# Sufficient conditions, lower bounds and trade-off relations for quantumness
in Kirkwood-Dirac quasiprobability

Agung Budiyono<EMAIL_ADDRESS>Research Center for Quantum Physics,
National Research and Innovation Agency, South Tangerang 15314, Republic of
Indonesia

###### Abstract

Kirkwood-Dirac (KD) quasiprobability is a quantum analog of classical phase
space probability. It offers an informationally complete representation of
quantum state wherein the quantumness associated with quantum noncommutativity
manifests in its nonclassical values, i.e., the nonreal and/or negative values
of the real part. This naturally raises a question: how does such form of
quantumness comply with the uncertainty principle which also arise from
quantum noncommutativity? Here, first, we obtain sufficient conditions for the
KD quasiprobability defined relative to a pair of PVM (projection-valued
measure) bases to have nonclassical values. Using these nonclassical values,
we then introduce two quantities which capture the amount of KD quantumness in
a quantum state relative to a single PVM basis. They are defined respectively
as the nonreality, and the classicality which captures both the nonreality and
negativity, of the associated KD quasiprobability over the PVM basis of
interest, and another PVM basis, and maximized over all possible choices of
the latter. We obtain their lower bounds, and derive trade-off relations
respectively reminiscent of the Robertson and Robertson-Schrödinger
uncertainty relations but with lower bounds maximized over the convex sets of
Hermitian operators whose complete sets of eigenprojectors are given by the
PVM bases. We discuss their measurement using weak value measurement and
classical optimization. We then suggest an information theoretical
interpretation of the KD nonreality relative to a PVM basis as a lower bound
to the maximum total root-mean-squared error in an optimal estimation of the
PVM basis, and thereby obtain a lower bound and a trade-off relation for the
root-mean-squared error. Finally, we suggest an interpretation of the KD
nonclassicality relative to a PVM basis as a lower bound to the total state
disturbance caused by a nonselective projective binary measurement associated
with the PVM basis, and derive a lower bound and a trade-off relation for the
disturbance.

Kirkwood-Dirac quasiprobability, quantum noncommutativity, nonreality,
negativity, quantumness, sufficient conditions, lower bounds, trade-off
relations

###### pacs:

03.65.Ta, 03.65.Ca

##  I Introduction

Heisenberg uncertainty principle is a basic tenet of quantum mechanics which
sets down a radical conceptual demarcation from classical mechanics Heisenberg
UR . It stipulates a fundamental restriction, in the forms of trade-off
relations, on the simultaneous predictability of outcomes of measurement of
two physical quantities. Formally, the trade-off relations arise from the
noncommutativity of operators representing quantum measurements Kennard UR ;
Weyl UR ; Robertson UR . From the very beginning, the uncertainty principle
has led to the foundational debate about the deep nature of randomness arising
in quantum measurement EPR paradox  and the intimately related conceptual
issue on the meaning of quantum correlation Bell's theorem . In recent
decades, attempts to better understand the meaning of uncertainty relation and
quantum randomness in general, has opened an avenue for fruitful applications
in different areas of quantum science and quantum technology Coles entropic
uncertainty relation review . It is thus important to study the uncertainty
principle from various perspectives to appreciate its rich and multi-faceted
nature and to conceive further implications.

The earliest uncertainty relations are developed based on the quantification
of the measurement uncertainty in terms of variance of measurement outcomes
Kennard UR ; Weyl UR ; Robertson UR ; Schroedinger UR . Certain drawbacks of
variance for characterizing unpredictability motivated the construction of
uncertainty relations based on the Shannon entropy of the measurement outcomes
Hirschman UR based on entropy ; Everett question on the UR based on entropy ;
Bialynicki-Birula UR based on entropy for phase space ; Deutsch entropic UR ;
Maassen entropic UR ; Berta entropic UR with side information ; Coles entropic
UR ; Hall quantum-classical decomposition ; Wehner entropic UR review ; Coles
entropic uncertainty relation review . Variance and Shannon entropy of
measurement outcomes however do not only quantify the genuine quantum
uncertainty originating from the noncommutativity between the quantum state
and the measurement operators. But, they also take into account the classical
uncertainty stemming from the agent’s ignorance about the preparation, either
due to classical noise or lack of access of another system entangled with the
system of interest, leading to the preparation of mixed states. It is thus
instructive to ask if it is possible to develop uncertainty relations for the
intrinsic quantum uncertainty rather than for the total measurement
uncertainty. A notable result along this direction was reported in Ref. Luo
quantum Robertson-like uncertainty relation for WY skew information  where the
author derived a trade-off relation for an intrinsic quantum uncertainty
quantified by means of Wigner-Yanase skew information Wigner-Yanase skew
information , having a form similar to the Robertson uncertainty relation.
This result is generalized in Ref. Furuichi quantum Robertson-Schroedinger
uncertainty relation for WY skew information  to obtain a trade-off relation
similar to the Robertson-Schrödinger uncertainty relation. Another approach is
suggested in Refs Korzekwa quantum-classical decomposition ; Singh uncertainty
relation for coherence ; Yuan uncertainty relation for coherence ; Hall
quantum-classical decomposition  which used some measures of quantum coherence
to isolate the intrinsic quantum uncertainty and showed that they satisfy some
trade-off relations similar to the entropic uncertainty relations.

In the present study, we work with an informationally equivalent
representation of quantum states on a finite-dimensional Hilbert space using
Kirkwood-Dirac (KD) quasiprobability Kirkwood quasiprobability ; Dirac
quasiprobability ; Barut KD quasiprobability . KD quasiprobability is a
quantum analog of classical phase space probability wherein the quantumness
associated with noncommutativity manifests in its nonclassical values, i.e.,
non-real values and/or negative values of its real part. This prompts the
question on how the uncertainty principle imposes a restriction on such form
of quantumness. In order to answer this question, we first derive sufficient
conditions for the KD quasiprobability relative to a pair of rank-1 orthogonal
PVM (projection-valued measure) bases to have nonclassical values. We then
introduce two quantities which measure the KD quantumness in a quantum state
relative to a single PVM basis. The first quantity is defined as the
nonreality in the KD quasiprobability over the PVM basis of interest and
another PVM basis, and maximized over all possible choices of the latter. We
call it the KD nonreality in the quantum state relative the PVM basis. The
second quantity is defined similarly, but relative to the nonclassicality
which captures simultaneously both the nonreality and the negativity of the KD
quasiprobability. We call it the KD nonclassicality in the quantum state
relative the PVM basis. Both quantities have been proposed earlier in Refs.
Agung KD-nonreality coherence ; Agung KD-nonclassicality coherence  as
faithful quantifiers of quantum coherence relative to the incoherent
orthonormal basis corresponding to the rank-1 PVM basis. We obtain lower
bounds for the quantumness captured by the above defined KD nonreality and KD
nonclassicality in a state relative to a PVM basis.

We then proceed to derive trade-off relations for the KD nonreality in a state
relative to a PVM basis and that relative to another PVM basis, and similarly
for the KD nonclassicality in a state relative to a PVM basis and that
relative to another PVM basis. They are respectively reminiscent of the
Robertson Robertson UR  and the Robertson-Schrödinger uncertainty relations
Schroedinger UR , but with lower bounds that are optimized over the convex
sets of all pairs of Hermitian operators whose eigenprojectors are given by
the two PVM bases of interest. The lower bounds and the trade-off relations
for the KD nonreality and KD nonclassicality in a state relative to a rank-$1$
orthogonal PVM basis lead to similar lower bounds and trade-off relations for
the $l_{1}$-norm coherence of the state relative to the incoherent orthonormal
basis corresponding to the PVM basis Baumgratz quantum coherence measure . We
sketch a measurement scheme of the KD nonreality and KD nonclassicality
relative to a PVM basis based on weak value measurement and classical
optimization. We then suggest an information theoretical interpretation of the
KD nonreality in a state relative to a PVM basis as a lower bound to the root-
mean-squared error of an optimal estimation of the PVM basis based on
projective measurement in the worst case scenario. This allows us to derive a
lower bound and a trade-off relation for the root-mean-squared error of the
optimal estimation of a PVM basis in the worst case scenario. We further
suggest an operational interpretation of the KD nonclassicality in a state
relative to a PVM basis as a lower bound to the total state disturbance caused
by a nonselective projective binary measurement associated with the PVM basis,
and thereby derive a lower bound and a trade-off relation of such state
disturbance.

##  II Sufficient conditions for nonclassical Kirkwood-Dirac quasiprobability

KD quasiprobability is a specific quantum analog of phase space probability
distribution in classical statistical mechanics Kirkwood quasiprobability ;
Dirac quasiprobability . The KD quasiprobability associated with a quantum
state represented by a density operator $\varrho$ on a Hilbert space
$\mathcal{H}$ over a pair of orthonormal bases $\\{\ket{a}\\}$ and
$\\{\ket{b}\\}$ of $\mathcal{H}$, is defined as Kirkwood quasiprobability ;
Dirac quasiprobability ; Barut KD quasiprobability

$\displaystyle{\rm Pr}_{\rm KD}(a,b|\varrho):={\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\\},$ (1)

where $\Pi_{x}:=\ket{x}\bra{x}$, $x=a,b$. We note that $\\{\Pi_{x}\\}$
comprises a set of rank-1 orthogonal PVM, i.e., $\sum_{x}\Pi_{x}=\mathbb{I}$,
$\Pi_{x}\Pi_{x^{\prime}}=\delta_{xx^{\prime}}\Pi_{x}$, where $\mathbb{I}$ is
the identity operator on $\mathcal{H}$ and $\delta_{xx^{\prime}}$ is the
Kronecker delta. The PVM $\\{\Pi_{x}\\}$ describes a sharp projective
measurement with outcomes $x$ and probability ${\rm Pr}(x|\varrho)={\rm
Tr}\\{\Pi_{x}\varrho\\}$. Here on we shall thus refer to $\\{\Pi_{x}\\}$ as a
rank-1 PVM basis.

KD quasiprobability gives correct marginal probabilities, i.e., $\sum_{i}{\rm
Pr}_{\rm KD}(a,b|\varrho)={\rm Pr}(j|\varrho)$, $i\neq j$, $i,j=\\{a,b\\}$.
However, unlike conventional classical probability, KD quasiprobability may
take nonreal value, and its real part, called the Terletsky-Margenou-Hill
quasiprobability Terletsky TBMH quasiprobability ; Barut KD quasiprobability ;
Margenau TBMH quasiprobability , may be negative. Such nonreality and
negativity capture the quantum noncommutativity, that is, assuming two of its
three ingredients $\\{\varrho,\Pi_{a},\Pi_{b}\\}$ commute, e.g.,
$[\Pi_{a},\varrho]_{-}=0$, renders the KD quasiprobability ${\rm Pr}_{\rm
KD}(a,b|\varrho)$ real and nonnegative. Here and in what follows,
$[X,Y]_{\mp}:=XY\mp YX$ denotes the commutator and anticommutator between two
Hermitian operators $X$ and $Y$. In this sense, the nonreality or/and the
negativity of KD quasiprobability delineate some form of quantumness stemming
from quantum noncommutativity. The converse however is not necessarily true
Drori nonclassicality tighter and noncommutativity ; deBievre nonclassicality
in KD distribution . Remarkably, the real and imaginary parts of the KD
quasiprobability can be estimated in experiment without resorting to full
state tomography either using weak value measurement or other methods Aharonov
weak value ; Aharonov-Daniel book ; Wiseman weak value ; Lundeen measurement
of KD distribution ; Salvail direct measurement KD distribution ; Bamber
measurement of KD distribution ; Thekkadath measurement of density matrix ;
Johansen quantum state from successive projective measurement ; Lostaglio KD
quasiprobability and quantum fluctuation ; Hernandez-Gomez experimental
observation of TBMH negativity ; Wagner measuring weak values and KD
quasiprobability ; Vallone strong measurement to reconstruct quantum wave
function ; Cohen estimating of weak value with strong measurements ; Lundeen
complex weak value ; Jozsa complex weak value . This form of quantumness,
i.e., the nonreality or/and the negativity in the KD quasiprobability has thus
found applications in different areas of quantum science and technology
Lostaglio KD quasiprobability and quantum fluctuation ; Allahverdyan TBMH as
quasiprobability distribution of work ; Lostaglio TBMH quasiprobability
fluctuation theorem contextuality ; Levy quasiprobability distribution for
heat fluctuation in quantum regime ; Alonso KD quasiprobability witnesses
quantum scrambling ; Halpern quasiprobability and information scrambling ;
Arvidsson-Shukur quantum advantage in postselected metrology ; Lupu-Gladstein
negativity enhanced quantum phase estimation 2022 ; Das KD quasiprobability in
postselected metrology ; Pusey negative TBMH quasiprobability and
contextuality ; Kunjwal contextuality of non-real weak value ; Lostaglio
contextuality in quantum linear response ; Agung KD-nonreality coherence ;
Agung KD-nonclassicality coherence ; Agung translational asymmetry from
nonreal weak value ; Agung estimation and operational interpretation of trace-
norm asymmetry ; Agung KD general quantum correlation .

KD quasiprobability gives an informationally complete representation of an
arbitrary quantum state. That is, given a KD quasiprobability ${\rm Pr}_{\rm
KD}(a,b|\varrho)$ defined over a pair of orthonormal bases $\\{\ket{a}\\}$ and
$\\{\ket{b}\\}$ with $\braket{a}{b}\neq 0$ for all $(a,b)$, the associated
quantum state can be reconstructed as $\varrho=\sum_{a,b}{\rm Pr}_{\rm
KD}(a,b|\varrho)\frac{\ket{a}\bra{b}}{\braket{b}{a}}$. This important fact
naturally raises an intriguing question on how the KD quantumness capture
different yet interrelated nonclassical concepts associated with a quantum
state subjected to quantum measurements. To this end, we have argued
previously that the nonreality or simultaneously both the nonreality and
negativity of the KD quasiprobability can be used to quantitatively
characterize quantum coherence Agung KD-nonreality coherence ; Agung KD-
nonclassicality coherence , asymmetry Agung translational asymmetry from
nonreal weak value ; Agung estimation and operational interpretation of trace-
norm asymmetry , and general quantum correlation Agung KD general quantum
correlation . In the present article, we study how the quantumness in the KD
quasiprobability complies with the quantum uncertainty principle. Both the KD
quantumness and the uncertainty principle arise from the quantum
noncommutativity.

First, we summarize two mathematical objects for quantifying respectively the
nonreality and the total nonclassicality which captures simultaneously both
the nonreality and the negativity in the KD quasiprobability. To quantify the
nonreality in the KD quasiprobability, we use the following $l_{1}$-norm of
the nonreal part of the KD quasiprobability:

$\displaystyle{\rm NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$
$\displaystyle:=$ $\displaystyle\sum_{a,b}|{\rm Im}{\rm Pr}(a,b|\varrho)|$ (2)
$\displaystyle=$ $\displaystyle\sum_{a,b}|{\rm Im}{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\\}|.$

It vanishes if and only if the KD quasiprobability is real. Next, let us
define the following quantity Drori nonclassicality tighter and
noncommutativity ; Alonso KD quasiprobability witnesses quantum scrambling ;
Lostaglio KD quasiprobability and quantum fluctuation :

$\displaystyle{\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$
$\displaystyle:=$ $\displaystyle\sum_{a,b}|{\rm Pr}_{\rm KD}(a,b|\varrho)|-1$
(3) $\displaystyle=$ $\displaystyle\sum_{a,b}|{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\\}|-1.$

It is nonnegative by definition since $\sum_{a,b}|{\rm Pr}_{\rm
KD}(a,b|\varrho)|\geq|\sum_{a,b}{\rm Pr}_{\rm KD}(a,b|\varrho)|=1$, where the
equality follows from the fact that KD quasiprobability is always normalized,
i.e., $\sum_{a,b}{\rm Pr}_{\rm KD}(a,b|\varrho)=1$. Moreover, it vanishes only
when $|{\rm Pr}_{\rm KD}(a,b|\varrho)|={\rm Pr}_{\rm KD}(a,b|\varrho)$ for all
$a$ and $b$, i.e., only when ${\rm Pr}_{\rm KD}(a,b|\varrho)$ is real and
nonnegative. ${\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ defined in Eq
(3) thus quantifies the failure of the KD quasiprobability ${\rm Pr}_{\rm
KD}(a,b|\varrho)$ to be both real and nonnegative. We refer to ${\rm
NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ and ${\rm NCl}(\\{{\rm Pr}_{\rm
KD}(a,b|\varrho)\\})$ defined respectively in Eqs. (2) and (3) as the KD
nonreality and the KD nonclassicality in the quantum state $\varrho$ relative
to the pair of PVM bases $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$.

We obtain two simple sufficient conditions respectively for nonvanishing ${\rm
NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ and ${\rm NCl}(\\{{\rm Pr}_{\rm
KD}(a,b|\varrho)\\})$. Below, we use the notation $\|X\|_{\infty}$ to denote
the operator norm or the $\infty$-Schatten norm of an operator $X$.
$\|X\|_{\infty}$ is equal to the largest eigenvalue modulus or the spectral
radius of $X$. Using the operator norm of a Hermitian operator $X$, we then
define the corresponding normalized Hermitian operator as
$\tilde{X}:=X/\|X\|_{\infty}$.

First, we have the following result for the KD nonreality in a state relative
to a pair of PVM bases.
Lemma 1. Given a state $\varrho$ on a Hilbert space $\mathcal{H}$, the
nonreality in the associated KD quasiprobability over a pair of PVM bases
$\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$ of $\mathcal{H}$ defined in Eq. (2), is
lower bounded as

$\displaystyle{\rm NRe}(\\{{\rm Pr}_{\rm
KD}(a,b|\varrho)\\})\geq\frac{1}{2}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|},$ (4)

where $A$ and $B$ are any Hermitian operators with bounded spectrum whose
complete set of eigenprojectors are respectively given by $\\{\Pi_{a}\\}$ and
$\\{\Pi_{b}\\}$.
Proof. Let $A=\sum_{a}a\Pi_{a}$ be a Hermitian operator on $\mathcal{H}$ with
the complete set of eigenprojectors $\\{\Pi_{a}\\}$ and the associated
spectrum of eigenvalues $\\{a\\}$. Similarly, let $B=\sum_{b}b\Pi_{b}$ be a
Hermitian operator on $\mathcal{H}$ with the complete set of eigenprojectors
$\\{\Pi_{b}\\}$ and the associated spectrum of eigenvalues $\\{b\\}$. From the
definition of the KD nonreality in the quantum state $\varrho$ relative to a
pair of PVM bases $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$ in Eq. (2), we have

$\displaystyle{\rm NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$
$\displaystyle=$
$\displaystyle\frac{1}{\|A\|_{\infty}\|B\|_{\infty}}\sum_{a,b}\|A\|_{\infty}\|B\|_{\infty}\big{|}{\rm
Im}({\rm Tr}\\{\Pi_{b}\Pi_{a}\varrho\\})\big{|}$ (5) $\displaystyle\geq$
$\displaystyle\frac{1}{\|A\|_{\infty}\|B\|_{\infty}}\big{|}{\rm Im}{\rm
Tr}\\{BA\varrho\\}\big{|}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|},$

where the inequality in Eq. (5) is due to the fact that
$\|A\|_{\infty}=\max\\{|a|\\}$ and $\|B\|_{\infty}=\max\\{|b|\\}$ and triangle
inequality. ∎

As an immediate corollary of the Lemma 1, while noncommutativity of all pairs
of $A$, $B$, and $\varrho$ are not sufficient for the KD quasiprobability
${\rm Pr}_{\rm KD}(a,b|\varrho)$ associated with $\varrho$ defined over the
eigenbasis $\\{\ket{a}\\}$ of $A$ and the eigenbasis $\\{\ket{b}\\}$ of $B$ to
have nonreal value (or its real part is negative, or both) Drori
nonclassicality tighter and noncommutativity ; deBievre nonclassicality in KD
distribution ; deBievre incompatibility-uncertainty-KD nonclassicality ; Xu KD
classical pure states , a nonvanishing lower bound in Eq. (4), i.e., ${\rm
Tr}\\{B[A,\varrho]_{-}\\}\neq 0$ for a pair of Hermitian operators $A$ and
$B$, is sufficient for the corresponding KD quasiprobability ${\rm Pr}_{\rm
KD}(a,b|\varrho)$ to be nonreal for some $(a,b)$. It is interesting to remark
that the lower bound in Eq. (4) takes a form similar to that of the Robertson
uncertainty relation Robertson UR .

Next, we derive a lower bound for the KD nonclassicality in a state relative
to a pair of PVM bases defined in Eq. (3).
Lemma 2. Given a state $\varrho$ on a Hilbert space $\mathcal{H}$, the
nonclassicality in the associated KD quasiprobability associated with
$\varrho$ over a pair of PVM bases $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$ of
$\mathcal{H}$ defined in Eq. (3) is lower bounded as

$\displaystyle{\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}$ (6)
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}-1,$

where $\tilde{X}_{\varrho}:=\frac{X}{\|X-{\rm
Tr}\\{X\varrho\\}\mathbb{I}\|_{\infty}}$, $X=A,B$, and $A$ and $B$ are any
Hermitian operators with bounded spectrum whose complete set of
eigenprojectors are respectively given by $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$.
Proof. Let again $A=\sum_{a}a\Pi_{a}$ be a Hermitian operator on $\mathcal{H}$
with the complete set of eigenprojectors $\\{\Pi_{a}\\}$ and the associated
spectrum of eigenvalues $\\{a\\}$. Likewise, let $B=\sum_{b}b\Pi_{b}$ be a
Hermitian operator on $\mathcal{H}$ with the complete set of eigenprojectors
$\\{\Pi_{b}\\}$ and the associated spectrum of eigenvalues $\\{b\\}$. Then,
from the definition of the KD nonclassicality in $\varrho$ relative to a pair
of PVM bases $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$ in Eq. (3), we first have

$\displaystyle{\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$
$\displaystyle=$ $\displaystyle\sum_{a,b}\big{|}{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\\}\big{|}-1$ (7) $\displaystyle=$
$\displaystyle\frac{\sum_{a,b}\|A\|_{\infty}\|B\|_{\infty}\big{|}{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\\}\big{|}}{\|A\|_{\infty}\|B\|_{\infty}}-1$
$\displaystyle\geq$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho\tilde{B}\tilde{A}\\}\big{|}-1$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{|}{\rm
Tr}\\{\varrho[\tilde{B},\tilde{A}]_{-}\\}+{\rm
Tr}\\{\varrho[\tilde{A},\tilde{B}]_{+}\\}\big{|}-1,$

where to get the last line, we have used a decomposition:
$\tilde{B}\tilde{A}=\frac{1}{2}[\tilde{B},\tilde{A}]_{-}+\frac{1}{2}[\tilde{A},\tilde{B}]_{+}$.
Notice that ${\rm Tr}\\{\varrho[\tilde{B},\tilde{A}]_{-}\\}$ is pure imaginary
while ${\rm Tr}\\{\varrho[\tilde{A},\tilde{B}]_{+}\\}$ is real. Hence, the
modulus in Eq. (7) can be evaluated to give

$\displaystyle{\rm NCl}(\\{{\rm Pr}_{\rm
KD}(a,b|\varrho)\\})\geq\frac{1}{2}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{B},\tilde{A}]_{-}\\}\big{|}^{2}+\big{|}{\rm
Tr}\\{\varrho[\tilde{A},\tilde{B}]_{+}\\}\big{|}^{2}\big{)}^{1/2}-1.$ (8)

Next, note that the left-hand side in Eq. (8) does not depend on the spectrum
of eigenvalues of $A$ and $B$. Now, consider the following Hermitian operators
$A^{\prime}=\sum_{a}(a-{\rm Tr}\\{A\varrho\\})\Pi_{a}=A-{\rm
Tr}\\{A\varrho\\}\mathbb{I}$ and $B^{\prime}=\sum_{b}(b-{\rm
Tr}\\{B\varrho\\})\Pi_{b}=B-{\rm Tr}\\{B\varrho\\}\mathbb{I}$. Then, we have
${\rm Tr}\\{\varrho[A^{\prime},B^{\prime}]_{-}\\}={\rm
Tr}\\{\varrho[A,B]_{-}\\}$ and ${\rm
Tr}\\{\varrho[A^{\prime},B^{\prime}]_{+}\\}={\rm
Tr}\\{\varrho[A,B]_{+}\\}-2{\rm Tr}\\{A\varrho\\}{\rm Tr}\\{B\varrho\\}$.
Using these relations, replacing $A$ and $B$ in Eq. (8) respectively with
$A^{\prime}$ and $B^{\prime}$, we obtain Eq. (6). ∎

Lemma 2 shows that a nonvanishing lower bound in Eq. (6), i.e.,
$\frac{1}{2}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}+\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}-1>0$, provides a
sufficient condition for the associated KD quasiprobability ${\rm Pr}_{\rm
KD}(a,b|\varrho)$ to be nonreal, or its real part is negative, or both, for
some $(a,b)$. It is again interesting to note that the lower bound takes a
form similar to the lower bound of the Robertson-Schrödinger uncertainty
relation. Unlike the latter, however, the lower bound in Eq. (6) depends
nonlinearly on the state. Note that the sufficient condition in Lemma 2 is
stronger than that in Lemma 1 since the former can also detect negativity of
the KD quasiprobability.

##  III Lower bounds and trade-off relations for KD quantumness in a state
relative to a single rank-1 orthogonal PVM basis

We first stress that both ${\rm NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$
and ${\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ defined in Eqs. (2) and
(3) quantify the KD quantumness stemming from the failure of commutativity
between the state $\varrho$ and both of the rank-1 PVMs bases $\\{\Pi_{a}\\}$
and $\\{\Pi_{b}\\}$, and also between the pair of the PVMs bases. How does the
quantumness of the KD quasiprobability portray the noncommutativity between a
state and a single PVM basis, e.g., between $\varrho$ and the PVM basis
$\\{\Pi_{a}\\}$? Quantities which reliably capture the noncommutativity
between a state $\varrho$ and a single PVM basis $\\{\Pi_{a}\\}$ is desirable
as we discuss the relation between the quantumness of the KD quasiprobability
and the uncertainty in measurement described by the rank-1 PVM $\\{\Pi_{a}\\}$
over the state $\varrho$, and the associated uncertainty relations. To this
end we introduce the following two quantities.
Definition 1. The KD nonreality in a state $\varrho$ on a finite-dimensional
Hilbert space $\mathcal{H}$ relative to a PVM basis $\\{\Pi_{a}\\}$ of
$\mathcal{H}$ is defined as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle:=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}{\rm NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ (9)
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}{\rm Im}{\rm Pr}_{\rm
KD}(a,b|\varrho)\big{|},$

where the supremum is taken over the set $\mathcal{M}_{\rm
r1PVM}(\mathcal{H})$ of all the rank-1 PVM bases of $\mathcal{H}$.
Definition 2. The KD nonclassicality in a state $\varrho$ on a finite-
dimensional Hilbert space $\mathcal{H}$ relative to a PVM $\\{\Pi_{a}\\}$ of
$\mathcal{H}$ is defined as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle:=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}{\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ (10)
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}|{\rm Tr}\\{\Pi_{b}\Pi_{a}\varrho\\}|-1.$

Let us mention that $\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$
and $\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$ defined
respectively in Eqs. (9) and (10) have been introduced earlier in Refs. Agung
KD-nonreality coherence ; Agung KD-nonclassicality coherence . There, it is
argued that both quantities can be used as faithful quantifiers of coherence
of $\varrho$ relative to the incoherent orthonormal basis $\\{\ket{a}\\}$
corresponding to the rank-1 orthogonal PVM basis $\\{\Pi_{a}\\}$ possessing
certain desirable properties. In particular, one can show that both quantities
are vanishing if and only if the state and the measurement basis are
commuting: $[\Pi_{a},\varrho]_{-}=0$ for all $a$ so that $\varrho$ is
incoherent relative to the orthonormal basis $\\{\ket{a}\\}$.

In the following subsections, we will derive lower bounds and trade-off
relations for the KD nonreality and KD nonclassicality in a quantum state
relative to a PVM basis defined respectively in Eqs. (9) and (10). For this
purpose, we denote by $\mathbb{H}(\mathcal{H})$ the convex set of all bounded
Hermitian operators on the Hilbert space $\mathcal{H}$,
$\mathbb{H}(\mathcal{H}|\\{\Pi_{x}\\})$ is the convex set of all bounded
Hermitian operators on $\mathcal{H}$ having the complete set of
eigenprojectors $\\{\Pi_{x}\\}$, and $\mathbb{H}(\mathcal{H}|\\{x\\})$ denotes
the set of all bounded Hermitian operators with a spectrum of eigenvalues
$\\{x\\}$, $x\in\mathbb{R}$.

###  III.1 Lower bound and trade-off relation for the KD nonreality in a state
relative to a PVM basis

Using Lemma 1, we directly obtain a lower bound for the quantumness associated
with the KD nonreality in a quantum state relative to a PVM basis.
Proposition 1. The KD nonreality in a state $\varrho$ on a finite-dimensional
Hilbert space $\mathcal{H}$ relative to a PVM basis $\\{\Pi_{a}\\}$ of
$\mathcal{H}$ defined in Eq. (9) is lower bounded as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|}.$ (11)

Proof. Taking the supremum over the set $\mathcal{M}_{\rm r1PVM}(\mathcal{H})$
of all the rank-1 PVM bases $\\{\Pi_{b}\\}$ of $\mathcal{H}$ to both sides of
Eq. (4), and noting Eq. (9), we first have

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}{\rm NRe}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ (12)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{b\\})}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|}.$

Next, notice that the left-hand side of Eq. (12) depends only on the PVM basis
$\\{\Pi_{a}\\}$, i.e., it is independent of the spectrum of eigenvalues
$\\{a\\}$ of $A$ and the spectrum of eigenvalues $\\{b\\}$ of $B$. Hence, upon
further taking the supremum over all possible eigenvalues spectrum of $A$ and
that of $B$ on the right-hand side of Eq. (12), the inequality can be
strengthened as in Eq. (11). ∎

The lower bound in Eq. (11) can be further evaluated to have the following
result.
Proposition 2. The KD nonreality in a quantum state $\varrho$ on a finite-
dimensional Hilbert space $\mathcal{H}$ relative to a PVM basis
$\\{\Pi_{a}\\}$ of $\mathcal{H}$ is lower bounded by the maximum trace-norm
asymmetry of the state relative to the translation group generated by all
Hermitian operators with the complete set of eigenprojectors that is given by
$\\{\Pi_{a}\\}$ as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\|[A,\varrho]_{-}\|_{1}/2\|A\|_{\infty}.$
(13)

Here, $\|O\|_{1}={\rm Tr}\\{\sqrt{OO^{\dagger}}\\}$ is the Schatten 1-norm or
the trace-norm of operator $O$, and $\|[A,\varrho]_{-}\|_{1}/2$ is just the
trace-norm asymmetry of the state $\varrho$ relative to the group of
translation unitary generated by the Hermitian operator $A$ Marvian - Spekkens
speakable and unspeakable coherence .
Proof. See Appendix A.

We show in Appendix B that for two-dimensional Hilbert space
$\mathbb{H}\cong\mathbb{C}^{2}$, i.e., a system of a single qubit, both the
inequalities in Eqs. (11) and (13) become equalities for arbitrary state
$\varrho$ on $\mathbb{C}^{2}$ and arbitrary PVM basis $\\{\Pi_{a}\\}$ of
$\mathbb{C}^{2}$. In this case, both sides in Eqs. (11) and (13) are given by
the corresponding $l_{1}$-norm coherence of a state $\varrho$ relative to the
incoherent orthonormal basis $\\{\ket{a}\\}$ defined as
$C_{l_{1}}(\varrho;\\{\ket{a}\\}):=\sum_{a\neq
a^{\prime}}|\braket{a}{\varrho}{a^{\prime}}|$ Baumgratz quantum coherence
measure , directly quantifying the total magnitude of the off-diagonal terms
of the density matrix. Hence, for any state $\varrho$ on $\mathbb{C}^{2}$ and
any PVM basis $\mathbb{A}=\\{\ket{e}\bra{e},\ket{e_{\perp}}\bra{e_{\perp}}\\}$
of $\mathbb{C}^{2}$, where $\ket{e_{\perp}}$ is the orthonormal partner of
$\ket{e}$, we have

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\mathbb{A})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathbb{C}^{2}|\mathbb{A})}\sup_{B\in\mathbb{H}(\mathbb{C}^{2})}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|}$ (14) $\displaystyle=$
$\displaystyle\sup_{A\in\mathbb{H}(\mathbb{C}^{2}|\mathbb{A})}\|[\tilde{A},\varrho]_{-}\|_{1}/2$
$\displaystyle=$ $\displaystyle 2|\braket{e}{\varrho}{e_{\perp}}|$
$\displaystyle=$ $\displaystyle
C_{l_{1}}(\varrho;\\{\ket{e},\ket{e_{\perp}}\\}).$

Moreover, the eigenbasis of $B_{*}$, where
$B_{*}\in\mathbb{H}(\mathbb{C}^{2})$ is a Hermitian operator which attains the
supremum in Eq. (14), is mutually unbiased with the orthonormal reference
basis $\\{\ket{e},\ket{e_{\perp}}\\}$ and also with the eigenbasis of
$\varrho$.

We finally obtain the following trade-off relation.
Proposition 3. The KD nonreality in a quantum state $\varrho$ on a finite-
dimensional Hilbert space $\mathcal{H}$ relative to a rank-1 PVM basis
$\\{\Pi_{a}\\}$ of $\mathcal{H}$ and that relative to another rank-1 PVM basis
$\\{\Pi_{b}\\}$ of $\mathcal{H}$ satisfy the following trade-off relation:

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{b}\\})\geq\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}^{2}.$ (15)

Proof. We first write the inequality in Eq. (11) as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}.$ (16)

Next, exchanging the role of $A$ and $B$ in Eq. (16), we also have

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{b}\\})\geq\frac{1}{2}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\sup_{A\in\mathbb{H}(\mathcal{H})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|},$ (17)

where the supremum are now taken over the set
$\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})$ of all bounded Hermitian operators $B$
on $\mathcal{H}$ whose complete set of eigenprojectors is given by the PVM
basis $\\{\Pi_{b}\\}$, and over the set $\mathbb{H}(\mathcal{H})$ of all
bounded Hermitian operator $A$ on $\mathcal{H}$. Combining Eqs. (16) and (17),
we thus finally obtain

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{b}\\})$ (18) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}$ $\displaystyle\times$
$\displaystyle\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\sup_{A\in\mathbb{H}(\mathcal{H})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}$ $\displaystyle\geq$
$\displaystyle\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}^{2},$

where to get the inequality in Eq. (18), we have made use of the fact that
$\sup_{X\in\mathbb{H}(\mathcal{H})}\\{\cdot\\}\geq\sup_{X\in\mathbb{H}(\mathcal{H}|\\{\Pi_{x}\\})}\\{\cdot\\}$.
∎

One can see that the lower bound in the trade-off relation of Eq. (15) takes a
form similar to that of the Robertson uncertainty relation. Unlike the latter,
however, it involves optimizations over the convex sets
$\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})$ and
$\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})$ of all Hermitian operators on
$\mathcal{H}$ whose complete set of eigenprojectors are given respectively by
the PVM bases $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$ relative to which we define
the KD nonreality in the state $\varrho$: $\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})$ and $\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{b}\\})$. The trade-off relation shows that if there is a
pair of Hermitian operators $A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})$ and
$B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})$ such that ${\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\neq 0$, then the lower bound in Eq.
(15) is not vanishing. In this case, both the KD nonreality $\mathcal{Q}_{\rm
KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$ in $\varrho$ relative to the PVM basis
$\\{\Pi_{a}\\}$ and the KD nonreality $\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{b}\\})$ in $\varrho$ relative to the PVM basis
$\\{\Pi_{b}\\}$ cannot be vanishing, and their product must satisfy the trade
off relation of Eq. (15).

Let us proceed to show that the lower bounds in Eqs. (11) and (13) and the
trade-off relation of Eq. (15) for the KD nonreality in a state relative to a
rank-1 orthogonal PVM basis, imply lower bounds and trade-off relation for the
$l_{1}$-coherence of the state relative to the orthonormal basis corresponding
to the PVM basis. First, note that as shown in Ref. Agung KD-nonreality
coherence , the KD nonreality $\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})$ in the state $\varrho$ relative to the rank-1 PVM
basis $\\{\Pi_{a}\\}$ gives a lower bound to the $l_{1}$-norm coherence
$C_{l_{1}}(\varrho;\\{\ket{a}\\})$ of $\varrho$ relative to the orthonormal
basis $\\{\ket{a}\\}$ corresponding to $\\{\Pi_{a}\\}$ as

$\displaystyle C_{l_{1}}(\varrho;\\{\ket{a}\\})\geq\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\}).$ (19)

Moreover, for an arbitrary state of a single qubit and arbitrary orthonormal
basis $\\{\ket{a}\\}$ of $\mathbb{C}^{2}$, the inequality becomes equality
Agung KD-nonreality coherence .

Using Eqs. (11) and (19), we thus obtain the following result.
Corollary 1. The $l_{1}$-norm coherence of a quantum state $\varrho$ on a
finite-dimensional Hilbert space $\mathcal{H}$ relative to an incoherent
orthonormal basis $\\{\ket{a}\\}$ of $\mathcal{H}$ is lower bounded as:

$\displaystyle C_{l_{1}}(\varrho;\\{\ket{a}\\})$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|}.$ (20)

As shown in Appendix B, for two-dimensional Hilbert space $\mathbb{C}^{2}$,
the inequality in Eq. (20) becomes equality for arbitrary single qubit state
and arbitrary orthonormal basis, as expressed in Eq. (14).

Next, from Eqs. (13) and (19), we have the following result.
Corollary 2. The KD nonreality in a quantum state $\varrho$ on a finite-
dimensional Hilbert space $\mathcal{H}$ relative to a rank-1 orthogonal PVM
basis $\\{\Pi_{a}\\}$ of $\mathcal{H}$, the corresponding $l_{1}$-norm
coherence of $\varrho$ relative to the orthonormal basis $\\{\ket{a}\\}$, and
the trace-norm asymmetry of $\varrho$ relative to the translation group
generated by any Hermitian operator $A$ with a complete set of eigenprojectors
$\\{\Pi_{a}\\}$, obey the following ordering:

$\displaystyle C_{l_{1}}(\varrho;\\{\ket{a}\\})$ $\displaystyle\geq$
$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\|[\tilde{A},\varrho]_{-}\|_{1}/2.$
(21)

For two-dimensional Hilbert space $\mathbb{C}^{2}$, as shown in Appendix B,
both inequalities in Eq. (21) become equalities for arbitrary state $\varrho$
and arbitrary incoherent orthonormal basis $\\{\ket{a}\\}$.

Recall that whilst $C_{l_{1}}(\varrho;\\{\ket{a}\\})$ is a measure of quantum
coherence of $\varrho$ which is independent of its encoding in the reference
incoherent orthonormal basis $\\{\ket{a}\\}$, the trace-norm asymmetry
$\|[\tilde{A},\varrho]\|_{1}/2$ can also be seen as a measure of coherence (as
translational asymmetry) of $\varrho$ which depends on its encoding in the
reference incoherent eigenbasis $\\{\ket{a}\\}$ of the generator
$\tilde{A}=A/\|A\|_{\infty}$ of the translation group
$U_{\theta}=e^{-i\tilde{A}\theta}$, $\theta\in\mathbb{R}$. The former is
sometimes called speakable coherence while the latter is called unspeakable
coherence Marvian - Spekkens speakable and unspeakable coherence .

Finally, combining Eqs. (15) with (19), we obtain the following trade-off
relation.
Corollary 3. The $l_{1}$-norm coherence of a quantum state $\varrho$ on a
finite-dimensional Hilbert space $\mathcal{H}$ relative to an incoherent
orthonormal basis $\\{\ket{a}\\}$ of $\mathcal{H}$ and that relative to an
incoherent orthonormal basis $\\{\ket{b}\\}$ of $\mathcal{H}$ satisfy the
following trade-off relation:

$\displaystyle
C_{l_{1}}(\varrho;\\{\ket{a}\\})C_{l_{1}}(\varrho;\\{\ket{b}\\})\geq\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}^{2}.$ (22)

Next, from Eq. (11) of Proposition 1, we obtain the following additive trade-
off relation for the KD nonreality in a state $\varrho$ relative to the PVM
basis $\\{\Pi_{a}\\}$ and that relative to the PVM basis $\\{\Pi_{b}\\}$:

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})+\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{b}\\})\geq\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}.$ (23)

The proof follows exactly similar steps as the proof of Eq. (15) of
Proposition 3. It can also be proven by applying the inequality for the
arithmetic mean and geometric mean, i.e., $(a+b)/2\geq\sqrt{ab}$,
$a,b\in\mathbb{R}^{+}$, to Eq. (15). Since $\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})$ is a faithful measure of coherence, Eq. (23) has
a form of additive uncertainty relation for coherence measure reported in
Refs. Korzekwa quantum-classical decomposition ; Singh uncertainty relation
for coherence ; Yuan uncertainty relation for coherence ; Hall quantum-
classical decomposition . One can then check that the left-hand side is not
vanishing when the state is not totally mixed, i.e.,
$\varrho\neq\mathbb{I}/d$, and the PVM bases are noncommuting as stated in
Theorem 1 of Ref. Yuan uncertainty relation for coherence . In particular,
combining Eq. (23) with Eq. (19), we have

$\displaystyle
C_{l_{1}}(\varrho;\\{\ket{a}\\})+C_{l_{1}}(\varrho;\\{\ket{b}\\})\geq\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}.$ (24)

We note that unlike the standard entropic uncertainty relation Coles entropic
uncertainty relation review ; Wehner entropic UR review , the lower bound in
Eqs. (23) and (24) depends on the state as for the uncertainty relation for
coherence measures in Refs. Korzekwa quantum-classical decomposition ; Singh
uncertainty relation for coherence ; Yuan uncertainty relation for coherence ;
Hall quantum-classical decomposition . In particular, it is vanishing when the
state is maximally mixed $\varrho=\mathbb{I}/d$, in case of which, the left-
hand sides in Eqs. (23) and (24) are also vanishing. Hence, the uncertainty
relation depends on the purity of the state as expected Korzekwa quantum-
classical decomposition . It is interesting in the future to compare the type
of lower bound in Eqs. (23) and (24) to those reported in Refs. Korzekwa
quantum-classical decomposition ; Singh uncertainty relation for coherence ;
Yuan uncertainty relation for coherence ; Hall quantum-classical decomposition
. In the Appendix C, we evaluate the optimization in the lower bound
analytically for two-dimensional system, showing that it is determined by the
purity of the state, and three parameters that characterize the pairwise
noncommutativity among the two PVM bases and the eigenbasis of the state. We
furthermore show that for pure state in two dimensional Hilbert space, the
inequality becomes equality when the bases $\\{\ket{a}\\}$, $\\{\ket{b}\\}$,
and the eigenbasis of $\varrho$ comprise a set of three mutually unbiased
bases of $\mathbb{C}^{2}$.

###  III.2 Lower bound and uncertainty relation for the KD nonclassicality in
a state relative to a PVM

First, using Lemma 2, we obtain the following proposition.
Proposition 4. The KD nonclassicality in a state $\varrho$ on a finite-
dimensional Hilbert space $\mathcal{H}$ relative to a PVM basis
$\\{\Pi_{a}\\}$ of $\mathcal{H}$ defined in Eq. (10) is lower bounded as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}$ (25)
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}-1.$

Proof. Taking the supremum over the set $\mathcal{M}_{\rm r1PVM}(\mathcal{H})$
of all the rank-1 PVM bases $\\{\Pi_{b}\\}$ of $\mathcal{H}$ to both sides of
Eq. (6), and noting Eq. (10), we obtain

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}{\rm NCl}(\\{{\rm Pr}_{\rm KD}(a,b|\varrho)\\})$ (26)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{b\\})}\big{\\{}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}-1.$

Observe further that the left-hand side depends only on the PVM basis
$\\{\Pi_{a}\\}$, i.e., it is independent of the eigenvalues spectrum $\\{a\\}$
of $A$ and the eigenvalues spectrum $\\{b\\}$ of $B$. Noting this, the
inequality of Eq. (26) can be further strengthened to get Eq. (25). ∎

We then obtain the following trade-off relation.
Proposition 5. The KD nonclassicality in a quantum state $\varrho$ on a
finite-dimensional Hilbert space $\mathcal{H}$ relative to a PVM basis
$\\{\Pi_{a}\\}$ of $\mathcal{H}$ and that relative to a PVM basis
$\\{\Pi_{b}\\}$ of $\mathcal{H}$ satisfy the following trade-off relation:

$\displaystyle(\mathcal{Q}_{\rm KD}^{\rm
NCl}(\varrho;\\{\Pi_{a}\\})+1)(\mathcal{Q}_{\rm KD}^{\rm
NCl}(\varrho;\\{\Pi_{b}\\})+1)$ (27) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{\\{}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{\\}}.$

Proof. First, swapping the role of $A$ and $B$ in Eq. (25), we have

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{b}\\})+1$ (28)
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\sup_{A\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}.$

Hence, combining Eqs. (25) and (28), we obtain

$\displaystyle(\mathcal{Q}_{\rm KD}^{\rm
NCl}(\varrho;\\{\Pi_{a}\\})+1)(\mathcal{Q}_{\rm KD}^{\rm
NCl}(\varrho;\\{\Pi_{b}\\})+1)$ (29) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\Big{(}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{{\varrho}}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}\Big{)}$
$\displaystyle\times$
$\displaystyle\Big{(}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\sup_{A\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}\Big{)}$
$\displaystyle\geq$
$\displaystyle\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{\\{}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{\\}},$

where the last inequality in Eq. (29) is due to the fact that
$\sup_{X\in\mathbb{H}(\mathcal{H})}\\{\cdot\\}\geq\sup_{X\in\mathbb{H}(\mathcal{H}|\\{\Pi_{x}\\})}\\{\cdot\\}$.
∎

One can see that Eq. (27) takes a form analogous to the Robertson-Schrödinger
uncertainty relation for observables $\tilde{A}_{\varrho}$ and
$\tilde{B}_{\varrho}$. Unlike the Robertson-Schrödinger uncertainty relation,
however, the lower bound in Eq. (27) is nonlinear in the state $\varrho$.
Moreover, there are optimizations over a pair of convex sets
$\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})$ and
$\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})$ of Hermitian operators whose complete
sets of eigenprojectors are respectively $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$
relative to which we define the KD nonclassicality in $\varrho$:
$\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$ and $\mathcal{Q}_{\rm
KD}^{\rm NCl}(\varrho;\\{\Pi_{b}\\})$. Recall that the KD nonclassicality in a
state relative to a PVM basis quantifies the total quantumness, i.e., it
quantifies simultaneously the nonreality and negativity of the corresponding
KD quasiprobability, capturing the noncommutativity between the state and the
PVM basis. In this sense, the trade-off relation of Eq. (27) thus imposes a
restriction on a joint nonclassicality in a quantum state relative to a pair
of noncommuting PVM bases.

Let us proceed to show that the lower bound and the trade-off relation for the
KD nonclassicality relative to a PVM basis obtained above lead also to a lower
bound and a trade-off relation for the corresponding $l_{1}$-norm coherence.
First, as shown in Appendix D, the KD nonclassicality in a state $\varrho$ on
a finite-dimensional Hilbert space $\mathcal{H}$ relative to a rank-1
orthogonal PVM basis $\\{\Pi_{a}\\}$ of $\mathcal{H}$ gives a lower bound to
the $l_{1}$-norm coherence of the state $\varrho$ relative to the incoherent
orthonormal basis $\\{\ket{a}\\}$ corresponding to $\\{\Pi_{a}\\}$, i.e.,

$\displaystyle C_{l_{1}}(\varrho;\\{\ket{a}\\})\geq\mathcal{Q}_{\rm KD}^{\rm
NCl}(\varrho;\\{\Pi_{a}\\}).$ (30)

Combining Eq. (30) with Eq. (25), we thus obtain the following corollary.
Corollary 4. The $l_{1}$-norm coherence of a quantum state $\varrho$ on a
finite-dimensional Hilbert space $\mathcal{H}$ relative to an incoherent
orthonormal basis $\\{\ket{a}\\}$ of $\mathcal{H}$ is lower bounded as

$\displaystyle C_{l_{1}}(\varrho;\\{\ket{a}\\})$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$ (31)
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}-1.$

Next, combining Eq. (30) with Eq. (27), we obtain the following trade-off
relation.
Corollary 5. The $l_{1}$-norm coherence of a quantum state $\varrho$ on a
finite-dimensional Hilbert space $\mathcal{H}$ relative to an orthonormal
basis $\\{\ket{a}\\}$ of $\mathcal{H}$ and that relative to an orthonormal
basis $\\{\ket{b}\\}$ of $\mathcal{H}$ satisfy the following trade-off
relation:

$\displaystyle\big{(}C_{l_{1}}(\varrho;\\{\ket{a}\\})+1\big{)}\big{(}C_{l_{1}}(\varrho;\\{\ket{b}\\})+1\big{)}$
(32) $\displaystyle\geq$
$\displaystyle\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{\\{}\big{(}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}\big{\\}}.$

Following exactly similar steps as above, we can also prove the following
additive trade-off relation for the $l_{1}$-norm coherence of a state
$\varrho$ relative to an orthonormal basis $\\{\ket{a}\\}$ and that relative
to an orthonormal basis $\\{\ket{b}\\}$ as:

$\displaystyle
C_{l_{1}}(\varrho;\\{\ket{a}\\})+C_{l_{1}}(\varrho;\\{\ket{b}\\})$ (33)
$\displaystyle\geq$
$\displaystyle\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{\\{}\big{(}|{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}|^{2}$
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}-2.$

##  IV Operational and statistical meaning

In this section we discuss operational and information theoretical
interpretations of the KD nonreality and KD nonclassicality in a state
relative to a PVM basis in terms of transparent laboratory operations. One is
based on the representation of the KD quasiprobability in terms of weak value
which can be obtained using various methods in experiment, and the other is
based on the decomposition of the KD quasiprobability in terms of real and
nonnegative joint probability and quantum modification terms obtained via two
successive projective measurements.

First, one observes that the KD nonreality and the KD nonclassicality in a
state $\varrho$ relative to a PVM basis $\\{\Pi_{a}\\}$ defined respectively
in Eqs. (9) and (10) can be expressed as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}{\rm Im}\pi_{a}^{\rm
w}(\Pi_{b}|\varrho)\big{|}{\rm Tr}\\{\Pi_{b}\varrho\\},$ (34)
$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}\pi_{a}^{\rm
w}(\Pi_{b}|\varrho)\big{|}{\rm Tr}\\{\Pi_{b}\varrho\\}-1.$ (35)

Here, $\pi_{a}^{\rm w}(\Pi_{b}|\varrho):=\frac{{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\\}}{{\rm Tr}\\{\Pi_{b}\varrho\\}}$ is known as the
weak value of $\Pi_{a}$ with the preselected state $\varrho$ and postselected
state $\ket{b}$ Aharonov weak value ; Aharonov-Daniel book ; Wiseman weak
value . It is in general complex and its real part may lie outside $[0,1]$.
Remarkably, the real and imaginary parts of the weak value can be estimated in
experiment without recourse to state tomography either using weak measurement
with postselection Aharonov weak value ; Aharonov-Daniel book ; Wiseman weak
value ; Lundeen complex weak value ; Jozsa complex weak value ; Lundeen
measurement of KD distribution ; Salvail direct measurement KD distribution ;
Bamber measurement of KD distribution ; Thekkadath measurement of density
matrix  or different methods without weak measurement Johansen quantum state
from successive projective measurement ; Vallone strong measurement to
reconstruct quantum wave function ; Cohen estimating of weak value with strong
measurements ; Lostaglio KD quasiprobability and quantum fluctuation ; Wagner
measuring weak values and KD quasiprobability ; Hernandez-Gomez experimental
observation of TBMH negativity . Noting this, the KD nonreality and KD
nonclassicality in a state relative to a PVM basis of Eqs. (34) and (35) can
thus be directly operationally estimated using weak value measurement together
with the classical optimization over the set $\mathcal{M}_{\rm
r1PVM}(\mathcal{H})$ of all the rank-1 orthogonal PVM bases of the Hilbert
space $\mathcal{H}$. This estimation scheme should in principle be
implementable in terms of variational quantum circuits using the currently
available NISQ hardware Cerezo VQA review .

The above operational interpretation suggests the following information
theoretical meaning of the KD nonreality $\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})$ in a state $\varrho$ relative to a PVM basis
$\\{\Pi_{a}\\}$ defined in Eq. (9) and the associated trade-off relation
expressed in Eq. (15). First, applying the Jensen inequality to Eq. (34), we
have

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})^{2}$
$\displaystyle=$ $\displaystyle\big{(}\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}{\rm Im}\pi_{a}^{\rm
w}(\Pi_{b}|\varrho)\big{|}{\rm Tr}\\{\Pi_{b}\varrho\\}\big{)}^{2}$ (36)
$\displaystyle\leq$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}{\rm Im}\pi_{a}^{\rm
w}(\Pi_{b}|\varrho)\big{|}^{2}{\rm Tr}\\{\Pi_{b}\varrho\\}$ $\displaystyle:=$
$\displaystyle\epsilon^{2}_{\\{\Pi_{a}\\}}(\varrho),$

where $\epsilon^{2}_{\\{\Pi_{a}\\}}(\varrho)$ is the total sum of the variance
of the imaginary part of the weak value $\pi_{a}^{\rm w}(\Pi_{b}|\varrho)$
over the probability ${\rm Pr}(b|\varrho)={\rm Tr}\\{\Pi_{b}\varrho\\}$
maximized over the set $\mathcal{M}_{\rm r1PVM}(\mathcal{H})$ of all the PVM
bases $\\{\Pi_{b}\\}$ of $\mathcal{H}$. On the other hand, it was argued by
Johansen and Hall in Refs. Johansen weak value best estimation ; Hall prior
information , that the variance of the imaginary part of the weak value
$\pi^{\rm w}_{a}(\Pi_{b}|\varrho)$ over the probability ${\rm
Pr}(b|\varrho)={\rm Tr}\\{\Pi_{b}\varrho\\}$ can be interpreted as the mean-
squared error of the optimal estimation of $\Pi_{a}$ based on the outcomes of
measurement described by the PVM basis $\\{\Pi_{b}\\}$ when the preparation is
represented by the state $\varrho$. Noting this,
$\epsilon^{2}_{\\{\Pi_{a}\\}}(\varrho)$ defined in Eq. (36) may thus be
statistically interpreted as the total mean-squared error of the optimal
estimation of the PVM basis $\\{\Pi_{a}\\}$ based on projective measurement,
given the preparation $\varrho$, in the worst case scenario. Equation (36)
thus shows that the total root-mean-squared error of the optimal estimation of
the PVM basis $\\{\Pi_{a}\\}$ given $\varrho$ in the worst case scenario is
lower bounded by the corresponding KD nonreality in $\varrho$ relative to the
PVM basis $\\{\Pi_{a}\\}$.

Combining Eq. (36) with Eqs. (11) and (13), we thus obtain the following
results.
Corollary 5a. The total root-mean-squared error of the optimal estimation of a
PVM basis $\\{\Pi_{a}\\}$ of a finite-dimensional Hilbert space $\mathcal{H}$
given a preselected state $\varrho$ on $\mathcal{H}$, based on projective
measurement described by a PVM basis in $\mathcal{M}_{\rm
r1PVM}(\mathcal{H})$, in the worst case scenario, is lower bounded as

$\displaystyle\epsilon_{\\{\Pi_{a}\\}}(\varrho)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|}.$ (37)

Moreover, it can also be lower bounded in terms of trace-norm asymmetry as

$\displaystyle\epsilon_{\\{\Pi_{a}\\}}(\varrho)$ $\displaystyle\geq$
$\displaystyle\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\|[A,\varrho]_{-}\|_{1}/2\|A\|_{\infty}.$
(38)

Next, combining Eqs. (36) with (15), we have the following uncertainty
relation.
Corollary 5b. Given a preparation represented by a density operator $\varrho$
on a finite-dimensional Hilbert space $\mathcal{H}$, the total root-mean-
squared errors of the optimal estimation of $\\{\Pi_{a}\\}$ of $\mathcal{H}$
based on projective measurement described by a PVM basis in $\mathcal{M}_{\rm
r1PVM}(\mathcal{H})$, and that of the optimal estimation of $\\{\Pi_{b}\\}$ of
$\mathcal{H}$, in the worst case scenario, satisfy the following trade-off
relation:

$\displaystyle\epsilon_{\\{\Pi_{a}\\}}(\varrho)\epsilon_{\\{\Pi_{b}\\}}(\varrho)\geq\frac{1}{4}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H}|\\{\Pi_{b}\\})}\big{|}{\rm
Tr}\\{[\tilde{A},\tilde{B}]_{-}\varrho\\}\big{|}^{2}.$ (39)

Let us proceed to discuss an operational interpretation of the KD
nonclassicality in a state relative to a rank-1 PVM basis in terms of a
sequence of two strong projective measurements. First, it has been shown by
Johansen in Ref. Johansen quantum state from successive projective measurement
that the KD quasiprobability associated with a state $\varrho$ over a pair of
rank-1 PVM bases $\\{\Pi_{a}\\}$ and $\\{\Pi_{b}\\}$ can be expressed as

$\displaystyle{\rm Pr}_{\rm KD}(a,b|\varrho)$ $\displaystyle=$
$\displaystyle{\rm Tr}\\{\Pi_{b}\Pi_{a}\varrho\Pi_{a}\\}+\frac{1}{2}{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b}\\}$ (40) $\displaystyle-$
$\displaystyle i\frac{1}{2}{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b|a}^{\pi/2}\\}.$

Here,
$\varrho_{\Pi_{a}}:=\Pi_{a}\varrho\Pi_{a}+(\mathbb{I}-\Pi_{a})\varrho(\mathbb{I}-\Pi_{a})$
is the state after a nonselective binary projective measurement described by
$\\{\Pi_{a},\mathbb{I}-\Pi_{a}\\}$, and
$\Pi_{b|a}^{\pi/2}=e^{i\Pi_{a}\pi/2}\Pi_{b}e^{-i\Pi_{a}\pi/2}$. The first term
on the right-hand side of Eq. (40), i.e., ${\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\Pi_{a}\\}={\rm
Tr}\\{\Pi_{b}\frac{\Pi_{a}\varrho\Pi_{a}}{{\rm Tr}\\{\varrho\Pi_{a}\\}}\\}{\rm
Tr}\\{\varrho\Pi_{a}\\}$, is just the joint probability to get $a$ in the
measurement described by $\\{\Pi_{a}\\}$ and then to get $b$ afterward in the
measurement described by $\\{\Pi_{b}\\}$, so that it is always real and
nonnegative. In this sense, the other two terms are called the quantum
modification terms responsible for the negativity and nonreality of the KD-
quasiprobability. One can then see that the negativity and the nonreality
capture different forms of state disturbance due to the nonselective binary
projective measurement $\\{\Pi_{a},\mathbb{I}-\Pi_{a}\\}$ as captured by the
expectation values of $\Pi_{b}$ and $\Pi_{b|a}^{\pi/2}$, respectively.

Using the decomposition of the KD quasiprobability in Eq. (40), the KD
nonclassicality in $\varrho$ relative to the PVM $\\{\Pi_{a}\\}$ defined in
Eq. (10) can then be upper bounded as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\Pi_{a}\\}+\frac{1}{2}{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b}\\}$ (41) $\displaystyle-$
$\displaystyle i\frac{1}{2}{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b|a}^{\pi/2}\\}\big{|}-1$
$\displaystyle\leq$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b}\\}\big{|}:=\delta_{\\{\Pi_{a}\\}}(\varrho).$

Here, to get Eq. (41), we have used triangle inequality, the normalization
$\sum_{a,b}|{\rm Tr}\\{\Pi_{b}\Pi_{a}\varrho\Pi_{a}\\}|=\sum_{a,b}{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\Pi_{a}\\}=1$ for any
$\\{\Pi_{b}\\}\in\mathcal{M}_{\rm r1PVM}(\mathcal{H})$, and also
$\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm r1PVM}(\mathcal{H})}\sum_{a,b}|{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b|a}^{\pi/2}\\}|=\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}|{\rm
Tr}\\{(\varrho-\varrho_{\Pi_{a}})\Pi_{b}\\}|$, where the equality is due to
the fact that $\\{\Pi_{b|a}^{\pi/2}\\}$ comprises again a rank-1 PVM basis of
$\mathcal{H}$, and the set of the PVM basis $\\{\Pi_{b|a}^{\pi/2}\\}$ is the
same as the set of the PVM basis $\\{\Pi_{b}\\}$ given by $\mathcal{M}_{\rm
r1PVM}(\mathcal{H})$. Hence, the KD nonclassicality $\mathcal{Q}_{\rm KD}^{\rm
NCl}(\varrho;\\{\Pi_{a}\\})$ in $\varrho$ relative to the rank-1 PVM basis
$\\{\Pi_{a}\\}$ gives a lower bound to the total disturbance
$\delta_{\\{\Pi_{a}\\}}(\varrho)$ in the state $\varrho$ caused by the
nonselective projective binary measurement $\\{\Pi_{a},\mathbb{I}-\Pi_{a}\\}$
associated with the PVM basis $\\{\Pi_{a}\\}$.

Combining Eq. (41) and Eq. (25), we first have the following corollary.
Corollary 6a. The total disturbance $\delta_{\\{\Pi_{a}\\}}(\varrho)$ in the
state $\varrho$ caused by the nonselective projective binary measurement
$\\{\Pi_{a},\mathbb{I}-\Pi_{a}\\}$ associated with the PVM basis
$\\{\Pi_{a}\\}$ of a finite-dimensional Hilbert space $\mathcal{H}$ is lower
bounded as

$\displaystyle\delta_{\\{\Pi_{a}\\}}(\varrho)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}$ (42)
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}-1.$

From Corollary 6a, we finally obtain the following trade-off relation, the
proof of which follows similar steps to that of Proposition 5.
Corollary 6b. Given a preparation represented by $\varrho$ on a finite-
dimensional Hilbert space $\mathcal{H}$, the total disturbance
$\delta_{\\{\Pi_{a}\\}}(\varrho)$ in the state $\varrho$ caused by the
nonselective projective binary measurement $\\{\Pi_{a},\mathbb{I}-\Pi_{a}\\}$
associated with the PVM basis $\\{\Pi_{a}\\}$, and the total disturbance
$\delta_{\\{\Pi_{b}\\}}(\varrho)$ in the state $\varrho$ caused by the
nonselective projective binary measurement $\\{\Pi_{b},\mathbb{I}-\Pi_{b}\\}$
associated with the PVM basis $\\{\Pi_{b}\\}$, satisfy the following trade-off
relation:

$\displaystyle\delta_{\\{\Pi_{a}\\}}(\varrho)\delta_{\\{\Pi_{b}\\}}(\varrho)$
$\displaystyle\geq$
$\displaystyle\frac{1}{4}\Big{(}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{H}(\mathcal{H})}\big{\\{}\big{(}\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{-}\\}\big{|}^{2}$ (43)
$\displaystyle+$ $\displaystyle\big{|}{\rm
Tr}\\{\varrho[\tilde{A}_{\varrho},\tilde{B}_{\varrho}]_{+}\\}-2{\rm
Tr}\\{\tilde{A}_{\varrho}\varrho\\}{\rm
Tr}\\{\tilde{B}_{\varrho}\varrho\\}\big{|}^{2}\big{)}^{1/2}\big{\\}}-1\Big{)}^{2}.$

##  V Summary and Discussion

We have first derived lower bounds for the KD nonreality and the KD
nonclassicality relative to a pair of rank-1 PVM bases, respectively in Eqs.
(4) and (6). Nonvanishing lower bounds thus provide sufficient conditions for
the KD quasiprobability to be nonclassical, i.e., its value is nonreal or its
real part is negative, or both. We then defined the KD nonreality and KD
nonclassicality in a state relative to a single PVM basis by taking the
supremum over the other basis as in Eqs. (9) and (10). They can be interpreted
as quantifying the amount of the quantumness in the state relative to the PVM
basis manifesting their noncommutativity. We obtained lower bounds for the KD
nonreality and KD nonclassicality in a state relative to a single PVM basis,
given respectively in Eqs. (11) and (25). A lower bound for the KD-nonreality
in a state relative a rank-1 PVM in terms of extremal trace-norm asymmetry is
given in Eq. (13). The same lower bounds also apply to the corresponding
$l_{1}$-norm coherence.

We proceeded to derive trade-off relations for the KD nonreality and the KD
nonclassicality relative to a PVM basis and those relative to another PVM
basis given in Eqs. (15) and (27), having similar forms respectively to the
Robertson and Roberston-Schrödinger uncertainty relations. The lower bounds
for the trade-off relations involve optimization over two convex sets of
Hermitian operators whose complete set of eigenprojectors are given by the
corresponding PVM bases. We then showed that the trade-off relations imply
similar trade-off relations for the $l_{1}$-norm coherence. The trade-off
relations thus restrict simultaneous quantumness associated with a state
$\varrho$ relative to two noncommuting rank-1 PVM basis. More detailed
comparison of the uncertainty relations to the uncertainty relation for
intrinsic quantum randomness presented in Refs. Korzekwa quantum-classical
decomposition ; Singh uncertainty relation for coherence ; Yuan uncertainty
relation for coherence ; Hall quantum-classical decomposition  is left for
future study.

We further briefly discussed a hybrid quantum-classical variational scheme for
a direct measurement of the KD nonreality and KD nonclassicality in a state
relative a PVM basis by means of weak value measurement for the reconstruction
of the KD quasiprobability, combined with a classical optimization scheme for
searching the supremum over the set of rank-1 PVM bases of the Hilbert space.
This operational interpretation leads to an information theoretical
interpretation for the KD nonreality in a state relative a PVM basis as a
lower bound for the total root-mean-squared error of the optimal estimation of
the PVM basis based on the outcomes of projective measurement, in the worst
case scenario. Moreover, it also leads to an uncertainty relation between the
root-mean-squared error of the optimal estimation of the PVM basis and that of
the optimal estimation of the other PVM basis, based on projective
measurement, in the worst case scenario. We further applied the decomposition
of the KD quasiprobability obtained via two successive projective
measurements, into a real and nonnegative joint probability and two quantum
modification terms which are responsible for the negativity and nonreality of
the KD quasiprobability. Using this decomposition, the KD nonclassicality in a
state relative to a PVM basis can be shown to give a lower bound to the total
disturbance to the state caused by a nonselective projective binary
measurement associated with the PVM basis. This further implies similar lower
bound and trade-off relation for such total disturbance as those for the KD
nonclassicality relative to a PVM basis.

In this article, we have based all of our discussion on the standard KD
quasiprobability associated with a density operator over a pair of rank-1
orthogonal PVM bases as in Eq. (1). This suggests directions for further
investigation in the future. First, it is natural to ask if one can extend the
methods and results of the present work to more general POVM (positive-
operator-valued measure) bases. Next, recently, motivated by certain
interesting physical problems such as quantum metrology with postselection
Arvidsson-Shukur quantum advantage in postselected metrology ; Lupu-Gladstein
negativity enhanced quantum phase estimation 2022  and detection of OTOC (out-
of-time-order correlation) in many body chaotic system Halpern
quasiprobability and information scrambling ; Alonso KD quasiprobability
witnesses quantum scrambling , there are proposals to extend the KD
quasiprobability by extending the number of PVM basis. Within the
representation of the KD quasiprobability via weak value, this extension means
that we increase the number of weak measurements before making strong
projective postselection measurement. This extension of the KD
quasiprobability too shares the properties of the standard KD
quasiprobability. In particular, its negativity and nonreality signal
quantumness associated with quantum noncommutativity. It is therefore
interesting to apply the methods and reasoning developed in the present
article and also in Refs. Agung KD-nonreality coherence ; Agung KD-
nonclassicality coherence ; Agung KD general quantum correlation  to use the
extended KD quasiprobability to probe quantum coherence, general quantum
correlation and to see the restriction imposed by the uncertainty principle.
Such an approach might in turn help clarify the roles of coherence and general
correlation in quantum metrology with postselection and OTOC.

###### Acknowledgements.

##  Appendix A Proof of Proposition 2

Notice that the left-hand side of Eq. (11) does not depend on
$\tilde{B}=B/\|B\|_{\infty}$. Hence, the inequality in Eq. (11) can be
strengthened to obtain

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\sup_{B\in\mathbb{O}(\mathcal{H})}\big{|}{\rm
Tr}\\{\tilde{B}[\tilde{A},\varrho]_{-}\\}\big{|},$ (44)

where $\mathbb{O}(\mathcal{H})$ is the set of all bounded operators on
$\mathcal{H}$. Next, since $\|\tilde{B}\|_{\infty}=1$, then one can further
strengthen the inequality in Eq. (44) as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\frac{1}{2}\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\big{\\{}\sup_{X\in\mathbb{O}(\mathcal{H}|\|X\|_{\infty}\leq
1)}\big{|}{\rm Tr}\\{X^{\dagger}[\tilde{A},\varrho]_{-}\\}\big{|}\big{\\}},$
(45)

where $\mathbb{O}(\mathcal{H}|\|X\|_{\infty}\leq 1)$ is the set of all bounded
operators $X$ with $\|X\|_{\infty}\leq 1$. One then observes that the term on
the right-hand side of Eq. (45) inside the bracket $\\{\dots\\}$ is just the
variational definition of the Schatten $p=1$ norm via its conjugate norm
$p_{*}=\infty$ Watrous book on quantum Shannon theory , so that one has

$\displaystyle\sup_{X\in\mathbb{O}(\mathcal{H}|\|X\|_{\infty}\leq
1)}\big{|}{\rm
Tr}\\{X^{\dagger}[\tilde{A},\varrho]_{-}\\}\big{|}=\|[\tilde{A},\varrho]_{-}\|_{1}.$
(46)

Inserting this into the right-hand side of Eq. (45), we obtain Eq. (13).

As an alternative scheme of proof, first, from Proposition 4 of Ref. Agung
estimation and operational interpretation of trace-norm asymmetry , we have
(see Eq. (22) of Ref. Agung estimation and operational interpretation of
trace-norm asymmetry )

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\|[A,\varrho]_{-}\|_{1}/2\|A\|_{\infty},$ (47)

where $\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\\{\Pi_{a}\\})$ in this article
is denoted by $C_{\rm KD}(\varrho;\\{\ket{a}\\})$ in Ref. Agung estimation and
operational interpretation of trace-norm asymmetry . This can be proven by
using the equality between the trace-norm asymmetry Marvian - Spekkens
speakable and unspeakable coherence  and the measure of asymmetry based on the
maximum mean absolute value of the imaginary part of the weak value of the
generator $A$ of the translation group proposed in Agung translational
asymmetry from nonreal weak value  as expressed in Proposition 1 of Ref. Agung
estimation and operational interpretation of trace-norm asymmetry  (see Eq.
(5) of Ref. Agung estimation and operational interpretation of trace-norm
asymmetry ). Observe that the left-hand side of Eq. (47) is independent of the
eigenvalues spectrum of the Hermitian operator $A$ which appears on the right-
hand side. Hence, the inequality can be strengthened as

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\\{\Pi_{a}\\})\geq\sup_{A\in\mathbb{O}(\mathcal{H}|\\{\Pi_{a}\\})}\|[A,\varrho]_{-}\|_{1}/2\|A\|_{\infty}.$
(48)

∎

##  Appendix B Propositions 1 and 2 for two-dimensional Hilbert space

First, assume without loss of generality, that the PVM basis of the two-
dimensional Hilbert space $\mathcal{H}\cong\mathbb{C}^{2}$ relative to which
we define the KD nonreality in a state on the left-hand side of Eq. (11) is
given by the complete set of eigenprojectors of the Pauli operator
$\sigma_{z}$, i.e., $\mathbb{A}:=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\}$. All the
Hermitian operators on $\mathbb{C}^{2}$ with the eigenprojectors $\mathbb{A}$
thus take the general form as:

$\displaystyle A=a_{0}\ket{0}\bra{0}+a_{1}\ket{1}\bra{1},$ (49)

where $a_{0},a_{1}\in\mathbb{R}$ are the eigenvalues. We denote the set of all
such Hermitian operators by $\mathbb{H}(\mathbb{C}^{2}|\mathbb{A})$. Moreover,
the general form of all Hermitian operators on the Hilbert space
$\mathbb{C}^{2}$ reads as

$\displaystyle
B(\alpha,\beta)=b_{+}\ket{b_{+}(\alpha,\beta)}\bra{b_{+}(\alpha,\beta)}+b_{-}\ket{b_{-}(\alpha,\beta)}\bra{b_{-}(\alpha,\beta)},$
(50)

with the eigenvalues $b_{+},b_{-}\in\mathbb{R}$, and the corresponding
orthonormal eigenvectors
$\\{\ket{b_{+}(\alpha,\beta)},\ket{b_{-}(\alpha,\beta)}\\}$ can be expressed
using the Bloch sphere parameterization as

$\displaystyle\ket{b_{+}(\alpha,\beta)}$ $\displaystyle=$
$\displaystyle\cos\frac{\alpha}{2}\ket{0}+e^{i\beta}\sin\frac{\alpha}{2}\ket{1};$
$\displaystyle\ket{b_{-}(\alpha,\beta)}$ $\displaystyle=$
$\displaystyle\sin\frac{\alpha}{2}\ket{0}-e^{i\beta}\cos\frac{\alpha}{2}\ket{1},$
(51)

where $\alpha\in[0,\pi]$, $\beta\in[0,2\pi)$. Let us denote the set of all
Hermitian operators on $\mathbb{C}^{2}$ by $\mathbb{H}(\mathbb{C}^{2})$.

We further assume, without loss of generality, that the singular values of $A$
and $B$ have the following orderings: $|a_{0}|\geq|a_{1}|$ and
$|b_{+}|\geq|b_{-}|$, so that we have $\|A\|_{\infty}=|a_{0}|$ and
$\|B\|_{\infty}=|b_{+}|$. Then, computing the lower bound in Eq. (11), we
obtain

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\mathbb{A})$
$\displaystyle\geq$
$\displaystyle\sup_{A\in\mathbb{H}(\mathbb{C}^{2}|\mathbb{A})}\sup_{B\in\mathbb{H}(\mathbb{C}^{2})}\frac{|{\rm
Tr}\\{B[A,\varrho]_{-}\\}|}{2\|A\|_{\infty}\|B\|_{\infty}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\max_{\\{a_{0},a_{1}\\}\in\mathbb{R}^{2}}\max_{\\{b_{+},b_{-}\\}\in\mathbb{R}^{2}}\max_{\\{\alpha,\beta\\}\in[0,\pi]\times[0,2\pi)}\Big{\\{}|\sin\alpha\sin(\beta-\phi_{01})|$
$\displaystyle\times$
$\displaystyle\frac{|b_{+}-b_{-}|}{|b_{+}|}\frac{|a_{0}-a_{1}|}{|a_{0}|}|\braket{1}{\varrho}{0}|\Big{\\}}$
(53) $\displaystyle=$
$\displaystyle\frac{1}{2}\max_{\\{a_{0},a_{1}\\}\in\mathbb{R}^{2}}\max_{\\{b_{+},b_{-}\\}\in\mathbb{R}^{2}}\Big{\\{}\frac{|b_{+}-b_{-}|}{|b_{+}|}\frac{|a_{0}-a_{1}|}{|a_{0}|}|\braket{0}{\varrho}{1}|\Big{\\}}$
(54) $\displaystyle=$ $\displaystyle
2|\braket{0}{\varrho}{1}|=C_{l_{1}}(\varrho;\mathbb{A}).$ (55)

Here, $\phi_{01}=-\arg\braket{0}{\varrho}{1}$, the maximum is obtained for
Hermitian operator $A$ of Eq. (49) with $|a_{0}-a_{1}|=2|a_{0}|$ and for
Hermitian operator $B(\alpha,\beta)$ having the form of (50) with
$\alpha=\pi/2$ and $\beta=\phi_{01}+\pi/2$ and $|b_{+}-b_{-}|=2|b_{+}|$, and
$C_{l_{1}}(\varrho;\mathbb{A})=2|\braket{0}{\varrho}{1}|$ is the $l_{1}$-norm
coherence of $\varrho$ relative to the orthonormal basis
$\mathbb{A}=\\{\ket{0},\ket{1}\\}$. Note that, to get Eq. (55), we have used
the fact that for any pair $x,y\in\mathbb{C}$ with $|x|\geq|y|$, we always
have $|x-y|\leq 2|x|$, and equality is attained for $x=-y$. On the other hand,
as shown in Ref. Agung KD-nonreality coherence , for arbitrary state of a
single qubit and any PVM basis of $\mathbb{C}^{2}$, the left-hand side of Eq.
(B) is also given by the $l_{1}$-norm coherence, i.e.:

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm
NRe}(\varrho;\mathbb{A})=2|\braket{0}{\varrho}{1}|=C_{l_{1}}(\varrho;\mathbb{A}).$
(56)

Hence, the inequality in Eq. (B) indeed becomes equality. Moreover, one can
see from the values of $\alpha$ and $\beta$ that achieve the maximum in Eq.
(B), that the eigenbasis of $B_{*}$ expressed in Eq. (51) which attains the
supremum in Eq. (B) are mutually unbiased with
$\mathbb{A}=\\{\ket{0},\ket{1}\\}$ and also with the eigenbasis of $\varrho$.
This proves Proposition 1 for the case of two-dimensional Hilbert space.

Next, one can see from the proof of Proposition 2 in Appendix A that the lower
bound in Eq. (11) is less than or equal to the lower bound in Eq. (13), and
the left-hand sides of the two equations are the same. Hence, when the
inequality in Eq. (11) becomes equality, the inequality in Eq. (13) must also
become equality. This combined with the above result means that for two-
dimensional Hilbert space, the inequality in Eq. (13) must also become
equality. Indeed, computing the trace-norm asymmetry of the state $\varrho$
relative to the translation group generated by $A$ having the form of Eq.
(49), one has $\|[A,\varrho]\|_{1}/2=|a_{0}-a_{1}||\braket{0}{\varrho}{1}|$.
Upon inserting into the lower bound in Eq. (13), we have

$\displaystyle\sup_{A\in\mathbb{H}(\mathcal{H}|\\{\Pi_{a}\\})}\|[A,\varrho]_{-}\|_{1}/2\|A\|_{\infty}$
(57) $\displaystyle=$
$\displaystyle\max_{\\{a_{0},a_{1}\\}\in\mathbb{R}^{2}}\frac{|a_{0}-a_{1}|}{|a_{0}|}|\braket{0}{\varrho}{1}|$
$\displaystyle=$ $\displaystyle 2|\braket{0}{\varrho}{1}|$ $\displaystyle=$
$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NRe}(\varrho;\mathbb{A}),$

where the last equality in Eq. (57) is just Eq. (56) and the maximum is
obtained when $|a_{0}-a_{1}|=2|a_{0}|$. This proves Proposition 2 for the case
of two-dimensional Hilbert space.

Combining Eqs. (56) and (57), we thus obtain Eq. (14) of the main text. ∎

##  Appendix C Trade-off relation of Eq. (24) for a single qubit

Without loss of generality, one can proceed as in Appendix B, but now the
optimization is over the set $\mathbb{H}(\mathbb{C}^{2}|\mathbb{A})$ of all
Hermitian operators on $\mathbb{C}^{2}$ with the complete set of
eigenprojectors $\mathbb{A}=\\{\ket{0}\bra{0},\ket{1}\bra{1}\\}$ having the
form of Eq. (49), and over the set $\mathbb{H}(\mathbb{C}^{2}|\mathbb{B})$ of
all Hermitian operators on $\mathbb{C}^{2}$ with the complete set of
eigenprojectors
$\mathbb{B}=\\{\ket{b_{+}(\alpha,\beta)}\bra{b_{+}(\alpha,\beta)},\ket{b_{-}(\alpha,\beta)}\bra{b_{+}(\alpha,\beta)}\\}$
having the form of Eq. (50). Evaluating the lower bound in Eq. (24) for two-
dimensional Hilbert space, we have

$\displaystyle C_{l_{1}}(\varrho;\mathbb{A})+C_{l_{1}}(\varrho;\mathbb{B})$
(58) $\displaystyle\geq$
$\displaystyle\sup_{A\in\mathbb{H}(\mathbb{C}^{2}|\mathbb{A})}\sup_{B(\alpha,\beta)\in\mathbb{H}(\mathbb{C}^{2}|\mathbb{B})}\frac{|{\rm
Tr}\\{[A,B(\alpha,\beta)]\varrho\\}|}{\|A\|_{\infty}\|B\|_{\infty}}$
$\displaystyle=$
$\displaystyle\max_{\\{b_{+},b_{-}\\}\in\mathbb{R}^{2}}\max_{\\{a_{0},a_{1}\\}\in\mathbb{R}^{2}}\frac{|a_{0}-a_{1}|}{|a_{0}|}\frac{|b_{+}-b_{-}|}{|b_{+}|}|\braket{0}{\varrho}{1}|$
$\displaystyle\times$ $\displaystyle|\sin\alpha\sin(\beta-\phi_{01})|$
$\displaystyle=$ $\displaystyle
4|\braket{0}{\varrho}{1}||\sin\alpha||\sin(\beta-\phi_{01})|$ $\displaystyle=$
$\displaystyle 2\sqrt{r^{2}-r_{z}^{2}}|\sin\alpha||\sin(\beta-\phi_{01})|$
$\displaystyle=$ $\displaystyle
2r|\sin\phi_{z}||\sin\alpha||\sin(\beta-\phi_{01})|.$

Here, the maximum is obtained when $|a_{0}-a_{1}|=2|a_{0}|$ and
$|b_{+}-b_{-}|=2|b_{+}|$ (see Appendix B), we have used the expression for the
qubit state
$\varrho=\frac{1}{2}(\mathbb{I}+r_{x}\sigma_{x}+r_{y}\sigma_{y}+r_{z}\sigma_{z})$,
$r_{x}^{2}+r_{y}^{2}+r_{z}^{2}=r^{2}$ so that
$2|\braket{0}{\varrho}{1}|=|r_{x}-ir_{y}|=\sqrt{r^{2}-r_{z}^{2}}$, and
$\phi_{z}$ is the angle between the Bloch vector of the state and the positive
$z$-axis. One can see that the lower bound decreases as the purity of the
state given by ${\rm Tr}(\varrho^{2})=(1+r^{2})/2$ decreases. Moreover, it
also decreases when the noncommutativity between the two PVM bases, i.e.,
$\mathbb{A}$ and $\mathbb{B}$, quantified by $|\sin\alpha|$, decreases. In
particular, the lower bound vanishes for $r=0$, i.e., for maximally mixed
state $\varrho=\mathbb{I}/2$ with minimum purity, and it also vanishes when
$\sin\alpha=0,\pi$, i.e., when the two PVM bases $\mathbb{A}$ and $\mathbb{B}$
are commuting. This result is in accord with that obtained in Ref. Yuan
uncertainty relation for coherence . Note that $|\sin\phi_{z}|$ and
$|\sin(\beta-\phi_{01})|$ on the right-hand side characterize respectively the
noncommutativity between the state $\varrho$ and the PVM basis $\mathbb{A}$
and between $\varrho$ and the PVM basis $\mathbb{B}$. They vanish respectively
when $\varrho$ commutes with $\mathbb{A}$ and $\varrho$ commutes with
$\mathbb{B}$, as expected. As an example consider the case when the state is
pure so that $r=1$, and take $\alpha=\pi/2$ and $\phi_{01}+\pi/2=\beta$ so
that $\sin\alpha=\sin(\beta-\phi_{01})=1$. Then, taking $\phi_{z}=\pi/2$, we
have $C_{l_{1}}(\varrho;\mathbb{A})=C_{l_{1}}(\varrho;\mathbb{B})=1$ and the
inequality in Eq. (58) becomes equality, i.e., both sides are equal to $2$.
Note that in this case, the triple $\mathbb{A}$, $\mathbb{B}$ and the
eigenbasis of $\varrho$ comprise a three mutually unbiased bases of
$\mathbb{C}^{2}$.

##  Appendix D Proof of Eq. (30)

One first has, from the definition of the KD nonclassicality in a state
$\varrho$ relative to a PVM $\\{\Pi_{a}\\}$ in Eq. (10),

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$ (59)
$\displaystyle=$ $\displaystyle\sup_{\\{\Pi_{b}\\}\in\mathcal{M}_{\rm
r1PVM}(\mathcal{H})}\sum_{a,b}\big{|}\sum_{a^{\prime}}{\rm
Tr}\\{\Pi_{b}\Pi_{a}\varrho\Pi_{a^{\prime}}\\}\big{|}-1$ $\displaystyle\leq$
$\displaystyle\sum_{a,a^{\prime}}\big{|}\braket{a}{\varrho}{a^{\prime}}|\sum_{b_{*}}|\braket{a^{\prime}}{b_{*}}\braket{b_{*}}{a}\big{|}-1,$

where we have used a completeness relation
$\sum_{a^{\prime}}\Pi_{a^{\prime}}=\mathbb{I}$, triangle inequality, and
$\\{\Pi_{b_{*}}\\}\in\mathcal{M}_{\rm r1PVM}(\mathcal{H})$ is a PVM basis
which achieves the supremum. On the other hand, using the Cauchy-Schwartz
inequality, we have
$\sum_{b_{*}}|\braket{b_{*}}{a}\braket{a^{\prime}}{b_{*}}|\leq(\sum_{b_{*}}|\braket{b_{*}}{a}|^{2}\sum_{b_{*}^{\prime}}|\braket{a^{\prime}}{b^{\prime}_{*}}|^{2})^{1/2}=1$,
where we have used a completeness relation
$\sum_{b_{*}}\ket{b_{*}}\bra{b_{*}}=\mathbb{I}$. Inserting this into Eq. (59),
we finally obtain

$\displaystyle\mathcal{Q}_{\rm KD}^{\rm NCl}(\varrho;\\{\Pi_{a}\\})$
$\displaystyle\leq$
$\displaystyle\sum_{a,a^{\prime}}\big{|}\braket{a}{\varrho}{a^{\prime}}|-1$
(60) $\displaystyle=$ $\displaystyle\sum_{a\neq
a^{\prime}}\big{|}\braket{a}{\varrho}{a^{\prime}}|$ $\displaystyle=$
$\displaystyle C_{l_{1}}(\varrho;\\{\ket{a}\\}),$

where we have used the normalization $\sum_{a}\braket{a}{\varrho}{a}=1$. ∎

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