# Standard symmetrized variance with applications to coherence, uncertainty
and entanglement

Ming-Jing Zhao  School of Science, Beijing Information Science and Technology
University, Beijing, 100192, PR China  State key Laboratory of Networking and
Switching Technology (Beijing University of Posts and Telecommunications),
Beijing, 100876, PR China  Lin Zhang  E-mail<EMAIL_ADDRESS>Institute of
Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China  Shao-Ming Fei
School of Mathematical Sciences, Capital Normal University, Beijing 100048,
China  Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig,
Germany

###### Abstract

Variance is a ubiquitous quantity in quantum information theory. Given a
basis, we consider the averaged variances of a fixed diagonal observable in a
pure state under all possible permutations on the components of the pure state
and call it the symmetrized variance. Moreover we work out the analytical
expression of the symmetrized variance and find that such expression is in the
factorized form where two factors separately depends on the diagonal
observable and quantum state. By shifting the factor corresponding to the
diagonal observable, we introduce the notion named the standard symmetrized
variance for the pure state which is independent of the diagonal observable.
We then extend the standard symmetrized variance to mixed states in three
different ways, which characterize the uncertainty, the coherence and the
coherence of assistance, respectively. These quantities are evaluated
analytically and the relations among them are established. In addition, we
show that the standard symmetrized variance is also an entanglement measure
for bipartite systems. In this way, these different quantumness of quantum
states are unified by the variance.

##  I Introduction

Measuring statistically the deviation of measurement outcomes from the ideal
value of quantum measurements on given quantum states, the variance plays an
important role in quantum physics and quantum information theory. The first
uncertainty relation known as the Heisenberg’s uncertainty relation was given
in terms of the variance W. Heisenberg ; H. P. Robertson , which describes the
restrictions on the accuracy of measurement results of two or more
noncommutative observables. Since then, many efforts are made to find tighter
and state-independent lower bounds on such kind of uncertainty relations
Pati2014 ; S. Friedland ; Schwonnek2017 . In order to study the restriction
between the uncertainties for two or more observables, the notion of the
uncertainty region is also put forward and characterized Werner2015 ;
Zhang2021 .

For any given mixed state and observable, it has been shown that the minimal
averaged variance among all possible pure-state decompositions is exactly the
quantum Fisher information and the maximal averaged variance is the variance
itself Yu2013 ; Toth2013 ; S. Braunstein ; G. Toth2017 . Concerning the
relation between the variance and quantum Fisher information, the improved
uncertainty relations have been derived in terms of quantum Fisher information
and variance S. Luo2005pra ; Toth2021 ; Gessner2021 . Moreover, the variance-
based uncertainty relation has been incorporated into quantum multiparameter
estimation by investigating the quantum Fisher information matrix and the
classical Fisher information matrix from measurements Lu2021 . It is well
known that the quantum Fisher information places the fundamental limit to the
accuracy for estimating an unknown parameter and plays an important role in
quantum metrology. The uncertainty relation and quantum metrology are then
closely connected by the variance and quantum Fisher information.

More than that, the variance can also be used to detect quantum entanglement
Schwonnek2017 ; H. F. Hofmann ; O. Guhne ; N. Li2013 ; Y. Hong  and quantify
quantum coherence Y. Zhang2020 ; Li2021 ; Xu2021 . In view of the importance
of the variance in quantum information theory, we aim to explore the role
played by the variance, as a unifying tool, in characterizing the quantumness.
In a $d$-dimensional system with fixed basis
$\\{|i\rangle:i=0,1,\ldots,d-1\\}$, any pure state can be expressed as
$|\psi\rangle=\sum_{i=0}^{d-1}\psi_{i}|i\rangle$. Since the quantumness of the
pure state is invariant under relabeling the coefficients $\\{\psi_{i}\\}$, in
order to study the quantumness of the pure state $|\psi\rangle$, one may
consider the averaged quantumness in the set of pure states
$\\{\bm{P}_{\pi}|\psi\rangle\\}$, where $\bm{P}_{\pi}$ are all possible
permutation matrices corresponding to the permutations $\pi$ of the
permutation group $S_{d}$. In this context, we propose the averaged variances
of any diagonal observable over the set of $\\{\bm{P}_{\pi}|\psi\rangle\\}$
and named as symmetrized variance
$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$. By analysis and
calculation, we obtain the analytical expression of the symmetrized variance
and find that the information of the pure state and the diagonal observable
are factorized. By shifting the information of the diagonal observable, we
introduce the core notion of this paper called the standard symmetrized
variance $\widehat{V}(|\psi\rangle)$.

Although the standard symmetrized variance is in terms of variance, we find
that it is independent of the diagonal observable and just depends on the pure
state itself. Namely, the standard symmetrized variance describes the
quantumness in the pure state. We generalize the standard symmetrized variance
from the pure states to mixed states based on three different extensions. The
first extension is the standard symmetrized variance $\widehat{V}(\rho)$
obtained by replacing the pure state with mixed ones directly. We get the
analytical formula of $\widehat{V}(\rho)$ and show that it is a measure of
uncertainty associated to the state $\rho$. The second extension is the convex
roof extension $\widehat{V}_{c}(\rho)$. The lower bound of
$\widehat{V}_{c}(\rho)$ is derived in terms of the eigenvalues and
eigenvectors of the mixed state, which can be attained for qubit quantum
states. The convex roof extension $\widehat{V}_{c}(\rho)$ is a coherence
measure for $\rho$. The third extension is the concave bottom extension
$\widehat{V}_{a}(\rho)$, which, dual to $\widehat{V}_{c}(\rho)$, is the
coherence of assistance. These three extensions are depicted in Bloch ball for
any qubit state. The standard symmetrized variance can also give rise to an
entanglement measure in bipartite systems. The schematic on the relations
among these quantities is given in Fig. 1.

Figure 1: The diagram of the standard symmetrized variances.
$\operatorname{Var}(|\psi\rangle,\bm{A})$ is the variance of the diagonal
observable $\bm{A}$ in the pure state $|\psi\rangle$ defined in Eq. (1).
$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$ is the symmetrized
variance of the diagonal observable $\bm{A}$ in the pure state $|\psi\rangle$
defined in Eq. (II.1). $\widehat{V}(|\psi\rangle)$ is the standard symmetrized
variance of the pure state $|\psi\rangle$ defined in Eq. (4).
$\widehat{V}(\rho)$ is the standard symmetrized variance of the mixed state
$\rho$ defined in Eq. (6), which is an uncertainty measure.
$\widehat{V}_{c}(\rho)$ is the convex roof extension of the standard
symmetrized variance defined in Eq. (9), which is a coherence measure.
$\widehat{V}_{a}(\rho)$ is the concave bottom extension of the standard
symmetrized variance defined in Eq. (12), which is the coherence of
assistance.

##  II The standard symmetrized variance

For any quantum state $\rho$ and an observable $\bm{A}$ in a $d$-dimensional
system, the variance

$\displaystyle\operatorname{Var}(\rho,\bm{A})=\operatorname{Tr}\left(\bm{A}^{2}\rho\right)-[\operatorname{Tr}\left(\bm{A}\rho\right)]^{2}$

is a measure of uncertainty of the observable $\bm{A}$ in the state $\rho$. In
this paper, we fix the reference basis as $\\{|i\rangle:i=0,1,\ldots,d-1\\}$
and just consider the variance of diagonal observables $\bm{A}$ in quantum
state $\rho$ throughout this paper.

###  II.1 Standard symmetrized variance for pure states

For any pure state $|\psi\rangle=\sum_{i=0}^{d-1}\psi_{i}|i\rangle$, the
vector
$\left(r_{0},r_{1},\ldots,r_{d-1}\right)^{\scriptscriptstyle\mathsf{T}}$ with
$r_{i}:=\left\lvert\mspace{1.0mu}\psi_{i}\mspace{1.0mu}\right\rvert^{2}$ is
called the _coherence vector_ of a pure state $|\psi\rangle$ S. Du ; H. Zhu .
Based on the variance

$\displaystyle\operatorname{Var}(|\psi\rangle,\bm{A})$ $\displaystyle=$
$\displaystyle\left\langle\psi\left|\bm{A}^{2}\right|\psi\right\rangle-\left\langle\psi\left|\bm{A}\right|\psi\right\rangle^{2},$
(1)

we define the symmetrized variance of the observable $\bm{A}$ in the pure
state $|\psi\rangle$ as

$\displaystyle\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$
$\displaystyle=$ $\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}\operatorname{Var}(\bm{P}_{\pi}|\psi\rangle,\bm{A})$ $\displaystyle=$
$\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}\operatorname{Var}(|\psi\rangle,\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi}).$

$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$ is the averaged variances
of the observable $\bm{A}$ in the set of pure states
$\\{\bm{P}_{\pi}|\psi\rangle\\}$.

The symmetrized variance $\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$
satisfies the following properties: (i)
$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$ is symmetric under
relabeling the entries of the vector $\\{\psi_{i}\\}$; (ii)
$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$ is a concave function of
the state $|\psi\rangle$; (iii)
$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})=\widehat{\operatorname{Var}}(|\psi\rangle,\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi})$
for any permutation matrix $\bm{P}_{\pi}$.

Theorem 1. For any pure state $|\psi\rangle=\sum_{i=0}^{d-1}\psi_{i}|i\rangle$
and diagonal observable $\bm{A}$, the symmetrized variance
$\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$ is given by

$\displaystyle\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})=\kappa(\bm{A})\left[1-\sum^{d-1}_{i=0}|\psi_{i}|^{4}\right].$
(3)

with
$\kappa(\bm{A})=\frac{d\operatorname{Tr}\left(\bm{A}^{2}\right)-[\operatorname{Tr}\left(\bm{A}\right)]^{2}}{d(d-1)}$.

The proof is in the Appendix A. Theorem 1 provides an analytical expression of
the symmetrized variance. This formula shows that the information about the
diagonal observable $\bm{A}$ and the pure state $|\psi\rangle$ given in the
symmetrized variance $\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$ are
factorized. Inspired by this, we define the standard symmetrized variance of
the pure state $|\psi\rangle$ as

$\displaystyle\widehat{V}(|\psi\rangle)=\frac{1}{\kappa(\bm{A})}\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A}),$
(4)

which can be expressed as

$\widehat{V}(|\psi\rangle)=1-\sum^{d-1}_{i=0}\left\lvert\mspace{1.0mu}\psi_{i}\mspace{1.0mu}\right\rvert^{4}.$
(5)

This standard symmetrized variance $\widehat{V}(|\psi\rangle)$ is independent
of the observable $\bm{A}$ and is only given by the pure state $|\psi\rangle$.
The standard symmetrized variance $\widehat{V}(|\psi\rangle)$ has the
following properties. First, $\widehat{V}(|\psi\rangle)$ is symmetric and
concave with respect to the coherence vector
$(r_{0},r_{1},\cdots,r_{d-1})^{\scriptscriptstyle\mathsf{T}}$. Second,
$0\leqslant\widehat{V}(|\psi\rangle)\leqslant 1-\frac{1}{d}$. Moreover,
$\widehat{V}(|\psi\rangle)=0$ if and only if $|\psi\rangle$ is incoherent,
namely, $|\psi\rangle\langle\psi|$ is diagonal under the reference basis.
$\widehat{V}(|\psi\rangle)=1-\frac{1}{d}$ if and only if $|\psi\rangle$ is
maximally coherent,
$|\psi\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}e^{i\theta_{j}}|j\rangle$.

###  II.2 The extension to mixed states

Now we extend the standard symmetrized variance $\widehat{V}(|\psi\rangle)$
from pure states to mixed states in three different ways.

(1) The standard symmetrized variance $\widehat{V}(\rho)$ and uncertainty. For
mixed states, we define the standard symmetrized variance $\widehat{V}(\rho)$
as

$\displaystyle\widehat{V}(\rho)=\frac{1}{\kappa(\bm{A})}\frac{1}{d!}\sum_{\pi\in
S_{d}}\operatorname{Var}(\bm{P}_{\pi}\rho\bm{P}^{\dagger}_{\pi},\bm{A}).$ (6)

Let $\Pi=\left\\{\Pi_{i}=|i\rangle\langle i|:i=0,1,\cdots,d-1\right\\}$ be the
projective measurements, $\Pi(\rho)=\sum_{i}\Pi_{i}\rho\Pi_{i}$ be the post-
measurement state, and $S_{L}(\rho)=1-\operatorname{Tr}\left(\rho^{2}\right)$
be the linear entropy which measures the mixedness of $\rho$. Then the
standard symmetrized variance $\widehat{V}(\rho)$ actually gives the mixedness
of postmeasurement state $\Pi(\rho)$.

Theorem 2. For any quantum state $\rho$, the standard symmetrized variance
$\widehat{V}(\rho)$ is given by

$\displaystyle\widehat{V}(\rho)=S_{L}(\Pi(\rho)).$ (7)

The proof is in the Appendix B. The standard symmetrized variance
$\widehat{V}(\rho)$ has the following properties. (i) It is a concave function
of $\rho$; (ii) It is invariant under the permutation $\bm{P}_{\pi}$,
$\widehat{V}(\rho)=\widehat{V}(\bm{P}_{\pi}\rho\bm{P}^{\dagger}_{\pi})$; (iii)
$0\leqslant\widehat{V}(\rho)\leqslant 1-\frac{1}{d}$. $\widehat{V}(\rho)=0$ if
and only if $\rho$ is pure and incoherent. $\widehat{V}(\rho)=1-\frac{1}{d}$
if and only if $\rho$ is maximally coherent; (iv) For
$\rho=p_{1}\rho_{1}\oplus p_{2}\rho_{2}\oplus\cdots\oplus p_{n}\rho_{n}$, it
satisfies $\widehat{V}(p_{1}\rho_{1}\oplus p_{2}\rho_{2}\oplus\cdots\oplus
p_{n}\rho_{n})=p_{1}^{2}\widehat{V}(\rho_{1})+p_{2}^{2}\widehat{V}(\rho_{2})+\cdots+p_{n}^{2}\widehat{V}(\rho_{n})+\sum_{i\neq
j}p_{i}p_{j}$, which is proved in the Appendix C.

Since the standard symmetrized variance $\widehat{V}(\rho)$ is a concave
function of the vector
$(\rho_{00},\ldots,\rho_{d-1,d-1})^{\scriptscriptstyle\mathsf{T}}$, where
$\rho_{ii}=\left\langle i\left|\rho\right|i\right\rangle$, and is invariant
under the permutation $\bm{P}_{\pi}$, $\widehat{V}(\rho)$ is a measure of
uncertainty which quantifies the uncertainty associated with the quantum state
$\rho$ in the framework of uncertainty measure S. Friedland . In Refs. S.
Luo2005 ; S. Luo2017 ; Y. Sun2021  the authors proposed the idea to split the
uncertainty into quantum part and classical part. In Refs. S. Luo2005 ; S.
Luo2017  the total uncertainty is specified to the variance of observable
$\bm{A}$ in state $\rho$, while the quantum uncertainty is specified to the
Wigner-Yanase skew information. Here we can decompose the standard symmetrized
variance $\widehat{V}(\rho)$ into the quantum uncertainty $Q(\rho)$ and the
classical uncertainty $C(\rho)$,

$\displaystyle\widehat{V}(\rho)=C(\rho)+Q(\rho),$ (8)

where $C(\rho)=S_{L}(\rho)$ is the mixedness of $\rho$,
$Q(\rho)=S_{L}(\Pi(\rho))-S_{L}(\rho)$ is just the genuine coherence of $\rho$
Y. Sun2021 ; J. Vicente .

Theorem 2 can also be viewed as an uncertainty relation in summation form if
we reexpress the standard symmetrized variance $\widehat{V}(\rho)$ in the form
of variance. Since $\widehat{V}(\rho)$ can be equivalently rewritten as
$\sum_{\pi\in
S_{d}}\operatorname{Var}(\rho,\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi})=d!\kappa(\bm{A})S_{L}(\Pi(\rho))$,
it can be viewed as an uncertainty relation among the observables
$\\{\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi}\\}_{\pi}$ in the state $\rho$.
For example, for qubit systems this uncertainty relation reduces to
$\operatorname{Var}(\rho,\bm{A})+\operatorname{Var}(\rho,\sigma_{1}\bm{A}\sigma_{1})=[2\operatorname{Tr}\left(\bm{A}^{2}\right)-[\operatorname{Tr}\left(\bm{A}\right)]^{2}]\widehat{V}(\rho)$,
which gives an uncertainty relation for the commutative diagonal observables
$\bm{A}$ and $\sigma_{1}\bm{A}\sigma_{1}$ in the form of equality, with
$\sigma_{1}=|0\rangle\langle 1|+|1\rangle\langle 0|$.

(2) The convex roof extension $\widehat{V}_{c}(\rho)$ and coherence measure.
Now we extend the standard symmetrized variance to mixed states by convex roof
extension,

$\displaystyle\widehat{V}_{c}(\rho)$ $\displaystyle=$
$\displaystyle\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\widehat{V}(|\psi_{k}\rangle),$
(9)

where the minimum runs over all possible pure state ensemble decompositions of
$\rho=\sum_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$.

Since the standard symmetrized variance $\widehat{V}(|\psi\rangle)$ is a real
symmetric concave function of the coherence vector
$\left(r_{0},r_{1},\ldots,r_{d-1}\right)^{\scriptscriptstyle\mathsf{T}}$, and
is zero if and only if $|\psi\rangle$ is incoherent, $\widehat{V}_{c}(\rho)$
is a coherence measure S. Du ; H. Zhu . The convex roof extension
$\widehat{V}_{c}(\rho)$ satisfies the following properties. (i)
$\widehat{V}_{c}(\rho)$ is convex; (ii)
$0\leqslant\widehat{V}_{c}(\rho)\leqslant 1-\frac{1}{d}$. Moreover,
$\widehat{V}_{c}(\rho)=0$ if and only if $\rho$ is incoherent, and
$\widehat{V}_{c}(\rho)=1-\frac{1}{d}$ if and only if $\rho$ is maximally
coherent; (iii) For $\rho=p_{1}\rho_{1}\oplus p_{2}\rho_{2}$, one has
$\widehat{V}_{c}(p_{1}\rho_{1}\oplus
p_{2}\rho_{2})={p_{1}}\widehat{V}_{c}(\rho_{1})+{p_{2}}\widehat{V}_{c}(\rho_{2})$
X. Yu .

Now we evaluate the convex roof extension $\widehat{V}_{c}(\rho)$
analytically. Here we employ the following equality about the minimum average
invariance over all pure state decomposition and the quantum Fisher
information
$F(\rho,\bm{A})=\sum_{k,l}2\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}\left\lvert\mspace{1.0mu}A_{kl}\mspace{1.0mu}\right\rvert^{2}$,
where $A_{kl}=\left\langle\phi_{k}\left|\bm{A}\right|\phi_{l}\right\rangle$
with $\\{\lambda_{k},|\phi_{k}\rangle\\}$ the eigenvalues and the
corresponding eigenstates of the mixed state $\rho$ Yu2013 ; Toth2013 ; S.
Braunstein ; G. Toth2017 , i.e.,

$F(\rho,\bm{A})=4\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\operatorname{Var}(|\psi_{k}\rangle,\bm{A}).$
(10)

Theorem 3. For any mixed state $\rho$, the convex roof extension
$\widehat{V}_{c}(\rho)$ is bounded from below by

$\displaystyle\widehat{V}_{c}(\rho)\geqslant\frac{1}{2}\sum_{k,l}\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}\sum_{i=0}^{d-1}\left\lvert\mspace{1.0mu}\phi^{(k)}_{i}\mspace{1.0mu}\right\rvert^{2}\left\lvert\mspace{1.0mu}\phi^{(l)}_{i}\mspace{1.0mu}\right\rvert^{2},$
(11)

where $\lambda_{k}$ and $|\phi_{k}\rangle=\sum_{i}\phi^{(k)}_{i}|i\rangle$ are
the eigenvalues and the corresponding eigenstates of the mixed state $\rho$,
respectively.

In particular, for qubit systems we have the following conclusion.

Theorem 4. For any qubit state $\rho$, the convex roof extension
$\widehat{V}_{c}(\rho)=\frac{1}{8}F(\rho,\sigma_{3})=\frac{1}{2}(\lambda_{1}-\lambda_{2})^{2}\left\lvert\mspace{1.0mu}\left\langle\phi_{1}\left|\sigma_{3}\right|\phi_{2}\right\rangle\mspace{1.0mu}\right\rvert^{2}$,
where $\\{\lambda_{k},|\phi_{k}\rangle\\}_{k=1,2}$ are the eigenvalues and the
corresponding eigenstates of $\rho$, $\sigma_{3}=|0\rangle\langle
0|-|1\rangle\langle 1|$ is the Pauli matrix.

The proofs of Theorems 3 and 4 are in the Appendices D and E respectively. By
Theorem 4 we can see that the quantum Fisher information is a coherence
measure in qubit systems. But this is not true for high dimensional systems,
as a counterexample has been given in Refs. X. Feng ; H. Kwon .

(3) The concave bottom extension $\widehat{V}_{a}(\rho)$ and coherence of
assistance. Now we extend the standard symmetrized variance to mixed states by
the concave bottom extension,

$\displaystyle\widehat{V}_{a}(\rho)$ $\displaystyle=$
$\displaystyle\max_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\widehat{V}(|\psi_{k}\rangle),$
(12)

where the maximum runs over all possible pure state decompositions of $\rho$.

The concave bottom extension $\widehat{V}_{a}(\rho)$ is dual to the convex
roof extension $\widehat{V}_{c}(\rho)$. Since the convex roof extension
$\widehat{V}_{c}(\rho)$ is a coherence measure, the concave bottom extension
$\widehat{V}_{a}(\rho)$ can be interpreted operationally in the following way.
Suppose Alice holds a state $\rho^{A}$ with coherence $C(\rho^{A})$. Bob holds
another part of the purified state of $\rho^{A}$. The joint state between
Alice and Bob is $\sum_{k}p_{k}|\psi_{k}\rangle_{A}\otimes|k\rangle_{B}$ with
$\rho^{A}=\sum_{k}p_{k}|\psi_{k}\rangle\\!\langle\psi_{k}|$. Bob performs
local measurements $\\{|k\rangle\\}$ and informs Alice the measurement
outcomes by classical communication. Alice’s quantum state will be in a pure
state ensemble $\\{p_{k},|\psi_{k}\rangle\\}$ with average coherence
$\sum_{k}p_{k}C(|\psi_{k}\rangle\\!\langle\psi_{k}|)$. This process enables
Alice to increase the coherence from $C(\rho^{A})$ to the average coherence
$\sum_{k}p_{k}C(|\psi_{k}\rangle\\!\langle\psi_{k}|)$ due to the convexity of
the coherence measure, and it is called the assisted coherence distillation.
The maximum average coherence is called the coherence of assistance and
quantifies the one way coherence distillation rate E. Chitambar . In this
context, the concave bottom extension $\widehat{V}_{a}(\rho)$ is a coherence
of assistance corresponding to the coherence measure standard symmetrized
variance.

The above three extensions of the standard symmetrized variance satisfy the
following relation,

$\displaystyle\widehat{V}_{c}(\rho)\leqslant\widehat{V}_{a}(\rho)\leqslant\widehat{V}(\rho).$
(13)

The right inequality becomes equality if and only if there is a pure state
decomposition $\\{p_{k},|\psi_{k}\rangle\\}$ of $\rho$ such that
$\Pi(|\psi_{k}\rangle\\!\langle\psi_{k}|)=\Pi(\rho)$ for all $k$ M. Zhao .
Especially, the equality $\widehat{V}_{a}(\rho)=\widehat{V}(\rho)$ holds true
for all mixed states in two and three dimensional systems according to the
results in Ref. M. Zhao .

For qubit systems, we have the following relation satisfied by
$\widehat{V}_{a}(\rho)$ and $\widehat{V}_{c}(\rho)$.

Theorem 5. For any qubit state $\rho$,

$\widehat{V}_{a}(\rho)-\widehat{V}_{c}(\rho)=\frac{3}{8}S_{L}(\Pi(\rho))+\frac{1}{4}S_{L}(\rho).$

The proof is in the Appendix F. Theorem 5 shows the convex roof extension
$\widehat{V}_{c}(\rho)$ coincides with the concave bottom extension
$\widehat{V}_{a}(\rho)$ if and only if $\rho$ is an incoherent pure state.
Hence, for all qubit mixed states, the concave bottom extension
$\widehat{V}_{a}(\rho)$ is strictly greater than the convex roof extension
$\widehat{V}_{c}(\rho)$. This not only gives an affirmative answer to the
conjecture that the concave bottom extension is strictly greater than the
convex roof extension for all mixed states and all coherence measures M. Zhao
, but also implies that the coherence of all mixed states in qubit systems can
be increased in the assisted coherence distillation M. Zhao .

It is worthy to point out that the standard symmetrized variance
$\widehat{V}(\rho)$ as well as the convex roof extension
$\widehat{V}_{c}(\rho)$ and the concave bottom extension
$\widehat{V}_{a}(\rho)$ are all observable-independent. Therefore, in order to
evaluate these quantities theoretically or experimentally, one may choose
arbitrarily an appropriate diagonal observable. For example, if we choose the
diagonal observable $\bm{A}=\Pi_{i}$, $0\leqslant i\leqslant d-1$, then the
standard symmetrized variance
$\widehat{V}(\rho)=\frac{1}{d}\sum_{i=0}^{d-1}\operatorname{Var}(\rho,\Pi_{i})$
reduces to the uncertainty of $\rho$ proposed in Ref. Y. Sun2021 , while the
convex roof extension
$\widehat{V}_{c}(\rho)=\frac{1}{d}\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\sum_{i=0}^{d-1}\operatorname{Var}(|\psi_{k}\rangle,\Pi_{i})$
reduces to the coherence measure proposed in Ref. Li2021 , and the concave
bottom extension is given by
$\widehat{V}_{c}(\rho)=\frac{1}{d}\max_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\sum_{i=0}^{d-1}\operatorname{Var}(|\psi_{k}\rangle,\Pi_{i})$
correspondingly.

###  II.3 The standard symmetrized variance in qubit systems

For qubit states, the density matrix can be expressed as
$\rho=\frac{1}{2}(I+\bm{r}\cdot\bm{\sigma})$, where
$\bm{r}=r(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ is the Bloch
vector such that
$r=\left\lvert\mspace{1.0mu}\bm{r}\mspace{1.0mu}\right\rvert\in[0,1]$ on Bloch
ball, $\bm{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ with the Pauli matrices

$\sigma_{1}=\left(\begin{array}[]{cccccccc}0&1\\\
1&0\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cccccccc}0&-\mathrm{i}\\\
\mathrm{i}&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cccccccc}1&0\\\
0&-1\end{array}\right).$

Without loss of generality, we assume $\theta,\phi\in[0,\pi/2]$, which means
that the Bloch vector $\bm{r}$ is in the first octant of the Bloch ball. By
direct calculation, we have the convex roof extension of the standard
symmetrized variance $\widehat{V}_{c}(\rho)$,

$\widehat{V}_{c}(\rho)=\frac{1}{2}r^{2}\sin^{2}\theta.$

The corresponding optimal pure state decomposition of $\rho$ which gives rise
to $\widehat{V}_{c}(\rho)$ is
$\left\\{p_{k}^{(m)},|\chi_{k}^{(m)}\rangle\right\\}_{k=1}^{2}$, where

$|\chi_{1}^{(m)}\rangle=\cos\frac{\theta_{m}}{2}|0\rangle+e^{i\phi}\sin\frac{\theta_{m}}{2}|1\rangle$

with probability

$p_{1}^{(m)}=\frac{-1+r^{2}\sin^{2}{\theta}-r\cos{\theta}\sqrt{1-r^{2}\sin^{2}{\theta}}}{-2+2r^{2}\sin^{2}{\theta}},$

and

$|\chi_{2}^{(m)}\rangle=\sin\frac{\theta_{m}}{2}|0\rangle+e^{\mathrm{i}\phi}\cos\frac{\theta_{m}}{2}|1\rangle$

with probability

$p_{2}^{(m)}=\frac{-1+r^{2}\sin^{2}{\theta}+r\cos{\theta}\sqrt{1-r^{2}\sin^{2}{\theta}}}{-2+2r^{2}\sin^{2}{\theta}},$

whose Bloch vectors are
$\bm{r}(|\chi_{1}^{(m)}\rangle)=(\sin\theta_{m}\cos\phi,\sin\theta_{m}\sin\phi,\cos\theta_{m})$,
$\bm{r}(|\chi_{2}^{(m)}\rangle)=(\sin(\pi-\theta_{m})\cos\phi,\sin(\pi-\theta_{m})\sin\phi,\cos(\pi-\theta_{m}))$
with $\theta_{m}=\arccos{\sqrt{1-r^{2}\sin^{2}{\theta}}}$.

The standard symmetrized variance $\widehat{V}(\rho)$ and the concave bottom
extension of the standard symmetrized variance $\widehat{V}_{a}(\rho)$ are
given by

$\widehat{V}(\rho)=\widehat{V}_{a}(\rho)=\frac{1}{2}+\frac{1}{2}r^{2}\cos^{2}\theta.$

The corresponding optimal pure state decomposition of $\rho$ which attains the
value of $\widehat{V}_{a}(\rho)$ is
$\left\\{p_{k}^{(M)},|\chi_{k}^{(M)}\rangle\right\\}_{k=1}^{2}$, where

$|\chi_{1}^{(M)}\rangle=\cos\frac{\theta_{M}}{2}|0\rangle+e^{\mathrm{i}\phi}\sin\frac{\theta_{M}}{2}|1\rangle$

with probability

$p_{1}^{(M)}=\frac{1}{2}\left(1+\frac{r\sin\theta}{\sin\theta_{M}}\right),$

and

$|\chi_{2}^{(M)}\rangle=\cos\frac{\theta_{M}}{2}|0\rangle-e^{\mathrm{i}\phi}\sin\frac{\theta_{M}}{2}|1\rangle$

with probability

$p_{2}^{(M)}=\frac{1}{2}\left(1-\frac{r\sin\theta}{\sin\theta_{M}}\right),$

of which the Bloch vectors are
$\bm{r}(|\chi_{1}^{(M)}\rangle)=(\sin\theta_{M}\cos\phi,\sin\theta_{M}\sin\phi,\cos\theta_{M})$,
$\bm{r}(|\chi_{2}^{(M)}\rangle)=(\sin\theta_{M}\cos(\pi+\phi),\sin\theta_{M}\sin(\pi+\phi),\cos\theta_{M})$
with $\theta_{M}=\arccos(r\cos\theta)$, see Fig. 2.

Figure 2: (Color Online) The optimal pure state decompositions of $\rho$ for
the convex roof extension $\widehat{V}_{c}(\rho)$ and the concave bottom
extension $\widehat{V}_{a}(\rho)$. The pure state decomposition
$\left\\{p_{k}^{(m)},|\chi_{k}^{(m)}\rangle\right\\}_{k=1}^{2}$ (dashed lines)
is optimal for the convex roof extension $\widehat{V}_{c}(\rho)$. The pure
state decomposition
$\left\\{p_{k}^{(M)},|\chi_{k}^{(M)}\rangle\right\\}_{k=1}^{2}$ (solid lines)
is optimal for the concave bottom extension $\widehat{V}_{a}(\rho)$.

##  III Connections with quantum entanglement

To have an intuitive illustration on the relations between the standard
symmetrized variance and the quantum entanglement, let us consider bipartite
pure states in Schmidt decomposition form,
$|\psi_{AB}\rangle=\sum_{i=0}^{d-1}\lambda_{i}|ii\rangle$ with
$\lambda_{i}\geq 0$, $\sum_{i}\lambda_{i}=1$ and local diagonal observable
$\bm{A}\otimes\bm{B}=\sum_{i}a_{i}|i\rangle\\!\langle
i|\otimes\sum_{i}b_{i}|i\rangle\\!\langle i|$ acting on $C^{d}\otimes C^{d}$.
The variance of the local diagonal observable $\bm{A}\otimes\bm{B}$ in
bipartite pure state $|\psi_{AB}\rangle$ is
$\operatorname{Var}(|\psi_{AB}\rangle,\bm{A}\otimes\bm{B})=\left\langle\psi_{AB}\left|\bm{A}^{2}\otimes\bm{B}^{2}\right|\psi_{AB}\right\rangle-\left\langle\psi_{AB}\left|\bm{A}\otimes\bm{B}\right|\psi_{AB}\right\rangle^{2}$.
Similarly we have the symmetrized variance
$\widehat{\operatorname{Var}}(|\psi_{AB}\rangle,\bm{A}\otimes\bm{B})$,

$\begin{array}[]{rcl}\widehat{\operatorname{Var}}(|\psi_{AB}\rangle,\bm{A}\otimes\bm{B})=\frac{1}{d!}\sum_{\pi\in
S_{d}}\operatorname{Var}(\bm{P}_{\pi}\otimes\bm{P}_{\pi}|\psi_{AB}\rangle,\bm{A}\otimes\bm{B}).\end{array}$
(14)

By direct calculation we have

$\displaystyle\widehat{\operatorname{Var}}(|\psi_{AB}\rangle,\bm{A}\otimes\bm{B})=\kappa(\bm{A}\bm{B})\left[1-\sum^{d-1}_{i=0}\left\lvert\mspace{1.0mu}\lambda_{i}\mspace{1.0mu}\right\rvert^{4}\right].$

Then the standard symmetrized variance $\widehat{V}(|\psi_{AB}\rangle)$ is
given by

$\displaystyle\widehat{V}(|\psi_{AB}\rangle)$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa(\bm{A}\bm{B})}\widehat{\operatorname{Var}}(|\psi_{AB}\rangle,\bm{A}\otimes\bm{B})$
$\displaystyle=$ $\displaystyle
1-\sum^{d-1}_{i=0}\left\lvert\mspace{1.0mu}\lambda_{i}\mspace{1.0mu}\right\rvert^{4}.$

It is obvious that $\widehat{V}(|\psi_{AB}\rangle)$ is a real symmetric
concave function of the Schmidt vector
$(\lambda_{0},\lambda_{1},\cdots,\lambda_{d-1})^{\scriptscriptstyle\mathsf{T}}$.
Therefore, $\widehat{V}(|\psi_{AB}\rangle)$ is an entanglement measure for
pure state $|\psi_{AB}\rangle$ G. Vidal . Then the convex roof extension of
the standard symmetrized variance
$\widehat{V}_{c}(\rho_{AB})=\min_{\\{p_{k},|\psi^{(k)}_{AB}\rangle\\}}\sum_{k}p_{k}\widehat{V}(|\psi^{(k)}_{AB}\rangle)$,
where the minimum runs over all possible pure state decompositions of
$\rho_{AB}=\sum_{k}p_{k}|\psi^{(k)}_{AB}\rangle\langle\psi^{(k)}_{AB}|$, is an
entanglement measure for bipartite quantum states. Recall that the well-known
entanglement measure concurrence of a pure state $|\psi_{AB}\rangle$ is
defined by
$\mathrm{E}(|\psi_{AB}\rangle)=\sqrt{2(1-\operatorname{Tr}\left(\rho_{A}^{2}\right))}$
with $\rho_{A}=Tr_{B}(|\psi_{AB}\rangle\langle\psi_{AB}|)$, and
$E(\rho_{AB})=\min_{\\{p_{k},|\psi^{(k)}_{AB}\rangle\\}}\sum_{k}p_{k}\mathrm{E}(|\psi_{AB}^{(k)}\rangle)$
with the minimum running over all possible pure state decompositions of
$\rho_{AB}=\sum_{k}p_{k}|\psi^{(k)}_{AB}\rangle\langle\psi^{(k)}_{AB}|$ W.
Wooters ; P. Rungta . By straightforward derivation we find that
$\widehat{V}(|\psi_{AB}\rangle)=\frac{1}{2}\mathrm{E}^{2}(|\psi_{AB}\rangle)$
and
$\widehat{V}_{c}(\rho_{AB})=\frac{1}{2}\min_{\\{p_{k},|\psi^{(k)}_{AB}\rangle\\}}\sum_{k}p_{k}\mathrm{E}^{2}(|\psi_{AB}^{(k)}\rangle)$.
Equivalently,
$E(\rho_{AB})=\min_{\\{p_{k},|\psi^{(k)}_{AB}\rangle\\}}\sum_{k}p_{k}\sqrt{2\widehat{V}(|\psi_{AB}^{(k)}\rangle)}$,
namely, the entanglement measure concurrence can be expressed in terms of the
variance.

Since the standard symmetrized variance $\widehat{V}(|\psi_{AB}\rangle)$ is
observable-independent, we can choose the projective measurement onto the
Schmidt basis of the pure state
$|\psi_{AB}\rangle=\sum_{i=0}^{d-1}\lambda_{i}|ii\rangle$,
$\Pi^{A}_{i}=|i\rangle\langle i|$ and $\Pi^{B}_{i}=|i\rangle\langle i|$. In
this way, we obtain

$\displaystyle\widehat{V}(|\psi_{AB}\rangle)$ $\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{i=0}^{d-1}[\left\langle\psi_{AB}\left|\Pi^{A}_{i}\otimes\Pi^{B}_{i}\right|\psi_{AB}\right\rangle-\left\langle\psi_{AB}\left|\Pi^{A}_{i}\otimes\Pi^{B}_{i}\right|\psi_{AB}\right\rangle^{2}].$

Therefore, in order to measure the entanglement of a pure state
experimentally, one only needs to measure the expectation values
$\left\langle\psi_{AB}\left|\Pi^{A}_{i}\otimes\Pi^{B}_{i}\right|\psi_{AB}\right\rangle$
for $i=0,1,\cdots,d-1$.

Different from the uncertainty and coherence, entanglement characterizes the
quantum correlations between the subsystems and is basis-independent. Here we
have supposed that the pure state $|\psi_{AB}\rangle$ is in Schmidt
decomposition form and the observables of the subsystems $A$ and $B$ are
diagonal in the Schmidt basis. Then the standard symmetrized variance
$\widehat{V}(|\psi_{AB}\rangle)$ as well as the convex roof extension
$\widehat{V}_{c}(\rho_{AB})$ are basis independent. They are indeed
entanglement measures.

##  IV Conclusions

In summary, we have proposed the standard symmetrized variance for pure states
and extended it to mixed states in three different ways. The first extension
quantifies the uncertainty in quantum states. The second convex roof extension
quantifies the coherence in quantum states. The third concave bottom extension
quantifies the coherence of assistance. These quantities have been estimated
in detail. Besides, the standard symmetrized variance also quantifies the
entanglement in bipartite systems and gives an experimental way to evaluate
the entanglement by measurements. In this way, the quantum uncertainty,
coherence and entanglement are closely connected by the standard symmetrized
variance.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under grant Nos. 12171044, 11971140, and 12075159, Open Foundation of State
key Laboratory of Networking and Switching Technology (Beijing University of
Posts and Telecommunications) (SKLNST-20**-2-0*), Beijing Natural Science
Foundation (Z190005), Academy for Multidisciplinary Studies, Capital Normal
University, Shenzhen Institute for Quantum Science and Engineering, Southern
University of Science and Technology (SIQSE202001), and the Academician
Innovation Platform of Hainan Province.

##  V Appendices

Appendix A. The proof of Theorem 1.

###### Proof.

First, Let $\delta_{\pi(i),j}$ be the Kronecker delta symbol defined by
$\delta_{\pi(i),j}=1$ if $\pi(i)=j$ and $\delta_{\pi(i),j}=0$ if $\pi(i)\neq
j$. We denote $\bm{P}_{\pi}=(\delta_{\pi(i),j})$ the permutation matrix
representation of $\pi\in S_{d}$, the set of all possible permutations on the
set $\\{0,1,\ldots,d-1\\}$. Then for any diagonal observable $\bm{A}$ with
diagonal entries $a_{i}$, we have the following relations

$\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}a^{2}_{\pi(i)}=\frac{\operatorname{Tr}\left(\bm{A}^{2}\right)}{d}$ (16)

for any $i$, and

$\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}a_{\pi(i)}a_{\pi(j)}=\frac{[\operatorname{Tr}\left(\bm{A}\right)]^{2}-\operatorname{Tr}\left(\bm{A}^{2}\right)}{d(d-1)}$
(17)

for any $i\neq j$, where the summation goes over all possible permutations.

Now we are ready to prove Theorem 1. From Eq. (16), we have

$\displaystyle\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$
$\displaystyle=$ $\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}\left[\left\langle\psi\left|(\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi})^{2}\right|\psi\right\rangle-\left\langle\psi\left|\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi}\right|\psi\right\rangle^{2}\right]$
$\displaystyle=$ $\displaystyle\left\langle\psi\left|\frac{1}{d!}\sum_{\pi\in
S_{d}}\left(\bm{A}^{\pi}\right)^{2}\right|\psi\right\rangle-\frac{1}{d!}\sum_{\pi\in
S_{d}}\left\langle\psi\left|\bm{A}^{\pi}\right|\psi\right\rangle^{2}$
$\displaystyle=$
$\displaystyle\frac{\operatorname{Tr}\left(\bm{A}^{2}\right)}{d}-\frac{1}{d!}\sum_{\pi\in
S_{d}}\left\langle\psi\left|\bm{A}^{\pi}\right|\psi\right\rangle^{2},$

where $\bm{A}^{\pi}=\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi}$. Note that
$\left\langle\psi\left|\bm{A}^{\pi}\right|\psi\right\rangle^{2}=\sum^{d-1}_{i,j=0}a_{\pi(i)}a_{\pi(j)}r_{i}r_{j}$.
We have

$\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}\left\langle\psi\left|\bm{A}^{\pi}\right|\psi\right\rangle^{2}$
$\displaystyle=$ $\displaystyle\sum^{d-1}_{i=0}\left(\frac{1}{d!}\sum_{\pi\in
S_{d}}a^{2}_{\pi(i)}\right)r^{2}_{i}+\sum^{d-1}_{i\neq
j}\left(\frac{1}{d!}\sum_{\pi\in S_{d}}a_{\pi(i)}a_{\pi(j)}\right)r_{i}r_{j}.$

Using Eqs. (16) and (17), we have further

$\displaystyle\frac{1}{d!}\sum_{\pi\in
S_{d}}\left\langle\psi\left|\bm{A}^{\pi}\right|\psi\right\rangle^{2}$
$\displaystyle=$
$\displaystyle\frac{\operatorname{Tr}\left(\bm{A}^{2}\right)}{d}\sum^{d-1}_{i=0}r^{2}_{i}+\frac{[\operatorname{Tr}\left(\bm{A}\right)]^{2}-\operatorname{Tr}\left(\bm{A}^{2}\right)}{d(d-1)}\left(1-\sum^{d-1}_{i=0}r^{2}_{i}\right).$

Therefore

$\displaystyle\widehat{\operatorname{Var}}(|\psi\rangle,\bm{A})$
$\displaystyle=$
$\displaystyle\frac{d\operatorname{Tr}\left(\bm{A}^{2}\right)-[\operatorname{Tr}\left(\bm{A}\right)]^{2}}{d(d-1)}\left[1-\sum^{d-1}_{i=0}r^{2}_{i}\right].$

This completes the proof. ∎

Appendix B. The proof of Theorem 2.

###### Proof.

Analogous to the proof of the Theorem 1, one gets

$\displaystyle\widehat{V}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa(\bm{A})}\frac{1}{d!}\sum_{\pi\in
S_{d}}\left[\operatorname{Tr}\left(\bm{A}^{2}\bm{P}_{\pi}\rho\bm{P}^{\dagger}_{\pi}\right)-\left[\operatorname{Tr}\left(\bm{A}\bm{P}_{\pi}\rho\bm{P}^{\dagger}_{\pi}\right)\right]^{2}\right]$
$\displaystyle=$ $\displaystyle 1-\sum^{d-1}_{i=0}\rho_{ii}^{2}$
$\displaystyle=$ $\displaystyle S_{L}(\Pi(\rho)).$

This completes the proof. ∎

Appendix C. The proof of the equation

$\displaystyle\widehat{V}(p_{1}\rho_{1}\oplus p_{2}\rho_{2}\oplus\cdots\oplus
p_{n}\rho_{n})$ $\displaystyle=$ $\displaystyle
p_{1}^{2}\widehat{V}(\rho_{1})+p_{2}^{2}\widehat{V}(\rho_{2})+\cdots+p_{n}^{2}\widehat{V}(\rho_{n})+\sum_{i\neq
j}p_{i}p_{j}.$

###### Proof.

If $\rho=p_{1}\rho_{1}\oplus p_{2}\rho_{2}\oplus\cdots\oplus p_{n}\rho_{n}$,
by Theorem 2, we have

$\displaystyle\widehat{V}(p_{1}\rho_{1}\oplus p_{2}\rho_{2}\oplus\cdots\oplus
p_{n}\rho_{n})$ $\displaystyle=$ $\displaystyle 1-Tr(\Pi(\rho))^{2}$
$\displaystyle=$ $\displaystyle 1-Tr(\Pi(p_{1}\rho_{1}\oplus
p_{2}\rho_{2}\oplus\cdots\oplus p_{n}\rho_{n}))^{2}$ $\displaystyle=$
$\displaystyle 1-Tr(\Pi(p_{1}\rho_{1}))^{2}-Tr(\Pi(p_{2}\rho_{2}))^{2}-\cdots-
Tr(\Pi(p_{n}\rho_{n}))^{2}$ $\displaystyle=$
$\displaystyle[p_{1}^{2}-p_{1}^{2}Tr(\Pi(\rho_{1}))^{2}]+[p_{2}^{2}-p_{2}^{2}Tr(\Pi(\rho_{2}))^{2}]$
$\displaystyle+\cdots+[p_{n}^{2}-p_{n}^{2}Tr(\Pi(\rho_{n}))^{2}]+\sum_{i\neq
j}p_{i}p_{j}$ $\displaystyle=$ $\displaystyle
p_{1}^{2}\widehat{V}(\rho_{1})+p_{2}^{2}\widehat{V}(\rho_{2})+\cdots+p_{n}^{2}\widehat{V}(\rho_{n})+\sum_{i\neq
j}p_{i}p_{j}.$

∎

Appendix D. The proof of Theorem 3.

###### Proof.

For any mixed state $\rho$, by the relation about the minimum average
invariance over all pure state decomposition and the quantum Fisher
information Yu2013 ; Toth2013 ; S. Braunstein ; G. Toth2017

$F(\rho,\bm{A})=4\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\operatorname{Var}(|\psi_{k}\rangle,\bm{A}),$
(18)

we have

$\displaystyle\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\widehat{\operatorname{Var}}(|\psi_{k}\rangle,\bm{A})$
$\displaystyle\geqslant\frac{1}{d!}\sum_{\pi\in
S_{d}}\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\operatorname{Var}(\bm{P}_{\pi}|\psi_{k}\rangle,\bm{A})$
$\displaystyle=\frac{1}{d!}\sum_{\pi\in
S_{d}}\frac{1}{4}F(\bm{P}_{\pi}\rho\bm{P}^{\dagger}_{\pi},\bm{A})$
$\displaystyle=\frac{1}{2}\sum_{k,l}\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}\frac{1}{d!}\sum_{\pi\in
S_{d}}|\left\langle\phi_{k}\left|\bm{P}^{\dagger}_{\pi}\bm{A}\bm{P}_{\pi}\right|\phi_{l}\right\rangle|^{2}$
$\displaystyle=\frac{1}{2}\sum_{k,l}\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}\sum_{i,j}\phi^{(k)*}_{i}\phi^{(l)}_{i}\phi^{(l)*}_{j}\phi^{(k)}_{j}\left(\frac{1}{d!}\sum_{\pi\in
S_{d}}a_{\pi(i)}a_{\pi(j)}\right).$

For $i=j$, we have by Eq. (16),

$\displaystyle\sum_{i}\left\lvert\mspace{1.0mu}\phi^{(k)}_{i}\mspace{1.0mu}\right\rvert^{2}\left\lvert\mspace{1.0mu}\phi^{(l)}_{i}\mspace{1.0mu}\right\rvert^{2}\left[\frac{1}{d!}\sum_{\pi\in
S_{d}}a_{\pi(i)}^{2}\right]$ $\displaystyle=$
$\displaystyle\frac{\operatorname{Tr}\left(\bm{A}^{2}\right)}{d}\sum_{i}\left\lvert\mspace{1.0mu}\phi^{(k)}_{i}\mspace{1.0mu}\right\rvert^{2}\left\lvert\mspace{1.0mu}\phi^{(l)}_{i}\mspace{1.0mu}\right\rvert^{2}.$

For $i\neq j$, from Eq. (17) and the orthogonality
$\langle\phi_{k}|\phi_{l}\rangle=0$ for $k\neq l$ we have

$\displaystyle\sum_{i\neq
j}\phi^{(k)*}_{i}\phi^{(l)}_{i}\phi^{(l)*}_{j}\phi^{(k)}_{j}\left[\frac{1}{d!}\sum_{\pi\in
S_{d}}a_{\pi(i)}a_{\pi(j)}\right]$ $\displaystyle=$
$\displaystyle-\frac{[\operatorname{Tr}\left(\bm{A}\right)]^{2}-\operatorname{Tr}\left(\bm{A}^{2}\right)}{d(d-1)}\sum_{i}\left\lvert\mspace{1.0mu}\phi^{(k)}_{i}\mspace{1.0mu}\right\rvert^{2}\left\lvert\mspace{1.0mu}\phi^{(l)}_{i}\mspace{1.0mu}\right\rvert^{2}.$

Therefore,

$\displaystyle\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\widehat{\operatorname{Var}}(|\psi_{k}\rangle,\bm{A})$
$\displaystyle\geqslant$
$\displaystyle\frac{\kappa(\bm{A})}{2}\sum_{k,l}\frac{(\lambda_{k}-\lambda_{l})^{2}}{\lambda_{k}+\lambda_{l}}\sum_{i}|\phi^{(k)}_{i}|^{2}|\phi^{(l)}_{i}|^{2},$

which completes the proof. ∎

Appendix E. The proof of Theorem 4.

###### Proof.

In the permutation group $S_{2}$, it has two permutations: $(1)$ and $(1,2)$.
For any qubit mixed state $\rho$, one has
$\bm{P}^{\dagger}_{\pi}\sigma_{3}\bm{P}_{\pi}=-\sigma_{3}$ for $\pi=(1,2)$.
Utilizing the Eq. (18), we have

$\displaystyle\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\widehat{\operatorname{Var}}(|\psi_{k}\rangle,\sigma_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\sum_{\pi\in
S_{2}}\operatorname{Var}(\bm{P}_{\pi}|\psi_{k}\rangle,\sigma_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\sum_{\pi\in
S_{2}}\operatorname{Var}(|\psi_{k}\rangle,\bm{P}^{\dagger}_{\pi}\sigma_{3}\bm{P}_{\pi})$
$\displaystyle=$
$\displaystyle\min_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}\operatorname{Var}(|\psi_{k}\rangle,\sigma_{3})$
$\displaystyle=$ $\displaystyle\frac{1}{4}F(\rho,\sigma_{3})$ $\displaystyle=$
$\displaystyle(\lambda_{1}-\lambda_{2})^{2}\left\langle\phi_{1}\left|\sigma_{3}\right|\phi_{2}\right\rangle^{2}.$

Combining with $\kappa(\sigma_{3})=2$, we get
$\widehat{V}_{c}(\rho)=\frac{1}{8}F(\rho,\sigma_{3})=\frac{1}{2}(\lambda_{1}-\lambda_{2})^{2}\left\langle\phi_{1}\left|\sigma_{3}\right|\phi_{2}\right\rangle^{2}$.
This completes the proof. ∎

Appendix F. The proof of Theorem 5.

###### Proof.

In qubit systems, we have
$\widehat{V}_{c}(\rho)=\frac{1}{8}F(\rho,\sigma_{3})$ and
$\widehat{V}_{a}(\rho)=\widehat{V}(\rho)=\frac{1}{2}\operatorname{Var}(\rho,\sigma_{3})$.
For any rank two mixed state $\rho$ and observable $\bm{A}$, one has
$\operatorname{Var}(\rho,\bm{A})-F(\rho,\bm{A})=\frac{1}{2}[1-\operatorname{Tr}\left(\rho^{2}\right)](\omega_{1}-\omega_{2})^{2}$
with $\omega_{k}$ the eigenvalues of $\bm{A}$ G. Toth2017 . Therefore, we have

$\displaystyle\widehat{V}_{a}(\rho)-\widehat{V}_{c}(\rho)$ $\displaystyle=$
$\displaystyle\frac{3}{8}\operatorname{Var}(\rho,\sigma_{3})+\frac{1}{8}[\operatorname{Var}(\rho,\sigma_{3})-F(\rho,\sigma_{3})]$
$\displaystyle=$
$\displaystyle\frac{3}{8}[1-\operatorname{Tr}\left(\Pi(\rho)^{2}\right)]+\frac{1}{4}[1-\operatorname{Tr}\left(\rho^{2}\right)].$

This completes the proof. ∎

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