The uncertainty of quantum states with respect to the projective measurement

Ming-Jing Zhao1,∗, Yuanhong Tao2,†

1School of Science, Beijing Information Science and Technology University,
Beijing, 100192, P. R. China
2Department of Big Data, School of Science, Zhejiang University of Science and
Technology, Hangzhou 310023, P. R. China

Abstract The uncertainty relation is a distinctive characteristic of quantum
theory. The uncertainty is essentially rooted in quantum states. In this work
we regard the uncertainty as an intrinsic property of quantum state and
characterize it systematically with respect to given projective measurement.
Some basic concepts about uncertainty are reformulated in this context. We
prove and get the form of the uncertainty preserving operations. The quantum
states with maximal uncertainty are characterized. A universal decomposition
of uncertainty into classical uncertainty and quantum uncertainty is provided.
Furthermore, a unified and general relation among uncertainty, coherence and
coherence of assistance is established. These results are independent of any
explicit uncertainty measure. At last, we propose a new uncertainty measure
called the geometric uncertainty based on the fidelity and link it with the
geometric coherence.

Emails:∗<EMAIL_ADDRESS>†<EMAIL_ADDRESS>
Keywords Uncertainty, Quantum uncertainty, Classical uncertainty, Quantum
coherence

##  I Introduction

The uncertainty relation is first introduced by Heisenberg to describe the
restrictions on the accuracy of measurement results of two or more
noncommutative observables W. Heisenberg ; H. P. Robertson . Formally the
uncertainty relations are established in forms of variance H. P. Robertson ,
Shannon entropy D. Deutsch ; H. Maassen ; Coles. P.J. , Rényi entropy H.
Maassen , conditional entropy Renes. J.M ; Gour. G , mutual information A.
Grudka , etc. A universal uncertainty relation is then discovered with
majorization techniques S. Friedland ; Pucha . As the applications, the
uncertainty relation is factually the central ingredient in the security
analysis of almost all quantum cryptographic protocols, such as quantum key
distribution and two-party quantum cryptography Tomamichel. M. . The
uncertainty relation has also been shown to be a powerful tool in the quantum
random number generation G. Vallone ; Z. Cao , entanglement witness M. Berta ,
EPR steering S. P. Walborn ; J. Schneeloch , and quantum metrology V.
Giovannetti .

However one fundamental question is where does the uncertainty stem from? It
is from quantum states. In other words, the uncertainty is an intrinsic
property of quantum states. Since it is displayed by means of the measurement
results of observables, so it also relied on the observables. For example,
under the projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with
$\Pi_{i}=|i\rangle\langle i|$, the uncertainty of quantum state $\rho$ is
reflected by the probability vector $(\langle 0|\rho|0\rangle,\langle
1|\rho|1\rangle,\cdots,\langle d-1|\rho|d-1\rangle)^{T}$. More specifically,
let us consider the mixed state $\rho=\frac{1}{2}(|0\rangle\langle
0|+|1\rangle\langle 1|)$ and pure state
$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ in qubit system. Under
the projective measurement $\\{\Pi_{i}\\}=\\{|i\rangle\langle i|\\}_{i=0,1}$,
the measurement results are ${\rm
Tr}(\Pi_{i}\rho\Pi_{i})=\langle\psi|\Pi_{i}|\psi\rangle=\frac{1}{2}$ for
$i=0,1$. Therefore the uncertainties existed in $\rho$ and $|\psi\rangle$ are
equal under the the projective measurement $\\{\Pi_{i}\\}$.

In fact, the uncertainty of quantum states is manifested in two forms,
classical uncertainty and quantum uncertainty S. Luo2005 ; S. Luo2005-1 ; S.
Luo2017 ; Y. Sun2021-1 ; Y. Sun2021 . The classical uncertainty arises from
classical mixing while the quantum uncertainty arises from the superposition.
In this sense, the uncertainty in mixed state
$\rho=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)$ is classical
uncertainty and the uncertainty in pure state
$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ is quantum uncertainty
S. Luo2005 ; S. Luo2017 ; Y. Sun2021-1 ; Y. Sun2021 . Generally speaking, the
uncertainty is an abundant resource existed in quantum state.

However, compared with other quantumness like entanglement Horodecki ,
correlation Modi  and coherence revStreltsov ; revHu ; Marvian ; Hillery , the
study of uncertainty in quantum states is rather inadequate. Our aim is to
characterize the uncertainty in quantum states unambiguously and formally and
link the uncertainty with other quantumness tightly. Here we fix the
projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with
$\Pi_{i}=|i\rangle\langle i|$ and explore the uncertainty of quantum states
under this measurement systematically.

The remainder of the paper is arranged as follows. In section II, some basic
concepts including the free states, free operations and measures about
uncertainty in the framework of resource theory are presented. Here we
reformulate the uncertainty in terms of the intrinsic property of quantum
state with respect to the projective measurement. We show a necessary and
sufficient condition for the uncertainty preserving operation. In section III,
we characterize the quantum states with maximal uncertainty. In section IV, we
show a universal decomposition of uncertainty into classical uncertainty and
quantum uncertainty both formally and quantitatively. In section V, we
establish a unified and general relation among uncertainty, coherence and
coherence of assistance quantitatively. In section VI, we recall two
uncertainty measures and propose a new uncertainty measure named geometric
uncertainty. The decompositions of uncertainty into classical part and quantum
part are also made respectively. In section VII, we conclude with some
outlooks that remain to be answered in uncertainty theory.

##  II The uncertainty of quantum states

For any $d$-dimensional Hilbert space $\mathcal{H}_{d}$, suppose
$\\{|i\rangle\\}_{i=0}^{d-1}$ is any orthonormal basis of $\mathcal{H}_{d}$.
For any quantum state $\rho$ in $\mathcal{H}_{d}$, we refer to the uncertainty
of the quantum state $\rho$ as the uncertainty of the probability vector
$(\langle 0|\rho|0\rangle,\langle 1|\rho|1\rangle,\cdots,\langle
d-1|\rho|d-1\rangle)^{T}$ with respect to the projective measurement
$\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with $\Pi_{i}=|i\rangle\langle i|$ throughout
this paper. In this sense the uncertainty of quantum state is basis-dependent.
In the framework of the uncertainty resource theory, we specify the free
states as certain states and free operations as certain operations. Next we
characterize the free states and free operations and present a necessary and
sufficient condition for free operations.

First, for any quantum state $\rho$, we call it certain state as the free
state in the uncertainty resource theory if $\rho$ is rank-one diagonal under
the reference basis, i.e. $\rho=|i\rangle\langle i|$ for $i=0,1,\cdots,d-1$.
Or else it is uncertain. Let $\mathcal{C}$ denote the set of certain states,
$\mathcal{C}=\\{|0\rangle\langle 0|,|1\rangle\langle
1|,\cdots,|d-1\rangle\langle d-1|\\}$. All quantum states are classified into
certain states and uncertain states. The uncertain states contains
uncertainty, that is classical uncertainty or quantum uncertainty or both S.
Luo2005 ; S. Luo2017 ; Y. Sun2021-1 ; Y. Sun2021 . For pure states
$|\psi\rangle=\sum_{i}\psi_{i}|i\rangle$, it has no classical uncertainty and
its uncertainty is attributed to the quantum uncertainty. For incoherent
states which is diagonal under the reference basis,
$\rho=\sum_{i}\rho_{ii}|i\rangle\langle i|$, it has no quantum uncertainty and
its uncertainty is attributed to the classical uncertainty. For general
coherent mixed states, it contains both the classical uncertainty and quantum
uncertainty.

Furthermore, for any quantum operation $\Lambda$, we call it certain operation
as the free operation in the uncertainty resource theory if $\Lambda(\rho)$
keeps certain for all certain quantum states $\rho$. We show the certain
operations are in the following form.

######  Theorem 1.

All certain operations $\Lambda$ are in form with Kraus representation
$\Lambda=\sum_{l}K_{l}(\cdot)K_{l}^{\dagger}$ with
$K_{l}=\sum_{i}\sqrt{p_{il}}e^{{\rm i}\theta_{il}}|g(i)\rangle\langle i|$,
$\sum_{l}p_{il}=1$ for all $i$, and
$\sum_{l}\sqrt{p_{i^{\prime}l}p_{il}}e^{{\rm
i}(\theta_{il}-\theta_{i^{\prime}l})}\langle g(i^{\prime})|g(i)\rangle=0$ for
$i\neq i^{\prime}$, $g$ is a map from $\\{i\\}_{i=0}^{d-1}$ to
$\\{i\\}_{i=0}^{d-1}$.

The proof is in Appendix X.1. Theorem 1 shows that for certain operations, the
positions of the nonzero entries of all Kraus operators are the same and there
are at most one nonzero entries in each row for every Kraus operator.
Explicitly, all permutation operators $P_{\pi}=\sum_{i}|\pi(i)\rangle\langle
i|$ with $\pi$ any permutation in the permutation group are certain
operations. But certain operations may be not permutations. For example, in
qubit system, let $K_{1}=|0\rangle\langle 0|$ and $K_{2}=|0\rangle\langle 1|$
satisfying $K_{1}^{\dagger}K_{1}+K_{2}^{\dagger}K_{2}=I$, then the quantum
operation $\Lambda=K_{1}(\cdot)K_{1}^{\dagger}+K_{2}(\cdot)K_{2}^{\dagger}$ is
also certain because $\Lambda(|0\rangle\langle 0|)=\Lambda(|1\rangle\langle
1|)=|0\rangle\langle 0|$. For diagonal matrices $D_{l}$ with
$\sum_{l}D_{l}^{\dagger}D_{l}=I$, the quantum operations
$\Lambda=\sum_{l}D_{l}(\cdot)D_{l}^{\dagger}$ are also certain operations.
Furthermore the composition of the certain operations is still certain.
Therefore for any permutation $P_{\pi}$, the quantum operation
$\Lambda=\sum_{l}P_{\pi}D_{l}(\cdot)D_{l}^{\dagger}P_{\pi}$ is also certain.

For the quantification of the uncertainty, an axiomatic approach has been
proposed which requires that any reasonable measure of uncertainty is a
function of the probability vector satisfying (1) invariant under permutations
of its elements, (2) nondecreasing under a random relabeling of its argument
S. Friedland . Mathematically, the uncertainty measure should be a non-
negative Schur-concave function that takes the value zero on the vector
$\vec{x}=(1,0,\cdots,0)^{T}$ S. Friedland . Since all symmetric concave
functions are Schur-concave function, here we restrict to the nonnegative
symmetric concave functions [See Appendix X.2] as the uncertainty measure.

######  Proposition 1.

For any nonnegative symmetric concave function $f$ and any quantum state
$\rho$, $U(\rho)=f(\langle 0|\rho|0\rangle,\langle
1|\rho|1\rangle,\cdots,\langle d-1|\rho|d-1\rangle)$ is an uncertainty measure
of $\rho$.

For any given uncertainty measure $U$ and any quantum operations $\Lambda$, we
call $\Lambda$ uncertainty preserving if it preserves uncertainty for every
quantum state $\rho$, $U(\Lambda(\rho))=U(\rho)$. The uncertainty preserving
operations can be characterized as follows.

######  Theorem 2.

For any quantum operation $\Lambda=\sum_{l}K_{l}(\cdot)K_{l}^{\dagger}$, it is
an uncertainty preserving operation if and only if the Kraus operators
$K_{l}=D_{l}P_{\pi}$ with diagonal matrices $D_{l}$ such that
$\sum_{l}D_{l}^{\dagger}D_{l}=I$ and some permutation $P_{\pi}$.

The proof is in the Appendix X.3. The uncertainty preserving operation is
factually a special certain operation. Under the action of uncertainty
preserving operations, the uncertainty is kept, but the quantum part does not
increase. So the uncertainty preserving operation transfers the quantum
uncertainty into classical uncertainty. For example, the projective
measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with $\Pi_{i}=|i\rangle\langle i|$
is an uncertainty preserving operation, it projects any quantum state
$\rho=\sum_{ij}\rho_{ij}|i\rangle\langle j|$ into diagonal states
$\Pi(\rho)=\sum_{i}\Pi_{i}\rho\Pi_{i}=\sum_{i}\rho_{ii}|i\rangle\langle i|$.
Initially $\rho$ contains quantum uncertainty and classical uncertainty, after
the projective measurement $\Pi$, the resulted state $\Pi(\rho)$ just contains
classical uncertainty.

##  III Maximally uncertain states

In this section, we shall study and characterize the quantum states with
maximal uncertainty under the projective measurement
$\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with $\Pi_{i}=|i\rangle\langle i|$.

######  Definition 1.

For any quantum state $\rho$ and any uncertainty measure $U$, $\rho$ is called
maximally uncertain if the uncertainty of $\rho$ reaches the maximum of
$U(\rho)$ over all quantum states.

Under the projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with
$\Pi_{i}=|i\rangle\langle i|$, the maximally uncertain states can be
characterized definitely.

######  Theorem 3.

For $d$-dimensional quantum state $\rho=\sum_{ij}\rho_{ij}|i\rangle\langle
j|$, it is maximally uncertain if and only if $\rho_{ii}=1/d$ for
$i=0,1,\cdots,d-1$.

The proof is in Appendix X.4. Theorem 3 shows that although there are many
uncertainty measures and the maximums of the uncertainty measures are probably
different, the maximally uncertain states are definite. In a qubit system, all
quantum states can be represented in the Bloch sphere. All maximally uncertain
states are distributed in the largest horizontal disc, and the pure states
$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{{\rm i}\theta}|1\rangle)$ with
maximal quantum uncertainty are distributed in the largest horizontal circle.
The spherical center is the quantum state $I/2$ with maximal classical
uncertainty (See Fig. 1). Overall the maximally uncertain states are the
furthest from the certain states $|0\rangle$ and $|1\rangle$ vertically.

Figure 1: (Color Online) The distribution of maximally uncertain states in
Bloch sphere. All maximally uncertain states are distributed in the largest
horizontal disc. The boundary is the quantum states
$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{{\rm i}\theta}|1\rangle)$ with
maximal quantum uncertainty and the spherical center is the quantum state
$I/2$ with maximal classical uncertainty.

In fact all quantum states can be transformed into maximally coherent states.
For any mixed state $\rho$, let the unitary operator
$Q=\sum_{i=0}^{d-1}|i\rangle\langle i+1\mod{d}|$, first, we perform the
quantum operation $\Lambda_{1}=\frac{1}{d}\sum_{t=1}^{d}Q^{t}(\cdot)Q^{t}$ on
quantum state $\rho$ with $Q^{t}$ the power $t$ of $Q$. By directly
calculating we get the diagonal entries of $\Lambda_{1}(\rho)$ are all $1/d$,
which means $\Lambda_{1}(\rho)$ is maximally uncertain. Compared with the
initial quantum state $\rho$, the uncertainty has been increased, but the
quantum uncertainty is decreased as the operation $\Lambda_{1}$ is incoherent
operation. So the increased uncertainty of $\rho$ is classical uncertainty.
Second, we perform the projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$
with $\Pi_{i}=|i\rangle\langle i|$ on $\Lambda_{1}(\rho)$ and get the output
state $I/d$ which can be decomposed as the equal weight of the maximally
coherent pure states. Third, with the one-way assisted coherence distillation
$\Lambda_{3}$ E. Chitambar , the quantum state $I/d$ is transformed into the
maximally coherent pure state
$|\Psi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{i}|i\rangle$. This process is

$\displaystyle\rho\xrightarrow{\Lambda_{1}}\Lambda_{1}(\rho)\xrightarrow{\Pi}I/d\xrightarrow{\Lambda_{3}}|\Psi_{d}\rangle.$

Therefore, with these three steps, any quantum state can be transformed into
maximally coherent states.

From another point of view, any quantum state $\rho$ is maximally uncertain
under some projective measurement
$\\{\Pi_{\alpha}\\}=\\{|\alpha\rangle\langle\alpha|\\}$ with
$\\{|\alpha\rangle\\}$ some orthonormal basis of $\mathcal{H}_{d}$. This is
because any matrix is unitarily equivalent to a matrix with constant main
diagonal fillmore . Therefore, there exist some unitary matrix $U$ such that
the diagonal entries of $U^{\dagger}\rho U$ are all $1/d$. Then under the
projective measurement $\\{\Pi_{\alpha}\\}=\\{|\alpha\rangle\langle\alpha|\\}$
consisted by the eigenvectors of $U$, we have
$\langle\alpha|\rho|\alpha\rangle=1/d$ for all $\alpha$. So $\rho$ is maximal
uncertain under this projective measurement $\\{\Pi_{\alpha}\\}$.

##  IV A universal decomposition of uncertainty into quantum uncertainty and
classical uncertainty

Generally the uncertainty can be decomposed into quantum uncertainty and
classical uncertainty formally,

Uncertainty=Quantum Uncertainty + Classical Uncertainty. (1)

Hence the uncertainty is termed total uncertainty when necessary to avoid any
confusion with quantum uncertainty and classical uncertainty. Originally,
quantum uncertainty arises from the superposition while classical uncertainty
arises from the classical mixing. So the quantum uncertainty is also
recognized as coherence and the classical uncertainty is recognized as
mixedness.

For quantum uncertainty, let us recall the coherence theory briefly. Under the
fixed reference basis $\\{|i\rangle\\}$, the quantum state is incoherent if it
is diagonal, $\rho=\sum_{i=0}^{d-1}\rho_{ii}|i\rangle\langle i|$. This set of
incoherent states is labeled by ${\cal I}$. The quantum state is called
coherent if it is not incoherent. Quantum operation
$\Lambda=\sum_{l}K_{l}(\cdot)K_{l}^{\dagger}$ is an incoherent operation if it
fulfills $K_{l}\sigma K_{l}^{{\dagger}}/{\rm Tr}(K_{l}\sigma
K_{l}^{{\dagger}})\in{\cal I}$ for all $\sigma\in{\cal I}$ and for all $l$.
Any coherence measure $C(\rho)$ is a convex extension of $C(|\psi\rangle)$
from pure states to mixed states with
$C(|\psi\rangle)=f(|\psi_{0}|^{2},|\psi_{1}|^{2},\cdots,|\psi_{d-1}|^{2})$,
where $f$ is some nonnegative symmetric concave function, and
$|\psi\rangle=\sum_{i}\psi_{i}|i\rangle$ H. Zhu ; S. Du . Some common used
coherence measures are $l_{1}$ norm coherence Baumgratz , relative entropy
coherence Baumgratz , coherence concurrence TGao  and so on. The gap between
total uncertainty and quantum uncertainty is classical uncertainty. One well-
known classical uncertainty quantifier is linear entropy $S_{2}(\rho)=1-{\rm
Tr}\rho^{2}$. It is worthy to point out that the uncertainty and quantum
uncertainty are both basis dependent, while the classical uncertainty is
independent of the measurement basis.

In fact, the total uncertainty can be decomposed quantitatively into quantum
uncertainty and classical uncertainty. For any given nonnegative symmetric
concave function $f$ and any quantum state
$\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$, we have

$\displaystyle U(\rho)=C(\rho)+D(\rho),$ (2)

where $U(\rho)=f(\langle 0|\rho|0\rangle,\langle
1|\rho|1\rangle,\cdots,\langle d-1|\rho|d-1\rangle)$ quantifies the total
uncertainty existed in $\rho$, $C(\rho)$ is any convex extension of
$C(|\psi\rangle)$ from pure states to mixed states with
$C(|\psi\rangle)=f(|\psi_{0}|^{2},|\psi_{1}|^{2},\cdots,|\psi_{d-1}|^{2})$ for
$|\psi\rangle=\sum_{i}\psi_{i}|i\rangle$ and quantifies the quantum
uncertainty existed in $\rho$, and $D(\rho)=U(\rho)-C(\rho)$ quantifies the
classical uncertainty existed in $\rho$. Therefore, the equality (2) gives
rise to an operational decomposition corresponding to the equality (1). We
should emphasize that the decomposition (2) is universal because it applies
for all uncertainty measures. Additionally, the decomposition (2) is not
unique because the coherence measure $C(\rho)$ as the convex extensions of
$C(|\psi\rangle)$ is not unique. The comparison among the total uncertainty,
quantum uncertainty and classical uncertainty are made in Table I.

Total uncertainty $U(\rho)$ Quantum uncertainty $C(\rho)$ Classical
uncertainty $D(\rho)$ Nullity Certain states $|i\rangle\langle i|$ Incoherent
states $\sum_{i}\rho_{ii}|i\rangle\langle i|$ Pure states Maximizer
$\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$ with $\rho_{ii}=1/d$
$\frac{1}{\sqrt{d}}\sum_{i}e^{{\rm i}\theta_{i}}|i\rangle$ $I/d$ Convexity
Concave Convex Concave

Table 1: The comparison among the total uncertainty, quantum uncertainty and
classical uncertainty in $d$-dimensional systems.

##  V Uncertainty, coherence and coherence of assistance

For any coherence measure $C$, there is another quantifier called the
coherence of assistance $C_{a}$ corresponding to it, which is defined as

$\displaystyle
C_{a}(\rho)=\max_{\\{p_{k},|\psi_{k}\rangle\\}}\sum_{k}p_{k}C(|\psi_{k}\rangle)$
(3)

with the maximum running over all possible pure state decompositions of
$\rho=\sum_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$. This is the concave
bottom extension of $C(|\psi\rangle)$ from pure states to mixed states. It 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 is a
special case of assisted coherence distillation to generate the maximal
possible coherence on the target system in bipartite systems by only
incoherent operations on the target system and local quantum operations on the
auxiliary system. It 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. The maximum average coherence is known as the coherence
of assistance E. Chitambar .

For any given nonnegative symmetric concave function $f$ and any quantum state
$\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$, any convex extension of
$f(|\psi_{0}|^{2},|\psi_{1}|^{2},\cdots,|\psi_{d-1}|^{2})$ for any pure state
$|\psi\rangle=\sum_{i}\psi_{i}|i\rangle$ to mixed states gives rise to a
coherence measure $C(\rho)$ and the concave bottom extension in Eq. (3) gives
rise to the corresponding coherence of assistance $C_{a}(\rho)$. Since we can
also derive an uncertainty measure by
$U(\rho)=f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})$, so the uncertainty
measure $U$, coherence measure $C$ and the coherence of assistance $C_{a}$ are
connected by the real symmetric concave function $f$ (See Fig. 2).
Furthermore, they are also comparable quantitatively.

$f$$C_{a}$$C$$U$ Figure 2: Here $f$ is any nonnegative symmetric concave
function. It deduces uncertainty measure $U$, coherence measure $C$ and
coherence of assistance $C_{a}$. The quantum part of uncertainty $U$ is the
coherence $C$. By local operation and one-way classical communication with the
assistance of another party, the coherence $C$ can be increased into $C_{a}$,
which is no more than the uncertainty $U$.

######  Theorem 4.

For any quantum state $\rho$, the relation among its uncertainty, coherence,
and coherence of assistance is

$\displaystyle C(\rho)\leq C_{a}(\rho)\leq U(\rho).$ (4)

The proof is in Appendix X.5. In the process of assisted coherence
distillation above, the quantum coherence can be increased. It lies in that
the classical uncertainty of mixed state $\rho$ is transformed not necessary
totally into the quantum uncertainty with the help of Bob in the process of
purification. Since the purification does not affect the total uncertainty,
the coherence of assistance can not beyond the total uncertainty. In fact the
coherence of assistance in qubit and qutrit systems can reach the total
uncertainty, that is, $C_{a}(\rho)=U(\rho)$ for two and three dimensional
quantum states M. Zhao . But it fails in high dimensional systems.

##  VI Three uncertainty measures

In this section, we consider three uncertainty measures which correspond to
the nonnegative symmetric concave functions as (1)
$f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=1-\sum_{i}\rho_{ii}^{2}$; (2)
$f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=-\sum_{i}\rho_{ii}\log\rho_{ii}$;
and (3) $f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=1-\max_{i}\rho_{ii}$.
They are originated from the variance, entropy and fidelity respectively.

Example 1. The uncertainty measure of quantum states based on variance with
$f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=1-\sum_{i}\rho_{ii}^{2}$.

Under the projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with
$\Pi_{i}=|i\rangle\langle i|$, the total uncertainty in quantum state
$\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$ in terms of variance is

$\displaystyle U_{var}(\rho)=\sum_{i}V(\Pi_{i},\rho)=1-\sum_{i}\rho_{ii}^{2},$
(5)

where the variance of observable $A$ in the quantum state $\rho$ is
$V(A,\rho)={\rm Tr}(A^{2}\rho)-[{\rm Tr}(A\rho)]^{2}$. The total uncertainty
$U_{var}(\rho)$ can be decomposed as S. Luo2005 ; Y. Sun2021

$\displaystyle U_{var}(\rho)=C(\rho)+I(\rho),$ (6)

where $I(\rho)=\sum_{i}I(\rho,\Pi_{i})=\sum_{i}-\frac{1}{2}{\rm
Tr}[\sqrt{\rho},\Pi_{i}]^{2}=1-\sum_{i}\langle i|\sqrt{\rho}|i\rangle^{2}$ is
the quantum uncertainty quantified by the skew information and
$C(\rho)=[\sum_{i}(\langle i|\sqrt{\rho}|i\rangle^{2}-\rho_{ii}^{2})]$ is the
classical uncertainty. The total uncertainty $U_{var}(\rho)$ can be also
decomposed as

$\displaystyle U_{var}(\rho)=S_{2}(\rho)+Q(\rho),$ (7)

where $S_{2}(\rho)=1-{\rm Tr}\rho^{2}$ is the classical uncertainty quantified
by the linear entropy and $Q(\rho)=S_{2}(\Pi(\rho))-S_{2}(\rho)$ with
$\Pi(\rho)=\sum_{i}\Pi_{i}\rho\Pi_{i}$ is the quantum uncertainty Y. Sun2021-1
.

Example 2. The uncertainty measure of quantum states based on entropy with
$f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=-\sum_{i}\rho_{ii}\log\rho_{ii}$.

Under the projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with
$\Pi_{i}=|i\rangle\langle i|$, the total uncertainty in quantum state
$\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$ in terms of entropy is

$\displaystyle U_{s}(\rho)=S(\Pi(\rho))=-\sum_{i}\rho_{ii}\log\rho_{ii},$ (8)

which can be decomposed as

$\displaystyle U_{s}(\rho)=S(\rho)+C_{r}(\rho)$ (9)

with entropy $S(\rho)$ quantifying the classical uncertainty and the relative
entropy $C_{r}(\rho)=S(\rho\|\Pi(\rho))=S(\Pi(\rho))-S(\rho)$ quantifying the
quantum uncertainty T. Baumgratz ; X. Yuan2017 .

Example 3. The uncertainty measure of quantum states based on fidelity with
$f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=1-\max_{i}\rho_{ii}$.

Now we propose a new uncertainty measure based on the fidelity
$F(\rho,\sigma)=[{\rm Tr}(\sqrt{\rho}\sigma\sqrt{\rho})^{\frac{1}{2}}]^{2}$
Jozsa . Let

$\displaystyle U_{f}(\rho)=1-\max_{\sigma\in\mathcal{C}}F(\rho,\sigma),$ (10)

where the maximum is taken over all certain states. We call it geometric
uncertainty and show it is a well-defined uncertainty measure.

######  Theorem 5.

For any quantum state $\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$, the
geometric uncertainty $U_{f}(\rho)$ is an uncertainty measure and

$\displaystyle U_{f}(\rho)=1-\max_{i}\rho_{ii}.$ (11)

The proof is in Appendix X.6. The geometric uncertainty $U_{f}$ has the
following properties.

  1. (1)

$0\leq U_{f}(\rho)\leq 1-\frac{1}{d}$. $U_{f}(\rho)=0$ if and only if $\rho$
is certain, $U_{f}(\rho)=1-\frac{1}{d}$ if and only if $\rho$ is maximally
uncertain.

  2. (2)

$U_{f}(\rho)$ is a concave function of $\rho$,
$U_{f}(p\rho_{1}+(1-p)\rho_{2})\geq pU_{f}(\rho_{1})+(1-p)U_{f}(\rho_{2})$ for
$0\leq p\leq 1$.

  3. (3)

$U_{f}(\rho)$ is invariant under the action of unitary operation $W$,
$U_{f}(W\rho W^{\dagger})=U_{f}(\rho)$.

  4. (4)

$U_{f}(\rho)$ is decreasing under partial trace in the sense that
$U_{f}(\rho_{A(B)})\leq U_{f}(\rho_{AB})$ for bipartite quantum state
$\rho_{AB}$ with reduced states $\rho_{A(B)}={\rm Tr}_{B(A)}\rho_{AB}$.

  5. (5)

$U_{f}(\rho_{1})U_{f}(\rho_{2})=U_{f}(\rho_{1})+U_{f}(\rho_{2})-U_{f}(\rho_{1}\otimes\rho_{2})$
for any quantum states $\rho_{1}$ and $\rho_{2}$.

Now we prove these properties item by item. The first item is by
$\frac{1}{d}\leq\max_{i}{\rho_{ii}}\leq 1$ for any quantum state
$\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$. The second item is because the
maximum function is a convex function. The third item is because under the
projective measurement $\Pi=\\{\Pi_{i}\\}_{i=0}^{d-1}$ with
$\Pi_{i}=|i\rangle\langle i|$, $\Pi(W\rho W^{\dagger})=\Pi(\rho)$ for any
unitary operation $W$ and quantum state $\rho$. The fourth item is because
$U_{f}(\rho_{A})=1-\max_{i}\sum_{j}\rho_{ij,ij}\leq
1-\max_{i,j}\rho_{ij,ij}=U_{f}(\rho_{AB})$ for any bipartite quantum state
$\rho=\sum_{i,j,k,l}\rho_{ij,kl}|ij\rangle\langle kl|$. The fifth item is
because
$U_{f}(\rho_{1})U_{f}(\rho_{2})=(1-\max_{\sigma_{1}\in\mathcal{C}}F(\rho_{1},\sigma_{1}))(1-\max_{\sigma_{2}\in\mathcal{C}}F(\rho_{2},\sigma_{2}))=(1-\max_{\sigma_{1}\in\mathcal{C}}F(\rho_{1},\sigma_{1}))+(1-\max_{\sigma_{2}\in\mathcal{C}}F(\rho_{2},\sigma_{2}))-(1-\max_{\sigma_{1}\in\mathcal{C}}\max_{\sigma_{2}\in\mathcal{C}}U_{f}(\rho_{1},\sigma_{1})U_{f}(\rho_{2},\sigma_{2}))=U_{f}(\rho_{1})+U_{f}(\rho_{2})-U_{f}(\rho_{1}\otimes\rho_{2})$.

Recall that the geometric coherence defined as

$\displaystyle C_{f}(\rho)=1-\max_{\sigma\in\mathcal{I}}F(\rho,\sigma),$ (12)

with the maximum running over all incoherent states, is a coherence measure A.
Streltsov2015 . Let
$D_{f}(\rho)=\max_{\sigma\in\mathcal{I}}F(\rho,\sigma)-\max_{\sigma\in\mathcal{C}}F(\rho,\sigma)$,
then the geometric uncertainty can be decomposed as

$\displaystyle U_{f}(\rho)=D_{f}(\rho)+C_{f}(\rho),$ (13)

with $D_{f}(\rho)$ quantifying the classical uncertainty. For any pure state
$|\psi\rangle$, the classical uncertainty $D_{f}(|\psi\rangle)$ vanishes and
the total uncertainty coincides with quantum uncertainty
$U_{f}(|\psi\rangle)=C_{f}(|\psi\rangle)$. For any incoherent state
$\rho=\sum_{i}\rho_{ii}|i\rangle\langle i|$, the quantum uncertainty
$C_{f}(\rho)$ vanishes and the total uncertainty coincides with classical
uncertainty $U_{f}(\rho)=D_{f}(\rho)$.

##  VII Conclusions and discussions

To conclude, we have studied the uncertainty as the intrinsic property of
quantum states systematically with respect to the projective measurement. In
this context, some basic concepts are reformulated and some new results are
obtained. We have proved a necessary and sufficient condition for the
uncertainty preserving operation and characterized the quantum states with
maximal uncertainty. We have also shown a universal decomposition of total
uncertainty into quantum uncertainty and classical uncertainty for all
uncertainty measures both formally and quantitatively. The unified relation
among uncertainty, coherence, coherence of assistance is further established.
Last but not the least, we have proposed a new uncertainty measure called the
geometric uncertainty. Overall, this study of the uncertainty as the intrinsic
property of quantum states may shed light on the resource theory and its
application. On the other hand, it may be also useful for the interpretation
of the uncertainty relation.

Here we should emphasize that in this paper we consider the uncertainty of
quantum states with respect to the projective measurement. For instance, in
qubit systems, under the reference basis $\\{|0\rangle,|1\rangle\\}$, the pure
state $|0\rangle$ is certain and contains null uncertainty. But under the
orthonormal basis $\\{|+\rangle,|-\rangle\\}$ with
$|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and
$|-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$, the pure state
$|0\rangle=\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$ is maximally uncertain and
contains the maximal uncertainty in qubit systems. Therefore the results
derived here are basis dependent. It also deduces the famous uncertainty
relation which refers to the constraints on the uncertainty of the quantum
state under different measurement bases.

Furthermore, there are some interesting problems which can be further
considered. The first question is the relation between uncertainty and other
quantumness of quantum states. The relations between the uncertainty and the
entanglement M. Berta , the partial coherence (quantum uncertainty) and
quantum correlation Xiong  have been revealed previously. More than that, the
uncertainty, coherence and entanglement can be quantified by geometric
uncertainty, geometric coherence, and geometric entanglement T.C.Wei ;
Streltsov2010  consistently in terms of the fidelity. And the geometric
coherence amounts to the maximum bipartite geometric entanglement that can be
generated via incoherent operations applied to the system and an incoherent
ancilla A. Streltsov2015 . So the relation among uncertainty, coherence and
entanglement will be more clear if the conversion between geometric
uncertainty and geometric coherence is clarified. Second, instead of the
projective measurements, one may consider the uncertainty with respect to the
Lüders measurement or POVM, which may unify coherence and uncertainty in a
more extensive context. Third, we have proved that the coherence of assistance
is no more than the uncertainty. This implies the coherence obtained in the
one-way coherence distillation in one-copy setting is no more than the total
uncertainty. Does the regularized coherence of assistance
$C_{a}^{\infty}(\rho)=\lim_{n\rightarrow\infty}\frac{1}{n}C_{a}(\rho^{\otimes
n})$ reaches the total uncertainty, that is $C_{a}^{\infty}(\rho)=U(\rho)$?
The physical explanation is whether the coherence obtained in the one-way
coherence distillation in many-copy setting can reach the uncertainty or not.

##  VIII Acknowledgement

We thank Shunlong Luo and Nan Li for discussions and valuable suggestions.
This work is supported by the National Natural Science Foundation of China
under grant Nos. 12171044.

##  IX Data availability statement

All data that support the findings of this study are included within the
article (and any supplementary files).

##  X Appendices

###  X.1 The proof of Theorem 1.

Proof: For any certain quantum operations
$\Lambda=\sum_{l}K_{l}(\cdot)K_{l}^{\dagger}$ with
$\sum_{l}K_{l}^{\dagger}K_{l}=I$, and for any given certain state
$|i\rangle\langle i|$, it has $\Lambda(|i\rangle\langle i|)\in{\cal C}$.
Without loss of generality, we suppose $\Lambda(|i\rangle\langle
i|)=|g(i)\rangle\langle g(i)|$ with $g(i)\in\\{i\\}$. Therefore
$K_{l}(|i\rangle\langle i|)K_{l}^{\dagger}=p_{il}|g(i)\rangle\langle g(i)|$
and $K_{l}|i\rangle=\sqrt{p_{il}}e^{{\rm i}\theta_{il}}|g(i)\rangle$ for any
$l$. When $i$ runs all over $\\{i\\}_{i=0}^{d-1}$, we have
$K_{l}=\sum_{i}\sqrt{p_{il}}e^{{\rm i}\theta_{il}}|g(i)\rangle\langle i|$.
Since $\sum_{l}K_{l}^{\dagger}K_{l}=I$, so
$\sum_{l}\sum_{i^{\prime}}\sum_{i}\sqrt{p_{i^{\prime}l}p_{il}}e^{{\rm
i}(\theta_{il}-\theta_{i^{\prime}l})}\langle
g(i^{\prime})|g(i)\rangle|i^{\prime}\rangle\langle i|=I$. So we have
$\sum_{l}p_{il}=1$ for all $i$, and
$\sum_{l}\sqrt{p_{i^{\prime}l}p_{il}}e^{{\rm
i}(\theta_{il}-\theta_{i^{\prime}l})}\langle g(i^{\prime})|g(i)\rangle=0$ for
$i\neq i^{\prime}$.

###  X.2 The nonnegative symmetric concave function

The nonnegative symmetric concave function defined on the probability simplex
$\Omega=\\{\vec{x}=(x_{0},x_{1},\cdots,x_{d-1})^{T}|\sum_{i=0}^{d-1}x_{i}=1\
\text{and}\ x_{i}\geq 0\\}$ is a function such that it is

  * •

$f((1,0,\cdots,0)^{T})=0$.

  * •

invariant under any permutation transformation $P_{\pi}$,
$f(P_{\pi}\vec{x})=f(\vec{x})$.

  * •

concave, $f(\lambda\vec{x}+(1-\lambda)\vec{y})\geq\lambda
f(\vec{x})+(1-\lambda)f(\vec{y})$ for any $0\leq\lambda\leq 1$ and
$\vec{x},\vec{y}\in\Omega$.

To guarantee the uncertainty (coherence) in the uncertain (coherent) states is
positive, it is reasonable to demand the

  * •

$f(\vec{x})=0$ if and only if $\vec{x}=P_{\pi}(1,0,\cdots,0)^{T}$ for some
permutation transformation $P_{\pi}$.

Therefore $U(\rho)=0$ ($C(\rho)=0$) if and only if $\rho$ is certain
(incoherent). Throughout this paper, when we say the nonnegative symmetric
concave function, we default it satisfying this additional condition.

###  X.3 The proof of Theorem 2.

Proof: On one hand, suppose the quantum operation $\Lambda$ preserving
uncertainty for every quantum state $\rho$, $U(\Lambda(\rho))=U(\rho)$. First
we assume $\rho$ is certain, then $U(\rho)=0$. Therefore $U(\Lambda(\rho))=0$,
which implies $\Lambda(\rho)$ is a certain state. This requires that
$\Lambda(\rho)$ is certain for all certain states $\rho$, so $\Lambda$ is a
certain operation. By Theorem 1, we suppose
$\Lambda=\sum_{l}K_{l}(\cdot)K_{l}^{\dagger}$ with
$K_{l}=\sum_{i}\sqrt{p_{il}}e^{{\rm i}\theta_{il}}|g(i)\rangle\langle i|$. For
quantum state $\rho=\sum_{i,j=0}^{1}\rho_{ij}|i\rangle\langle j|$, its
uncertainty is $U(\rho)=f(\rho_{00},\rho_{11},0,\cdots,0)$. After the action
of quantum operation $\Lambda$, the resulted state is
$\Lambda(\rho)=\sum_{i,j=0}^{1}\sum_{l}\sqrt{p_{il}p_{jl}}e^{{\rm
i}(\theta_{il}-\theta_{jl})}\rho_{ij}|g(i)\rangle\langle g(j)|$. Due to the
arbitrariness of $\rho$, we have $g(i)\neq g(j)$ for any $i\neq j$. Therefore
$g$ is a permutation in $\\{i\\}$. This implies the Kraus operators $K_{l}$ is
in form of $K_{l}=D_{l}P_{\pi}$ with some diagonal matrices $D_{l}$ and
permutation $P_{\pi}$.

On the other hand, for the quantum operation
$\Lambda=\sum_{l}K_{l}(\cdot)K_{l}^{\dagger}$ with Kraus operators
$K_{l}=D_{l}P_{\pi}$ with diagonal matrices $D_{l}$ such that
$\sum_{l}D_{l}^{\dagger}D_{l}=I$ and some permutation $P_{\pi}$, it is easy to
verify that
$\Pi(\Lambda(\rho))=\sum_{i}\rho_{ii}|\pi(i)\rangle\langle\pi(i)|$, therefore
$U(\Lambda(\rho))=U(\rho)$.

###  X.4 The proof of Theorem 3

Proof: For any quantum state $\rho=\sum_{ij}\rho_{ij}|i\rangle\langle j|$, it
satisfies
$(1/d,1/d,\cdots,1/d)\prec(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})$. Hence
$f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})\leq f(1/d,1/d,\cdots,1/d)$ for
any Schur concave function $f$ Bhatia , which implies $U(\rho)\leq
f(1/d,1/d,\cdots,1/d)$. Therefore, $\rho$ is maximally uncertain if and only
if $\rho_{ii}=1/d$ for $i=0,1,\cdots,d-1$.

###  X.5 Proof of Theorem 4.

Proof: Here we only need to prove the right hand side inequality
$C_{a}(\rho)\leq U(\rho)$. In fact, for any nonnegative symmetric concave
function $f$ and any pure state $|\psi\rangle$, the uncertainty coincides with
quantum coherence as
$U(|\psi\rangle)=C(|\psi\rangle)=f(|\psi_{0}|^{2},|\psi_{1}|^{2},\cdots,|\psi_{d-1}|^{2})$.
For any mixed state $\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$, suppose
$\rho=\sum_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$ with
$|\psi_{k}\rangle=\sum_{i}\psi_{i}^{(k)}|i\rangle$ is any pure-state
decomposition of $\rho$. Then we have

$\begin{array}[]{rcl}C(\rho)&\leq&\sum_{k}p_{k}C(|\psi_{k}\rangle)\\\
&=&\sum_{k}p_{k}f(|\psi_{0}^{(k)}|^{2},|\psi_{1}^{(k)}|^{2},\cdots,|\psi_{d-1}^{(k)}|^{2})\\\
&\leq&f(\sum_{k}p_{k}|\psi_{0}^{(k)}|^{2},\sum_{k}p_{k}|\psi_{1}^{(k)}|^{2},\cdots,\sum_{k}p_{k}|\psi_{d-1}^{(k)}|^{2})\\\
&=&f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})\\\ &=&U(\rho),\end{array}$
(14)

where the first inequality is the convexity of coherence measures and the
second inequality is the concavity of function $f$. So we get $C_{a}(\rho)\leq
U(\rho)$.

###  X.6 The proof of Theorem 5.

Proof: For any quantum state $\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$, it
has $F(\rho,|i\rangle\langle i|)=[Tr(\sqrt{\rho}|i\rangle\langle
i|\sqrt{\rho})^{\frac{1}{2}}]^{2}=[\frac{1}{\sqrt{\rho_{ii}}}Tr\sqrt{\rho}|i\rangle\langle
i|\sqrt{\rho}]^{2}=[\sqrt{\rho_{ii}}]^{2}=\rho_{ii}$, therefore

$\displaystyle U_{f}(\rho)=1-\max_{i}F(\rho,|i\rangle\langle
i|)=1-\max_{i}\rho_{ii}.$ (15)

Since $f(\rho_{00},\rho_{11},\cdots,\rho_{d-1,d-1})=1-\max_{i}\rho_{ii}$ is a
nonnegative symmetric concave function, it demonstrates that $U_{f}$ is a bona
fide uncertainty measure.

## References

  * (1) W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43, 172 (1927).
  * (2) H.P. Robertson, The Uncertainty Principle, Phys. Rev. 34, 163 (1929).
  * (3) H. Maassen, and J. Uffink, Generalized Entropic Uncertainty Relations, Phys. Rev. Lett. 60, 1103(1988).
  * (4) D. Deutsch, Uncertainty in Quantum Measurements. Phys. Rev. Lett. 50, 631 (1983).
  * (5) P. J. Coles, M. Berta, M. Tomamichel, S., Wehner, Entropic uncertainty relations and their applications. Rev. Mod. Phys. 89, 015002 (2017).
  * (6) J. M. Renes, J. C. Boileau, Conjectured strong complementary information tradeoff. Phys. Rev. Lett. 103, 020402 (2009).
  * (7) G. Gour, A. Grudka, M. Horodecki, W. Kłobus, J. Łodyga, V. Narasimhachar, Conditional uncertainty principle. Phys. Rev. A. 97, 042130 (2018).
  * (8) A. Grudka, M. Horodecki, P. Horodecki, R. Horodecki, W. Kłobus, Ł. Pankowski, Conjectured strong complementary-correlations tradeoff. Phys. Rev. A. 88, 032106 (2013).
  * (9) S. Friedland, V. Gheorghiu, and G. Gour, Universal Uncertainty Relations, Phys. Rev. Lett. 111, 230401 (2013).
  * (10) Z. Puchała, Ł. Rudnicki, Ł, K. Życzkowski, Majorization entropic uncertainty relations. J. Phys. A: Math. Theor. 46, 272002 (2013).
  * (11) M. Tomamichel, R. Renner, Uncertainty Relation for Smooth Entropies. Phys. Rev. Lett. 106, 110506 (2011).
  * (12) G. Vallone, D. G. Marangon, M. Tomasin, P. Villoresi, Quantum randomness certified by the uncertainty principle. Phys. Rev. A 90, 052327 (2014).
  * (13) Z. Cao, H. Zhou, X. Yuan, X. Ma, Source-Independent Quantum Random Number Generation. Phys. Rev. X 6, 011020 (2016).
  * (14) M. Berta, P. J. Coles, S. Wehner, Entanglement-assisted guessing of complementary measurement outcomes. Phys. Rev. A 90, 062127 (2014).
  * (15) S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, P. H. S. Ribeiro, Revealing Hidden Einstein-Podolsky-Rosen Nonlocality. Phys. Rev. Lett. 106, 130402 (2011).
  * (16) J. Schneeloch, C. J. Broadbent, S. P. Walborn, E. G. Cavalcanti, J. C. Howell, Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations. Phys. Rev. A 87, 062103 (2013).
  * (17) V. Giovannetti, S. Lloyd, L. Maccone, Advances in quantum metrology. Nat. Photon. 5, 222 (2011).
  * (18) S. Luo, Quantum versus classical uncertainty. Theor. Math. Phys. 143, 681 (2005).
  * (19) S. Luo, and Y. Sun, Quantum coherence versus quantum uncertainty, Phys. Rev. A 96, 022130 (2017).
  * (20) Y. Sun, and S. Luo, Coherence as uncertainty, Phys. Rev. A 103, 042423 (2021).
  * (21) Y. Sun and N. Li, The uncertainty of quantum channels in terms of variance, Quan. Inf. Process 20, 25 (2021).
  * (22) S. Luo, Heisenberg uncertainty relation for mixed states, Phys. Rev. A 72, 042110 (2005).
  * (23) R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865, (2019).
  * (24) K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: discord and related measures, Rev. Mod. Phys. 84, 1655 (2012).
  * (25) A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  * (26) M. L. Hu, X. Hu, J. Wang, Y. Peng, Y. R. Zhang, and H. Fan, Quantum coherence and geometric quantum discord, Phys. Rep. 762, 1 (2018).
  * (27) I. Marvian and R. W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94, 052324 (2016).
  * (28) M. Hillery, Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and avariation, Phys. Rev. A 93, 012111 (2016).
  * (29) P. A. Fillmore, On Similarity and the Diagonal of a Matrix, The American Mathematical Monthly, 76, 167 (1969).
  * (30) H. Zhu, Z. Ma, Z. Cao, S. M. Fei, and V. Vedral, Operational one-to-one mapping between coherence and entanglement measures, Phys. Rev. A 96, 032316 (2017).
  * (31) S. Du, Z. Bai, and X. Qi, Coherence measures and optimal conversion for coherent states. Quantum Inf. Comput. 15, 1307 (2015).
  * (32) T. Baumgratz, M. Cramer, M. B. Plenio, Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014).
  * (33) X. Qi, T. Gao, F. L. Yan, Measuring coherence with entanglement concurrence. J. Phys. A 50, 285301 (2017).
  * (34) E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, Assisted Distillation of Quantum Coherence. Phys. Rev. Lett. 116, 070402 (2016).
  * (35) M.J. Zhao, T. Ma, and R. Pereira, Average quantum coherence of pure-state decomposition, Phys. Rev. A 103, 042428 (2021).
  * (36) X. Yuan, G. Bai, T. Peng, and X. Ma, Quantum uncertainty relation using coherence, Phys. Rev. A 96, 032313 (2017).
  * (37) T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying Coherence, Phys. Rev. Lett. 113, 140401 (2014).
  * (38) R. Jozsa, Fidelity for Mixed Quantum States, J. Mod. Opt. 41, 2315 (1994).
  * (39) A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Measuring Quantum Coherence with Entanglement, Phys. Rev. Lett. 115, 020403 (2015).
  * (40) C. Xiong, A. Kumar, M. Huang, S. Das, U. Sen and J. Wu, Partial coherence and quantum correlation with fidelity and affinity distances, Phys. Rev. A 99, 032305 (2019).
  * (41) T. C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68, 042307 (2003).
  * (42) A. Streltsov, H. Kampermann, and D. Bruß, Linking a distance measure of entanglement to its convex roof, New J. Phys. 12, 123004 (2010).
  * (43) R. Bhatia, Matrix Analysis, Springer-Verlag, (New York) (1997).


# Finite Action Principle and wormholes.

Jan Chojnacki1 1Faculty of Physics, University of Warsaw, ul. Pasteura 5,
02-093 Warsaw, Poland  Jan Kwapisz1,2 1Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
2CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M,
Denmark

###### Abstract

In this work, we elaborate on the finite action for wormholes in higher
derivative theories. Both non-traversable and traversable wormholes in
theories with higher curvature invariants posses finite action.

###### keywords:

Horava gravity; Wormholes; finite action

##  1 Introduction

The Finite Action Principle (FAP) proposing that the physical spacetimes are
those for which the action is finite. This comes reasoning comes from the fact
that in Euclidian path integral the weighting factor is $e^{-S[\Phi]}$. Hence
lesser the action the more it contributes to the path integral. On the other
hand in the Lorentztian signature the more varying actions contribute less.
Hence for the actions neighbouring the infinite one the variation is
significant and the neighbouring ones do cancel each other, see [1].
This notion has recently received significant attention in the literature, see
[2, 3, 4] for the cosmological considerations and [1, 4] for the black hole
ones. Since it is expected that the quantum gravity should resolve the black-
hole singularity problem, one may ask which of the microscopic actions remain
finite for non-singular black holes and conversely interfere destructively for
the singular ones. This we shall call the finite action selection principle.
Only after the inclusion of higher-curvature operators, beyond the Einstein-
Hilbert term, such selection principle can be satisfied [1]. Furthermore in
asymptotic safety, the quantum corrections to the Newtonian potential
eliminate the classical-singularity [5].
These findings suggest that by taking into account the higher curvatures one
can resolve the singularities in black holes. Yet, an issue with the higher-
curvature theory of quantum gravity is the existence of the particles with the
negative mass-squared spectrum, known as ghosts, which makes the theory non-
unitary. In this article, we explore possible resolution, namely, we
investigate Horava-Lifshitz (H-L) gravity [6], where the Lorentz Invariance
(LI) is broken at the fundamental level (see [7] for a comprehensive progress
report on this subject). Kinetic terms are first order in the time
derivatives, while higher spatial curvature scalars regulate the UV behavior
of the gravity. In our investigation we follow Horava [6] and assume that Wick
rotation is well defined. This can be motivated by the lack of the higher
time-like curvature invariants. In the usual higher derivative constructions
the existence of massive poles in the propagator makes the Wick rotation
troublesome [8]. Conversely to the latter, the new poles are massless and non-
tachyonic for suitable choice of parameters [9]. Furthermore, the existence of
the foliation supports that claim. For the discussion of Minkowski path
integral and FAP, see [1].
In the previous article, we discussed the possibility that the Finite Action
arguments applied to the projectable H-L gravity result in a flat,
homogeneous, UV-complete, and ghost-free beginning of the universe [4].
However there is no known regular black hole solutions in HL gravity. Here we
discuss the wormholes solutions in context of the Finite Action Principle in
HL gravity as well as in the $f(R)$ theories. Interestingly the stable,
traversable wormholes solutions are known only in the higher derivative
gravities [10] (without exotic matter), so there seems to be a wormhole/non-
singular BH trade-off after taking into account the Finite Action Principle.

##  2 Horava-Lifshitz gravity

In the Horava-Lifshitz gravity, space and time are scaled in a non-equivalent
way. Diffeomorphism invariance is broken by the foliation of the 4-dimensional
spacetime into 3-dimensional hypersurfaces of constant time, called leaves,
making the theory power-counting renormalizable (see also the renormalization
group studies of the subject [11, 12, 13]). The remaining symmetry respects
transformations:

$\displaystyle t\xrightarrow{}\xi_{0}(t),\quad
x^{i}\xrightarrow{}\xi^{i}(t,x^{k}),$ (1)

and is often referred to as the foliation-preserving diffeomorphism, denoted
by Diff$(M,\mathcal{F})$. The diffeomorphism invariance is still present on
the leaves. The four-dimensional metric may be expressed in the Arnowitt-
Deser-Misner (ADM) [14] variables:

$\displaystyle(N,N^{i},\,^{(3)}g_{ij}),$ (2)

where $N,\,N^{i},\,\,^{(3)}g_{ij}$ denote respectively the lapse function,
shift vector, and 3-dimensional induced metric on the leaves. The theory is
constructed from the following quantities:

$\,{}^{(3)}R_{ij},\quad K_{ij},\quad a_{i},\quad\,^{(3)}\nabla_{i},$ (3)

where $\,{}^{(3)}R_{ij}$ is the 3-dimensional Ricci curvature tensor,
$\,{}^{(3)}\nabla_{i}$ is the covariant derivative constructed from the
3-dimensional metric $\,{}^{(3)}g_{ij}$, and $a_{i}:=\frac{N,_{i}}{N}$.
Extrinsic curvature $K_{ij}$ is the only object, invariant under general
spatial diffeomorphisms containing exactly one time derivative of the metric
tensor $\,{}^{(3)}g_{ij}$:

$\displaystyle K_{ij}=\frac{1}{2N}\left(\frac{\partial\,^{(3)}g_{ij}}{\partial
t}-\,^{(3)}\nabla_{i}N_{j}-\,^{(3)}\nabla_{j}N_{i}\right).$ (4)

Quantities (2) are tensor/vectors with respect to Diff($M,\mathcal{F}$)
possessing the following mass dimensions:

$\displaystyle[\,^{(3)}R_{ij}]=2,\quad[K_{ij}]=3,\quad[a_{i}]=1,\quad[\,^{(3)}\nabla_{i}]=1.$
(5)

One may use (2) to construct, order by order, scalar terms appearing in the
Lagrangian of the theory. Following [7, 15] the action of the Horava gravity
takes the form:

$\displaystyle S_{g}=\zeta^{2}\int
dtdx^{3}N\sqrt{{}^{(3)}g}\left(\mathcal{K}-V\right),$ (6)

where $\mathcal{K}=K_{ij}K^{ij}-\lambda K^{2}$ with $K=K_{ij}\,^{(3)}g^{ij}$,
$\,{}^{(3)}g$ denotes the determinant of the 3-dimensional metric and
$\zeta^{2}=1/16\pi G$. It may be expressed as the difference of the kinetic
and potential part $\mathcal{L}=\mathcal{K}-V$ with
$\mathcal{K}=\left(K_{ij}K^{ij}-\lambda K^{2}\right)$. At the 6th order, the
potential part of the lagrangian contains over 100 terms [7]. The immense
number of invariants is limited by imposing further symmetries. One possible
restriction for the potential comes from the projectability condition
$N=N(t)$, then terms proportional to $a_{i}\equiv 0$ vanish. Up to the sixth
order (compatible with power counting renormalizability), the potential $V$
restricted by the projectability condition is given by:

$\displaystyle
V=2\Lambda\zeta^{2}-\,^{(3)}R+\frac{1}{\zeta^{2}}\left(g_{2}\,^{(3)}R^{2}+g_{3}\,^{(3)}R^{ij}\,{}^{(3)}R_{ij}\right)$
$\displaystyle+\frac{1}{\zeta^{4}}\left(g_{4}{}^{(3)}R^{3}+g_{5}\,^{(3)}R\,^{(3)}R^{ij}\,{}^{(3)}R_{ij}+g_{6}\,^{(3)}R^{i}_{j}\,{}^{(3)}R^{j}_{k}\,{}^{(3)}R^{k}_{i}\right),$
$\displaystyle+\frac{1}{\zeta^{4}}\left(g_{7}{}^{(3)}R\nabla^{2}{}^{(3)}R+g_{8}(\nabla_{i}{}^{(3)}R_{jk})(\nabla^{i}{}^{(3)}R^{jk})\right),$
(7)

where $\Lambda$ is the cosmological constant and $\alpha_{ij}$ are the
coupling constants. For our purposes, we drop terms containing covariant
derivatives $\,{}^{(3)}\nabla_{i}$. One should also mention that this _minimal
theory_ [16] suffers from the existence of spin 0 graviton, which is unstable
in the IR. Various solutions to this problem have been proposed. One can add
the additional local $U(1)$ symmetry [7, 17]. Then by the introduction of new
fields prevents the zero-mode from propagating. On the other hand, one can
drop the projectability condition $a_{i}=0$ and include the terms containing
$a_{i}$ in the potential term:

$\displaystyle
V=2\Lambda\zeta^{2}-\,^{(3)}R-\beta_{0}a_{i}a^{i}+\sum_{n=3}^{6}\mathcal{L}_{V}^{(n)},$
(8)

then for the spin-0 mode to be stable one requires $0<\beta_{0}<2$ [18, 19].

##  3 Black holes and wormholes

In this section, we show that H-L gravity satisfies the Finite Action
selection principle for the microscopic action of quantum gravity [1]. We
study both the solutions of H-L gravity and the known, off-shell BH
spacetimes, due to the fact that there are no known regular BH solutions in
the HL gravity. Keep in mind that a metric does not need to be a solution to
the equations of motion to enter the path integral. We require singular black-
hole metrics to interfere destructively, while the regular ones with finite
action contribute to the probability amplitudes. We broaden this analysis by
studying the wormhole solutions.

##  4 Wormholes and finite action principle

Here, we take the first step in the direction of the investigations of the
consequences of the Finite Action Principle in the context of wormholes (WH).
The wormholes may be characterized in two classes: traversable and non-
traversable. The traversable WH, colloquially speaking, are such that one can
go through it to the other side, see [20] for specific conditions. The
pioneering Einstein-Rosen bridge has been found originally as a non-static,
non-traversable solution to GR. The traversable solutions are unstable,
however, they might be stabilized by an exotic matter or inclusion of the
higher curvature scalar gravity [10]. This is important in the context of
finite action since usually the divergences of black holes do appear in the
curvature squared terms. Hence, due to the inclusion of the higher-order terms
in the actions, the traversable wormholes are solutions to the equations of
motions without the exotic matter. The exemplary wormhole spacetimes
investigated here are the Einstein-Rosen bridge proposed in [21], the Morris-
Thorne (MT) wormhole [20], the traversable exponential metric wormhole [22]
and the wormhole solution discussed in the H-L gravity [23]. All of them have
a finite action. Here, we shall discuss the exponential metric WH. The
conclusions for the other possible wormholes are similar and we discuss them
in the Appendix. For the exponential metric WH, the line element is given by:

$\displaystyle
ds^{2}=-e^{-\frac{2M}{r}}dt^{2}+e^{\frac{2M}{r}}\left(dr^{2}+r^{2}d\Omega^{2}\right).$
(9)

This spacetime consists two regions: “our universe” with $r>M$ and the “other
universe” with $r<M$. $r=M$ corresponds to the wormhole’s throat. The spacial
volume of the “other universe” is infinite when $r\xrightarrow{}0$. Such
volume divergence is irrelevant to our discussion since it describes large
distances in the ”other universe”. Hence, we further consider only $r\geq M$.
The resulting Ricci and Kretschmann scalars calculated in [22] and the measure
are non-singular everywhere:

$\displaystyle R$ $\displaystyle=-\frac{2M^{2}}{r^{4}}e^{\frac{-2M}{r}},$
$\displaystyle R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$
$\displaystyle=\frac{4M^{2}(12r^{2}-16Mr+7M^{2})}{r^{8}}e^{-4\frac{M}{r}},$
(10)

resulting in the finite action for the Stelle gravity. Similarly for the H-L
gravity:

$\displaystyle{}^{(3)}R=R,\quad K^{2}=K_{ij}K^{ij}=0,$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}=\frac{2M^{2}(M^{2}-2Mr+3r^{2})}{r^{8}}e^{-\frac{4M}{r}}.$
(11)

### Einstein-Rosen bridge

The Einstein-Rosen (E-R) bridge smoothly glues together two copies of the
Schwarzschild spacetime: black-hole and the white-hole solutions corresponding
to the positive and negative coordinate $u$. Metric tensor of the Einstein-
Rosen wormhole proposed in [21] and discussed in e.g. [24] is given by:

$\displaystyle
ds^{2}=\frac{-u^{2}}{u^{2}+4M}dt^{2}+(u^{2}+4M)du^{2}+\frac{1}{4}(u^{2}+4M)d\Omega^{2}.$
(12)

The E-R bridge is non-traversable and geodesically incomplete in $u=0$. This
fact, however, does not impact the regularity of the curvature scalars. The
4-dimensional Ricci scalar is:

$\displaystyle
R=\frac{2\left(64M^{2}+32Mu^{2}+4u^{4}+u^{2}\right)}{\left(4M+u^{2}\right)^{3}}.$
(13)

The second order curvature scalar $R_{\mu\nu}R^{\mu\nu}$ is:

$\displaystyle\frac{4\left(48M^{2}+8\left(4M+u^{2}\right)^{4}+(32M-1)\left(4M+u^{2}\right)^{2}\right)}{\left(4M+u^{2}\right)^{6}}.$
(14)

Both of which integrated with the measure are non-singular:

$\displaystyle\sqrt{g}=\frac{1}{4}u(4M+u^{2}).$ (15)

The wormhole solutions analyzed in this paper generally yield the finite
action in both GR and H-L. The finite Action Principle suggests, that in the
quantum UV regime, singular black-hole spacetimes may be replaced with the
regular wormhole solutions.

### Morris-Thorne wormhole

The MT wormhole is defined in the spherically symmetric, Lorentzian spacetime
by the line element:

$\displaystyle
ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-\frac{b(r)}{r}}+r^{2}d\Omega^{2}$
(16)

where $\Phi(r)$ is known as the redshift and there are no horizons if it is
finite. Function $b(r)$ determines the wormhole’s shape. We choose
$\Phi(r),\,b(r)$ to be:

$\displaystyle\Phi(r)$ $\displaystyle=0,\quad
b(r)=2M\left(1-e^{r_{0}-r}\right)+r_{0}e^{r_{0}-r},$ (17)

where $r_{0}$ is the radius of the throat of the wormhole, such that
$b(r_{0})=r_{0}$. 4-dimensional curvature scalars for this spacetime have been
calculated in [25]. The Ricci curvature scalar is singular at $r=0$, however,
the radial coordinate $r$ varies between $r_{0}>0$ and infinity:

$\displaystyle R=-2\left(2M-r_{0}\right)\frac{e^{r_{0}-r}}{r^{2}}.$ (18)

The resulting $S_{s}=\int_{r_{UV}=r_{0}}^{r_{IR}}\sqrt{g}R$ function is
divergent as $r_{UV}\xrightarrow{}r_{0}$ and cannot be expressed in terms of
simple functions:

$\displaystyle
2(2M-r_{0})\int_{r_{UV}=r_{0}}^{r_{IR}}\sqrt{\frac{r}{r-2M(1-e^{r_{0}-r})+r_{0}e^{r_{0}-r}}}e^{r_{0}-r}dr.$
(19)

However, this is only a coordinate singularity and one may get rid of it with
a proper transformation.
Higher-order curvature scalars for Morris-Thorne wormhole are:

$\displaystyle{}^{(3)}R$ $\displaystyle=\frac{2b^{\prime}(r)}{r^{2}}$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{3r^{2}b^{\prime}(r)^{2}-2rb(r)b^{\prime}(r)+3b(r)^{2}}{2r^{6}},$
$\displaystyle{}^{(3)}R^{i}_{j}{}^{(3)}R^{j}_{k}{}^{(3)}R^{k}_{i}$
$\displaystyle=\frac{-9r^{2}b(r)b^{\prime}(r)^{2}+5r^{3}b^{\prime}(r)^{3}+15rb(r)^{2}b^{\prime}(r)-3b(r)^{3}}{4r^{9}},$
(20)

and integrated give action that is finite.

### H-L wormhole

Static spherically traversable symmetric wormholes have been constructed in
[23] in the H-L theory through the modification of the Rosen-Einstein
spacetime:

$\displaystyle
ds^{2}=-N^{2}(\rho)dt^{2}+\frac{1}{f(\rho)}d\rho^{2}+(r_{0}+\rho^{2})^{2}d\Omega^{2},$
(21)

with additional $\mathbf{Z}_{2}$ symmetry with respect to the wormhole’s
throat. There are solutions with $\lambda=1$ asymptotically corresponding to
the Minkowski vacuum. Explicitly we have:

$\displaystyle f=N^{2}$ $\displaystyle=1+\omega(r_{0}+\rho^{2})^{2}$
$\displaystyle-\sqrt{(r_{0}+\rho^{2})\left(\omega^{2}(r_{0}+\rho^{2})^{3}+4\omega
M\right)}.$ (22)

Radius of the wormhole’s throat is given by $r_{0}$. The parameters $\omega,$
and $M$ are connected to the coupling constants in H-L action. See [23] for
their explicit form. Ricci scalar of the H-L wormhole invariants are given by

$\displaystyle{}^{(3)}R$
$\displaystyle=-\frac{1}{{(\rho^{2}+r_{0}){}^{2}}}\Big{[}2(-10\rho^{2}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}$
$\displaystyle-4r_{0}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}+16\rho^{6}\omega+8\rho^{2}$
$\displaystyle+36\rho^{4}r_{0}\omega+24\rho^{2}r_{0}^{2}\omega+4r_{0}^{3}\omega+4r_{0}-1)\Big{]},$
$\displaystyle{}^{(3)}R_{ij}{}^{(3)}R^{ij}$
$\displaystyle=\frac{1}{(\rho^{2}+r_{0})^{4}}\left\\{2(-7\rho^{2}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}-2r_{0}\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}\right.$
$\displaystyle+10\rho^{6}\omega+6\rho^{2}+22\rho^{4}r_{0}\omega+14\rho^{2}r_{0}^{2}\omega+2r_{0}^{3}\omega+2r_{0}-1){}^{2}+4M$
$\displaystyle+\frac{4(\rho^{2}+r_{0})}{\omega(4M+\omega(r_{0}+\omega^{2}){}^{3})}\left[\rho^{2}\omega(-4(\rho^{2}+r_{0})\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}+\omega(r_{0}+\omega^{2}){}^{3})\right]$
$\displaystyle\left.-2\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}\left[-\sqrt{\omega(\rho^{2}+r_{0})(4M+\omega(r_{0}+\omega^{2}){}^{3})}+\omega(\rho^{2}+r_{0}){}^{2}+1)\right]{}^{2}\right\\}$
(23)

The kinetic terms with $K_{ij}=0$ are vanishing, while the spacial Ricci
scalar and higher curvature terms are finite.
From the point of view of the Finite Action Principle, all of the investigated
wormhole spacetimes are included in the gravitational path integral.

##  5 Conclusions and discussion

The Finite Action Principle is a powerful tool to study quantum gravity
theories and also QFTs in general. From the point of view of finite action
selection principle [1, 4] they are equally good theories, resolving the black
holes singularities, assuming that ghost issue is resolved in the latter case.
Yet none of the regular B-H solutions have been found in the context of H-L
gravity [26] and the ones found for $f(R)$ suffer from mass inflation issue
[27, 28, 29]. Hence it is a strong suggestion that the wormholes may appear in
the UV regime of H-L gravity and can serve as a “cure” for singularities [30,
31, 32].
In the case of wormholes, both traversable and non-traversable wormholes are
on equal footing in the case of the Finite Action Principle, yet maybe the
finite amplitude principle could distinguish between those [3], since the
amplitude to cross non-traversable wormhole should be different than the
traversable one. However, this principle suggests that there is a trade-off
between the resolution of black-hole singularities and the appearance of
wormhole spacetimes due to higher curvature invariants. The wormhole solutions
will remain in both the LI and H-L path integrals. The higher-order curvature
scalars, generically present in the quantum gravity, stabilise the wormhole
solutions without the need for an exotic matter.

## Acknowledgements

We thank J. N. Borrisova, A. Eichhorn and A. Wang for inspiring discussions
and careful reading of the manuscript. J.H.K. was supported by the Polish
National Science Centre grant 2020/39/B/ST2/01279. J.H.K. would like to thank
the CP3-Origins for the extended hospitality during this work. J.H.K
acknowledges the NAWA Iwanowska scholarship PPN/IWA/2019/1/00048.

## References

  * Borissova and Eichhorn [2021] J. N. Borissova and A. Eichhorn, Universe 7, 48 (2021), arXiv:2012.08570 [gr-qc] .
  * Barrow and Tipler [1988] J. D. Barrow and F. J. Tipler, Nature 331, 31 (1988).
  * Jonas  _et al._ [2021] C. Jonas, J.-L. Lehners, and J. Quintin, Phys. Rev. D 103, 103525 (2021), arXiv:2102.05550 [hep-th] .
  * Chojnacki and Kwapisz [2021] J. Chojnacki and J. Kwapisz, Phys. Rev. D 104, 103504 (2021), arXiv:2102.13556 [gr-qc] .
  * Bosma  _et al._ [2019] L. Bosma, B. Knorr, and F. Saueressig, Phys. Rev. Lett. 123, 101301 (2019), arXiv:1904.04845 [hep-th] .
  * Hořava [2009] P. Hořava, Physical Review D 79 (2009), 10.1103/physrevd.79.084008.
  * Wang [2017] A. Wang, International Journal of Modern Physics D 26, 1730014 (2017).
  * Donoghue and Menezes [2019] J. F. Donoghue and G. Menezes, Phys. Rev. D 100, 105006 (2019), arXiv:1908.02416 [hep-th] .
  * Bemfica and Gomes [2011] F. S. Bemfica and M. Gomes, Phys. Rev. D 84, 084022 (2011), arXiv:1108.5979 [hep-th] .
  * Duplessis and Easson [2015] F. Duplessis and D. A. Easson, Physical Review D 92 (2015), 10.1103/physrevd.92.043516.
  * D’Odorico  _et al._ [2014] G. D’Odorico, F. Saueressig, and M. Schutten, Phys. Rev. Lett. 113, 171101 (2014), arXiv:1406.4366 [gr-qc] .
  * D’Odorico  _et al._ [2015] G. D’Odorico, J.-W. Goossens, and F. Saueressig, JHEP 10, 126 (2015), arXiv:1508.00590 [hep-th] .
  * Barvinsky  _et al._ [2017] A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov, and C. F. Steinwachs, Phys. Rev. Lett. 119, 211301 (2017), arXiv:1706.06809 [hep-th] .
  * Arnowitt  _et al._ [2008] R. Arnowitt, S. Deser, and C. W. Misner, General Relativity and Gravitation 40, 1997–2027 (2008).
  * Maier and Soares [2017] R. Maier and I. D. Soares, Physical Review D 96 (2017), 10.1103/physrevd.96.103532.
  * Horava [2011] P. Horava, Class. Quant. Grav. 28, 114012 (2011), arXiv:1101.1081 [hep-th] .
  * Horava and Melby-Thompson [2010] P. Horava and C. M. Melby-Thompson, Phys. Rev. D 82, 064027 (2010), arXiv:1007.2410 [hep-th] .
  * Blas  _et al._ [2011] D. Blas, O. Pujolas, and S. Sibiryakov, JHEP 04, 018 (2011), arXiv:1007.3503 [hep-th] .
  * Carloni  _et al._ [2011] S. Carloni, E. Elizalde, and P. J. Silva, Class. Quant. Grav. 28, 195002 (2011), arXiv:1009.5319 [hep-th] .
  * Morris and Thorne [1988] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
  * Einstein and Rosen [1935] A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).
  * Boonserm  _et al._ [2018] P. Boonserm, T. Ngampitipan, A. Simpson, and M. Visser, Physical Review D 98 (2018), 10.1103/physrevd.98.084048.
  * Botta Cantcheff  _et al._ [2010] M. Botta Cantcheff, N. E. Grandi, and M. Sturla, Physical Review D 82 (2010), 10.1103/physrevd.82.124034.
  * Katanaev [2014] M. O. Katanaev, Modern Physics Letters A 29, 1450090 (2014).
  * Mattingly  _et al._ [2020] B. Mattingly, A. Kar, W. Julius, M. Gorban, C. Watson, M. Ali, A. Baas, C. Elmore, B. Shakerin, E. Davis, and et al., Universe 6, 11 (2020).
  * [26] A. Wang, “private communication,” .
  * Carballo-Rubio  _et al._ [2018] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, JHEP 07, 023 (2018), arXiv:1805.02675 [gr-qc] .
  * Bonanno  _et al._ [2021] A. Bonanno, A.-P. Khosravi, and F. Saueressig, Phys. Rev. D 103, 124027 (2021), arXiv:2010.04226 [gr-qc] .
  * Bertipagani  _et al._ [2021] M. Bertipagani, M. Rinaldi, L. Sebastiani, and S. Zerbini, Phys. Dark Univ. 33, 100853 (2021), arXiv:2012.15645 [gr-qc] .
  * Lobo [2016] F. S. N. Lobo, Int. J. Mod. Phys. D 25, 1630017 (2016), arXiv:1604.02082 [gr-qc] .
  * Olmo  _et al._ [2017] G. J. Olmo, D. Rubiera-Garcia, and A. Sanchez-Puente, in  _14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories_, Vol. 2 (2017) pp. 1391–1396, arXiv:1601.00161 [gr-qc] .
  * Bambi  _et al._ [2016] C. Bambi, A. Cardenas-Avendano, G. J. Olmo, and D. Rubiera-Garcia, Phys. Rev. D 93, 064016 (2016), arXiv:1511.03755 [gr-qc] .