#  Asymmetry and tighter uncertainty relations for Rényi entropies via
quantum–classical decompositions of resource measures

Michael J. W. Hall  Department of Theoretical Physics, Research School of
Physics, Australian National University, Canberra, ACT 0200, Australia

###### Abstract

It is known that the variance and entropy of quantum observables decompose
into intrinsically quantum and classical contributions. Here a general method
of constructing quantum–classical decompositions of resources such as
uncertainty is discussed, with the quantum contribution specified by a measure
of the noncommutativity of a given set of operators relative to the quantum
state, and the classical contribution generated by the mixedness of the state.
Suitable measures of noncommutativity or ‘quantumness’ include quantum Fisher
information, and the asymmetry of a given set, group or algebra of operators,
and are generalised to nonprojective observables and quantum channels. Strong
entropic uncertainty relations and lower bounds for Rényi entropies are
obtained, valid for arbitrary discrete observables, that take the mixedness of
the state into account via a classical contribution to the lower bound. These
relations can also be interpreted without reference to quantum-classical
decompositions, as tradeoff relations that bound the asymmetry of one
observable in terms of the entropy of another.

##  I Introduction

A quantum observable such as position or spin has two generic sources of
uncertainty. The first arises when the observable does not commute with the
state of the quantum system: this implies that the state is not an eigenstate
of the observable, leading to a spread of measurement outcomes. The second
source of uncertainty arises when the state is mixed: a loss of classical
information due to mixing reduces the predictability of the measurement
outcome, leading to an increased spread. It follows that the two sources are,
respectively, intrinsically quantum and classical in nature. The first can be
useful as a resource, e.g., in allowing a complementary observable to have a
small spread, whereas the second is usually not, as mixing contributes noise
to all observables.

Luo suggested quantifying such quantum and classical contributions to a given
measure of uncertainty, $M(X|\rho)$, for a Hermitian operator $X$ and state
described by density operator $\rho$, via a quantum-classical decomposition of
the form [1, 2]

$M(X|\rho)=Q(X|\rho)+C(X|\rho),$ (1)

and gave the particular example

${\rm Var}_{\rho}(X)=-\frac{1}{2}{\rm
Tr}\left[{{[X,\sqrt{\rho}]^{2}}}\right]+\left({\rm
Tr}\left[{{X\sqrt{\rho}X\sqrt{\rho}}}\right]-\langle
X\rangle_{\rho}^{2}\right)$ (2)

for the variance of a Hermitian operator. In this example the ‘quantum’
contribution is the skew information of $X$ with respect to $\rho$ [3], and is
clearly nonzero only if the observable does not commute with the state,
whereas the ‘classical’ contribution is nonzero only if the state is mixed, as
expected. A second example is a quantum–classical decomposition of the entropy
of a Hermitian operator, given by Korzekwa et al. [4].

A simple approach to unifying and generalising such quantum–classical
decompositions is discussed in Sec. II, based on the idea that the
intrinsically quantum contribution to a given resource measure is maximised
when the observer has access to a pure state of the system, and is degraded by
classical mixing for a local observer who only has access to a component of
such a system. For example, a maximally-entangled pure state of two qubits is
a useful quantum resource for dense coding and for estimation of a local
rotation, but this usefulness vanishes for an observer who only has access to
one of the two qubits (described by a maximally-mixed state).

The general approach is described in Sec. II.1. It starts with some given
measure $Q(X,Y,\dots|\rho)$ of the ‘quantumness’ of a resource, for operators
$X,Y,\dots$ and state $\rho$. The corresponding ‘maximum potential’
$M(X,Y,\dots|\rho)$ of the resource and its ‘classicality’ $C(X,Y,\dots|\rho)$
are then constructed so as to satisfy a decomposition analogous to Eq. (1).
Several examples are given in Sec. II.2, including the decomposition of
variance and covariance matrices with respect to quantum Fisher information;
decompositions of Shannon entropy with respect to both asymmetry and the
conditional entropy of self-dual communication channels; and, of particular
relevance to this paper, the decomposition of Rényi entropy with respect to
Rényi asymmetry, where the latter has applications to quantum coherence, time-
energy uncertainty relations, quantum information, open quantum systems and
quantum metrology [6, 7, 9, 5, 8]. It is also shown that measures of
quantumness can equivalently be defined via sets, groups or algebras of
operators, and generalised to arbitrary discrete observables and to quantum
channels. An alternative approach, based on convex and concave roofs, is
discussed in Appendix A, with corresponding examples.

The main technical results are given in Sec. III, including a strong
uncertainty relation for Rényi entropies of discrete observables $X$ and $Y$
represented by positive operator valued measures (POVMs) $\\{X_{x}\\}$ and
$\\{Y_{y}\\}$:

$\displaystyle H_{\alpha}(X|\rho)+H_{\beta}(Y|\rho)$
$\displaystyle\geq-\log\mu_{XY}$
$\displaystyle~{}\,~{}~{}+\max\\{C_{\alpha}(Y|\rho),C_{\beta}(X|\rho)\\}.$ (3)

Here the Rényi parameters $\alpha,\beta\in[\mbox{$\frac{1}{2}$},\infty)$ are
related by $\alpha^{-1}+\beta^{-1}=2$; $\mu_{XY}$ denotes the maximum
eigenvalue of $X_{x}^{1/2}Y_{y}X_{x}^{1/2}$ over $x$ and $y$; and the last
term takes the mixedness of $\rho$ into account via the classical components
of the Rényi entropies. The logarithm base is left arbitrary throughout,
corresponding to a choice of units (e.g., to bits for base 2 and to nats for
base $e$).

Equation (I) is clearly stronger than the standard uncertainty relation

$H_{\alpha}(X|\rho)+H_{\beta}(Y|\rho)\geq-\log\mu_{XY},$ (4)

for Rényi entropies [10, 11, 12], which has no classical mixing term. Further,
for rank-1 POVMs, with $X_{x}=|x\rangle\langle x|$ and $Y_{y}=|y\rangle\langle
y|$, Eq. (I) generalises the known uncertainty relation

$H(X|\rho)+H(Y|\rho)\geq-\log\max_{x,y}|\langle x|y\rangle|^{2}+H(\rho)$ (5)

for Shannon entropies [4, 13, 14] (which correspond to $\alpha=\beta=1$), that
takes the mixedness of the state into account via its von Neumann entropy,
$H(\rho):=-{\rm Tr}\left[{{\rho\log\rho}}\right]$. It is also shown in Sec.
III that Eq. (I) may be written as a direct tradeoff between the asymmetry of
one observable and the entropy of another, without any reference to quantum-
classical decompositions, leading to a simple lower bound for Rényi asymmetry,
as well as to entropic bounds such as

$H_{\alpha}(X|\rho)\geq-\log\mu_{\rho
X},\qquad\alpha\geq\mbox{$\frac{1}{2}$},$ (6)

for the Rényi entropy of an arbitrary discrete observable $X$. Some readers
may choose to skip directly to Sec. III, and refer back to Sec. II for
definitions.

Results and possible future work are discussed in Sec. IV.

##  II Building quantum–classical decompositions

###  II.1 A general approach

As noted in the Introduction, access to a maximally-entangled pure state of
two qubits is a useful resource for the estimation of a local rotation, but
this usefulness vanishes for an observer who only has access to one of the two
qubits, described by a maximally-mixed state. This motivates the more general
idea that the quantum contribution to the degree to which a given resource can
be exploited is maximised for a notional observer who has access to a pure
state of the system, with any classical component arising from the presence of
mixing when there is only partial access.

Accordingly, if the ‘quantumness’ of some resource is quantified by a measure
$Q(X,Y,\dots|\rho)$, for some set of operators $X,Y,\dots$ and state $\rho$,
the maximum potential of the resource, $M(X,Y,\dots|\rho)$, is defined by

$M(X,Y,\dots|\rho):=Q(X\otimes\mathbbm{1}_{a},Y\otimes\mathbbm{1}_{a},\dots|\rho_{\psi}),$
(7)

where $\rho_{\psi}\equiv|\psi\rangle\langle\psi|$ is a purification of state
$\rho$ on the tensor product $\mathcal{H}\otimes\mathcal{H}_{a}$ of the system
Hilbert space $\cal H$ with an ancillary Hilbert space $\mathcal{H}_{a}$ [15].
This immediately yields a corresponding quantum-classical decomposition of the
form

$\displaystyle M(X,Y,\dots|\rho)=Q(X,Y,\dots|\rho)+C(X,Y,\dots|\rho),$ (8)

generalising Eq. (1), with
$C(X,Y,\dots|\rho):=Q(X\otimes\mathbbm{1}_{a},Y\otimes\mathbbm{1}_{a},\dots|\rho_{\psi})-Q(X,Y,\dots|\rho)$
representing the classical component of the resource relative to an observer
who only has access to the mixed state $\rho$ of the system (rather than to
the purified state $\rho_{\psi}$).

Several examples and generalisations are given below, but first three natural
assumptions for $Q(X,Y,\dots|\rho)$ are identified (related requirements have
been previously discussed by Luo [1, 2] and Korzekwa et al. [4]).

  * (i)

It will be assumed that there is no potential ‘quantum’ resource to exploit if
the relevant operators commute with the state of the system, i.e.,

$\qquad Q(X,Y,\dots|\rho)=0~{}~{}{\rm if}~{}~{}[X,\rho]=[Y,\rho]=\dots=0.$ (9)

Hence, the quantumness acts as a measure of noncommutativity.

  * (ii)

The decomposition must be well-defined, with the values of the quantum and
classical components being independent of the choice of purification. This is
guaranteed by the assumption that $Q(X,Y,\dots|\rho)$ is invariant under
unitary transformations, i.e.,

$\qquad Q(X,Y,\dots|\rho)=Q(UXU^{\dagger},UYU^{\dagger},\dots|U\rho
U^{\dagger})$ (10)

for all unitary transformations $U$. In particular, noting that any two
purifications $\rho_{\psi}$ and $\rho_{\psi^{\prime}}$ can be related by a
local unitary transformation $U=\mathbbm{1}\otimes U_{a}$ acting on the tensor
product $\mathcal{H}\otimes\mathcal{H}_{a}$ [15], where this leaves operators
on $\mathcal{H}$ invariant, it follows that Eqs. (7) and Eq. (8) are
independent of the choice of purification as required.

  * (iii)

The classical component must be nonnegative, and vanish for pure states (i.e.,
when there is no classical mixing). This is equivalent, via Eqs. (7) and (8),
to the assumption that the quantumness increases under purification (e.g.,
when an observer gains access to the full quantum system), i.e., that

$\qquad Q(X,Y,\dots|\rho)\leq
Q(X\otimes\mathbbm{1}_{a},Y\otimes\mathbbm{1}_{a},\dots|\rho_{\psi})$ (11)

for any purification $\rho_{\psi}$ of $\rho$. In particular, this assumption
ensures that $C(X,Y,\dots|\rho)\geq 0$, with equality for pure states, as
desired.

The main advantage of the above construction method is that the maximum
potential value of a quantum resource accessible to an observer, $M$, and the
classical contribution due to mixing, $C$, are fully determined by the choice
of the quantumness (or noncommutativity) measure $Q$. Other advantages are the
applicability to arbitrary sets of operators and the straightforward extension
to nonprojective observables and quantum channels, as illustrated in the
examples below. An alternative method of constructing quantum-classical
decompositions, based on convex and concave roofs of pure-state resource
measures, is discussed in Appendix A.

###  II.2 Examples

It is convenient in what follows to call a given discrete observable $X$ a
projective observable if the corresponding POVM $\\{X_{x}\\}$ consists of
orthogonal projections, i.e., $X_{x}X_{x^{\prime}}=\delta_{xx^{\prime}}X_{x}$,
and a nonprojective observable otherwise. Note that a projective observable
$X$ is equivalently represented by the Hermitian operator
$X\equiv\sum_{x}xX_{x}$. The distinction between a projective observable and
its corresponding Hermitian operator will always be clear by context.

####  II.2.1 Skew information, variance, and the
Fisher information matrix

Luo’s decomposition of the variance of a projective observable into quantum
and classical contributions [1, 2],

${\rm Var}_{\rho}(X)=M_{\rm skew}(X|\rho)=Q_{\rm skew}(X|\rho)+C_{\rm
skew}(X|\rho)$ (12)

as per Eq. (2), corresponds to choosing the quantumness measure to be the skew
information of the observable with respect to the state [3], i.e.,

$Q_{\rm skew}(X|\rho):=\frac{1}{2}{\rm
Tr}\left[{{(i[X,\sqrt{\rho}])^{2}}}\right].$ (13)

The skew information may be physically interpreted as, for example, a measure
of the information content of an ensemble that conserves $X$ [3]; a variant of
quantum Fisher information in quantum metrology [1, 17, 16]; a measure of
asymmetry [18]; or (for nondegenerate observables) as a measure of coherence
[19, 20].

Note that assumptions (9) and (10) are trivially satisfied by the skew
information, while assumption (11) follows because $Q_{\rm
skew}(X\otimes\mathbbm{1}_{a}|\rho_{\psi})={\rm Var}_{\rho}(X)$ for any
purification of $\rho$, implying via identity (2) and Eq. (12) that

$C_{\rm skew}(X|\rho)={\rm Tr}\left[{{\left(\rho^{1/4}(X-\langle
X\rangle_{\rho})\rho^{1/4}\right)^{2}}}\right]\geq 0.$ (14)

Hence, properties (i)–(iii) of Sec. II.1 all hold as required.

The above decomposition can be generalised to the case of more than one
projective observable, and to other measures of quantum Fisher information,
using recent results by Kudo and Tajima [21]. In particular, let
$Q^{f}_{F}(X_{1},X_{2},\dots|\rho)$ denote the quantum Fisher information
matrix corresponding to a given monotone metric function $f$, for some set of
projective observables $X_{1},X_{2},\dots$ and density operator $\rho$, where
each choice of $f$ corresponds to a measure of the sensitivity of the state to
unitary transformations generated by these observables [16, 21]. Assumptions
(9)–(11) of Sec. II.1 hold for $Q^{f}_{F}$ as a direct consequence of Theorem
9 of [21], and the construction method yields the quantum-classical
decomposition

$M(X_{1},X_{2},\dots|\rho)={\rm
Cov}_{\rho}(X_{1},X_{2}\dots|\rho)=Q^{f}_{F}+C^{f}_{F}$ (15)

of the symmetrised covariance matrix, with coefficients defined by

$\displaystyle[{\rm Cov}_{\rho}(X_{1},X_{2},\dots|\rho)]_{jk}$
$\displaystyle:=\mbox{$\frac{1}{2}$}\langle
X_{j}X_{k}+X_{k}X_{j}\rangle_{\rho}$ $\displaystyle\qquad~{}~{}~{}-\langle
X_{j}\rangle_{\rho}\langle X_{k}\rangle_{\rho}.$ (16)

Thus, the covariance matrix has an infinite family of quantum-classical
decompositions, with Eqs. (12)–(14) providing just one particular example. It
is shown in Appendix A that, for the case of a single projective observable,
the alternative construction method given there picks out the particular
decomposition having the minimal Fisher information (corresponding to the
symmetric logarithmic derivative [16]).

####  II.2.2 Distance, Shannon entropy and standard asymmetry

Shannon entropy is a well known resource measure in various information
contexts, and Korzekwa et al. have given a corresponding quantum-classical
decomposition for the case of projective observables [4], as briefly described
below. An alternative decomposition of Shannon entropy is noted in Appendix
A.1, based on information properties of self-dual quantum communication
channels, i.e., communication channels which are invariant under a duality
mapping between the signal ensemble and the receiver measurement [22, 23].

The decomposition in [4] corresponds to choosing the quantumness to be a
particular measure of the ‘distance’ from the state of the system to a set of
postmeasurement states. More generally, let $d(\rho,\sigma)\geq 0$ be a
positive function that vanishes for $\rho=\sigma$, and define the
corresponding measure of quantumness for a projective observable $X$ with
associated projection valued measure $\\{X_{x}\\}$ via

$Q_{d}(X|\rho):=\inf_{\sigma}d(\rho,\sigma_{X})=\inf_{\sigma:[\sigma,X]=0}d(\rho,\sigma),$
(17)

where $\sigma$ is implicitly restricted to the set of density operators of the
system and

$\sigma_{X}:=\sum X_{x}\sigma X_{x}$ (18)

is the postmeasurement state for a projective measurement of $X$ on state
$\sigma$. Such distance-based measures of quantumness or noncommutativity are
commonly used to quantify asymmetry and coherence resources [18, 24].

Note that $Q_{d}$ satisfies assumption (9) of Sec. II.1 by construction, while
assumptions (10) and (11) are satisfied if $d(\rho,\sigma)$ is invariant under
unitary transformations and is nonincreasing under the partial trace
operation—and in particular if $d(\rho,\sigma)$ is nonincreasing under general
completely positive trace preserving (CPTP) maps [15]. These properties hold
in the case of the relative entropy distance function [15], defined by

$d(\rho,\sigma)=D_{1}(\rho\|\sigma):={\rm
Tr}\left[{{\rho(\log\rho-\log\sigma)}}\right],$ (19)

for which $Q_{d}$ evaluates to the standard measure of asymmetry [25, 26],

$Q_{1}(X|\rho)=D_{1}(\rho\|\rho_{X})=H(\rho_{X})-H(\rho),$ (20)

using the identity
$D_{1}(\rho\|\sigma_{X})=H(\rho_{X})-H(\rho)+D_{1}(\rho_{X}\|\sigma_{X})$.
Here the subscript ‘1’ provides the basis for a generalisation to Rényi
asymmetry in the next example. The standard asymmetry $Q_{1}(X|\rho)$ is equal
to the increase in system entropy due to a nonselective measurement of $X$,
and is a useful quantum resource measure in various contexts [18, 25, 26, 28,
24, 4, 29, 27] (these contexts typically have different sets of ‘free’ states
and operations, but the resource measure itself has the same form).

Korzekwa et al. simply postulate that the standard asymmetry, $Q_{1}$, is the
intrinsically quantum contribution to the Shannon entropy of the observable,
$H(X|\rho):=-\sum_{x}p(x|\rho)\log p(x|\rho)$ [4]. In contrast, an advantage
of the general approach in Sec. II.1 is that the link to Shannon entropy is
derived rather than postulated. In particular, the maximum potential of the
standard asymmetry, available to an observer with access to a purification
$|\psi\rangle\langle\psi|$ of $\rho$, follows from definition (7) and Eq. (20)
as

$\displaystyle M_{1}(X|\rho)$
$\displaystyle=H\big{(}\sum_{x}X_{x}\otimes\mathbbm{1}_{a}|\psi\rangle\langle\psi|\
X_{x}\otimes\mathbbm{1}_{a}\big{)}-H(|\psi\rangle\langle\psi|)$
$\displaystyle=H\big{(}\sum_{x}\langle\psi|X_{x}\otimes\mathbbm{1}_{a}|\psi\rangle|\psi_{x}\rangle\langle\psi_{x}|\big{)}$
$\displaystyle=H(X|\rho),$ (21)

where $\\{|\psi_{x}\rangle\\}$ is the set of orthonormal states with
$|\psi_{x}\rangle$ proportional to
$(X_{x}\otimes\mathbbm{1}_{a})|\psi\rangle$, and the last line follows via
$\langle\psi|X_{x}\otimes\mathbbm{1}_{a}|\psi\rangle={\rm
Tr}\left[{{|\psi\rangle\langle\psi|X_{x}\otimes\mathbbm{1}_{a}}}\right]={\rm
Tr}\left[{{\rho X_{x}}}\right]=p(x|\rho)$. Hence, the approach in Sec. II.1
constructively generates the quantum-classical decomposition

$\displaystyle H(X|\rho)=Q_{1}(X|\rho)+C_{1}(X|\rho)$ (22)

given in [4], with the classical contribution evaluating to

$\displaystyle C_{1}(X|\rho)$
$\displaystyle=H(\rho)-\sum_{x}p(x|\rho)H(X_{x}\rho X_{x}/p(x|\rho)).$ (23)

Note for any rank-1 projective observable $X$, with
$X_{x}\equiv|x\rangle\langle x|$, that the classical contribution reduces to
the von Neumann entropy $H(\rho)$ [4]. More generally, if $H(X|\rho)$ is
identified with the entropy of the classical measurement record, then the
classical contribution equals the average decrease of the system entropy due
to making such a record.

Korzekwa et al. use decomposition (22) to obtain a simple proof of entropic
uncertainty relation (5) for rank-1 projective observables [4]. The
decomposition, uncertainty relation and method of proof are generalised below
to Rényi entropies and to nonprojective observables. A rather different
decomposition of Shannon entropy, based on a quantumness measure for self-dual
communication channels, is discussed in Appendix A.1.

####  II.2.3 Rényi asymmetry and Rényi entropy

A particular choice of interest for the distance function $d(\rho,\sigma)$ in
Eq. (17) is the quantum generalisation of the classical Rényi relative entropy
[30] to the sandwiched Rényi divergence [31, 32],

$d(\rho,\sigma)=D_{\alpha}(\rho\|\sigma):=\frac{1}{\alpha-1}\log{\rm
Tr}\left[{{\big{(}\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\big{)}^{\alpha}}}\right],$
(24)

where the index $\alpha$ ranges over $[0,\infty)$. This generalises the
relative entropy function in Eq. (19) (which corresponds to the limit
$\alpha\rightarrow 1$), and introductory expositions of its basic properties
may be found, for example, in [31, 33]. The most important of these properties
for the purposes of this paper is that it is nonincreasing under CPTP maps
when $\alpha\geq\mbox{$\frac{1}{2}$}$, i.e, the data-processing inequality

$D_{\alpha}(\phi(\rho)\|\phi(\sigma))\leq
D_{\alpha}(\rho\|\sigma),\qquad\alpha\geq\mbox{$\frac{1}{2}$},$ (25)

holds for any CPTP map $\phi$ [33, 34].

For projective observables the above choice of $d(x,y)$ generates the
corresponding quantumness measure [6]

$Q_{\alpha}(X|\rho):=\inf_{\sigma:[\sigma,X]=0}D_{\alpha}(\rho\|\sigma)=\inf_{\sigma}D_{\alpha}(\rho\|\sigma_{X})$
(26)

via Eq. (17). This measure generalises the standard asymmetry in Eq. (20), and
is known as the Rényi asymmetry of $X$ with respect to $\rho$. It has found
applications in the areas of quantum coherence, quantum information, time-
energy uncertainty relations, quantum metrology and open quantum systems [6,
7, 9, 5, 8].

Assumptions (9)–(11) hold for the Rényi asymmetry $Q_{\alpha}(X|\rho)$ when
$\alpha\geq\mbox{$\frac{1}{2}$}$ as a consequence of data-processing
inequality (25), as may easily be checked. Further, the maximum potential of
the Rényi asymmetry is directly related to the Rényi entropy [30]

$H_{\beta}(X|\rho):=\frac{1}{1-\beta}\log\sum_{x}p(x|\rho)^{\beta}$ (27)

of observable $X$ for state $\rho$, via

$M_{\alpha}(X|\rho)=Q_{\alpha}(X\otimes\mathbbm{1}_{a}|\rho_{\psi})=H_{\beta}(X|\rho),~{}~{}\frac{1}{\alpha}+\frac{1}{\beta}=2.$
(28)

Here the first equality follows via definition (7) and the second equality by
showing, via direct minimisation in Eq. (26), that
$Q_{\alpha}(X|\,|\psi\rangle\langle\psi|)=H_{\beta}(|\psi\rangle\langle\psi|_{X})$
for any pure state $|\psi\rangle$, from which
$M_{\alpha}(X|\rho)=H_{\beta}((\rho_{\psi})_{X\otimes\mathbbm{1}_{a}})=H_{\beta}(X|\rho)$
follows by direct calculation (see also the proof of Theorem 3 in [9]). Hence,
the general approach in Sec. II.1) yields the quantum-classical decomposition

$H_{\beta}(X|\rho)=Q_{\alpha}(X|\rho)+C_{\alpha}(X|\rho),~{}~{}\frac{1}{\alpha}+\frac{1}{\beta}=2,$
(29)

of Rényi entropy, generalising the case of Shannon entropy in Eq. (22) (which
corresponds to the limit $\alpha\rightarrow 1$). It follows that the Rényi
asymmetry $Q_{\alpha}(X|\rho)$ is upper bounded by the Rényi entropy
$H_{\beta}(X|\rho)$, with equality for pure states [9]. In contrast, the
classical component $C_{\alpha}(X|\rho)$ vanishes for pure states by
construction, as befits a measure of mixedness, and reaches the upper bound of
$H_{\beta}(X|\rho)$ when $X$ is ‘classical’ with respect to $\rho$, i.e., when
$[X,\rho]=0$. Decomposition (29) will be used in Sec. III to obtain strong
uncertainty relation (I) for Rényi entropies.

####  II.2.4 Generalising to nonprojective observables

The previous examples are restricted to projective observables. However, it is
not difficult to generalise to arbitrary observables, as required for the
general results in Sec. III.

The idea is to exploit Naimark’s extension theorem, that an observable $X$
with POVM $\\{X_{x}\\}$ on Hilbert space $\cal H$, can always be extended to a
projective observable $\tilde{X}$ with projection valued measure
$\\{\tilde{X}_{x}$} on a larger Hilbert space $\tilde{\cal H}$, with

$X_{x}=P\tilde{X}_{x}P,$ (30)

where $P$ denotes the projection from $\tilde{\cal H}$ onto $\cal H$ [35, 36,
37, 38, 39]. It follows that any given measure of quantumness $Q(X|\rho)$ for
projective observables can be extended to nonprojective observables via
(noting $\tilde{\rho}\equiv\rho$ on $\tilde{\cal H}$ since
$P\tilde{\rho}P=\rho$)

$Q(X|\rho):=Q(\tilde{X}|\tilde{\rho})\equiv Q(\tilde{X}|\rho).$ (31)

While this definition will typically depend on the choice of extension mapping
$X\rightarrow\tilde{X}$, the results obtained in this paper are valid for any
choice. Hence, the mapping can and will be left unspecified in what follows.

As a simple example, the extension of the skew information in Eq. (13) to
general observables follows, using $P\rho P=\rho$ and
$\tilde{X}_{x}\tilde{X}_{x^{\prime}}=\delta_{xx^{\prime}}\tilde{X}_{x}$, as

$Q_{\rm skew}(X|\rho)=\frac{1}{2}{\rm
Tr}\left[{{(i[\,\overline{X},\sqrt{\rho}])^{2}}}\right]+{\rm
Tr}\left[{{\rho(\overline{X^{2}}-\overline{X}^{2})}}\right],$ (32)

in terms of the moment operators $\overline{X^{m}}:=\sum_{x}x^{m}X_{x}$. This
is independent of the mapping $X_{x}\rightarrow\tilde{X}_{x}$ and reduces to
Eq. (13) for projective observables. Note that the second term is typically
nonzero for nonprojective observables, corresponding to an additional
information resource if the extended observable is physically accessible.
Remarkably, the maximum potential of the resource is given by the variance for
both projective and nonprojective observables, with definition (7) leading to

$M_{\rm skew}(X|\rho)={\rm
Var}_{\rho}(X)=\langle\overline{X^{2}}\rangle_{\rho}-\langle\overline{X}\rangle_{\rho}^{2}.$
(33)

Thus, the general decomposition of variance has the same form as Eq. (12),
with $X$ replaced by $\bar{X}$ in Eq. (14).

As a second example, the Rényi asymmetry in Eq. (26) generalises to

$Q_{\alpha}(X|\rho)=\inf_{\sigma:[\sigma,\tilde{X}]=0}D_{\alpha}(\rho\|\sigma)=\inf_{\sigma}D_{\alpha}(\rho\|\sigma_{\tilde{X}})$
(34)

where, as always, $\sigma$ is implicitly restricted to range over the density
operators of the system (hence, $P\sigma P=\sigma$). Thus, the asymmetry
quantifies the distance from the state of the system to the set of
postmeasurement states following a measurement of $\tilde{X}$. Such
postmeasurement states do not typically lie in the Hilbert space $\cal H$ of
the system, and hence the asymmetry will typically be nonvanishing for
nonprojective observables (see also Corollary 2), corresponding to the
sensitivity of $\rho$ to unitary displacements generated by $\tilde{X}$ on the
extended Hilbert space. If such displacements are not physically accessible
then the generalised asymmetry might more appropriately be referred to as
hidden asymmetry (or perhaps super-asymmetry — bazinga!).

Note from Eq. (30) and the property $P\rho P=\rho$ that ${\rm
Tr}[\rho\tilde{X}_{x}]={\rm Tr}\left[{{\rho X_{x}}}\right]$, implying that
$H_{\beta}(\tilde{X}|\rho)=H_{\beta}(X|\rho)$. It follows immediately that Eq.
(34) generates a quantum-classical decomposition of Rényi entropy having
precisely the same form as Eq. (29) for projective observables. Further, it is
straightforward to show, using definition (34) and
$P|\psi\rangle=|\psi\rangle$, that for a pure state
$\rho=|\psi\rangle\langle\psi|$ the generalised asymmetry is given by

$Q_{\alpha}(X|\,|\psi\rangle\langle\psi|)=H_{\beta}(X|\,|\psi\rangle\langle\psi|),~{}~{}~{}\frac{1}{\alpha}+\frac{1}{\beta}=2,$
(35)

similarly to the case of projective observables. Hence, Eq. (29) for the
general case implies that the classical contribution to the Rényi entropy
vanishes for pure states for both projective and nonprojective observables,
i.e.

$C_{\alpha}(X|\,|\psi\rangle\langle\psi|)=0,\qquad\alpha\geq\mbox{$\frac{1}{2}$}.$
(36)

Finally, it is of interest to note that the general definition of asymmetry in
Eq. (34) reduces to Eq. (26) for the case of projective observables,
irrespective of the choice of extension mapping
$X_{x}\rightarrow\tilde{X}_{x}$. In particular, if $PEP$ is a projection for
two projections $P$ and $E$, then writing $P^{\perp}=\mathbbm{1}-P$ one has
$(PEP^{\perp})(PEP^{\perp})^{\dagger}=PEP-(PEP)^{2}=0$, and so
$PEP^{\perp}=0=P^{\perp}EP$, yielding
$E=(P+P^{\perp})E(P+P^{\perp})=PEP+P^{\perp}EP^{\perp}$, from which $[P,E]=0$
follows. Hence, if $X$ is a projective observable then $[P,\tilde{X}_{x}]=0$
via Eq. (30). But $\rho$ and $\sigma$ in Eq. (34) similarly commute with $P$
(since $P\rho P=\rho$ and $P\sigma P=\sigma$), and evaluation of the right
hand side then leads directly to Eq. (26) as claimed.

####  II.2.5 Asymmetry of sets vs groups vs algebras vs channels

It is straighforward to generalise the class of distance-based quantumness
measures in Eq. (17) from the case of a single projective observable $X$ to
any set $S$ of bounded operators, via

$Q_{d}(S|\rho):=\inf_{\sigma:[\sigma,X]=0\,\forall X\in
S}d(\rho,\sigma)=\inf_{\sigma\in S^{\prime}}d(\rho,\sigma),$ (37)

where $\sigma$ again is implicitly restricted to the set of density operators
of the system and

$S^{\prime}:=\\{Y:[X,Y]=0~{}\forall X\in S\\}$ (38)

denotes the commutant of $S$, i.e., the set of operators that commute with all
members of $S$. This generalised measure is, therefore, the distance from the
state of the system to the set of states that commute with the operators in
$S$. Such measures are commonly used to quantify asymmetry resources, i.e.,
the degree to which the state is noninvariant with respect to members of $S$,
when $S$ is either (i) a unitary representation of some group [25, 26, 28, 18,
29, 7, 9], or (ii) an operator algebra [6, 8, 40].

Remarkably, while the cases of arbitrary sets, unitary group representations
and operator algebras may appear to represent prima facie significant
distinctions, this is in fact not so, provided that either $S$ or $S^{\prime}$
in Eq. (37) is closed under the adjoint operation (as is usually the case in
practice [25, 26, 28, 18, 6, 29, 7, 40, 8, 9]). For example, the standard
asymmetry (20) of a projective observable $X$ is equivalently represented in
Eq. (37) via any of the set $\\{X\\}$, the group of unitary operators
$\\{e^{iaX}\\}$, and the algebra of operators that commute with $X$. Links
between these cases are therefore briefly clarified below for the interested
reader, including their unification via the extension of quantumness to
quantum channels, and applications to the standard asymmetry measure (other
readers may wish to skip directly to Sec. III).

######  Proposition 1.

If either $S$ or $S^{\prime}$ is closed under the adjoint operation, then the
quantumness measure $Q_{d}(S|\rho)$ in definition (37) is invariant under the
replacement of the set $S$ by the von Neumann algebra $S^{\prime\prime}$, or
by the group $G_{S^{\prime\prime}}$ of unitary operators in $S^{\prime\prime}$
(where both replacements are closed under the adjoint operation).

###### Proof.

Note that $S^{\prime}$ is closed under the adjoint operation under either of
the conditions of the Proposition, since the closure of $S$ under the adjoint
implies that $[X^{\dagger},Y]=0$ for all $X\in S$ and $Y\in S^{\prime}$, which
is equivalent to $[X,Y^{\dagger}]=0$. Further, $S^{\prime}$ is closed under
multiplication and addition (since $[X,Y]=[X,Z]=0$ implies
$[X,YZ]=[X,Y+Z]=0$), and contains the unit operator (since
$[X,\mathbbm{1}]\equiv 0)$. Hence, $S^{\prime}$ is an algebra, and the closure
of $S^{\prime}$ under the adjoint operation guarantees that it is a von
Neumann algebra [41]. Now, any von Neumann algebra $\mathfrak{a}$ is equal to
its double commutant $\mathfrak{a}^{\prime\prime}$, and its commutant
$\mathfrak{a}^{\prime}$ is also a von Neumann algebra [41]. Hence,
$S^{\prime}=(S^{\prime})^{\prime\prime}=(S^{\prime\prime})^{\prime}$, implying
that $Q_{d}(S|\rho)=Q_{d}(S^{\prime\prime}|\rho)$ as claimed. Moreover, any
element of a von Neumann algebra can be written as a linear combination of (at
most four of) its unitaries [41], and hence an operator commutes with the
members of the von Neumann algebra $S^{\prime\prime}$ if and only if it
commutes with the members of the group $G_{S^{\prime\prime}}$ of unitary
operators in $S^{\prime\prime}$. Thus,
$S^{\prime}=(S^{\prime\prime})^{\prime}=(G_{S^{\prime\prime}})^{\prime}$,
implying
$Q_{d}(S|\rho)=Q_{d}(S^{\prime\prime}|\rho)=Q_{d}(G_{S^{\prime\prime}}|\rho)$
as claimed. Note each replacement is closed under the adjoint operation, since
$S^{\prime\prime}$ is a von Neumann algebra and $U^{-1}=U^{\dagger}$ for $U\in
G_{S^{\prime\prime}}$. ∎

Thus, for example, a distance-based measure of rotational asymmetry has the
same value irrespective of whether $S$ in Eq. (37) is chosen to be (a) the set
of rotation operators $\\{J_{x},J_{y},J_{z}\\}$, or (b) the algebra generated
by these operators, or (c) the unitary representation
$\\{e^{i\bm{J}\cdot\bm{n}}\\}$ of the rotation group [42]. This general link
between sets, groups and algebras of operators is further illuminated by
extending the definition of $Q_{d}$ to quantum channels.

In particular, for a given distance function $d(\rho,\sigma)$ and CPTP map
$\phi$, define

$Q_{d}(\phi|\rho):=\inf_{\sigma}d(\rho,\phi(\sigma)).$ (39)

Thus, $Q_{d}(\phi|\rho)$ is the distance from $\rho$ to the set of output
states of the channel, and may be interpreted as a measure of how closely
$\rho$ can be prepared via the channel. Further, this measure reduces to
$Q_{d}(X|\rho)$ in Eq. (17) for the channel $\phi_{X}(\sigma):=\sigma_{X}$. It
is therefore natural to ask, for the general case of an arbitrary set of
operators $S$, whether there is a corresponding channel $\phi_{S}$ such that

$Q_{d}(S|\rho)=Q_{d}(\phi_{S}|\rho)$ (40)

i.e., such that the range of $\phi_{S}$ is equal to the commutant $S^{\prime}$
of $S$? It turns out that, under the closure assumption of Proposition 1, the
answer is largely affirmative.

First, if $S=\\{X\\}$ for some projective operator $X$, then

$\phi_{S}(\sigma)=\phi_{X}(\sigma):=\sum_{x}X_{x}\sigma X_{x}=\sigma_{X}$ (41)

as noted above. Second, if the group $G_{S^{\prime\prime}}=\\{U_{g}\\}$ of
unitary operators on $S^{\prime\prime}$ is compact, with normalised measure
$dg$, then $\phi_{S}$ is the twirling map [25, 26]

$\phi_{S}(\sigma)=\phi_{G_{S^{\prime\prime}}}(\sigma):=\int dgU_{g}\sigma
U_{g}^{\dagger}.$ (42)

Third, if the system Hilbert space is finite dimensional, then

$\phi_{S}(\sigma):=\mathfrak{E}_{S^{\prime}}(\sigma)$ (43)

where, for a given von Neumann algebra $\mathfrak{a}$,
$\mathfrak{E}_{\mathfrak{a}}$ denotes the conditional expectation map defined
via [6, 41]

${\rm Tr}\left[{{XY}}\right]\equiv{\rm
Tr}\left[{{X\mathfrak{E}_{a}(Y)}}\right]$ (44)

for $X\in\mathfrak{a}$ and arbitrary $Y$. These maps agree on their common
domains, and cover most situations of interest.

Finally, these maps allow the explicit evaluation of the standard asymmetry
$Q_{1}(S|\rho)$ in most cases.

######  Proposition 2.

If $S=\\{X\\}$ for a projective observable $X$ and/or $G_{S^{\prime\prime}}$
is compact and/or the system Hilbert space is finite dimensional, for some set
of operators $S$, and $d(\rho,\sigma)$ is the relative entropy distance
function in Eq. (19), then the standard asymmetry measure $Q_{1}(S|\rho)$
defined in Eq. (37) is given by

$Q_{1}(S|\rho)=D_{1}(\rho\|\phi_{S}(\rho))=H(\phi_{S}(\rho))-H(\rho).$ (45)

###### Proof.

The result relies on the composition properties $\phi\circ\phi=\phi$,
$\phi\circ f=f\circ\phi$ for any numerical function $f$, and the duality
property $\phi^{*}=\phi$, which may be checked to hold when $\phi$ is any of
the maps in Eqs. (41)–(43). In particular, these three properties, used in
turn, yield ${\rm Tr}\left[{{\rho\log\phi(\sigma)}}\right]={\rm
Tr}\left[{{\rho\log\phi\circ\phi(\sigma)}}\right]={\rm
Tr}\left[{{\rho\phi(\log\phi(\sigma))}}\right]={\rm
Tr}\left[{{\phi(\rho)\log\phi(\sigma)}}\right]$. Hence, definitions (19) and
(39) give

$\displaystyle D_{1}(\rho\|\phi(\sigma))-D_{1}(\rho\|\phi(\rho))$
$\displaystyle={\rm
Tr}\left[{{\phi(\rho)(\log\phi(\rho)-\log\phi(\sigma))}}\right]$
$\displaystyle=D_{1}(\phi(\rho)\|\phi(\sigma))\geq 0,$

implying
$Q_{1}(\phi|\rho)=D_{1}(\rho\|\phi(\rho))+\inf_{\sigma}D_{1}(\phi(\rho)\|\phi(\sigma))=D_{1}(\rho\|\phi(\rho))$.
Thus, since Eq. (40) holds under the stated conditions,

$\displaystyle Q_{1}(S|\rho)$ $\displaystyle=Q_{1}(\phi|\rho)={\rm
Tr}\left[{{\rho\log\rho}}\right]-{\rm Tr}\left[{{\rho\log\phi(\rho)}}\right]$
$\displaystyle=-H(\rho)-{\rm
Tr}\left[{{\phi(\rho)\log\phi(\rho)}}\right]=H(\phi(\rho))-H(\rho)$

as required. ∎

Proposition 2 implies that $\phi_{S}(\rho)$ is the output state of $\phi_{S}$
that is closest to $\rho$, when ‘distance’ is measured via relative entropy,
and significantly generalises Eq. (20) for the standard asymmetry of single
observables. It has been previously given by Gao et al. for the case that $S$
is a von Neumann algebra on a finite Hilbert space [6]. Note also that
Corollary 2.3 of Gao et al. for strong subadditivity generalises to (using
$(S\cup T)^{\prime}=S^{\prime}\cap T^{\prime}$ and Proposition 1) the
uncertainty relation

$Q_{1}(S|\rho)+Q_{1}(T|\rho)\geq Q_{1}(S\cup T|\rho),$ (46)

for the standard asymmetry of any two sets $S$ and $T$ for which
$\phi_{S}\circ\phi_{T}=\phi_{T}\circ\phi_{S}$. The following section is
concerned with uncertainty relations for Rényi asymmetry.

##  III Tradeoff relations for asymmetry and entropy

###  III.1 Rényi asymmetry vs Rényi entropy

The Rényi asymmetry $Q_{\alpha}(X|\rho)$ in Eqs. (26) and (34) represents the
distance between the state of the system and a set of postmeasurement states
for $X$, as measured via the sandwiched Rényi divergence in Eq. (24). It is
also a measure of the sensitivity of the state to transformations generated by
$X$ (or its Naimark extension), and is the quantum component in the quantum-
classical decomposition of Rényi entropy in Eq. (29). It reduces to the
standard asymmetry given by the quantum relative entropy in Eq. (20) for the
case $\alpha=1$, and the general case has recently been found to have useful
applications in the areas of quantum information, quantum metrology, quantum
coherence, open quantum systems, and time-energy uncertainty relations [6, 7,
9, 5, 8].

As the intrinsically quantum contribution to Rényi entropy, Rényi asymmetry
should be expected to play a fundamental role in entropic uncertainty
relations. This expectation is supported by uncertainty relation (46) of Gao
et al. for the standard asymmetry. In particular, for two conjugate rank-1
projective observables $X$ and $Y$ on a $d$-dimensional Hilbert space, with
$X_{x}=|x\rangle\langle x|$, $Y_{y}=|y\rangle\langle y|$ and $|\langle
x|y\rangle|^{2}=d^{-1}$, one has
$(\sigma_{X})_{Y}=(\sigma_{Y})_{X}=d^{-1}\mathbbm{1}$ from Eq. (18), implying
$\phi_{\\{X,Y\\}}(\sigma)=d^{-1}\mathbbm{1}$ from Eqs. (37) and (40), and Eq.
(46) simplifies to

$Q_{1}(X|\rho)+Q_{1}(Y|\rho)\geq Q_{1}(\\{X,Y\\}|\rho)=\log d-H(\rho)$ (47)

with $Q_{1}(\\{X,Y\\}|\rho)$ evaluated via Proposition 2. Noting that
$H(\rho_{X})=H(X|\rho)$ and $H(\rho_{Y})=H(Y|\rho)$ for such observables, and
applying Eq. (20), this is equivalent to the tradeoff relation,

$Q_{1}(X|\rho)+H(Y|\rho)\geq\log d,$ (48)

between the standard asymmetry and Shannon entropy of the observables (and to
the strong entropic uncertainty relation $H(X|\rho)+H(Y|\rho)\geq\log
d+H(\rho)$ for the Shannon entropies of such observables [4, 13]).

A second example is the analogous tradeoff relation

$Q_{\alpha}(J_{z}|\rho)+H_{\alpha}(\Phi|\rho)\geq\log
2\pi,\qquad\alpha\geq\mbox{$\frac{1}{2}$},$ (49)

for angular momentum and rotation angle [9]. Noting the upper bound
$Q_{\alpha}(J_{z}|\rho)\leq H_{\beta}(J_{z}|\rho)$ from Eq. (29), this
immediately implies and so is stronger than the standard uncertainty relation
$H_{\alpha}(J_{z}|\rho)+H_{\beta}(\Phi|\rho)\geq\log 2\pi$ for the Rényi
entropies of these observables [43].

It is shown here that similar tradeoff relations hold for arbitrary pairs of
discrete observables, whether projective or nonprojective. However, the case
of one projective and one arbitrary observable will be considered first, as
the derivation is particularly simple for this case.

######  Theorem 1.

For a discrete projective observable $X$ with projection valued measure
$\\{X_{x}\\}$, and an arbitrary discrete observable $Y$ with POVM
$\\{Y_{y}\\}$, one has the tradeoff relation

$Q_{\alpha}(X|\rho)+H_{\alpha}(Y|\rho)\geq-\log\max_{x,y}\lambda_{\max}(X_{x}Y_{y}X_{x}),~{}~{}\alpha\geq\mbox{$\frac{1}{2}$},$
(50)

between Rényi asymmetry and entropy, where $\lambda_{\max}(A)$ is the maximum
eigenvalue of Hermitian operator $A$.

###### Proof.

The proof is a generalisation of the derivation given by Korzekwa et al. of
entropic uncertainty relation (5) for rank-1 projective observables [4].
First, define a ‘measure and discard’ CPTP map $\varphi_{Y}$, from the Hilbert
space of the system to the Hilbert space of a record system suitable for
registering the result of a measurement of $Y$, via

$\varphi_{Y}(\rho):=\sum_{y}{\rm Tr}\left[{{\rho
Y_{y}}}\right]|y\rangle\langle y|,$ (51)

where $\\{|y\rangle\\}$ is an orthonormal basis for the record system.
Equation (18) and data processing inequality (25) for the sandwiched Rényi
divergence then give

$\displaystyle D_{\alpha}(\rho\|\sigma_{X})$ $\displaystyle\geq
D_{\alpha}(\varphi_{Y}(\rho)\|\varphi_{Y}(\sigma_{X}))$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{y}{\rm Tr}\left[{{\rho
Y_{y}}}\right]^{\alpha}{\rm Tr}\left[{{\sigma_{X}Y_{y}}}\right]^{1-\alpha}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{y}{\rm Tr}\left[{{\rho
Y_{y}}}\right]^{\alpha}{\rm
Tr}\big{[}\sigma\sum_{x}X_{x}Y_{y}X_{x}\big{]}^{1-\alpha}.$

The final trace is over a sum of orthogonal subspaces, corresponding to the
set of projections $X_{x}$. Hence, this trace is maximised, for a given value
of $y$, when $\sigma$ is the eigenstate corresponding to the maximum possible
eigenvalue of $X_{x}Y_{y}X_{x}$ over all $x$, and so is upper-bounded by
$\mu=\max_{x,y}\lambda_{\max}(X_{x}Y_{y}X_{x})$. Thus, noting that
$\tfrac{1}{\alpha-1}\log\sum_{y}a(y)b(y)^{1-\alpha}$ is monotonic decreasing
in $b(y)$ for $a(y),b(y),\alpha\geq 0$, the Rényi asymmetry in Eq. (26) has
the lower bound

$\displaystyle Q_{\alpha}(X|\rho)$
$\displaystyle=\inf_{\sigma}D_{\alpha}(\rho\|\sigma_{X})$
$\displaystyle\geq\inf_{\sigma}\frac{1}{\alpha-1}\log\sum_{y}{\rm
Tr}\left[{{\rho Y_{y}}}\right]^{\alpha}\mu^{1-\alpha}$
$\displaystyle=-\frac{1}{1-\alpha}\log\sum_{y}p(y|\rho)^{\alpha}-\log\mu.$
(52)

Recalling the definition of Rényi entropy in Eq. (27), this is equivalent to
the statement of the theorem. ∎

Theorem 1 provides a strong link between asymmetry and entropy that
significantly extends tradeoff relations (48) and (49) for conjugate
observables. It further leads to strengthened uncertainty relations for Rényi
entropies, as discussed further below. Note that the proof of the Theorem is
relatively elementary, relying on a simple manipulation of data-processing
inequality (25) for the sandwiched Rényi divergence, suggesting it may also be
used to obtain analogous tradeoff relations for other distance-based measures
of quantumness or asymmetry.

Noting that the asymmetry $Q_{\alpha}(X|\rho)$ vanishes for $[X,\rho]=0$
(consistent with Eq. (9) for measures of quantumness), Theorem 1 gives a lower
bound for the Rényi entropy $H_{\alpha}(Y|\rho)$ in this case. A particular
choice of $X$ yields the following interesting albeit weak corollary.

######  Corollary 1.

For an arbitrary discrete observable $X$ with POVM $\\{X_{x}\\}$, and state
$\rho$ with spectral decomposition $\rho=\sum_{j}e_{j}E_{j}$ (i.e., $E_{j}$ is
the projection onto the eigenspace corresponding to eigenvalue $e_{j}$ of
$\rho$), one has the lower bound

$H_{\alpha}(X|\rho)\geq-\log\max_{x,j:e_{j}>0}\lambda_{\max}(E_{j}X_{x}E_{j}),~{}~{}~{}\alpha\geq\mbox{$\frac{1}{2}$},$
(53)

for Rényi entropy.

###### Proof.

Define the projection onto the support of $\rho$ by
$E:=\sum_{j:p_{j}>0}E_{j}$, implying $\rho=E\rho E$, and let $Y_{E}$ be the
projection of $Y$ onto the support of $\rho$, with POVM $\\{EY_{y}E\\}$.
Replacing $X$ by $\rho$ and $Y$ by $Y_{E}$ in Theorem 1, gives a vanishing
asymmetry $Q_{\alpha}(\rho|\rho)=0$ via definition (26) and a lower bound
$\mu_{\rho
Y_{E}}=\max_{y,j:p_{j}>0}\lambda_{\max}(E_{j}EY_{y}EE_{j})=\max_{y,j:p_{j}>0}\lambda_{\max}(E_{j}Y_{y}E_{j})$
via $EE_{j}=E_{j}~{}(=0)$ for $p_{j}>0~{}(=0)$. Noting
$H_{\alpha}(Y_{E}|\rho)=H_{\alpha}(Y|\rho)$ and changing notation from $Y$ to
$X$ then yields the Corollary as desired. ∎

Corollary 1 bounds the Rényi entropy of any observable in terms of its degree
of incompatibility with the system state. Note that it represents a slight
sharpening of Eq. (6) in the Introduction, where the latter does not limit the
maximisation to $e_{j}>0$, and implies a state-independent lower bound,
$-\log\lambda_{\max}(X_{x})$, via $E_{j}\leq\mathbbm{1}$. However, the
Corollary does not take the mixedness of the state into account, making it
relatively weak in comparison to bounds that do.

For example, Corollary 1 is weaker than the known classical lower bound

$H_{\alpha}(X|\rho)\geq H_{\infty}(X|\rho)=-\log\max_{x}p(x|\rho)$ (54)

for Rényi entropy, following from the monontonic decreasing property
$H_{\alpha}(X|\rho)\geq H_{\beta}(X|\rho)$ for $\alpha<\beta$ [44, 45], where
$H_{\infty}$ depends on mixedness via the eigenvalues $e_{j}$ of $\rho$. In
particular, the operator $E_{j}X_{x}E_{j}$ is nonzero only on the ${\rm
Tr}\left[{{E_{j}}}\right]$-dimensional unit eigenspace of $E_{j}$ and so has
at most ${\rm Tr}\left[{{E_{j}}}\right]$ nonzero eigenvalues, implying that
the sum of its eigenvalues, ${\rm Tr}\left[{{E_{j}X_{x}E_{j}}}\right]$, is
upper bounded by ${\rm
Tr}\left[{{E_{j}}}\right]\lambda_{\max}(E_{j}X_{x}E_{j})$. Hence, defining
$p_{j}:=e_{j}{\rm Tr}\left[{{E_{j}}}\right]$, then $\sum_{j:p_{j}>0}p_{j}=1$
and one has $p(x|\rho)=\sum_{j:e_{j}>0}e_{j}{\rm
Tr}\left[{{E_{j}X_{x}E_{j}}}\right]\leq\sum_{j:p_{j}>0}p_{j}\lambda_{\max}(E_{j}X_{x}E_{j})\leq\max_{j:e_{j}>0}\lambda_{\max}(E_{j}X_{x}E_{j})$.
Maximising over $x$ immediately implies the bound in Corollary 1 is never
greater than the bound $H_{\infty}(X|\rho)$ in Eq. (54), as claimed (with
equality for pure states).

A second example will be given in Corollary 5 below, which provides a strong
lower bound for Rényi entropy that depends on mixedness rather than
incompatibility, and which is stronger than both Corollary 1 and the classical
bound (54) for sufficiently mixed states.

The main technical result of this paper is a generalisation of Theorem 1 to
all discrete observables, whether projective or nonprojective, via the
generalised definition of asymmetry in Eq. (34).

######  Theorem 2.

For arbitary discrete observables $X$ and $Y$, with corresponding POVMs
$\\{X_{x}\\}$ and $\\{Y_{y}\\}$, one has the tradeoff relation

$Q_{\alpha}(X|\rho)+H_{\alpha}(Y|\rho)\geq-\log\mu_{XY},\qquad\alpha\geq\mbox{$\frac{1}{2}$},$
(55)

for asymmetry and entropy, where

$\mu_{XY}:=\max_{x,y}\lambda_{\max}(X_{x}^{1/2}Y_{y}X_{x}^{1/2})=\mu_{YX}$
(56)

denotes the maximum eigenvalue of $X_{x}^{1/2}Y_{y}X_{x}^{1/2}$ over $x$ and
$y$.

###### Proof.

It is first necessary to generalise $Y$ to an observable $\tilde{Y}$ on the
extended Hilbert space $\tilde{\cal H}$ for which the Naimark extension (30)
is defined for $X$. In particular, $\tilde{Y}$ is defined to correspond to the
POVM $\\{Y_{y}\\}\cup\\{P^{\perp}\\}$ on $\tilde{\cal H}$, with
$P^{\perp}:=\tilde{\mathbbm{1}}-P$, and $\varphi_{Y}$ in Eq. (51) is
generalised to the ‘measure and discard’ map

$\varphi_{\tilde{Y}}(\tilde{\rho}):=\sum_{y}{\rm
Tr}\left[{{\tilde{\rho}Y_{y}}}\right]|y\rangle\langle y|+{\rm
Tr}\left[{{\tilde{\rho}P^{\perp}}}\right]|y^{\perp}\rangle\langle y^{\perp}|,$
(57)

for general states $\tilde{\rho}$ on $\tilde{\cal H}$, where
$|y^{\perp}\rangle$ is the record state corresponding to POVM element
$P^{\perp}$. For system state $\rho$ on $\cal H$ and general state
$\tilde{\sigma}$ on $\tilde{\cal H}$ one then has, using data processing
inequality (25) and noting that $p(y^{\perp}|\rho)={\rm Tr}\left[{{\rho
P^{\perp}}}\right]=0$ via $\rho=P\rho P$,

$\displaystyle D_{\alpha}(\rho\|\tilde{\sigma}_{\tilde{X}})$
$\displaystyle\geq
D_{\alpha}(\varphi_{\tilde{Y}}(\rho)\|\varphi_{\tilde{Y}}(\tilde{\sigma}_{\tilde{X}}))$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{y}{\rm Tr}\left[{{\rho
Y_{y}}}\right]^{\alpha}{\rm
Tr}\left[{{\tilde{\sigma}_{\tilde{X}}Y_{y}}}\right]^{1-\alpha}$
$\displaystyle=\frac{1}{\alpha-1}\log\sum_{y}{\rm Tr}\left[{{\rho
Y_{y}}}\right]^{\alpha}{\rm
Tr}\big{[}\tilde{\sigma}\sum_{x}\tilde{X}_{x}Y_{y}\tilde{X}_{x}\big{]}^{1-\alpha}$

similarly to the proof of Theorem 1. Also similarly, noting that the
$\tilde{X}_{x}$ are orthogonal projections, the final trace is maximised for a
given value of $y$ by choosing $\tilde{\sigma}$ to be the eigenstate
corresponding to the maximum over $x$ of the maximum eigenvalue of
$\tilde{X}_{x}Y_{y}\tilde{X}_{x}$. Now,

$\displaystyle\lambda_{\max}(\tilde{X}_{x}Y_{y}\tilde{X}_{x})$
$\displaystyle=\lambda_{\max}[(\tilde{X}_{x}Y_{y}^{1/2})(\tilde{X}_{x}Y_{y}^{1/2})^{\dagger}]$
$\displaystyle=\lambda_{\max}[(\tilde{X}_{x}Y_{y}^{1/2})^{\dagger}(\tilde{X}_{x}Y_{y}^{1/2})]$
$\displaystyle=\lambda_{\max}(Y_{y}^{1/2}\tilde{X}_{x}Y_{y}^{1/2})]$
$\displaystyle=\lambda_{\max}(Y_{y}^{1/2}P\tilde{X}_{x}PY_{y}^{1/2})]$
$\displaystyle=\lambda_{\max}(Y_{y}^{1/2}X_{x}Y_{y}^{1/2})]$
$\displaystyle=\lambda_{\max}(X_{x}^{1/2}Y_{y}X_{x}^{1/2}),$ (58)

with the second and last lines following from
$\lambda_{max}(AA^{\dagger})=\lambda_{\max}(A^{\dagger}A)$, and the fourth and
fifth lines from $Y_{y}=PY_{y}P$ and Eq. (30), respectively. Hence,

$\displaystyle Q_{\alpha}(X|\rho)$
$\displaystyle=\inf_{\sigma}D_{\alpha}(\rho\|\sigma_{\tilde{X}})$
$\displaystyle\geq\inf_{\tilde{\sigma}}D_{\alpha}(\rho\|\tilde{\sigma}_{\tilde{X}})$
$\displaystyle\geq-\frac{1}{1-\alpha}\log\sum_{y}p(y|\rho)^{\alpha}-\log\mu_{XY}$
(59)

similarly to Eq. (52). This is equivalent to the statement of the Theorem,
noting that $\mu_{XY}=\mu_{YX}$ via the last two lines of Eq. (58). ∎

Theorem 2 reduces to Theorem 1 when $X$ is a projective observable, and
underpins the strong uncertainty relation for Rényi entropies given in Theorem
3 below. Note that $\mu_{XY}$ can also be written as

$\mu_{XY}=\max_{x,y}\|X_{x}^{1/2}Y_{y}^{1/2}\|_{\infty}^{2}=\max_{x,y}\|Y_{y}^{1/2}X_{x}^{1/2}\|_{\infty}^{2}$
(60)

where $\|A\|_{\infty}=[\lambda_{\max}(A^{\dagger}A)]^{1/2}$ denotes the
operator norm of $A$, i.e., the largest singular value of $A$.

Theorem 2 immediately provides a simple lower bound for Rényi asymmetry, as
per the following corollary.

######  Corollary 2.

For an arbitrary discrete observable $X$ with POVM $\\{X_{x}\\}$ one has the
lower bound

$Q_{\alpha}(X|\rho)\geq-\log\max_{x}\lambda_{\max}(X_{x}),\qquad\alpha\geq\mbox{$\frac{1}{2}$},$
(61)

for the Rényi asymmetry of $X$.

###### Proof.

Choose $Y=1$ in Theorem 2 and observe that $H_{\alpha}(1|\rho)\equiv 0$. ∎

The lower bound in Corollary 2 is trivial for projective observables, but
typically nonvanishing otherwise, corresponding to the ‘hidden’ asymmetry
discussed following Eq. (34). For example, the lower bound is
$\log\tfrac{3}{2}$ for the qubit trine observable $T$ discussed following
Corollary 5.

Finally, it is worth noting here, as a preview to the next Section, that the
standard uncertainty relation for Rényi entropies in Eq. (4) of the
Introduction has a simple direct derivation via Theorem 2 for the case of
Shannon entropy, i.e., $\alpha=\beta=1$, as per the following Corollary.

######  Corollary 3.

For arbitrary discrete observables $X$ and $Y$, with corresponding POVMs
$\\{X_{x}\\}$ and $\\{Y_{y}\\}$, one has the entropic uncertainty relation

$H(X|\rho)+H(Y|\rho)\geq-\log\mu_{XY}$ (62)

for Shannon entropy, where

$\mu_{XY}:=\max_{x,y}\lambda_{\max}(X_{x}^{1/2}Y_{y}X_{x}^{1/2})=\mu_{YX}$
(63)

denotes the maximum eigenvalue of $X_{x}^{1/2}Y_{y}X_{x}^{1/2}$ over $x$ and
$y$.

###### Proof.

The uncertainty relation follows immediately for any pure state
$\rho=|\psi\rangle\langle\psi|$, from Eq. (35) and Theorem 2. It then
immediately follows for general states via the concavity of Shannon entropy. ∎

Corollary 3 is given for rank-1 projective observables in [10] and for
arbitrary observables in [11], and generalised to uncertainty relation (4) for
Rényi entropies in [12]. However, the proofs of Theorem 2 and Corollary 3
ultimately rely on data-processing inequality (25), rather than on Riesz’s
theorem [10, 11, 12]. This has the advantage of leading to the stronger
uncertainty relation in Eq. (I) for Rényi entropies for very little additional
work, as will be shown next.

###  III.2 Strong uncertainty relations for Rényi entropies

It is straightforward to strengthen Corollary 3, so as to obtain uncertainty
relation (I) of the Introduction, which takes the mixedness of the state into
account.

######  Theorem 3.

For arbitrary discrete observables $X$ and $Y$, with corresponding POVMs
$\\{X_{x}\\}$ and $\\{Y_{y}\\}$, one has the entropic uncertainty relation

$\displaystyle H_{\alpha}(X|\rho)+H_{\beta}(Y|\rho)$
$\displaystyle\geq-\log\mu_{XY}$
$\displaystyle~{}\,~{}~{}+\max\\{C_{\alpha}(Y|\rho),C_{\beta}(X|\rho)\\}$ (64)

for Rényi entropies with $1/\alpha+1/\beta=2$, where $C_{\alpha}(Z|\rho)$ is
the classical contribution to $H_{\beta}(Z|\rho)$, defined via quantum-
classical decomposition (29), and

$\mu_{XY}:=\max_{x,y}\lambda_{\max}(X_{x}^{1/2}Y_{y}X_{x}^{1/2})=\mu_{YX}$
(65)

denotes the maximum eigenvalue of $X_{x}^{1/2}Y_{y}X_{x}^{1/2}$ over $x$ and
$y$.

###### Proof.

Substitution of the quantum-classical decomposition (29) of
$H_{\beta}(X|\rho)$ into Theorem 2 gives

$H_{\beta}(X|\rho)+H_{\alpha}(Y|\rho)\geq-\log\mu_{XY}+C_{\alpha}(X|\rho)$
(66)

for $1/\alpha+1/\beta=2$. Comparing the inequality resulting from swapping $X$
with $Y$ in this expression with the inequality resulting from swapping
$\alpha$ with $\beta$ then yields the Theorem as desired. ∎

Recalling that the classical component $C_{\alpha}(X|\rho)$ vanishes for pure
states as per Eq. (36), it is seen that Theorem 3 improves on the standard
uncertainty relation in Corollary 3 by taking the mixedness of the state into
account. For example, for the special case $\alpha=\beta=1$ one has the
following corollary.

######  Corollary 4.

For an arbitrary rank-1 discrete observable $X$ and an arbitrary discrete
observable $Y$, with respective POVM elements $\\{|x\rangle\langle x|\\}$ and
$\\{Y_{y}\\}$, the Shannon entropies of $X$ and $Y$ satisfy the uncertainty
relation

$H(X|\rho)+H(Y|\rho)\geq-\log\max_{x,y}\langle x|Y_{y}|x\rangle+H(\rho).$ (67)

###### Proof.

For projective observables this result is an immediate consequence of Eq. (23)
and Theorem 3 for $\alpha=\beta=1$. More generally, for any discrete
observable $X$, with Naimark extension $\tilde{X}$, Eqs. (23) and (31) yield
$C_{1}(X|\rho)=H(\rho)-\sum_{x}p(x|\rho)H(\tilde{\rho}_{x})$, with
$\tilde{\rho}_{x}:=\tilde{X}_{x}\rho\tilde{X}_{x}/p(x|\rho)$ defined on the
extended Hilbert space. Hence, $C_{1}(X|\rho)\leq H(\rho)$ for all observables
$X$ (including $X=Y$), with equality when $\tilde{X}$ is rank-1. But for a
rank-1 observable $X$ one can always choose $\tilde{X}$ to be rank-1. In
particular, expressing the projection $\tilde{X}_{x}$ as an orthogonal sum of
rank-1 projections, $\tilde{X}_{x}=\sum_{k}|x,k\rangle\langle x,k|$, the
Naimark extension property $X_{x}=P\tilde{X}_{x}P$ in Eq. (30) requires that
$P|x,k\rangle=0$ for all but one value of $k$, $k=k_{x}$ say, implying one can
replace $\tilde{X}$ by a rank-1 projective observable $\tilde{X}^{\prime}$ on
the extended Hilbert space $\tilde{H}^{\prime}$ generated by the span of
$\\{|x,k_{x}\rangle\\}$, with $\tilde{X}^{\prime}_{x}:=|x,k_{x}\rangle\langle
x,k_{x}|$. Hence, making such a choice, it follows that
$C_{1}(X|\rho)=H(\rho)\geq C_{1}(Y|\rho)$. Substitution into Theorem 3 with
$\alpha=\beta=1$ then gives the Corollary as desired. ∎

Corollary 4 generalises the known uncertainty relation for Shannon entropies
of rank-1 projective observables in Eq. (5) of the Introduction [13, 4], and
is given in Eq. (71) of [14] under the assumption of a finite system Hilbert
space. Note that $H(\rho)$ appears as the measure of mixedness in this case
because it is an upper bound for classicality when $\alpha=\beta=1$ and is
saturated for rank-1 observables.

Finally, Theorem 3 also yields a strong lower bound for the Rényi entropy of
discrete observables that takes mixedness into account.

######  Corollary 5.

For an arbitrary discrete observable $X$ with POVM $\\{X_{x}\\}$ one has the
lower bound

$H_{\alpha}(X|\rho)\geq-\log\max_{x}\lambda_{\max}(X_{x})+C_{\frac{\alpha}{2\alpha-1}}(X|\rho),$
(68)

for Rényi entropy.

###### Proof.

Choose $Y=1$ in Theorem 3, and note that $0\leq C_{\alpha}(1|\rho)\leq
H_{\beta}(1|\rho)=0$ via quantum-classical decomposition (29) (which is valid
for arbitrary observables as discussed in Sec. II.2.4). The Corollary also
follows directly from the asymmetry lower bound in Corollary 2, again using
Eq. (29) (but with $\alpha$ and $\beta$ swapped). ∎

For the case of a rank-1 observable $X$ and $\alpha=1$ one has
$C_{1}(X|\rho)=H(\rho)$ as per the proof of Corollary 4, and the lower bound
in Corollary 5 reduces to

$H(X|\rho)\geq-\log\max_{x}\langle x|x\rangle+H(\rho)$ (69)

This also corresponds to choosing $Y=1$ in Corollary 4, and is stronger than
the alternative choice $Y=X$, which replaces $H(\rho)$ by
$\mbox{$\frac{1}{2}$}H(\rho)$. For rank-1 projective observables (i.e,
$\langle x|x\rangle\equiv 1$), it reduces to the known property that the
Shannon entropy of such observables is never less than the von Neumann entropy
(since $p(x|\rho)=\sum_{j}S_{xj}p_{j}$ where $S_{xj}:=|\langle
x|\psi_{j}\rangle|^{2}$ is a doubly stochastic matrix for any orthogonal
decomposition $\rho=\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|$ of $\rho$).

It is of interest to compare Corollary 5, which gives a lower bound for Rényi
entropy that depends on the sharpness of $X$ and the mixedness of $\rho$, with
classical lower bound (54). As a simple example, consider the projective qubit
observable $X=\bm{\sigma}\cdot\bm{n}$ for spin direction $\bm{n}$, and state
$\rho=\mbox{$\frac{1}{2}$}(\mathbbm{1}+\bm{\sigma}\cdot\bm{r})$ with Bloch
vector $\bm{r}$. Corollary 5 then yields the lower bound

$H(X|\rho)\geq
H(\rho)=-\frac{1+r}{2}\log\frac{1+r}{2}-\frac{1-r}{2}\log\frac{1-r}{2},$ (70)

for the Shannon measurement entropy, with $r:=|\bm{r}|$. In contrast, the
corresponding classical lower bound in Eq. (54) gives

$H(X|\rho)\geq\log\frac{2}{1+|\bm{n}\cdot\bm{r}|}=\log\frac{2}{1+r|\cos\theta_{\bm{n},\bm{r}}|},$
(71)

where $\theta_{\bm{n},\bm{r}}$ is the angle between $\bm{n}$ and $\bm{r}$.
Both lower bounds range from 0 for $\bm{r}=\pm\bm{n}$ up to $\log 2$ for
$\bm{r}=\bm{0}$. However, as depicted in Fig. 1, the bound in Eq. (70) is
always stronger for sufficiently mixed states (as is easily proved noting that
Eq. (70) is concave in $r$ and Eq. (71) is convex in $r$, with equality at
$r=0$).

Figure 1: Lower bounds on measurement entropy for qubit states. The lower
bound in Eq. (70) for qubit observable $X=\bm{\sigma}\cdot\bm{n}$, following
from Corollary 5, is plotted as a function of the Bloch vector length $r$
(upper solid blue curve), while the classical lower bound (71) is plotted for
the cases $\theta_{\bm{n},\bm{r}}=0$, i.e., $\bm{n}\cdot\bm{r}=r$ (lower solid
red curve), and $\theta_{\bm{n},\bm{r}}=\frac{\pi}{3}$, i.e.,
$\bm{n}\cdot\bm{r}=\mbox{$\frac{1}{2}$}r$ (dashed black curve). Corresponding
plots of the lower bounds for the measurement entropy of the trine observable
$T$ in Eqs. (72) and (73) may be obtained by adding $\log\frac{3}{2}$ to each
lower bound. For both observables the classical lower bounds are seen to be
weaker than the bound from Corollary 5 for sufficiently mixed states, as
discussed more generally in the main text.

As an example for the case where $X$ is a nonprojective observable, consider
the qubit trine observable $X=T$ with POVM elements
$T_{j}=\tfrac{1}{3}(\mathbbm{1}+\bm{\sigma}\cdot\bm{m}^{j})$, where
$\bm{m}^{j}$ denotes the unit vector $(\cos(2\pi j/3),\sin(2\pi j/3),0)$ for
$j=0,1,2$ [37, 46, 39]. Corollary 5 then gives the lower bound

$H(T|\rho)\geq\log\frac{3}{2}+H(\rho),$ (72)

whereas the classical lower bound (54) gives

$H_{\alpha}(T|\rho)\geq\log\frac{3}{2}+\log\frac{2}{1+r\max_{j}\cos\theta_{\bm{m}^{j},\bm{r}}},$
(73)

where $\theta_{\bm{n},\bm{r}}$ is the angle between $\bm{m}^{j}$ and $\bm{r}$.
While both lower bounds range from a minimum value of $\log\tfrac{3}{2}$ for
$\bm{r}=\bm{m}^{j}$, up to the maximum possible value of
$H_{\alpha}(T|\rho)=\log 3$ for $\bm{r}$ orthogonal to the plane of trine
directions, the bound in Eq. (72) is always stronger for sufficiently mixed
states, as depicted in Fig. 1 (with the common term $\log\tfrac{3}{2}$
subtracted). For example, if $\bm{r}$ is anti-aligned with one of the trine
directions, then $\cos\theta_{\bm{m}^{j},\bm{r}}$ is maximised for
$\theta_{\bm{m}^{j},\bm{r}}=\tfrac{\pi}{3}$ and Corollary 5 outperforms
classical lower bound 54 (and hence also Corollary 1) for $r\lesssim 0.753$,
corresponding to the intersection of the upper solid and dashed curves in Fig.
1.

More generally, for the case of an arbitrary observable $X$ and $\alpha=1$,
Corollary 5 reduces to the lower bound

$\displaystyle H(X|\rho)$
$\displaystyle\geq-\log\max_{x}\lambda_{\max}(X_{x})+H(\rho)$
$\displaystyle\qquad-\sum_{x}p(x|\rho)H(\tilde{\rho}_{x}),$ (74)

for Shannon entropy, with
$\tilde{\rho}_{x}:=\tilde{X}_{x}\rho\tilde{X}_{x}/p(x|\rho)$, for any Naimark
extension $\tilde{X}$ of $X$ as per Eq. (30) (see also the proof of Corollary
4). This case is intriguingly similar in form to the lower bound

$\displaystyle H(X|\rho_{a})$
$\displaystyle\geq-\log\max_{x}\lambda_{\max}(X_{x})+H(\rho_{b})$
$\displaystyle\qquad-\sum_{x}p(x|\rho)H(\rho_{b|x})$ (75)

for Shannon entropy in Eq. (65) of [14] for any joint state $\rho_{ab}$, with
$\rho_{b|x}:={\rm tr}_{a}[\rho_{ab}X_{x}]/p(x|\rho_{a})$. It would be of
interest to explore the connection between these bounds in future work. This
is nontrivial given that $\rho_{b}=\sum_{x}p(x|\rho_{a})\rho_{b|x}$ whereas
$\rho\neq\sum_{x}p(x|\rho)\tilde{\rho}_{x}$. The key may lie in the related
asymmetry lower bound given in Corollary 2.

##  IV Discussion

The results weave two main threads together. The first is a general approach
to quantum-classical decompositions of resource measures for observables (and
sets of operators), based on the idea that pure states are the most
resourceful (Sec. II.1). The quantum contribution to the resource measure
arises from the noncommutativity or incompatibility of observables with the
state, and the classical contribution from the mixedness of the state. The
approach unifies previous decompositions of variance and entropy, and
generalises them to the decomposition of the symmetrised covariance matrix
with respect to measures of quantum Fisher information, and the decomposition
of Shannon entropy with respect to the conditional entropy of self-dual
communication channels (Secs. II.2.1, II.2.2 and Appendix A.1), and is applied
to Rényi entropy in particular (Sec. II.2.3). In the latter case the quantum
contribution is provided by the Rényi asymmetry, with its connection to Rényi
entropy arising from duality relation (35) for pure states.

The second thread is the generalisation and unification of measures of
quantumness and asymmetry to nonprojective observables (Sec. II.2.4), and to
sets, groups and algebras of operators and to quantum channels (Sec. II.2.5).
The Rényi asymmetry is a particular example of interest, as it generalises the
standard asymmetry measure based on relative entropy and has applications in
contexts as diverse as coherence, information, uncertainty relations,
metrology, and open quantum systems.

The weaving of the two threads is in two natural stages. In the first stage,
tradeoff relations between Rényi asymmetry and Rényi entropy are obtained, as
per Theorems 1 and 2 (Sec. III.1). These have relatively elementary
derivations, based on data processing inequality (25) for sandwiched Rényi
divergences, and significantly generalise known relations (48) and (49) for
the case of conjugate observables. Corollaries 1–3 are simple applications of
these Theorems that yield lower bounds for Rényi asymmetry and Rényi entropy,
and recover previously known uncertainty relation (4) for the case of Shannon
entropy. All these results are independent of any consideration of quantum-
classical decompositions.

It is in the second stage that the tradeoff relations for Rényi asymmetry are
combined with the quantum-classical decomposition of Rényi entropy in Eq.
(29), to obtain the strong uncertainty relation for Rényi entropies in Theorem
3 (Sec. III.2). This relation is valid for both projective and nonprojective
observables, and improves on known uncertainty relation (4) by taking the
mixedness of the system state into account via the classical contribution to
Rényi entropy. Corollary 4 recovers a general result from Theorem 3 for the
case of Shannon entropies, using the property that the classical contribution
is bounded by the von Neumann entropy of the system for this case, while
Corollary 5 gives a strong bound for Rényi entropy that improves on Corollary
1 and classical lower bound (54) for sufficiently mixed states.

It is worthwhile remarking here on a point of terminology. The term
‘quantumness’ has been used for the measure of noncommutativity
$Q(X,Y,\dots|\rho)$ introduced in Sec. II.1, given its fundamental role as the
inherently quantum contribution to uncertainty and other resource measures [1,
2, 4]. Note, however, that the nonvanishing of a quantum commutator has a
direct classical analogue: the nonvanishing of a classical Poisson bracket.
Hence, one can define classical analogues of asymmetry and the like (e.g., as
a measure of the distance between a given phase space distribution and the set
of phase space distributions invariant under canonical transformations
generated by a given classical observable). Nevertheless, this does not extend
to give a classical analogue of the approach to quantum-classical
decompositions in Sec. II.1. In particular, this approach relies on the
purification of a given quantum state on a larger Hilbert space, whereas there
is no analogous purification of classical phase space distributions to a
delta-distribution on a larger phase space. Hence, the term ‘quantumness’ may
be justified as reflecting an essentially quantum feature in the context of
quantum-classical decompositions.

There are various possible directions for future work, including the
following. First, it is known that the bound $\mu_{XY}$ in known uncertainty
relation (4) can be improved using majorisation properties of Schur-convex
functions such as Rényi entropy [48, 47]. This suggests looking for similar
improvements to the lower bounds in Theorems 1–3. Perhaps, for example, the
argument based on a monotonic decreasing function in the proof of Theorem 1
can be strengthened via a majorisation argument.

Second, it is of interest to seek analogous tradeoff relations and uncertainty
relations for continuous observables such as position and momentum. Here the
main technical issue is that the kets appearing in the spectral decompositions
of such observables are nonnormalisable, so that the analogue of $\sigma_{X}$
in Eq. (18) is not a density operator. However, it may be possible to take
suitable limits of discrete observables to obtain results for this case, as
has been done previously for Shannon entropic uncertainty relations [49, 50].

Third, whereas the measures of quantumness considered in the paper are focused
on measures of uncertainty and asymmetry, consideration of other resources
measures such as entanglement and Bell nonlocality, and related quantum-
classical decompositions, is also a possible avenue of exploration. One might,
for example, maximise some measure $Q(X_{1}\otimes X_{2},Y_{1}\otimes
Y_{2},\dots|\rho)$ or $u(X_{1}\otimes X_{2},Y_{1}\otimes Y_{2},\dots|\psi)$
over sets of local observables to obtain a measure of bipartite entanglement
for state $\rho$ or $\psi$, and apply the approach in Sec. II.1 or the
alternative approach in Appendix A.2 to obtain corresponding quantum-classical
decompositions. It would also be of interest to extend Proposition 2 for the
standard asymmetry measure, to sets $S$ of two or more (noncommuting)
operators on an infinite Hilbert space, for the case that the group of unitary
operators on the double commutant $S^{\prime\prime}$ is noncompact.

Fourth and finally, it should be possible to extend or relate the strong
uncertainty relation in Theorem 3 to the presence of quantum memory or quantum
side information, in which uncertainties are conditioned on systems with which
the system of interest is correlated [13, 14]. For example, the general method
given by Coles et al. in [51] is applicable to Rényi conditional entropies of
the form
$H_{\alpha}(a|b):=-\inf_{\sigma_{b}}D_{\alpha}(\rho_{ab}\|\mathbbm{1}_{a}\otimes\sigma_{b})$,
noting that the sandwiched Rényi divergence (24) satisfies the required
properties for $\alpha\geq\mbox{$\frac{1}{2}$}$. Hence, noting further the
duality property $H_{\alpha}(a|b)+H_{\beta}(a|c)=0$ for $1/\alpha+1/\beta=2$
and any pure state $\rho_{abc}$ [31, 33], Theorem 1 of [51] yields an
uncertainty relation of the form

$H_{\alpha}(X|b)+H_{\beta}(Y|c)\geq-\log\mu_{XY},~{}~{}~{}\frac{1}{\alpha}+\frac{1}{\beta}=2,$
(76)

where $X$ and $Y$ are observables on the first component of a tripartite state
$\rho_{abc}$, $\rho_{Xb}:=\oplus_{x}{\rm Tr}_{ac}[\rho_{abc}X_{x}\otimes
1_{b}\otimes 1_{c}]\equiv\varphi_{X}(\rho_{ab})$, and
$\rho_{Yc}:=\oplus_{y}{\rm Tr}_{ab}[\rho_{abc}Y_{y}\otimes 1_{b}\otimes
1_{c}]\equiv\varphi_{Y}(\rho_{ac})$ (see also Theorem 11 of [31]). It would be
of interest to find a connection between quantum memory and the classical
contribution to Rényi entropy via a comparison of this uncertainty relation
with Theorem 3 (see also the related discussion at the end of Sec. III.2).

###### Acknowledgements.

I am grateful to an anonymous referee for suggesting comparing Corollary 1
with classical lower bound (54). Having been recently diagnosed with
pancreatic cancer, this may be one of my last physics papers. I therefore take
this opportunity to thank the many colleagues I have interacted with over the
years, both locally and nonlocally :-)

##  Appendix A FURTHER EXAMPLES OF QUANTUM-CLASSICAL DECOMPOSITIONS

###  A.1 Decomposition of Shannon entropy via a quantumness measure for self-
dual channels

A quantum-classical decomposition of Shannon entropy is obtained here that is
quite different to the decomposition of Korzekwa et al. [4] discussed in Sec.
II.2.2, based on a choice of quantumness measure that arises in the context of
quantum information channels.

In particular, consider the communication channel defined by measuring
observable $X$ with POVM $\\{X_{x}\\}$ at the receiver, on members of an
ensemble of signal states with average density operator $\rho$, for the case
of the ‘dual’ ensemble $\mathcal{E}_{X,\rho}:=\\{p_{x};\rho_{x}\\}$ defined by
[22, 23]

$p_{x}:={\rm Tr}\left[{{\rho
X_{x}}}\right],\qquad\rho_{x}:=\frac{\rho^{1/2}X_{x}\rho^{1/2}}{{\rm
Tr}\left[{{\rho X_{x}}}\right]}$ (77)

(note that $\rho=\sum_{x}p_{x}\rho_{x}$, and the choice of $\rho_{x}$ is left
arbitrary if $p_{x}=0$). This ‘self-dual’ channel is invariant under source
duality, i.e., interchange of the the signal ensemble and the receiver
measurement [22, 23]. It provides a strong lower bound for the maximum
information obtainable from a measurement of $X$ on an ensemble with density
operator $\rho$ [23], and is closely related via the inverse map
$p_{x}\rho_{x}\rightarrow X_{x}$ to the ‘pretty good measurement’ used in the
proof of the Holevo-Schumacher-Westmoreland theorem for the classical capacity
of a quantum channel [15].

Each measurement result $x$ of the self-dual channel gives some information
about which signal state $\rho_{x^{\prime}}$ was transmitted, with an average
uncertainty quantified by the Shannon conditional entropy [15],

$H(X^{\prime}|X):=H(X,X^{\prime})-H(X)=H(X^{\prime})-I(X:\mathcal{E}_{X,\rho}).$
(78)

Here $I(X:\mathcal{E}_{X,\rho})$ is the mutual information of the channel, and
$H(X^{\prime})=-\sum_{x^{\prime}}p_{x^{\prime}}\log p_{x^{\prime}}=H(X|\rho)$
is the entropy of the signal-state distribution $p_{x^{\prime}}$ in Eq. (77).
Note that the joint distribution of $X$ and $X^{\prime}$ for state $\rho$
follows via Eq. (77) as

$p(x,x^{\prime}|\rho):={\rm
Tr}\left[{{\rho^{1/2}X_{x^{\prime}}\rho^{1/2}X_{x}}}\right].$ (79)

Now, if $X$ is a projective observable, then the conditional entropy of the
self-dual channel provides a measure of quantumness satisfying assumption (9)
of Sec. II.1 (since $[X,\rho]=0$ implies the signal states in Eq. (77) are
orthogonal, so that one has a noiseless classical channel with
$H(X,X^{\prime})=H(X)$). Further, assumptions (10) and (11) are also satisfied
(noting for a pure state $\rho=|\psi\rangle\langle\psi|$ that the signal
states are identical and hence carry no information). Hence, noting
$p(x,x^{\prime})$ in Eq. (79) is invariant under Naimark extension (30) (since
$P\rho P=\rho$), it follows that the conditional entropy of the self-dual
channel in Eq. (78) is a measure of quantumness for both projective and
nonprojective observables, corresponding to

$Q_{\rm sd}(X|\rho):=H(X|\rho)-I(X:\mathcal{E}_{X,\rho}).$ (80)

Recalling from Eq. (77) that the signal states are identical and carry no
information when $\rho$ is pure, it follows $M_{\rm
sd}(X|\rho)=H(\tilde{X}\otimes\mathbbm{1}_{a}|\rho_{\psi})=H(X|\rho)$ for any
Naimark extension $\tilde{X}$ of $X$ and purification $\rho_{\psi}$ of $\rho$,
and the approach of Sec. II.1 then yields the quantum-classical decomposition

$M_{sd}(X|\rho):=H(X|\rho)=Q_{\rm sd}(X|\rho)+C_{\rm sd}(X|\rho)$ (81)

of Shannon entropy, with the classical component given by the mutual
information of the self-dual channel, i.e.,

$C_{\rm
sd}(X|\rho):=I(X:\mathcal{E}_{X,\rho})=2H(X|\rho)-H(X,X^{\prime}|\rho).$ (82)

Here the second equality follows by direct calculation [23], where the last
term is the Shannon entropy of the joint probability distribution in Eq. (79).

The decompositions of Shannon entropy in Eqs. (17) and (81) are seen to be
rather different, although by construction they agree for projective
observables when $[\rho,X]=0$ or $\rho=|\psi\rangle\langle\psi|$. It would be
of interest to compare these decompositions in more detail elsewhere.

###  A.2 An alternative roof-based approach to quantum-classical
decompositions

The method of constructing quantum-classical decompositions in Sec. II.1 is
guided by the level of accessibility, and hence usefulness, of a given quantum
resource. In particular, the classical component arises due to lack of access
to a purified state of the system. Here an alternative approach is briefly
described, based on the direct decomposition of the system state into a
mixture of pure states. The approach is conceptually simple, but appears to be
less general and technically more challenging than the approach given in the
main text.

The starting point is a function $u(X,Y,\dots,|\psi)$ that quantifies a
quantum resource for the pure states of a given system. It is natural to
consider such functions in various contexts, e.g., the entropy of the Schmidt
coefficients for an entangled pure state. The function $u$ is assumed to
satisfies two minimal properties:

  * (a)

No ‘quantumness’ when the system is in a simultaneous eigenstate of
observables $X,Y,\dots$, i.e.,

$\qquad u(X,Y,\dots|\psi)=0~{}~{}{\rm
if}~{}~{}X\psi=x\psi,~{}Y\psi=y\psi,\dots;$ (83)

  * (b)

Invariance under unitary transformations, i.e.,

$\qquad u(UXU^{\dagger},UYU^{\dagger},\dots|U\psi)=u(X,Y,\dots|\psi)$ (84)

for any unitary transformation $U$.

Now, let $\mathcal{P}_{\rho}$ denote the set of pure-state decompositions of
density operator $\rho$, i.e.,

$\mathcal{P}_{\rho}:=\bigg{\\{}\\{p_{j};\psi_{j}\\}:p_{j}\geq
0,\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|=\rho\bigg{\\}},$ (85)

where $|\psi\rangle\langle\psi|$ denotes the density operator corresponding to
$\psi$. The quantum-classical decomposition corresponding to function $u$ is
then defined via

$\displaystyle M(X,Y,\dots|\rho)$
$\displaystyle:=\sup_{\\{p_{j};\psi_{j}\\}\in\mathcal{P}_{\rho}}\sum_{j}p_{j}u(X,Y,\dots|\psi_{j}),$
(86) $\displaystyle Q(X,Y,\dots|\rho)$
$\displaystyle:=\inf_{\\{p_{j};\psi_{j}\\}\in\mathcal{P}_{\rho}}\sum_{j}p_{j}u(X,Y,\dots|\psi_{j}),$
(87) $\displaystyle C(X,Y,\dots|\rho)$
$\displaystyle:=M(X,Y,\dots|\rho)-Q(X,Y,\dots|\rho).$ (88)

Thus, $M$ and $Q$ are the concave and convex roofs of $u$, with the classical
component given by their gap.

It is straightforward to check that the above construction satisfies the basic
requirements discussed in Sec. II.1. Assumption (9) follows from property
(83), noting that commutativity implies there is a pure-state decomposition
into simultaneous eigenstates, while assumptions 10 and (11) are direct
consequences of property (84) and definitions (86)–(88), respectively. It is
seen that the classical mixing contribution arises in this approach directly
via the mixing of pure states, in contrast to the approach in Sec. II in which
the mixing is generated via a partial trace.

####  A.2.1 Example: variance and minimal Fisher information

As a first example, choose $u$ to be the variance of a projective observable,
i.e.,

$u_{V}(X|\psi):={\rm Var}_{\psi}X=\langle X^{2}\rangle_{\psi}-\langle
X\rangle_{\psi}^{2}.$ (89)

The concave and convex roofs of this function are given in [52] and [53],
respectively (see also [54]), and the above ‘roof’ construction then yields
the corresponding quantum-classical decomposition

$M_{V}(X|\rho)={\rm Var}_{\rho}X=Q_{V}(X|\rho)+C_{V}(X|\rho),$ (90)

where

$Q_{V}(X|\rho)=\mbox{$\frac{1}{2}$}\sum_{j,k}\frac{|\langle
j|[X,\rho]|k\rangle|^{2}}{\langle j|\rho|j\rangle+\langle k|\rho|k\rangle}.$
(91)

Here $\\{|j\rangle\\}$ is the set of eigenstates of $\rho$ and the summation
runs over the eigenstates that generate nonzero denominators. The summation
may be recognised, up to a constant factor, as the minimal quantum Fisher
information, corresponding to the symmetric logarithmic derivative [16, 52,
53], and hence this decomposition of the variance is a special case of the
decomposition of the covariance matrix in Eq. (15) of Sec. II.2. Thus, the
’roof’ construction method picks out a preferred quantum Fisher information
for the decomposition, in comparison to the method in the main text.

####  A.2.2 Example: Shannon entropy and informational power

More generally, the convex and concave roofs of a given function will not be
amenable to analytic calculation, making this approach reliant on numerical
methods. Consider, for example, the case where the function $u$ is the Shannon
entropy of observable $X$, i.e.,
$u_{H}(X|\psi):=H(X|\psi)=-\sum_{x}\langle\psi|X_{x}|\psi\rangle\log\langle\psi|X_{x}|\psi\rangle$.
The roof construction then gives the decomposition

$M_{H}(X|\rho)=Q_{H}(X|\rho)+C_{H}(\rho)$ (92)

with

$\displaystyle M_{H}(X|\rho)$
$\displaystyle:=\sup_{\\{p_{j};\psi_{j}\\}\in\mathcal{P}_{\rho}}\sum_{j}p_{j}H(X|\psi_{j})$
$\displaystyle=H(X|\rho)-\inf_{\mathcal{E}\in\mathcal{P}_{\rho}}I(X:\mathcal{E})$
(93) $\displaystyle Q_{H}(X|\rho)$
$\displaystyle=\inf_{\\{p_{j};\psi_{j}\\}\in\mathcal{P}_{\rho}}\sum_{j}p_{j}H(X|\psi_{j})$
$\displaystyle=H(X|\rho)-\sup_{\mathcal{E}\in\mathcal{P}_{\rho}}I(X:\mathcal{E}),$
(94) $\displaystyle C_{H}(X|\rho)$
$\displaystyle:=\sup_{\mathcal{E}\in\mathcal{P}_{\rho}}I(X:\mathcal{E})-\inf_{\mathcal{E}\in\mathcal{P}_{\rho}}I(X:\mathcal{E}),$
(95)

where $I(X:\mathcal{E})$ is the Shannon mutual information for observable $X$
and signal ensemble $\mathcal{E}$. Hence the quantum component is directly
related to the maximum mutual information of the observable over the set of
pure-state signal ensembles for density operator $\rho$, also called the
‘informational power’ of the observable for state $\rho$ [55], while the
classical component is given by the gap between the maximum and minimum
possible mutual informations.

In general there are few analytic results for the maximum and minimum mutual
informations of a pure state channel with fixed $X$ and $\rho$, and so the
mutual information must be maximised or minimised numerically. However, upper
and lower bounds are provided by choosing particular pure state ensembles in
$\mathcal{P}_{\rho}$, such as the Scrooge ensemble [56] or (for rank-1
observables) the self-dual ensemble in Eq. (77) above, which can be used as
estimates of the minimum and maximum values, respectively. The von Neumann
entropy $H(\rho)$ also provides an estimate of the maximum value via the
Holevo bound for mutual information [15]. Hence, for example, for a rank-1
observable $X$ one has the inequalities

$H(X|\rho)-H(\rho)\leq Q_{H}(X|\rho)\leq Q_{\rm sd}(X|\rho),$ (96) $Q_{\rm
sd}(X|\rho)-q(\rho)\leq C_{H}(X|\rho)\leq H(\rho),$ (97)

where $Q_{\rm sd}(X|\rho)$ is the conditional entropy of the self-dual channel
in Eqs. (78) and (80), and $q(\rho)$ is the subentropy of $\rho$, i.e., the
entropy of the Scrooge ensemble for $\rho$ [56].

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