 58, 59, 60].
However, we leave relating this to entropic uncertainty for future work.

##  5 Additivity of bounds from multiplicativity of $(p,q)$-norms

In this section we will provide the proof of Prop. 1, i.e. the additivity of
linear uncertainty relations. By using Thm.1 from the section before we can
formulate the linear uncertainty in terms of the logarithm of a $(p,q)$-norm.
At this point, it is straightforward to check that the additivity of the
linear uncertainty is equivalent to the multiplicativity of the $(p,q)$-norm.
In fact, the following theorem Thm.2 provides that, for $p$ and $q$ coming
from the correct range: The $(p,q)$ norm of a transformation which admits a
product form $U_{AB}=U_{A}\otimes U_{B}$ is multiplicative.

######  Theorem 2 (Global bounds from local bounds).

Let $X_{AB}$ and $Y_{AB}$ be tensor-product bases of a Hilbert space
$\mathcal{H}_{AB}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, i.e. we have
$X_{AB}=\\{X_{A}^{i}\otimes X_{B}^{i}\\}_{i=1,\cdots,d}$ and
$Y_{AB}=\\{Y_{A}^{i}\otimes Y_{B}^{i}\\}_{i=1,\cdots,d}$, such as
$U_{AB}=U_{A}\otimes U_{B}$. Furthermore let $\eta_{A}$ and $\eta_{B}$ denote
the optimal constants for

$\displaystyle\|U_{A}\phi\|_{q}\leq\eta_{A}\|\phi\|_{p}\quad\forall\phi\in\mathcal{H}_{A}$
$\displaystyle\|U_{B}\phi\|_{q}\leq\eta_{B}\|\phi\|_{p}\quad\forall\phi\in\mathcal{H}_{B}\;.$
(55)

If $1\leq p\leq q$ then

$\displaystyle\|U_{AB}\;\phi\|_{q}\leq\eta_{A}\eta_{B}\|\phi\|_{p}\quad\forall\phi\in\mathcal{H}_{AB}$
(56)

holds with $\eta_{A}\eta_{B}=\eta_{AB}$ as optimal constant.

###### Proof.

We note that a related result, for pointwise positive maps between Lebesque
spaces, was discovered by Grey and Sinnamon [61].

The basic object of this proof will be the $p\otimes q$-norm which will be
defined immediately. The basic work of this proof is devoted to show some
properties of this norm from which the statement directly follows.

Let $\ket{\phi}\in\mathcal{H}$ with components $\phi=\\{\phi_{ij}\\}$ sorted
within the product base $X_{AB}$ by $\phi_{ij}=\braket{\phi}{e^{A}_{i}\otimes
e^{B}_{j}}$ and consider the norm

$\displaystyle\left\|\phi\right\|_{q\otimes
p}:=\left(\sum_{i}\left(\sum_{j}\left|\phi_{ij}\right|^{p}\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}\;.$
(57)

This norm shares the following properties

$\displaystyle(i)$ $\displaystyle\left\|\phi\right\|_{q\otimes q}$
$\displaystyle=\left\|\phi\right\|_{q}$ (58) $\displaystyle(ii)$
$\displaystyle\left\|(\mathbb{I}\otimes V\phi)\right\|_{r\otimes q}$
$\displaystyle\leq\left\|\phi\right\|_{r\otimes p}\eta_{V}$ (59)
$\displaystyle(iii)$ $\displaystyle\left\|\phi\right\|_{q\otimes p}$
$\displaystyle\leq\left\|\mathbb{F}\phi\right\|_{p\otimes q}$ (60) with
$\displaystyle\mathbb{F}\phi_{1}\otimes\phi_{2}=\phi_{2}\otimes\phi_{1}$
$\displaystyle\text{ and }p\leq q.$

We will show the validity of $(i-iii)$ in a moment. First notice that, if
$(i-iii)$ are valid we can easily conclude

$\displaystyle\|U_{AB}\phi\|_{q}$ $\displaystyle=\|U_{A}\otimes
U_{B}\phi\|_{q\otimes q}$ $\displaystyle=\|(\mathbb{I}\otimes
U_{B})(U_{A}\otimes\mathbb{I})\phi\|_{q\otimes q}$
$\displaystyle\leq\eta_{B}\|U_{A}\otimes\mathbb{I}\phi\|_{q\otimes p}$
$\displaystyle\leq\eta_{B}\|\mathbb{I}\otimes U_{A}\mathbb{F}\phi\|_{p\otimes
q}$ $\displaystyle\leq\eta_{B}\eta_{A}\|\mathbb{F}\phi\|_{p\otimes p}$
$\displaystyle=\eta_{B}\eta_{A}\|\phi\|_{p}\;.$ (61)

Furthermore, if we consider states that realize equality in (2), i.e. states
that belong to optimal $\eta_{A}$ and $\eta_{B}$. The tensor-product of two of
those states will realize, due to multiplicativity of the $p$-norm, equality
in (56) as well. Hence, (5) will prove the main statement of this Theorem.

Property $(i)$ follows directly by plugging $p=q$ in the definition of the
$p\otimes q$ norm, here is nothing more to prove. The property $(ii)$ follows
by expressing $\mathbb{I}\otimes V$ as $\delta_{ik}V_{jl}$ in $X$-Basis and

$\displaystyle\left\|(\mathbb{I}\otimes V\phi)\right\|_{r\otimes q}$
$\displaystyle=\left(\sum_{i}\left(\sum_{j}\left|\sum_{lk}\delta_{ik}V_{jl}\phi_{kl}\right|^{q}\right)^{\frac{r}{q}}\right)^{\frac{1}{r}}$
(62)
$\displaystyle=\left(\sum_{i}\left(\sum_{j}\left|\sum_{l}V_{jl}\phi_{il}\right|^{q}\right)^{\frac{r}{q}}\right)^{\frac{1}{r}}$
(63)
$\displaystyle=\left(\sum_{i}\|V\phi_{i}\|_{q}^{\;r}\right)^{\frac{1}{r}}$
(64)
$\displaystyle\leq\eta_{V}\left(\sum_{i}\left(\sum_{j}\left|\phi_{ij}\right|^{p}\right)^{\frac{r}{p}}\right)^{\frac{1}{r}}$
$\displaystyle=\eta_{V}\left\|\phi\right\|_{r\otimes p}\;.$ (65)

As a last step, $(iii)$ is a direct consequence of Minkowski’s inequality /
$l^{p}$-triangle inequality (see [62]), i.e. if $p\geq 1$ :

$\displaystyle\left(\sum_{y}\left|\sum_{x}a_{xy}\right|^{p}\
\right)^{\frac{1}{p}}\leq\sum_{x}\left(\sum_{y}\left|a_{xy}\right|^{p}\right)^{\frac{1}{p}}$
(66)

So, if $1\leq q/p$ we can use this inequality as follows

$\displaystyle\left\|\phi\right\|_{q\otimes p}$
$\displaystyle=\left(\sum_{i}\left(\sum_{j}\left|\phi_{ij}\right|^{p}\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}$
$\displaystyle=\left(\sum_{i}\left|\sum_{j}\left|\phi_{ij}\right|^{p}\right|^{\frac{q}{p}}\right)^{\frac{1}{q/p}\frac{1}{p}}$
$\displaystyle\leq\left(\sum_{j}\left(\sum_{i}\left|\phi_{ij}\right|^{p\frac{q}{p}}\right)^{\frac{p}{q}}\right)^{\frac{1}{p}}=\left\|\mathbb{F}\phi\right\|_{p\otimes
q}$ (67)

and show the validity of $(iii)$. ∎

######  Lemma 2 (Multiplicativity of the $(p,q)$-norm).

For $1\leq p\leq q$, the $(p,q)$-norm of a product unitary
$U_{AB}=U_{A}\otimes U_{B}$ is multiplicative, i.e. we have

$\displaystyle||U_{AB}||_{q,p}=||U_{A}||_{q,p}||U_{B}||_{q,p}.$ (68)

###### Proof.

This is a direct consequence of Thm. 2. Using the definition of the
$(p,q)$-norm we can parse $\eta_{A}=||U_{A}||_{q,p}$,
$\eta_{B}=||U_{B}||_{q,p}$ and $\eta_{AB}=||U_{AB}||_{q,p}$, if we consider
$\eta_{A}$, $\eta_{B}$ and $\eta_{AB}$ to be optimal bounds. ∎

Proof of Prop. 1

###### Proof.

For proving Prop. 1 it suffices to proof the additivity of the optimal case,
i.e. we will consider $c_{A}$, $c_{B}$ and $c_{AB}$ to already be constants
for the best linear uncertainty bound. If the additivity

$\displaystyle c_{AB}=c_{A}+c_{B}$ (69)

holds we can directly conclude that the sum of lower bounds on $c_{A}$ and
$c_{B}$ gives a valid lower bound on $c_{AB}$ as well.

Given measurements $X_{AB}$ and $Y_{AB}$, specified by a product unitary
$U_{AB}=U_{A}\otimes U_{B}$, we use Thm. 1 to rewrite $c_{A}$, $c_{B}$ and
$c_{AB}$ as the limit of logarithms of $(p,q)$-norms. We assume
$\lambda\leq\mu$ both to be finite and $N$ to be sufficiently large such that
we can use Thm. 1 part (ii) (here we needed $\lambda,\mu\leq N/2$), and get

$\displaystyle c_{A}$
$\displaystyle=-\lim_{N\rightarrow\infty}\;\log\left(||U_{A}||_{r,s}\right)$
(70) $\displaystyle c_{B}$
$\displaystyle=-\lim_{N\rightarrow\infty}\;\log\left(||U_{B}||_{r,s}\right)$
(71) $\displaystyle c_{AB}$
$\displaystyle=-\lim_{N\rightarrow\infty}\;\log\left(||U_{AB}||_{r,s}\right)$
(72)

Using $r,s$, as given in (20) it is straightforward to check that
$\lambda,\mu\leq N/2$ implies $1\leq r\leq s$. Therefore, we can use Lem. 2
and get

$\displaystyle c_{AB}$
$\displaystyle=-\lim_{N\rightarrow\infty}\;\log\left(||U_{A}||_{r,s}||U_{B}||_{r,s}\right)$
$\displaystyle=-\lim_{N\rightarrow\infty}\;\log\left(||U_{A}||_{r,s}\right)+\log\left(||U_{B}||_{r,s}\right)$
$\displaystyle=c_{A}+c_{B}\;.$ (73)

∎

## Outlook and conclusion

In this work we showed that linear uncertainty relations between product type
measurements in multipartions are additive. Prop. 1 gives some clear structure
to the problem of computing entropic uncertainty bounds. Especially in the
context of quantum-coding in cryptography, this result might turn out to be
useful, because now it is possible to compute uncertainty bounds in the limit
of infinite system sizes for block-coding schemes [63, 64, 6].

The generalization of the Maassen and Uffink bound for arbitrary weights
$(\lambda,\mu)$, provided in Lem. 1, can also be directly employed in a
multipartite setting in order to obtain valid state-independent uncertainty
relations for this case. However, this bound is easy computable, it is only a
lower bound and presumably only tight in high symmetrical cases (see [34] for
a characterization of tightness for the usual Maassen and Uffink bound). The
more general problem of providing a ’good’ method for computing the optimal
bound $c_{AB}$ remains open. We note that there are only few and special
cases, including angular momentum and mutual unbiased bases, where this
optimal bound is actually known. Thereby, the cases where the optimal bound
can be computed analytically are even fewer [65, 66, 34] and the known
numerical methods only work for very small dimensional problems [67]. Here the
proof of Thm. 1 might give a new ansatz for better numerics. Explicitly, the
minimization in (4) and maximization in (42) are giving rise to apply the
method of alternating minimization.

In Sec. 4 we presented an extension to the known connections between the
logarithm of $(p,q)$-norms and linear uncertainty relations in terms of the
shannon entropy. However, the technique used seems to apply only for the
special case we considered. An adaption of this technique to sets of more than
two local measurements is not possible without major modifications. As
mentioned in Rem. 7, this would require to incorporate generalizations of the
Golden-Thompson inequality which seems to be a fruitful topic for future work.
The technique from the proof of Thm. 1 might also fail if general POVMs
instead of projective measurements are considered. Moreover, it is not clear
if Prop. 1 will hold in this case. A third generalization, that does not hold,
arises by considering arbitrary Schur-concave functions. Here, the natural
question is to ask if at least any entanglement witness can be constructed. A
very recent result [68] shows that such witnesses, in fact, can be constructed
from Tsallis entropies.

## Acknowledgements

R.S. acknowledges K. Abdelkhalek, O. Gühne, A. Costa, I. Siemon, and R.F.
Werner for all the helpful discussions and comments. Furthermore, R.S.
acknowledges Coco, Inken and Lars for a careful reading and correcting of this
manuscript. R.S. also acknowledges financial support by the RTG 1991 and the
SFB DQ-Mat, both founded by the DFG, and the project Q.com-q founded by the
BMBF. Finally R.S. acknowledges the hospitality of the Centro de Ciencias de
Benasque Pedro Pascual and Petronilla, granted at the beginning of this
project. The publication of this article was funded by the Open Access Fund of
the Leibniz Universität Hannover.

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