# Adaptivity is not helpful for Pauli channel learning

Xuan Du Trinh and Nengkun Yu
The State University of New York at Stony Brook,
Stony Brook, NY 11794

(March 18, 2024)

###### Abstract

We observe that adaptive strategies do not offer additional advantages for
learning and testing Pauli channels with entangled input. We apply this
observation to different tasks. First, the tight query complexity of learning
Pauli channels with entangled input is established for the general norm
$l_{p}$. In particular, the complexities for the $l_{1}$, $l_{2}$ and
$l_{\infty}$ norms are improved or matched compared to previous results using
entanglement in the literature. We also settle the query complexity to test if
Pauli channels are white noise sources across $l_{p}$. Additionally, we
demonstrate that the query complexity of estimating the noise level of a Pauli
channel, characterized by the entropy of its error distribution and the count
of non-zero probabilities, is $\Theta(4^{n}/n)$. Further, $\Theta(4^{n}/n)$
queries are sufficient to estimate the diamond norm between two Pauli
channels.

##  1 Introduction

Quantum noise is a fundamental concept in quantum physics, challenging our
ability to demonstrate and harness the advantages of quantum information
processing. In quantum cryptography, noise primarily influences quantum
communication tasks by generating errors, thereby reducing the secret key rate
of quantum key distribution protocols. To build large-scale and fault-tolerant
quantum computers, mitigating quantum noise and performing error correction
play crucial roles. For these reasons, modeling and evaluating quantum noise
always remain a topic of interest.

When we address the problems of quantum error correction, bit flips, and phase
flips represent the most fundamental types of errors that illustrate the
impact of noise on qubit systems. Consequently, the Pauli channel emerges as a
standard noise model. Let $\rho$ denote the density operator of an $n$-qubit
system, the Pauli channel $\mathcal{P}$ is a completely positive and trace-
preserving (CPTP) map:

$\rho\longmapsto\sum_{i\in\\{0,1,2,3\\}^{n}}P(i)\cdot\tau_{i}\rho\tau_{i}^{\dagger}$
(1)

where
$\tau_{i}=\sigma_{i_{1}}\otimes\sigma_{i_{2}}\otimes...\otimes\sigma_{i_{n}}$
is a tensor product of Pauli operators from the set $\\{\sigma_{0}$,
$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}\\}$ of Pauli matrices and $P(i)$
represents the probability of the error denoted by $\tau_{i}$. This
probability distribution characterizes the channel $\mathcal{P}$. If a quantum
state $\rho$ passes through a noisy quantum device modeled by the Pauli
channel, one can collect information about $P$ through measurements on the
output state $\mathcal{P}(\rho)$. Employing a set of known input states and
appropriately choosing measurement settings, which can be projective or
generalized, allows the collection of a sample of size $N$ after repeating the
experiment $N$ times.

One might consider a specific parameterization for $P$ while analyzing data
samples based on physical observations. In particular, various simple Pauli
channels with only a single parameter have been studied, such as the bit flip
channel, the dephasing channel and the depolarizing channel [GEZ21]. However,
in our analysis, $P$ is maintained as generally as possible. Learning and
testing a probability distribution without presupposing its form presents a
challenging and contemporary problem in statistics and learning theory. This
paper explores the complexity of various related tasks, including learning the
Pauli channel and testing some aspects of this model.

For $\epsilon,\hskip 2.84526pt\delta\in(0,1)$, we define the learning
complexity as the number of times we query the Pauli channel and output an
estimation $\hat{P}$ that is $\epsilon$-close to $P$ with error probability at
most $\delta$. To simplify the analysis, in this study, if the value of the
soundness $\delta$ is not specified, we understand that it is fixed as a
constant, namely $\delta=1/3$. In literature, the learning complexity has been
investigated in both senses: the number of the Pauli channel applications and
the number of measurements. In the findings of this paper, distinguishing
between these two notions is deemed unnecessary.

Having already established the definition of learning complexity, we need a
scheme to discriminate between the actual parameters of the quantum channel
and the distribution learned from data samples. The distance between two
probability distributions considered here will be $l_{p}$ with $p\geq 1$,
defined by

$||\hat{P}-P||_{p}=\left(\sum_{i}^{n}|\hat{P}(i)-P(i)|^{p}\right)^{\frac{1}{p}}.$
(2)

It is noteworthy that $l_{1}=\sum_{i}^{n}|\hat{P}(i)-P(i)|$ represents twice
the total variation distance, while $l_{\infty}=\max_{i}{|\hat{P}(i)-P(i)|}$
signifies the maximal difference between two distributions at a single
coordinate, and $l_{2}$ corresponds to the common Euclidean distance.

p | This study | With entanglement | Without entanglement
---|---|---|---
$p=1$ | $\Theta(4^{n}/\epsilon^{2})$ |  $\tilde{O}(4^{n}/\epsilon^{2})$[CZSJ22, FOF23] | $O(n2^{3n}/\epsilon^{2})\cite[cite]{[\@@bibref{}{flammia2020paulichannel}{}{}]}$
$p=2$ | $\Theta(1/\epsilon^{2})$ | $None$ | $O(n2^{n}/\epsilon^{2})\cite[cite]{[\@@bibref{}{flammia2020paulichannel}{}{}]}$
$p=\infty$ | $\Theta(1/\epsilon^{2})$ |  $\Theta(1/\epsilon^{2})\cite[cite]{[\@@bibref{}{flammia2021pauli_populationrecovery}{}{}]}$111Flammia and O’Donnell connected the problem of learning $\mathcal{P}$ to the estimation of the discrete distribution $P$ in [FO21], citing the classical sample complexity of $\Theta(1/\epsilon^{2})$ for learning $P$ as reported in [Can20]. However, they did not provide a proven lower bound for the quantum query complexity. |  $\Omega(1/\epsilon^{2}),\hskip 2.84526ptO(1/\epsilon^{2})\log(n/\epsilon)$[FO21]
Table 1: Comparison of learning complexities obtained in Corollary 12 with
existing sample complexity bounds in literature for different common values of
$p$. The two last columns list the results previously obtained with and
without entanglement resources in terms of the number of measurement rounds or
the number of Pauli channel applications.

0^k & [wires=2]A [wires=2]A …….[wires=2]A [wires=2]A
0^n P P ……. ……. P

Figure 1: Adaptive strategy for the sampling task: The main $n$-qubit system
and a possible ancilla of $k$ qubits are initialized respectively in the state
$\ket{0^{n}}$ and $\ket{0^{k}}$. The operation $A$ denotes an adaptive
procedure comprising quantum measurements and state transformations based on
measurement outcomes in any possible order. The gate $\mathcal{P}$ processes
the main system in order to extract information from $P$. The circuit outputs
a sample from $P$.

Previous research has yielded remarkable results in estimating Pauli channels
under various conditions, including scenarios with and without entanglement
and employing different metrics for evaluation. In the entanglement-free
scheme, Flammia and Wallman introduced an algorithm requiring
$O(n2^{n}/\epsilon^{2})$ measurements to estimate Pauli channels in norm
$l_{2}$ in [FW20]. Recently, Fawzi et al. refined this algorithm and
established an upper bound of $O(n2^{3n}/\epsilon^{2})$ measurements in the
context of norm $l_{1}$ [FOF23]. While focusing on the $l_{\infty}$ norm and
without-entanglement condition, Flammia and O’Donnell related the learning of
Pauli channels to the Population Recovery problem. They provided an algorithm
that uses the channel $O(1/\epsilon^{2})\log(n/\epsilon)$ times [FO21]. In
another study in the same context, Chen et al. demonstrated that
$\Theta(2^{n}/\epsilon^{2})$ measurements are required for learning each
eigenvalue $P(i)$ [COZ+23]. Earlier, Chen et al. showed that entanglement
provides an exponential advantage, with an algorithm that applies the channel
$O(n/\epsilon^{2})$ times [CZSJ22], leading to an upper bound of
$O(4^{n}/\epsilon^{2})$ in the $l_{1}$ distance via the Parseval–Plancherel
identity [FOF23]. To the learning of the Pauli channel, Flammia and O’Donnell
highlighted that $\Theta(1/\epsilon^{2})$ samples of $P$ are both necessary
and sufficient for classically learning $P$ in the $l_{\infty}$ norm. They
further stated that Bell measurements can obtain these samples while ancilla,
i.e., entanglement, is available [FO21]. These studies, alongside others
[WE16, CLO+23, RF23], highlight the complexity and diversity of techniques in
quantum channel discrimination. Motivated by this diversity, our paper aims to
establish tight bounds for the learning problem across any $l_{p}$ norm, where
$p\geq 1$.

In addition to quantum channel discrimination, adaptive quantum learning
algorithms play a significant role. These refer to the strategy of dynamically
adjusting the choice of quantum measurements and input states based on the
outcomes of previous measurements (Figure 1). This approach contrasts with
non-adaptive strategies, where the measurement bases are predetermined before
experimentation. Notably, the algorithms in [FW20] employ adaptivity as a
crucial component. Subsequently, Fawzi et al. in their recent work explored
lower bounds for adaptive and non-adaptive strategies in the context of
$l_{1}$ and $l_{\infty}$ and specifically in scenarios devoid of entanglement
resources [FOF23]. Their results show gaps between the aforementioned
algorithms’ lower and upper bounds. This raises an essential question
regarding the actual efficacy of adaptivity in enhancing the learning process.

We present the main theorems of our study, addressing the utility of
adaptivity in the learning process of the Pauli channel when entanglement
resources and ancillary registers are accessible. This theorem facilitates the
exploration of additional aspects of Pauli channels.

######  Theorem 1.

Given a Pauli channel $\mathcal{P}$, the associated random variable $P$, and
the capability to prepare maximally entangled two-qubit states, it is
established that the behaviors of $\mathcal{P}$ and $P$ exhibit a bisimulation
relationship. Specifically, the following statements hold

  * •

The Pauli channel $\mathcal{P}$ can be precisely simulated utilizing the
random variable $P$.

  * •

Obtaining each sample from $P$ necessitates exactly one application of the
channel $\mathcal{P}$.

This theorem gives an insight that, in terms of information, accessing the
channel $\mathcal{P}$ is equivalent to accessing the random variable $P$. The
intuition behind the first point is straightforward since $P$ effectively
characterizes $\mathcal{P}$, and the second point will be demonstrated by
establishing a lower bound and introducing a sampling algorithm. The
equivalence signifies that tasks such as learning or estimating properties of
$\mathcal{P}$ can be efficiently executed by sampling from the random variable
$P$ and then applying a classical processing algorithm. Another important
implication here is that if the sample complexity for a task is $\mathcal{C}$,
then the query complexity on $\mathcal{P}$ directly aligns with $\mathcal{C}$.
For instance, achieving an $\epsilon$-close approximation in learning the
distribution $P$ within the $l_{p}$ norm requires
$\Theta(poly(\frac{1}{\epsilon},2^{n}))$ samples, then an equal number of
queries to the channel $\mathcal{P}$ is required. The upper and lower bounds
are all translated from sample complexity to query complexity.

The proof of Theorem 1 is presented in Section 2. Subsequent sections will
discuss its applications in learning and testing problems, providing
corollaries of this result while referencing earlier groundbreaking studies.

######  Theorem 2.

In estimating any property of $\mathcal{P}$ using samples drawn from the
distribution $P$, adaptivity does not improve the efficiency compared to non-
adaptive strategies.

Proof. Considering the adaptive strategy shown in Figure 1 in the context of
Theorem 1, a redundancy is evident. The theorem indicates that just one
application of the gate $\mathcal{P}$ suffices to produce a sample from the
distribution $P$. After this initial use, the entire system may undergo some
possible quantum operations. Subsequently, it is subjected to quantum
measurement. Afterward, no further quantum operations are necessary. Anything
done at that step to obtain the sample falls into classical processing. The
possible quantum operations before the measurement cannot be adaptive because
no measurement outcome was available. Hence, the sample is obtained without
adaptivity. In summary, given the capability to prepare a maximally entangled
state between qubit pairs, adaptivity offers no advantage over the non-
adaptive approach in this scenario $\square$.

p value |  Regime |  Necessary |  Sufficient
---|---|---|---
$1\leq p\leq 2$ |  $1>\epsilon\geq 2^{-\frac{2n(p-1)}{p}}$ |  $\epsilon^{-\frac{p}{2(p-1)}}$ |  $\epsilon^{-\frac{p}{2(p-1)}}$
$\epsilon<2^{-\frac{2n(p-1)}{p}}$ |  $2^{\frac{n(4-3p)}{p}}\epsilon^{-2}$ |  $2^{\frac{n(4-3p)}{p}}\epsilon^{-2}$
$2<p<\infty$ |  $\Theta\left(\frac{n}{4^{n}}\right)\geq\epsilon$ |  $n4^{-n}\epsilon^{-2}$ |  $2^{-n}\epsilon^{-2}$
$\Theta\left(\frac{n}{4^{n}}\right)<\epsilon\leq 2^{-n}$ |  $\epsilon^{-1}$ |  $2^{-n}\epsilon^{-2}$
$\epsilon>2^{-n}$ |  $\epsilon^{-1}$ |  $\epsilon^{-1}$
$p=\infty$ |  $\Theta\left(\frac{n}{4^{n}}\right)\geq\epsilon$ |  $n4^{-n}\epsilon^{-2}$ |  $n4^{-n}\epsilon^{-2}$
$\Theta\left(\frac{n}{4^{n}}\right)<\epsilon$ |  $\epsilon^{-1}$ |  $\epsilon^{-1}$
Table 2: Uniformity testing sample complexity bounds for different values of
$p$.

##  2 Sampling with Bell state measurement

In the theory of quantum operations, Choi’s isomorphism offers a
representation of quantum channels. In this representation, the channel
$\mathcal{P}$ acting on the $n$-qubit system $A$ is fully characterized by the
quantum state

$\Lambda=(\mathbb{1}\otimes\mathcal{P})(\ket{\Phi_{n}}\bra{\Phi_{n}})$ (3)

where $\ket{\Phi_{n}}=\frac{1}{\sqrt{2^{n}}}\sum_{i=0}^{2^{n}-1}\ket{ii}$ is
the maximally entangled state in the extension $A\otimes A^{\prime}$ with
$A^{\prime}$ isomorphic to $A$. The fact that $\Lambda$ encodes all
information about $\mathcal{P}$ implies that extracting information from
$\Lambda$ is equivalent to "measuring" the quantum channel. It suggests a
method for detecting Pauli errors using entangled states. Nevertheless,
entangled measurement on the total system $A\otimes A^{\prime}$ is generally
consuming. So, we will employ $n$ Bell measurements on $n$ pairs of qubits.

A simple example will give an intuition of our approach. We start with the
preparation of a pair of qubits in a maximally entangled state
$\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$. One of the two qubits
is then subjected to a Pauli channel $\mathcal{P}$. Subsequently, a Bell
measurement is performed on the qubits (Figure 2).

$\ket{0}$ & H 1 1 H
$\ket{0}$ P

Figure 2: The first Hadamard and CNOT gates create a maximally entangled
state. Then, the second qubit is subject to a single-qubit Pauli channel. The
second CNOT and Hadamard gates help to project Bell measurement to the
measurement in the computational basis.

Effect of Pauli Errors: If a Pauli gate error occurs on one qubit among an EPR
pair, it would transform the initial entangled state $\ket{\Phi^{+}}$ into
another Bell state. To summarize, the measurement outcomes in a Bell
measurement provide information about the types of Pauli errors that may have
affected the qubits:

  * •

$\ket{\sigma_{0}}=\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})=(\mathbb{1}_{2}\otimes\sigma_{0})\ket{\Phi^{+}}$
outcome indicates no errors ($\sigma_{0}=\mathbb{1}_{2}$).

  * •

$\ket{\sigma_{1}}=\ket{\Psi^{+}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})=(\mathbb{1}_{2}\otimes\sigma_{1})\ket{\Phi^{+}}$
outcome indicates a Pauli-X ($\sigma_{1}$) error.

  * •

$\ket{\sigma_{2}}=\ket{\Psi^{-}}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})=(\mathbb{1}_{2}\otimes\sigma_{2})\ket{\Phi^{+}}$
outcome indicates a Pauli-Y ($\sigma_{2}$) error.

  * •

$\ket{\sigma_{3}}=\ket{\Phi^{-}}=\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})=(\mathbb{1}_{2}\otimes\sigma_{3})\ket{\Phi^{+}}$
outcome indicates a Pauli-Z ($\sigma_{3}$) error.

It is quite interesting that the summary remains true regardless of which
qubit in each pair is subjected to Pauli errors. Such a strategy can be
generalized to measure the $n$-qubit quantum channel $\mathcal{P}$ using $n$
pairs of maximally entangled qubits for $n\geq 2$ thanks to the fact that

$\ket{\Phi_{n}}=\ket{\Phi^{+}}^{\otimes n}.$ (4)

From each pair, we pick one qubit to form a set. Then, we have two sets of $n$
qubits. Subsequently, one set passes through the channel $\mathcal{P}$. Bell
measurements detect errors on each qubit on its corresponding pairs. For the
simplest case $n=2$, the circuit in Figure 3 shows how the method is employed.

0 & H 1 1 H
0 [wires=2]P
0 H 1 1 H
0

Figure 3: The Pauli channel is applied on the second and third qubits. Then,
the errors on these two qubits are detected by Bell measurements on each
initial pair.

Intuitively, if the tuple of errors
$(\sigma_{i_{1}},\sigma_{i_{2}},...,\sigma_{i_{n}})$ is the output of the
measurements on $n$ qubit pairs, we get a sample of the distribution $P$ with
the value $i=(i_{1},i_{2},...,i_{n})\in\\{0,1,2,3\\}^{n}$. Then, one
measurement experiment exploiting one use of $\mathcal{P}$ results in one
sample, providing an upper bound for the query complexity of the sampling
task. To make the upper bound clear, the formal proof is given.

######  Lemma 1.

Resulting from the Bell measurements on $n$ pairs of qubits by the strategy
above, the tuple of errors
$(\sigma_{i_{1}},\sigma_{i_{2}},...,\sigma_{i_{n}})$ constitutes a sample
corresponding to the value $i=(i_{1},i_{2},...,i_{n})\in\\{0,1,2,3\\}^{n}$
from the distribution $P$.

Proof. We denote the post-measurement state corresponding to the tuple of
error $\tau_{i}=(\sigma_{i_{1}},\sigma_{i_{2}},...,\sigma_{i_{n}})$ by
$\ket{\tau_{i}}=\ket{\sigma_{i_{1}}}\ket{\sigma_{i_{2}}}...\ket{\sigma_{i_{n}}}$.
Considering the image of the Pauli channel under Choi isomorphism with the
remark that
$\ket{\sigma_{i_{k}}}=(\mathbb{1}_{2}\otimes\sigma_{i_{k}})\ket{\Phi^{+}}$ and
$\ket{\Phi_{n}}\bra{\Phi_{n}}=(\ket{\Phi^{+}}\bra{\Phi^{+}})^{\otimes n}$, we
have

$\displaystyle(\mathbb{1}_{A}\otimes\mathcal{P})(\ket{\Phi_{n}}\bra{\Phi_{n}})$
$\displaystyle=\sum_{i\in\\{0,1,2,3\\}}P(i)\cdot(\mathbb{1}_{A}\otimes\tau_{i})\ket{\Phi_{n}}\bra{\Phi_{n}}(\mathbb{1}_{A}\otimes\tau_{i}^{\dagger})$
(5)
$\displaystyle=\sum_{i\in\\{0,1,2,3\\}}P(i)\cdot\bigotimes_{k=1}^{n}(\mathbb{1}_{2}\otimes\sigma_{i_{k}})\ket{\Phi^{+}}\bra{\Phi^{+}}(\mathbb{1}_{2}\otimes\sigma^{\dagger}_{i_{k}})$
(6)
$\displaystyle=\sum_{i\in\\{0,1,2,3\\}}P(i)\cdot\bigotimes_{k=1}^{n}\ket{\sigma_{i_{k}}}\bra{\sigma_{i_{k}}}$
(7) $\displaystyle=\sum_{i\in\\{0,1,2,3\\}}P(i)\ket{\tau_{i}}\bra{\tau_{i}}.$
(8)

So with probability $P(i)$, the Bell measurements on $n$ pairs of qubits
detect the tuple error $\tau_{i}$ $\square$.

Proof of Theorem 1. We start from the first point: the Pauli channel
$\mathcal{P}$ can be precisely simulated utilizing the random variable $P$.
Consider a quantum system consisting of $n$ qubits initially in the state
$\rho$, on which we aim to simulate the action of the Pauli channel
$\mathcal{P}$, given access to the random variable $P$. This process is
straightforward. A sample of $P$, denoted by $i=({i_{1}},{i_{2}},...,{i_{n}})$
is taken automatically by a classical computer and then the Pauli operation
$\tau_{i}=(\sigma_{i_{1}},\sigma_{i_{2}},...,\sigma_{i_{n}})$ is applied to
the $n$ qubits. The outcome will be the quantum state
$\tau_{i}\rho\tau_{i}^{\dagger}$, with probability $P(i)$, i.e. the quantum
state is presented by the density operator

$\mathcal{P}(\rho)=\sum_{i\in\\{0,1,2,3\\}^{n}}P(i)\cdot\tau_{i}\rho\tau_{i}^{\dagger}.$
(9)

We now consider the second point: obtaining each sample from $P$ necessitates
exactly one application of the channel $\mathcal{P}$. The lower bound is
evident because, given the Pauli channel $\mathcal{P}$, to draw one sample
from the distribution $P$, the quantum circuit must query $\mathcal{P}$ at
least once. Lemma 1 confirms that one query is enough. The upper bound and the
lower bound match. Therefore, Theorem 1 is proven $\square$.

##  3 Learning and testing Pauli channels

Revisiting the exploration of learning and testing discrete distributions
based on $l_{p}$ norms reveals a longstanding research interest. Waggoner has
solidified the definitive bounds for the complexities of learning and
uniformity testing [Wag15]. Leveraging his main results allows us to infer the
learning complexity and the white noise testing for Pauli channels,
determining if $P$ represents a completely random variable of size $2^{n}$.
The logic is very simple. As discussed above, Theorem 1 identifies the number
of samples required to estimate any properties of $P$ with the necessary
number of queries to $\mathcal{P}$. Corollaries 12 and 2 are the consequences
of the main theorem in [Wag15] and the translation between the two kinds of
complexities.

###  3.1 Learning algorithm

We now consider the classical processing part of the learning channel
$\mathcal{P}$. The algorithm proposed in the reference [Wag15] is employed to
learn the distribution $P$. The procedure for drawing samples has been
described in the previous section.

Algorithm 1 Learning the discrete distribution $P$

1:Input: channel  $\mathcal{P}$; number of qubits $n$ ; distance parameter
$p\in[1,\infty]$ and $\epsilon,\delta\in(0,1)$

2:Choose $N$ to be sufficient for $p$, $n$, $\varepsilon$, $\delta$ according
to proven upper bounds in Lemma 10.

3:Draw $N$ samples.

4:Let $X_{i}$ be the number of samples drawn of each coordinate
$i\in\\{0,1,2,3\\}^{n}$.

5:Compute each $\hat{P}_{i}=\frac{X_{i}}{N}$.

6:Output $\hat{P}$.

We have the exact sufficient numbers of samples to learn the discrete
distribution $P$ using Algorithm 1 according to Theorem 5.2 in [Wag15],
applied in the context of distribution size of $4^{n}$, as written in Lemma
10. In addition, Theorem 5.4 in [Wag15] shows the lower bounds up to constant
factors of Algorithm 1, presented in Lemma 11.

######  Lemma 2 (Exact upper bounds of learning sample).

For learning $P$ with error $\epsilon$ in distance $l_{p}$ and constant
success probability $1-\delta$, it suffices to run Algorithm 1 with the
following number of samples

$N_{O}=\frac{1}{\delta}\begin{cases}4^{\frac{n(2-p)}{p}}\epsilon^{-2}&\text{if
}1\leq p\leq 2\hskip 2.84544pt\text{ and }\epsilon\leq 2\cdot
4^{-\frac{n(p-1)}{p}}\\\
\frac{1}{4}\left(\frac{2}{\epsilon}\right)^{\frac{p}{p-1}}&\text{if }1\leq
p\leq 2\hskip 2.84544pt\text{ and }\epsilon\geq 2\cdot
4^{-\frac{n(p-1)}{p}}\\\ \epsilon^{-2}&\text{if }2\leq
p\leq\infty.\end{cases}$ (10)

######  Lemma 3.

For learning $P$ with error $\epsilon$ in distance $l_{p}$ and a constant
success probability $1-\delta$, the required number of samples is at least

$N_{\Omega}=\begin{cases}\Omega\left(4^{\frac{n(2-p)}{p}}\epsilon^{-2}\right)&\text{if
}1\leq p\leq 2\text{ and }\epsilon\leq 4^{-\frac{n(p-1)}{p}}\\\
\Omega\left((\frac{1}{\epsilon})^{\frac{p}{p-1}}\right)&\text{if }1\leq p\leq
2\text{ and }\epsilon\geq 4^{-\frac{n(p-1)}{p}}\\\
\Omega\left({\epsilon^{-2}}\right)&\text{if }2\leq p\leq\infty\par\end{cases}$
(11)

The lower bounds and upper bounds in Lemma 10 and 11 match up to constant
factors. We conclude the tight bounds for learning Pauli channel
$\mathcal{P}$.

######  Corollary 1.

Let $\epsilon\in(0,1)$ and $\mathcal{P}$ be a Pauli channel of $n$ qubits.
Learning $\mathcal{P}$ to $\epsilon$-close in distance $l_{p}$ with a constant
success probability $1-\delta$ requires number $N$ of Pauli channel
applications up to constant factors depending on $\delta$ and $p$ as

$N=\begin{cases}\Theta(4^{\frac{n(2-p)}{p}}\epsilon^{-2})&\textit{ for }1\leq
p\leq 2\textit{ and }\epsilon\leq 4^{-\frac{n(p-1)}{p}}\\\
\Theta(4^{n})&\textit{ for }1\leq p\leq 2\textit{ and
}4^{-\frac{n(p-1)}{p}}\leq\epsilon\leq 2\cdot 4^{-\frac{n(p-1)}{p}}\\\
\Theta(\left(\frac{1}{\epsilon}\right)^{\frac{p}{p-1}})\hskip
31.2982pt&\textit{ for }1\leq p\leq 2\textit{ and }\epsilon\geq 2\cdot
4^{-\frac{n(p-1)}{p}}\\\ \Theta(\epsilon^{-2})&\textit{ for }2\leq
p\leq\infty\end{cases}$ (12)

Proof. The translation from distribution sample complexities to channel query
complexities is automatic. Except the tightly bound $\Theta(4^{n})$ in the
regime where $1\leq p\leq 2$ and $4^{-\frac{n(p-1)}{p}}\leq\epsilon\leq 2\cdot
4^{-\frac{n(p-1)}{p}}$, the derivation of the other values of $N$ directly
follows from the matching, up to constant factors, between the bounds in Lemma
10 and 11. Considering this distinct regime, given the fact that
$\epsilon=\Theta(4^{-\frac{n(p-1)}{p}})$, the lower bound and the upper bound
converge to the same order, i.e.
$(\frac{1}{\epsilon})^{\frac{p}{p-1}}=\Theta(4^{n})$ and
$4^{\frac{n(2-p)}{p}}\epsilon^{-2}=\Theta(4^{n})$ $\square$.

At the critical value $p=2$, all the tight bounds in the corollary converge to
$\Theta(1/\epsilon^{2})$. Interestingly, for all $p\geq 2$, and specifically
for the $l_{\infty}$ norm, the required number of channel applications remains
the same at $\Theta(1/\epsilon^{2})$. When $p=1$, we consistently find
ourselves within the regime $\epsilon\leq 2^{-\frac{2n(p-1)}{p}}=1$, leading
to $N=\Theta(4^{n}/\epsilon^{2})$. Remarkably, the complexities presented in
the corollary are optimal up to only constant factors, effectively covering
all possible scenarios for general $l_{p}$ norms.

The algorithm employing maximally entangled qubit pairs and Bell measurements
demonstrates significant quantum advantages in examining the comparisons in
Table 1. For the $l_{\infty}$ distance, it refines strategies from previous
entanglement-free settings, reducing their complexities from
$\Theta(2^{n}/\epsilon^{2})$ in [COZ+23] and
$O(1/\epsilon^{2})\log(n/\epsilon)$ in [FO21] to $\Theta(1/\epsilon^{2})$. The
bounds for the case of learning in the distance $l_{1}$ also benefit from
quantum advantage, improved from $O(n2^{3n}/\epsilon^{2})$ to
$\Theta(4^{n}/\epsilon^{2})$, and tighter than the previous one using
entanglement $\tilde{O}(4^{n}/\epsilon^{2})$. Moreover, for the $l_{2}$
distance, the algorithm reduces the upper bound from $O(n2^{n}/\epsilon^{2})$
to merely $\Theta(1/\epsilon^{2})$, eliminating the dependency on the number
of qubits. Thus, this part of the study’s principal contribution is
establishing tight bounds for learning Pauli channels with minimal reliance on
the Pauli channel across any $l_{p}$ distance. Furthermore, the effectiveness
of our non-adaptive algorithm further confirms that adaptivity does not
provide any additional benefit in learning the Pauli channel when entanglement
resources are deployed.

###  3.2 White noise testing

We now consider the problem of testing if the Pauli channel is a white noise
source, i.e., $P(i)=U(i)=1/4^{n}$ for all $i\in\\{0,1,2,3\\}^{n}$. Thanks to
Theorem 1, it is equivalent to testing the uniformity of the discrete
probability distribution. Regarding the general figure of merit $l_{p}$ and
the distribution $P$, we accept a constant error $\epsilon$ in the distance
and the error probability $\delta$. If $||P-U||_{p}<\epsilon$, the
distribution is concluded to be uniform; otherwise, we say that it is not
uniform. We combine the results on sample complexity of uniformity testing
from the work in [Wag15] with Theorem 1 to infer Corollary 2.

######  Corollary 2.

Given $\epsilon\in(0,1)$ and the discrete distribution $P$ that characterizes
$\mathcal{P}$, a Pauli channel of $n$ qubits. Up to constant factors, Table 2
shows the necessary and sufficient number of Pauli channel applications for
testing, whether $P$ is $\epsilon$-close to the uniform distribution in the
distance $l_{p}$ with a constant success probability $1-\delta$.

###  3.3 Estimating noise level

The fundamental measure of uncertainty in a probability distribution is
entropy. A high value of entropy indicates a greater level of
unpredictability. We will focus on Shannon entropy, denoted by $H$,
quantifying the expected amount of information in a random variable. In the
context of the Pauli channel, the Shannon entropy of the distribution $P$ over
the domain $\\{0,1,2,3\\}^{n}$ ranges from 0 to $2n$ bits, providing insights
into the diversity and randomness of errors in the $n$-qubit system. This
leads to a characterization of the noise level in the Pauli channel. However,
a challenge arises: even if we can estimate the probability of errors in an
arbitrary $l_{p}$ distance with $l\geq 1$ up to $\epsilon$-closeness, the
resulting estimate $\hat{P}$ may not provide an accurate estimation of
entropy, that is, $|H(P)-H(\hat{P})|$ could significantly exceed $1$. This
discussion aims to design an optimal algorithm for estimating $H(P)$.

Another aspect of the Pauli channel is that it can be sparse, meaning not all
possible error $\tau_{i}\in\\{0,1,2,3\\}^{n}$ occur with non-zero probability,
and the errors might be centralized on a particular subpart of the qubit
systems or be specific to certain types among bit flips or phase flips. Given
the exponential growth of the distribution’s size, i.e., $4^{n}$, collecting
sufficiently large samples to observe sparsity requires substantial resources.
However, if we could estimate the actual support size $S(P)$ of the
distribution $P$ with a relatively small sample, it would provide an insight
into the nature of the noise, in addition to the Shannon entropy $H(P)$
estimation. From now on, we call the entropy and the support size the two
measures of the Pauli channel’s noise level.

We learned about remarkable research by Valiant et al., as referenced in
[VV11, VV17], focusing on estimating the unseen aspects and characteristics of
probability distributions with only a limited sample size. We employ the
algorithm “Estimate Unseen” in [VV17], adapt it to the context of the
distribution $P$, and state the result in Lemma 4. For an in-depth
understanding of the algorithm, we encourage the readers to delve into these
papers directly. This algorithm plays the role of classical information
processing for testing the noise level of $\mathcal{P}$. The Bell measurement
strategy is again used to draw samples from $P$, leading to Corollay 3 in the
same logic of complexity translation offered by Theorem 1.

######  Lemma 4.

There exists an algorithm (estimator) satisfying the following property:
$\exists$ $\alpha$, $\gamma>0$ so that $\forall\epsilon\in(0,1)$, taking a
sample of size $\frac{\gamma}{\epsilon^{2}}\frac{4^{n}}{n}$ drawn from the
distribution $P$ as input, the algorithm outputs with probability at least
$1-e^{-4^{n\alpha}}$

  * •

an estimation $\hat{H}(P)$ of $H(P)$ satisfying $|\hat{H}(P)-H(P)|<\epsilon$
and

  * •

an estimation $\hat{S}(P)$ of $S(P)$ satisfying
$\frac{|\hat{S}(P)-S(P)|}{4^{n}}<\epsilon$ if we are provided that no value
$P(i)$ lies in the interval $(0,\frac{1}{4^{n}})$.

Furthermore, the sample complexity of the algorithm is optimal up to constant
factors, i.e. $\Theta(\frac{4^{n}}{n\epsilon^{2}})$ is the necessary and
sufficient size of the sample to estimate Shannon entropy and the support size
of $P$ with provided condition $P(i)\notin(0,\frac{1}{4^{n}})\forall i$.

######  Corollary 3.

(Informal) To estimate the noise level of the Pauli channel $\mathcal{P}$, the
necessary and sufficient number of Pauli channel applications is
$\Theta(\frac{4^{n}}{n\epsilon^{2}})$.

###  3.4 Estimating diamond distance between two Pauli channels

In this subsection, we explore distinguishing between two Pauli channels by
employing the diamond norm. We begin by considering a system composed of $n$
qubits, represented by the Hilbert space $A$, alongside an ancillary system
characterized by a finite-dimensional Hilbert space $B$. We represent the
general channel of interest, $\mathcal{P}$, as a CPTP map acting on
$\mathcal{L}(A)$, which is the space of all linear homomorphisms on $A$.
Concurrently, the map $\mathcal{I}_{d}$ signifies the identity operation on
$\mathcal{L}(B)$.

For a density matrix $\chi$ that operates on the joint space $A\otimes B$, the
diamond norm of the CPTP map $\mathcal{P}$ is rigorously defined as

$\|\mathcal{P}\|_{\diamond}=\max_{\chi}\|(\mathcal{I}_{d}\otimes\mathcal{P})\chi\|_{1}$
(13)

where $\|.\|_{1}$ denotes the trace norm for operators on
$\mathcal{L}(A\otimes B)$. This measure effectively quantifies the maximum
deviation between the actions of $\mathcal{P}$ and an ideal channel under the
most extreme cases, such as of quantum entanglement between the systems $A$
and $B$. Based on this norm, the diamond distance between two CPTP maps
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ is introduced as

$\|\mathcal{P}_{1}-\mathcal{P}_{2}\|_{\diamond}=\max_{\chi}\|\left[\mathcal{I}_{d}\otimes(\mathcal{P}_{1}-\mathcal{P}_{2})\right]\chi\|_{1}.$
(14)

This definition of distance is helpful as it captures the difference between
two quantum channels even when they act on part of an arbitrarily larger
system. This fact is particularly relevant in evaluating the accuracy of
quantum operations, where small differences between ideal and actual
operations may significantly impact the outcomes of quantum circuits. Hence,
the diamond distance is a crucial metric for verifying the precision and
reliability of quantum information processing.

In the context of Pauli channels, where two CPTP maps $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ are characterized by the probability distributions $P_{1}$
and $P_{2}$, respectively, an insightful observation is that the diamond
distance between these two maps is equal to twice the total variation distance
(or the $l_{1}$ distance) between $P_{1}$ and $P_{2}$. Mathematically, this
relationship is expressed as

$\|\mathcal{P}_{1}-\mathcal{P}_{2}\|_{\diamond}=\sum_{i\in\\{0,1,2,3\\}^{n}}|P_{1}(i)-P_{2}(i)|=\|P_{1}-P_{2}\|_{1}.$
(15)

According to Theorem 1, achieving an $\epsilon$-close approximation in $l_{1}$
distance for $P_{1}$ and $P_{2}$, yielding estimates $\hat{P}_{1}$ and
$\hat{P}_{2}$, necessitates a complexity of $\Theta(4^{n}/\epsilon^{2})$. This
finding suggests that to approximate their diamond distance,
$O(4^{n}/\epsilon^{2})$ uses of the Pauli channels are sufficient, as
demonstrated by the inequalities

$|l_{1}(P_{1},P_{2})-l_{1}(\hat{P}_{1},\hat{P}_{2})|\leq
l_{1}(P_{1},\hat{P}_{1})+l_{1}(P_{2},\hat{P}_{2})<2\epsilon.$ (16)

However, the question arises regarding the optimality of this approach. By
exploiting the work of Valiant et al. in [VV11, VV17] and our Bell measurement
strategy, we find that the actual number of applications of the Pauli channels
needed to estimate $\|\mathcal{P}_{1}-\mathcal{P}_{2}\|_{\diamond}$
effectively reduces to $\Theta(\frac{4^{n}}{n\epsilon^{2}})$, indicating a
more efficient estimation process. We apply Theorem 1.13 of the reference
[VV17] to state Lemma 17. In the same manner of complexity translation, this
lemma directly leads to Theorem 4.

######  Lemma 5.

There exists an algorithm (estimator) satisfying the following property:
$\exists$ $\alpha$, $\gamma>0$ so that $\forall\epsilon\in(0,1)$, taking two
samples of size $\frac{\gamma}{\epsilon^{2}}\frac{4^{n}}{n}$ drawn
independently from the distribution $P_{1}$ and $P_{2}$ as input, the
algorithm outputs with probability at least $1-e^{-4^{n\alpha}}$ an estimate
$\hat{d}$ of $l_{1}(P_{1},P_{2})=||P_{1}-P_{2}||_{1}$ such that

$|\hat{d}-l_{1}(P_{1},P_{2})|<\epsilon.$ (17)

######  Corollary 4.

To estimate the diamond distance between the two Pauli channels
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, the necessary and sufficient number
of applications of each channel is $\Theta(\frac{4^{n}}{n\epsilon^{2}})$.

##  4 Conclusion

Our work has addressed the complexities involved in learning and testing Pauli
channels, introducing lower and upper bounds for these processes across any
$l_{p}$ norm; most are tight. By referencing foundational work on discrete
probability distributions and employing novel algorithms, such as “Estimate
Unseen”, we have brought a deeper understanding of quantum noise
characterization within Pauli channels. Our exploration highlights the cost of
accurately estimating Shannon entropy and support size, which are critical
measures of the noise level in error distributions. Furthermore, our findings
show that the utility of adaptivity in learning algorithms is unnecessary,
provided that maximally entangled qubit pairs and Bell measurements are
feasible. This work contributes to the theoretical framework for quantum
information processing and offers practical insights for developing more
efficient quantum computing systems. Future research may explore extending
these methodologies to other quantum channel models, potentially opening new
avenues for quantum error correction and noise mitigation strategies.

## References

  * [Can20] Clément L. Canonne.  A short note on learning discrete distributions.  arXiv preprint arXiv:2002.11457, 2020.
  * [CLO+23] S. Chen, Y. Liu, M. Otten, et al.  The learnability of pauli noise.  Nature Communications, 14:52, 2023.  Received: 4 July 2022; Accepted: 23 December 2022; Published: 4 January 2023.
  * [COZ+23] Senrui Chen, Changhun Oh, Sisi Zhou, Hsin-Yuan Huang, and Liang Jiang.  Tight bounds on pauli channel learning without entanglement.  https://arxiv.org/abs/2309.13461, 2023.
  * [CZSJ22] Senrui Chen, Sisi Zhou, Alireza Seif, and Liang Jiang.  Quantum advantages for pauli channel estimation.  Phys. Rev. A, 105:032435, Mar 2022.
  * [FO21] Steven T. Flammia and Ryan O’Donnell.  Pauli error estimation via population recovery.  https://arxiv.org/abs/2105.02885, 2021.
  * [FOF23] Omar Fawzi, Aadil Oufkir, and Daniel Stilck Franc¸a.  Lower bounds on learning pauli channels.  arXiv preprint arXiv:2301.09192, 2023.
  * [FW20] Steven T. Flammia and Joel J. Wallman.  Efficient estimation of pauli channels.  arXiv preprint arXiv:1907.12976, 2020.
  * [GEZ21] Konstantinos Georgopoulos, Clive Emary, and Paolo Zuliani.  Modeling and simulating the noisy behavior of near-term quantum computers.  Phys. Rev. A, 104:062432, Dec 2021.
  * [RF23] Cambyse Rouzé and Daniel Stilck França França.  Efficient learning of the structure and parameters of local pauli noise channels.  arXiv preprint arXiv:2307.02959, 2023.
  * [VV11] G. Valiant and P. Valiant.  Estimating the unseen: An n/log(n)-sample estimator for entropy and support size, shown optimal via new clts.  In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC ’11, pages 685–694, 2011.
  * [VV17] G. Valiant and P. Valiant.  Estimating the unseen: Improved estimators for entropy and other properties.  Journal of the ACM, 64(6), 2017.
  * [Wag15] Bo Waggoner.  $l_{p}$ testing and learning of discrete distributions.  https://arxiv.org/abs/1412.2314, 2015.
  * [WE16] Joel J. Wallman and Joseph Emerson.  Noise tailoring for scalable quantum computation via randomized compiling.  Phys. Rev. A, 94:052325, Nov 2016.


# Foliated structure of weak nearly Sasakian manifolds

Vladimir Rovenski111Department of Mathematics, University of Haifa, 3498838
Haifa, Israel.
e-mail<EMAIL_ADDRESS>. ORCID: 0000-0003-0591-8307

###### Abstract

Weak contact metric manifolds, i.e., the linear complex structure on the
contact distribution is replaced by a nonsingular skew-symmetric tensor,
defined by the author and R. Wolak, allowed a new look at the theory of
contact manifolds. In this paper we study the new structure of this type,
called the weak nearly Sasakian structure. We find conditions that are
automatically satisfied by almost contact manifolds and under which the
contact distribution is curvature invariant and the weak nearly Sasakian
structure foliates into two types of totally geodesic foliations. Our main
result generalizes the theorem by B. Cappelletti-Montano and G. Dileo (2016)
about the foliated structure of nearly Sasakian manifolds.

Keywords: weak nearly Sasakian manifold, Killing vector field, Riemannian
curvature, totally geodesic foliation.

Mathematics Subject Classifications (2010) 53C15, 53C25, 53D15

##  1 Introduction

Riemannian contact geometry plays an important role in both modern mathematics
and mechanics in explaining physical processes. A popular class of almost
contact metric manifolds $M^{\,2n+1}(\varphi,\xi,\eta,g)$ consists of Sasakian
manifolds characterized by the equality, see [2],

$(\nabla_{X}\,\varphi)Y=g(X,Y)\,\xi-\eta(Y)X,\quad X,Y\in\mathfrak{X}_{M}.$

Here, $g$ is a Riemannian metric, $\nabla$ is the Levi-Civita connection,
$\varphi$ is a $(1,1)$-tensor, $\xi$ is a Reeb vector field and $\eta$ is a
1-form, satisfying $\eta(\xi)=1$ and

$g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\,\eta(Y),\quad X,Y\in\mathfrak{X}_{M},$

and $\mathfrak{X}_{M}$ is the space of vector fields on $M$. Nearly Sasakian
structure $(\varphi,\xi,\eta,g)$ was defined by D. Blair, D. Showers and Y.
Komatu [3] using a similar condition for the symmetric part of
$\nabla\varphi$:

$(\nabla_{Y}\,\varphi)Y=g(Y,Y)\,\xi-\eta(Y)Y,\quad Y\in\mathfrak{X}_{M}.$ (1)

Any 3-dimensional nearly Sasakian manifold is Sasakian, see [8], and any
5-dimensional nearly Sasakian manifold has Einstein metric of positive scalar
curvature, see [4]; an example is a sphere $S^{\,5}$ with the almost contact
metric structure induced by the almost Hermitian structure of $S^{\,6}$.

The Reeb vector field $\xi$ of a nearly Sasakian structure is a unit Killing
vector field (an infinitesimal generator of isometries or symmetries). The
influence of constant-length Killing vector fields on the Riemannian geometry
has been studied by many authors, e.g., [1].

B. Cappelletti-Montano and G. Dileo proved in [4] that nearly Sasakian
manifolds foliate into two types of totally geodesic foliations. Totally
geodesic foliations (e.g., fibrations or submersions with totally geodesic
fibers) of Riemannian manifolds have the simplest extrinsic geometry of the
leaves and appear as kernels of degenerate tensors, e.g., [14, 15]. A.D.
Nicola, G. Dileo and I. Yudin in [6], using results of [4, 7], gave a
criterion for the almost contact metric structure to be Sasakian; namely, that
every nearly Sasakian manifold of dimension greater than 5 is Sasakian.

In [9, 10, 11] and [15, Section 5.3.8], we introduced and studies metric
structures on a smooth manifold that generalize the almost contact, Sasakian,
etc. metric structures. These so-called “weak” structures (the linear complex
structure on the contact distribution is replaced by a nonsingular skew-
symmetric tensor) made it possible to take a new look at the classical
structures and find new applications. In [12] we investigated new structures
of this type, called weak nearly Sasakian structure and weak nearly
cosymplectic structure.

In this paper we continue our study of the weak nearly Sasakian structure. We
find conditions (5) and (6) that are automatically satisfied by almost contact
manifolds and under which the Reeb vector field is Killing, the contact
distribution is curvature invariant and the weak nearly Sasakian structure
foliates into two types of totally geodesic foliations. Section 2, following
the introductory Section 1, reviews the basics of weak almost contact
manifolds.

Section 3 contains auxiliary results on the geometry of weak almost Sasakian
structure; in particular, we find the expression of the Ricci tensor in the
$\xi$-direction and show that the contact distribution $\ker\eta$ is curvature
invariant under conditions (5) and (6).

In Section 4, we prove that a weak almost Sasakian manifold satisfying (5) and
(6), is foliated. Our main result (Theorem 1) generalizes Theorems 3.3 and 3.5
of [4].

In Section 5, using the approach of [13] we prove an auxiliary proposition
necessary for the main result. Our proofs use the properties of new tensors,
as well as classical constructions.

##  2 Preliminaries

A weak almost contact structure on a smooth manifold $M^{\,2n+1}\ (n\geq 1)$
is a set $(\varphi,Q,\xi,\eta)$, where $\varphi$ is a $(1,1)$-tensor, $\xi$ is
a vector field (called Reeb vector field), $\eta$ is a 1-form and $Q$ is a
nonsingular $(1,1)$-tensor on $TM$, satisfying, see [9, 10],

$\varphi^{2}=-Q+\eta\otimes\xi,\quad\eta(\xi)=1,\quad Q\,\xi=\xi.$ (2)

By (2), $\eta$ defines a smooth $2n$-dimensional distribution $\ker\eta$. We
assume that $\ker\eta$ is $\varphi$-invariant, i.e.,
$\varphi(\ker\eta)\subset\ker\eta$ (as in the classical theory [2], where
$Q={\rm id}_{\,TM}$). By this and (2), $\ker\eta$ is $Q$-invariant, i.e.,
$Q(\ker\eta)\subset\ker\eta$, and the following is true:

$\varphi\,\xi=0,\quad\eta\circ\varphi=0,\quad\eta\circ
Q=\eta,\quad[Q,\,\varphi]:={Q}\circ\varphi-\varphi\circ{Q}=0.$

A “small” (1,1)-tensor $\widetilde{Q}=Q-{\rm id}_{\,TM}$ is a measure of the
difference between a weak almost contact structure and an almost contact one.
Note that

$[\widetilde{Q},\varphi]:=\widetilde{Q}\circ\varphi-\varphi\circ\widetilde{Q}=0,\quad\eta\circ\widetilde{Q}=0,\quad\widetilde{Q}\,\xi=0.$

If there is a Riemannian metric $g$ on $M$ such that

$\displaystyle g(\varphi X,\varphi Y)=g(X,Q\,Y)-\eta(X)\,\eta(Y),\quad
X,Y\in\mathfrak{X}_{M},$ (3)

then $(\varphi,Q,\xi,\eta,g)$ is called a weak almost contact metric structure
on $M$. A weak almost contact manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta)$
endowed with a compatible Riemannian metric $g$ is called a weak almost
contact metric manifold and is denoted by $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$.

For a weak almost contact metric structure, $\varphi$ is skew-symmetric and
$Q$ is self-adjoint. Taking covariant derivative of the equality $g(\varphi
Y,Z)=-g(Y,\varphi Z)$, we see that $\nabla_{X}\,\varphi$ is skew-symmetric:

$g((\nabla_{X}\,\varphi)Y,Z)=-g((\nabla_{X}\,\varphi)Z,Y).$

Setting $Y=\xi$ in (3), we get $\eta(X)=g(\xi,X)$, as in the classical theory.
By (3),

$g(X,Q\,X)=g(\varphi X,\varphi X)>0$

is true for any nonzero vector $X\in\ker\eta$; thus, the tensor $Q$ is
positive definite.

A weak contact metric structure is a weak almost contact metric structure
satisfying

$d\eta(X,Y)=g(X,\varphi Y),\quad X,Y\in\mathfrak{X}_{M},$

where the exterior derivative $d\eta$ of $\eta$ are given by (for example,
[2])

$\displaystyle
d\eta(X,Y)=\frac{1}{2}\,\\{X(\eta(Y))-Y(\eta(X))-\eta([X,Y])\\}.$

A 1-form $\eta$ on a manifold $M^{\,2n+1}$ is said to be contact if
$\eta\wedge(d\eta)^{n}\neq 0$, e.g., [2]. Recall that for a weak contact
metric structure $(\varphi,Q,\xi,\eta,g)$, the 1-form $\eta$ is contact, see
[13, Lemma 2.1].

######  Definition 1 (see [12]).

A weak almost contact metric manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ is
called weak nearly Sasakian if (1) is true, or, equivalently,

$(\nabla_{Y}\,\varphi)Z+(\nabla_{Z}\,\varphi)Y=2\,g(Y,Z)\,\xi-\eta(Z)Y-\eta(Y)Z,\qquad
Y,Z\in\mathfrak{X}_{M}.$ (4)

In addition to (4), the following two conditions for weak almost contact
manifolds that are automatically satisfied by almost contact manifolds
($\widetilde{Q}=0$) play the key role in this paper:

$(\nabla_{X}\,\widetilde{Q})\,Y=0,\quad X\in\mathfrak{X}_{M},\ Y\in\ker\eta,$
(5) $R_{\widetilde{Q}X,Y}Z\in\ker\eta,\quad X\in TM,\ \ Y,Z\in\ker\eta,$ (6)

where the curvature tensor $R$ is given by, e.g., [5],

$R_{{X},{Y}}Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,\quad
X,Y,Z\in\mathfrak{X}_{M}.$

By (6) and the first Bianchi identity, e.g., [5]:
$R_{\,X,Y}\,\widetilde{Q}Z\in\ker\eta$ for all $Z\in TM$ and $X,Y\in\ker\eta$.

######  Example 1.

Let $M(\varphi,Q,\xi,\eta,g)$ be a three-dimensional weak nearly Sasakian
manifold with the condition (5). By (2), the symmetric tensor $Q$ has on the
plane field $\ker\eta$ the form $\lambda\,{\rm id}_{\,\ker\eta}$ for some
positive $\lambda\in\mathbb{R}$. It was shown in [12] that this structure
reduces to the nearly Sasakian structure
$(\tilde{\varphi},\xi,\eta,\tilde{g})$ on $M$, where

$\displaystyle\tilde{\varphi}=\lambda^{\,-\frac{1}{2}}\,\varphi,\quad\tilde{g}|_{\,\ker\eta}=\lambda^{\,\frac{1}{2}}\,g|_{\,\ker\eta},\quad\tilde{g}(\xi,\,\cdot)={g}(\xi,\,\cdot).$

Since $\dim M=3$, the nearly Sasakian structure
$(\tilde{\varphi},\xi,\eta,\tilde{g})$ is Sasakian, see [8, Theorem 5.1].

Using the equalities $(\nabla_{\xi}\,\varphi)\,\xi=0$ and $\varphi\,\xi=0$ for
a weak nearly Sasakian manifold, we conclude that $\xi$ is a geodesic vector
field ($\nabla_{\xi}\,\xi=0$). Recall [12] that on a weak nearly Sasakian
manifold with the property (5) the unit vector field $\xi$ is Killing
($\pounds_{\xi}\,g=0$, e.g., [5]). Here $\pounds$ is the Lie derivative and
the following identity is true:

$(\pounds_{\xi}\,g)(X,Y)=g(\nabla_{X}\,\xi,Y)+g(\nabla_{Y}\,\xi,X),\quad
X,Y\in\mathfrak{X}_{M}.$

Using (5) and $\nabla_{\xi}\,\xi=0$, we obtain

$\nabla_{\xi}\,\widetilde{Q}=0.$ (7)

For a Riemannian manifold equipped with a Killing vector field ${\xi}$, we
get, see [16],

$\nabla_{X}\nabla_{Y}\,{\xi}-\nabla_{\nabla_{X}Y}\,{\xi}=R_{\,X,\,{\xi}}\,Y.$
(8)

######  Remark 1.

The contact distribution of a nearly Sasakian manifold is curvature invariant:

$R_{\,X,Y}Z\in\ker\eta,\quad X,Y,Z\in\ker\eta,$ (9)

see [7]. For example, a real space form with a 1-form $\eta$ and the tangent
bundle of any totally geodesic submanifold satisfy (9). We will prove, see
(17) of Proposition 1, that a weak nearly Sasakian manifold satisfies (9) if
we assume a weaker condition (6).

##  3 Auxiliary results

In this section we generalize some properties of almost Sasakian manifolds to
the case of weak almost Sasakian manifolds $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$
with the conditions (5) and (6).

Let us define a (1,1)-tensor field $h$ on $M$, as in the classical case, e.g.,
[4],

$h=\nabla\xi+\varphi.$ (10)

We get $\eta\circ h=0$ and $h(\ker\eta)\subset\ker\eta$. Since $\xi$ is a
geodesic vector field ($\nabla_{\xi}\,\xi=0$), we also get

$h\,\xi=0.$ (11)

Since $\xi$ is a Killing vector field and $\varphi$ is skew-symmetric, the
tensor $h$ is skew-symmetric:

$g(hX,\,X)=g(\nabla_{X}\,\xi,X)+g(\varphi
X,X)=\frac{1}{2}\,(\pounds_{\xi}\,g)(X,X)=0,$

and $\nabla_{X}\,\eta=g((h-\varphi)X,\,\cdot)$ is true. Therefore,
$\xi$-curves determine a 1-dimensional Riemannian geodesic foliation. Note
that $-\nabla\xi=\varphi-h$ is its splitting operator (or, co-nullity tensor),
see [15, Section 1.3.1]. Moreover, $\eta$ is a Killing 1-form:
$(\nabla_{X}\,\eta)(X)=0$ for all $X\in\mathfrak{X}_{M}$.

The distribution $\ker\eta$ is integrable, $[X,Y]\in\ker\eta\
(X,Y\in\ker\eta)$, if and only if $h=\varphi$:

$g([X,Y],\xi)=2\,g((h-\varphi)Y,X),\quad X,Y\in\ker\eta;$

and in this case, our manifold splits along $\xi$ and $\ker\eta$ (i.e., is
locally the metric product).

######  Lemma 1.

For a weak nearly Sasakian manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ we
obtain

$\displaystyle(\nabla_{X}\,h)\,\xi=-h(h-\varphi)X,$ (12)
$\displaystyle(\nabla_{X}\,\varphi)\,\xi=-\varphi(h-\varphi)X,$ (13)
$\displaystyle h\,\varphi+\varphi\,h=-2\,\widetilde{Q},$ (14) $\displaystyle
h\,Q=Q\,h\quad(h\ {\rm commutes\ with}\ Q).$ (15)

Moreover,

$\displaystyle h^{2}\varphi=\varphi h^{2},\quad
h\varphi^{2}=\varphi^{2}h,\quad h^{2}\varphi^{2}=\varphi^{2}h^{2}.$ (16)

###### Proof.

Differentiating the equality $h\,\xi=0$ and using the definition (10), we
obtain (12):

$0=\nabla_{X}\,(h\,\xi)=(\nabla_{X}\,h)\,\xi+h(\nabla_{X}\,\xi)=(\nabla_{X}\,h)\,\xi+h(h-\varphi)X.$

Using $\varphi\,\xi=0$ and the definition (10), we get (13):

$(\nabla_{X}\,\varphi)\,\xi=-\varphi(\nabla_{X}\,\xi)=-\varphi(h-\varphi)X.$

Differentiating the equality $g(\varphi\,Y,\xi)=0$ yields

$0=Xg(\varphi\,Y,\xi)=g((\nabla_{X}\,\varphi)Y,\xi)+g(\varphi\,Y,(h-\varphi)X).$

Summing this with the equality
$g((\nabla_{Y}\,\varphi)X,\xi)+g(\varphi\,X,(h-\varphi)Y)=0$ and applying (4),
gives (14). By (14) and (2), using the equalities (11), $\eta\circ h=0$ and
$[\widetilde{Q},\varphi]=0$, we get (15). Using (14) and $[Q,\,\varphi]=0$, we
obtain (16). ∎

######  Proposition 1.

If a weak nearly Sasakian manifold satisfies the condition (6), then

$\displaystyle g(R_{\,\xi,Z}\,\varphi X,\varphi Y)=0,\quad{\rm hence,}\
\ker\eta\ {\rm is\ a\ curvature\ invariant\ distribution}.$ (17)

The Ricci tensor ${\rm Ric}$ on $(M,g)$ is defined as the suitable trace of
the curvature tensor:

${\rm Ric}\,(X,Y)={\rm trace}_{\,g}(Z\to
R_{\,Z,X}\,Y)=\sum\nolimits_{\,i}g(R_{\,E_{i},X}\,Y,E_{i})$

when $(E_{i})$ is an orthonormal frame. The following lemma is similar to
Lemma 3.5 in [13].

######  Lemma 2.

For a weak nearly Sasakian manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ with
the condition (6), we obtain

$\displaystyle R_{\,\xi,X}Y=-(\nabla_{X}\,(h-\varphi)\,)\,Y,$ (18)
$\displaystyle(\nabla_{X}\,(h-\varphi))Y=g((h-\varphi)^{2}X,Y)\,\xi-\eta(Y)\,(h-\varphi)^{2}X,$
(19) $\displaystyle{\rm Ric}\,(\xi,Z)=-\eta(Z)\,({\rm
tr}\,(h^{2}+\widetilde{Q})-2\,n).$ (20)

In particular, ${\rm tr}(h^{2}+\widetilde{Q})=const$, ${\rm
Ric}\,(\xi,\xi)=const\geq 0$ and

$\displaystyle\nabla_{\xi}\,h=\nabla_{\xi}\,\varphi=\varphi h+\widetilde{Q}.$
(21)

###### Proof.

From (8) (since $\xi$ is a Killing vector field) and (10), we find (18).
Replacing $Y$ by $\varphi Y$ and $Z$ by $\varphi Z$ in

$g(R_{\,\xi,X}Y,Z)=g((\nabla_{X}\,(\varphi-h))Y,Z),$

see (18), and using (17), we get $g((\nabla_{X}\,(h-\varphi))\,\varphi
Y,\varphi Z)=0$, i.e.,

$g((\nabla_{X}\,(h-\varphi))Y,Z)=0\quad(Y,Z\in\ker\eta).$ (22)

Then, using (22), we find the $\xi$-component of $(\nabla_{X}\,(h-\varphi))Y$:

$\displaystyle g((\nabla_{X}\,(h-\varphi))Y,\xi)=g(\nabla_{X}(h\,Y-\varphi
Y),\,\xi)$
$\displaystyle=-g((h-\varphi)Y,\,\nabla_{X}\,\xi)=-g((h-\varphi)X,(h-\varphi)Y)=g((h-\varphi)^{2}X,Y),$

and $\ker\eta$-component of $(\nabla_{X}\,(h-\varphi))Y$:

$\displaystyle
g((\nabla_{X}\,(h-\varphi))Y,Z)=\eta(Y)\,g((\nabla_{X}\,(h-\varphi))\,\xi,\,Z)$
$\displaystyle=\eta(Y)\,g(h(\varphi-h)X,Z)+\eta(Y)\,g(\varphi(h-\varphi)X,Z)$
$\displaystyle=-\eta(Y)\,g((h-\varphi)^{2}X,Z)\quad(Z\in\ker\eta),$

from which (19) follows. From (19) with $X=\xi$ we find
$\nabla_{\xi}\,(h-\varphi)=0$.

Let $\\{e_{i}\\}\ (i=1,\ldots,2n+1)$ be a local orthonormal frame on $M$ with
$e_{\,2n+1}=\xi$. Putting $X=Y=e_{i}$ in (19), then using (9) and summing over
$i=1,\ldots,2n+1$, we get

${\rm Ric}\,(\xi,Z)=-\eta(Z)\,{\rm tr}\,((h-\varphi)^{2})$

and ${\rm Ric}\,(\xi,\xi)\geq 0$. From the above and
$(h-\varphi)^{2}=h^{2}+\varphi^{2}+2\,\widetilde{Q}$, we get (20).

Using ${\rm tr}(\varphi^{2})=-{\rm tr}\,(\widetilde{Q}+2\,n)$, we get $X({\rm
tr}(\varphi^{2}))=-X({\rm tr}\,\widetilde{Q})$. Similarly,

$\displaystyle X({\rm tr}(h^{2}))={\rm tr}(\nabla_{X}\,(h^{2}))={\rm
tr}(h\nabla_{X}\,h+(\nabla_{X}\,h)h)$ $\displaystyle={\rm
tr}(h\nabla_{X}(h-\varphi)+(\nabla_{X}(h-\varphi))h)+{\rm
tr}(h\nabla_{X}\,\varphi+(\nabla_{X}\,\varphi)h),$

where, in view of (14),

$\displaystyle{\rm tr}(h\nabla_{X}\,\varphi+(\nabla_{X}\,\varphi)h)$
$\displaystyle={\rm tr}(\nabla_{X}(h\varphi+\varphi h))-2\,{\rm
tr}\big{(}\varphi\nabla_{X}(h-\varphi)\big{)}{-}X({\rm tr}(\varphi^{2}))$
$\displaystyle=-X({\rm tr}\,\widetilde{Q})-2\,{\rm
tr}\big{(}\varphi\nabla_{X}(h{-}\varphi)\big{)}.$

By (19) we get

${\rm
tr}\big{(}h\nabla_{X}(h-\varphi)+(\nabla_{X}(h-\varphi))h\big{)}=0,\quad{\rm
tr}\big{(}\varphi\nabla_{X}(h-\varphi)\big{)}=0.$

Therefore, $X({\rm tr}(h^{2}))=-X({\rm tr}\,\widetilde{Q})$ for all
$X\in\mathfrak{X}_{M}$. This implies ${\rm tr}(h^{2}+\widetilde{Q})=const$. By
this and (18), we also get ${\rm Ric}\,(\xi,\xi)=-{\rm
tr}\,(h^{2}+\widetilde{Q})+2\,n=const$. By (19) with $X=\xi$ we get
$\nabla_{\xi}\,h=\nabla_{\xi}\,\varphi$. Then applying (4) to
$(\nabla_{\xi}\,\varphi)X\ (X\in\mathfrak{X}_{M})$, we complete the proof of
(21). ∎

######  Remark 2.

Note that $(h-\varphi)^{2}X=h^{2}X+\varphi^{2}X+2\,\widetilde{Q}X$. By
(18)–(19), we get

$\displaystyle
g(R_{\,\xi,X}Y,Z)=\eta(Y)\,g((h-\varphi)^{2}X,Z)-\eta(Z)\,g((h-\varphi)^{2}X,Y).$
(23)

##  4 Main results

Here, we prove that weak nearly Sasakian manifolds with conditions (5) and (6)
have a foliated structure. First, we will generalize Lemma 2.1 in [8], which
characterizes Sasakian manifolds among nearly Sasakian manifolds by the
condition $h=0$.

######  Proposition 2.

For a weak nearly Sasakian manifold, $h=0$ holds if and only if the manifold
is Sasakian.

###### Proof.

Let $h=0$, i.e., $\varphi=-\nabla\xi$, see (10). For every vector fields $X,Y$
in $\ker\eta$ we have

$2\,d\eta(X,Y)=g(\nabla_{X}\,\xi,Y)-g(\nabla_{Y}\,\xi,X)=2\,g(X,\varphi Y),$

thus, our manifold is contact. From (14) we find $\widetilde{Q}=0$, thus our
manifold is nearly Sasakian. Finally, by (23), we obtain
$g(R_{\,\xi,X}Y,\xi)=g(X,Y)$ for $X,Y\in\ker\eta$. Thus, the $\xi$-sectional
curvature is 1, and our manifold is Sasakian. ∎

The following proposition generalizes [4, Proposition 3.2].

######  Proposition 3.

For a weak nearly Sasakian manifold with conditions (5) and (6), the
eigenvalues (and their multiplicities) of the symmetric operators $h^{2}$,
$\varphi^{2}$ and $(h-\varphi)^{2}$ are constant.

###### Proof.

From (23), (2) and Lemma 2 we obtain the following generalization of (10) in
[4]:

$\displaystyle\quad(\nabla_{X}\,h^{2})Y=h(\nabla_{X}\,h)Y+(\nabla_{X}\,h)hY$
$\displaystyle=h(\nabla_{X}\,\varphi)Y+(\nabla_{X}\,\varphi)hY-\eta(Y)h\,(h-\varphi)^{2}X+g((h-\varphi)^{2}X,hY)\,\xi$
$\displaystyle=-\varphi(\nabla_{X}\,h)Y-(\nabla_{X}\,h)\,\varphi
Y-2(\nabla_{X}\,\widetilde{Q})Y-\eta(Y)h\,(h-\varphi)^{2}X+g((h-\varphi)^{2}X,hY)\,\xi$
$\displaystyle=-\varphi(\nabla_{X}\,\varphi)Y+\eta(Y)\varphi(h-\varphi)^{2}X-(\nabla_{X}\,\varphi)\,\varphi
Y-g((h-\varphi)^{2}X,\varphi Y)\,\xi$
$\displaystyle\quad-2(\nabla_{X}\,\widetilde{Q})Y-\eta(Y)h\,(h-\varphi)^{2}X+g((h-\varphi)^{2}X,hY)\,\xi$
$\displaystyle=(\nabla_{X}\,\widetilde{Q})Y-g((h-\varphi)X,Y)\,\xi-\eta(Y)(h-\varphi)X+\eta(Y)\varphi(h-\varphi)^{2}X$
$\displaystyle\quad-g((h-\varphi)^{2}X,\varphi
Y)\,\xi-2\,(\nabla_{X}\,\widetilde{Q})Y+g((h-\varphi)^{2}X,hY)\,\xi-\eta(Y)\,h(h-\varphi)^{2}X$
$\displaystyle=g\big{(}(h^{2}+\widetilde{Q})X,(h-\varphi)Y\big{)}\,\xi-\eta(Y)(h-\varphi)(h^{2}+\widetilde{Q})X-(\nabla_{X}\,\widetilde{Q})Y.$
(24)

Consider an eigenvalue $\mu$ of $h^{2}$ and a local unit vector field
$Y\bot\,\xi$ such that $h^{2}Y=\mu Y$. Applying (4) for any nonzero vector
fields $X\in\mathfrak{X}_{M}$ and using (5), we find
$g((\nabla_{X}\,h^{2})Y,Y)=0$, thus

$\displaystyle
0=g((\nabla_{X}\,h^{2})Y,Y)=g(\nabla_{X}\,(h^{2}Y),Y)-g(h^{2}(\nabla_{X}\,Y),Y)$
$\displaystyle\ \
=X(\mu)\,g(Y,Y)+\mu\,g(\nabla_{X}\,Y,Y)-g(\nabla_{X}\,Y,h^{2}Y)=X(\mu)\,g(Y,Y),$

which implies that $X(\mu)=0$ for all $X\in\mathfrak{X}_{M}$. ∎

Since $h$ is skew-symmetric, the nonzero eigenvalues of $h^{2}$ are negative.
By Proposition 3, the spectrum of the self-adjoint operator $h^{2}$ has the
form

$Spec(h^{2})=\\{0,-\lambda_{1}^{2},\ldots-\lambda_{r}^{2}\\},$ (25)

where $\lambda_{i}$ is a positive real number and $\lambda_{i}\neq\lambda_{j}$
for $i\neq j$.

In particular, ${\rm tr}\,(h^{2})=const\leq 0$, and by Lemma 2, ${\rm
tr}\,Q=const>0$.

Denote by $[\xi]$ the 1-dimensional distribution generated by $\xi$, and by
$D_{0}$ a smooth distribution of the eigenvectors of $h^{2}$ with eigenvalue 0
orthogonal to $\xi$. Denote by $D_{i}$ a smooth distribution of the
eigenvectors of $h^{2}$ with eigenvalue $-\lambda^{2}_{i}$. Note that the
distributions $D_{0}$ and $D_{i}\ (i=1,\ldots,r)$ belong to $\ker\eta$ and are
$\varphi$-invariant and $h$-invariant. In particular, the eigenvalue $0$ has
multiplicity $2p+1$ for some integer $p\geq 0$. If $X$ is a unit eigenvector
of $h^{2}$ with eigenvalue $-\lambda^{2}_{i}$, then by (14) and (15),
$X,\varphi X,hX$ and $h\,\varphi X$ are nonzero eigenvectors of $h^{2}$ with
eigenvalue $-\lambda^{2}_{i}$.

######  Lemma 3.

The tensors $h$ and $\widetilde{Q}$ of a weak nearly Sasakian manifold are
equal to zero on $D_{0}$.

###### Proof.

Let $h^{2}X=0$ for some $X\neq 0$. Since $g(hX,hX)=-g(h^{2}X,X)=0$, we get
$hX=0$. Similarly, using (16), we get

$g(h\varphi X,h\varphi X)=-g(h^{2}\varphi X,\varphi X)=-g(\varphi
h^{2}X,\varphi X)=0;$

hence, $h\varphi X=0$. Taking into account the above and (14), we obtain
$\widetilde{Q}X=0$. ∎

The following theorem generalizes [4, Theorems 3.3 and 3.5].

######  Theorem 1.

Let $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ be a weak nearly Sasakian manifold with
conditions (5) and (6), and let the spectrum of the self-adjoint operator
$h^{2}$ have the form (25), where the eigenvalue $0$ has multiplicity $2p+1$
for some integer $p\geq 0$. Then, the distribution $[\xi]\oplus D_{0}$ and
each distribution $[\xi]\oplus D_{i}\ (i=1,\ldots,r)$ are integrable with
totally geodesic leaves. If $p>0$, then

$(a)$ the distribution $[\xi]\oplus D_{1}\oplus\ldots\oplus D_{r}$ is
integrable and defines a Riemannian foliation with totally geodesic leaves;

$(b)$ the leaves of $[\xi]\oplus D_{0}$ are $(2p+1)$-dimensional Sasakian
manifolds.

###### Proof.

Consider a unit eigenvector $X$ of $h^{2}$ with eigenvalue $\mu$. Then, by
(10) and (16),

$h^{2}(\nabla_{X}\,\xi)=h^{2}(h-\varphi)X=(h-\varphi)h^{2}X=\mu(h-\varphi)X=\mu\nabla_{X}\,\xi,$

hence $\nabla_{X}\,\xi\in D_{\mu}$. On the other hand, (4) and (7) imply
$\nabla_{\xi}\,h^{2}=0$, and thus $\nabla_{\xi}X$ is also an eigenvector of
$h^{2}$ with eigenvalue $\mu$.

Taking unit vectors $X,Y\in D_{0}$ and applying (4) and $\widetilde{Q}X=0$, we
get $h^{2}(\nabla_{X}\,Y)=0$; thus $\nabla_{X}\,Y\in[\xi]\oplus D_{0}$. Hence,
the distribution $[\xi]\oplus D_{0}$ is integrable with totally geodesic
leaves.

If $\mu\neq 0$, then taking unit vectors $X,Y\in D_{\mu}$ and applying (4) and
$hY=\mu Y=\varphi Y$, we get $h^{2}(\nabla_{X}\,Y)=\mu\,\nabla_{X}Y$. Hence,
each distribution $[\xi]\oplus D_{i}\ (i=1,\ldots,r)$ is integrable with
totally geodesic leaves.

Let $p>0$. Using $h^{2}\varphi^{2}=\varphi^{2}h^{2}$, see (16), we get for
$\mu\neq 0$:

$h^{2}(\varphi^{2}\nabla_{X}Y)=\varphi^{2}(h^{2}\nabla_{X}Y)=\mu\,\varphi^{2}(\nabla_{X}Y).$

Thus $\varphi^{2}\nabla_{X}Y\in D_{\mu}$. Similarly, using (15), we get
$\widetilde{Q}\nabla_{X}Y\in D_{\mu}$. Using the above and (2), we get

$\nabla_{X}Y=-\widetilde{Q}\,\nabla_{X}Y-\varphi^{2}\nabla_{X}Y+\eta(\nabla_{X}Y)\,\xi,$

hence $\nabla_{X}Y$ belongs to the distribution $[\xi]\oplus D_{\mu}$ with
$\mu\neq 0$. By (4) we get

$\displaystyle g(\nabla_{X}Y,Z)$
$\displaystyle=-(1/\lambda_{j}^{2})\,g(\nabla_{X}(h^{2}Y),Z)=-(1/\lambda_{j}^{2})\,g((\nabla_{X}h^{2})Y+h^{2}\nabla_{X}Y,Z)$
$\displaystyle=-(1/\lambda_{j}^{2})\,g(\nabla_{X}Y,h^{2}Z)=0,\quad X\in
D_{i},\ Y\in D_{j},\ Z\in[\xi]\oplus D_{0},$

thus the distribution $[\xi]\oplus D_{1}\oplus\ldots\oplus D_{r}$ defines a
totally geodesic foliation.

First, for any $Z,Z^{\prime}\in D_{0}$, we have
$(\pounds_{\xi}\,g)(Z,Z^{\prime})=0$ since $\xi$ is Killing. Next, since the
distribution $[\xi]\oplus D_{0}$ is integrable with totally geodesic leaves,
for any $X\in D_{1}\oplus\ldots\oplus D_{r}$ we get

$g(\nabla_{Z}X,Z^{\prime})=-g(\nabla_{Z}Z^{\prime},X)=0.$

By the above, we conclude that for any $Z,Z^{\prime}\in D_{0}$ and any $X\in
D_{1}\oplus\ldots\oplus D_{r}$:

$(\pounds_{X}\,g)(Z,Z^{\prime})=g(\nabla_{Z}X,Z^{\prime})+g(\nabla_{Z^{\prime}}X,Z)=0.$

Thus, the distribution $[\xi]\oplus D_{1}\oplus\ldots\oplus D_{r}$ defines a
Riemannian foliation, hence, a).

If $X$ is an eingenvector of $h^{2}$ orthogonal to $\xi$ with eigenvalue
$\mu$, then also $\varphi X$ is an eingenvector with the same eigenvalue
$\mu$. Hence, the eigenvalue $0$ has odd multiplicity $2p+1$ for some integer
$p\geq 0$. If $p>0$, then the structure $(\varphi,Q,\xi,\eta,g)$ induces a
weak nearly Sasakian structure on the totally geodesic leaves of the
distribution $[\xi]\oplus D_{0}$ whose associated tensor $h$ vanishes. Thus,
$\varphi X=-\nabla_{X}\,\xi$ for all $X\in[\xi]\oplus D_{0}$, and, by Lemma 3,
$QX=X$ for all $X\in[\xi]\oplus D_{0}$. Therefore, the induced structure is
Sasakian, hence, b). ∎

##  5 Proof of Proposition 1

The curvature tensor of a weak nearly Sasakian manifold satisfies the equality

$\displaystyle g(R_{\,\varphi X,Y}Z,V)+g(R_{\,X,\varphi
Y}Z,V)+g(R_{\,X,Y}\,\varphi Z,V)+g(R_{\,X,Y}Z,\varphi V)$
$\displaystyle=g(Y,V)\,g((h-\varphi)X,Z)-g(X,Y)\,g(Z,(h-\varphi)V)+g(Y,Z)\,g(X,(h-\varphi)V)$
$\displaystyle-(1/2)\,g(Z,V)\,g((h-\varphi)X,Y)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)V)$
(26)

for all $X,Y,Z,V\in\mathfrak{X}_{M}$. The proof of (5) is similar to the proof
of equation (19) in [13] and we omit it. In the proof of (17), we also use the
following “small” tensor:

$\delta(X,Y,Z,V)=g(R_{\,X,Y}\widetilde{Q}Z,V)+g(R_{\,X,Y}Z,\widetilde{Q}V)-g(R_{\widetilde{Q}X,Y}Z,V)-g(R_{\,X,\widetilde{Q}Y}Z,V).$

The tensor $\delta$ of a weak nearly Sasakian manifold has some symmetries:

$\displaystyle\delta(Y,X,Z,V)=\delta(X,Y,V,Z)=\delta(Z,V,X,Y)=-\delta(X,Y,Z,V).$

If (6) is true, then by (23), we get

$\displaystyle\delta(X,Y,Z,\xi)=g(R_{\,X,Y}\,\widetilde{Q}Z,\xi)-g(R_{\widetilde{Q}X,Y}Z,\xi)-g(R_{\,X,\widetilde{Q}Y}Z,\xi)$
$\displaystyle\qquad=g(R_{\,\xi,\,\widetilde{Q}Z}Y,X)+g(R_{\,\xi,Z}\,\widetilde{Q}X,Y)+g(R_{\,\xi,Z}\,X,\widetilde{Q}Y)=0,$

hence,
$\delta(\xi,Y,Z,V)=\delta(X,\xi,Z,V)=\delta(X,Y,\xi,V)=\delta(X,Y,Z,\xi)=0$.

Replacing $X$ by $\varphi X$ in (5) and using (2), we have

$\displaystyle-g(R_{\,QX,Y}Z,V){+}\eta(X)g(R_{\,\xi,Y}Z,V){+}g(R_{\,\varphi
X,\varphi Y}Z,V){+}g(R_{\,\varphi X,Y}\,\varphi Z,V){+}g(R_{\,\varphi
X,Y}Z,\varphi V)$ $\displaystyle=g(V,Y)\,g((h-\varphi)\varphi X,Z)-g(\varphi
X,Y)\,g(Z,(h-\varphi)V)+g(Y,Z)\,g(\varphi X,(h-\varphi)V)$ $\displaystyle\
-(1/2)\,g(Z,V)\,g((h-\varphi)\varphi X,Y)+(1/2)\,g(\varphi
X,Z)\,g(Y,(h-\varphi)V).$ (27)

Exchanging $X$ and $Y$ in (5), we find

$\displaystyle g(R_{\,X,QY}Z,V){+}\eta(Y)g(R_{\,\xi,X}Z,V){-}g(R_{\,\varphi
X,\varphi Y}Z,V){+}g(R_{\,\varphi Y,X}\,\varphi Z,V){+}g(R_{\,\varphi
Y,X}Z,\varphi V)$ $\displaystyle=g(V,X)\,g((h-\varphi)\varphi Y,Z)-g(\varphi
X,Y)\,g(Z,(h-\varphi)V)-g(X,Z)\,g(\varphi(Y,h-\varphi)V)$ $\displaystyle\
-(1/2)\,g(Z,V)\,g(X,(h-\varphi)\varphi Y)+(1/2)\,g(Z,\varphi
Y)\,g(X,(h-\varphi)V).$ (28)

Subtracting (5) from (5), we obtain

$\displaystyle 2\,g(R_{\,\varphi X,\varphi
Y}Z,V)-2\,g(R_{\,X,Y}Z,V)+\eta(X)\,g(R_{\,\xi,Y}Z,V)-\eta(Y)\,g(R_{\,\xi,X}Z,V)$
$\displaystyle\ +g(R_{\,\varphi X,Y}\,\varphi Z,V)-g(R_{\,\varphi
Y,X}\,\varphi Z,V)+g(R_{\,\varphi X,Y}Z,\varphi V)-g(R_{\,\varphi
Y,X}Z,\varphi V)$ $\displaystyle\
-g(R_{\widetilde{Q}X,Y}Z,V)-g(R_{\,X,\widetilde{Q}Y}Z,V)$
$\displaystyle=g(Y,V)\,g((h-\varphi)\varphi
X,Z)+g(Y,Z)\,g((h-\varphi)V,\varphi X)+g(Z,V)\,g(\widetilde{Q}X,Y)$
$\displaystyle\ +(1/2)\,g(\varphi X,Z)\,g(Y,(h-\varphi)V)-(1/2)\,g(\varphi
Y,Z)\,g(X,(h-\varphi)V)$ $\displaystyle\ -g(X,V)\,g((h-\varphi)\varphi
Y,Z)+g(X,Z)\,g(Y,\varphi(h-\varphi)V).$ (29)

Then, replacing $Z$ by $\varphi Z$ and also $V$ by $\varphi V$ in (5) and
using (2), we get two equations

$\displaystyle-g(R_{\,X,Y}QZ,V)={-}\eta(Z)\,g(R_{\,X,Y}\,\xi,V){-}g(R_{\,X,Y}\,\varphi
Z,\varphi V){-}g(R_{\,X,\varphi Y}\,\varphi Z,V){-}g(R_{\,\varphi
X,Y}\,\varphi Z,V)$ $\displaystyle\
-g(Y,V)\,g(\varphi(h-\varphi)X,Z)-g(X,Y)\,g(\varphi
Z,(h-\varphi)V)+g(Y,\varphi Z)\,g(X,(h-\varphi)V)$ $\displaystyle\
-(1/2)\,g(\varphi Z,V)\,g((h-\varphi)X,Y)+(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V),$ (30)
$\displaystyle-g(R_{\,X,Y}Z,QV)=-\eta(V)\,g(R_{\,X,Y}Z,\,\xi){-}g(R_{\,X,Y}\,\varphi
Z,\varphi V){-}g(R_{\,\varphi X,Y}Z,\varphi V){-}g(R_{\,X,\varphi Y}Z,\varphi
V)$ $\displaystyle\ +g(Y,\varphi
V)\,g((h-\varphi)X,Z)-g(X,Y)\,g(Z,(h-\varphi)\varphi
V)+g(Y,Z)\,g(X,(h-\varphi)\varphi V)$ $\displaystyle\ -(1/2)\,g(Z,\varphi
V)\,g((h-\varphi)X,Y)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V).$ (31)

Adding (5) to (5), we get

$\displaystyle-2\,g(R_{\,X,Y}Z,V)=-2\,g(R_{\,X,Y}\,\varphi Z,\varphi
V)+g(R_{\,X,Y}\widetilde{Q}Z,V)+g(R_{\,X,Y}Z,\widetilde{Q}V)$ $\displaystyle\
-\,\eta(Z)\,g(R_{\,X,Y}\,\xi,V)-g(R_{\,X,\varphi Y}\,\varphi
Z,V)-g(R_{\,\varphi X,Y}\,\varphi Z,V)$ $\displaystyle\
-\,\eta(V)\,g(R_{\,X,Y}Z,\,\xi)-g(R_{\,\varphi X,Y}Z,\varphi
V)-g(R_{\,X,\varphi Y}Z,\varphi V)$ $\displaystyle\
-g(Y,V)\,g(\varphi(h-\varphi)X,Z)+g(Y,\varphi Z)\,g(X,(h-\varphi)V)$
$\displaystyle\ +g(Y,\varphi
V)\,g((h-\varphi)X,Z)+2\,g(X,Y)\,g(Z,\widetilde{Q}V)+g(Y,Z)\,g(X,(h-\varphi)\varphi
V)$ $\displaystyle\ +(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V).$

Substituting the above equation into (5), we have

$\displaystyle 2\,g(R_{\,\varphi X,\varphi Y}Z,V)-2\,g(R_{\,X,Y}\,\varphi
Z,\varphi V)-\eta(Z)\,g(R_{\,\xi,V}X,Y)+\eta(V)\,g(R_{\,\xi,Z}\,X,Y)$
$\displaystyle+\,\eta(X)\,g(R_{\,\xi,Y}Z,V)-\eta(Y)\,g(R_{\,\xi,X}Z,V)+\delta(X,Y,Z,V)$
$\displaystyle-g(Y,V)\,g(\varphi(h-\varphi)X,Z)+g(Y,\varphi
Z)\,g(X,(h-\varphi)V)$ $\displaystyle+g(Y,\varphi
V)\,g((h-\varphi)X,Z)+g(Y,Z)\,g(X,(h-\varphi)\varphi
V)+2\,g(X,Y)\,g(Z,\widetilde{Q}V)$ $\displaystyle+(1/2)\,g(X,\varphi
Z)\,g(Y,(h-\varphi)V)+(1/2)\,g(X,Z)\,g(Y,(h-\varphi)\varphi V)=0.$ (32)

Replacing $X$ by $\varphi X$ and also $Y$ by $\varphi Y$ in (5) and using (2),
we obtain

$\displaystyle 2\,g(R_{\,QX,QY}Z,V)-2\,g(R_{\,\varphi X,\varphi Y}\,\varphi
Z,\varphi V)-2\,\eta(X)\,g(R_{\,\xi,QY}Z,V)$
$\displaystyle+2\,\eta(Y)\,g(R_{\,\xi,QX}Z,V)-\eta(Z)\,g(R_{\,\xi,V}X,Y)+\eta(V)\,g(R_{\,\xi,Z}\,\varphi
X,\varphi Y)$ $\displaystyle+\delta(\varphi X,\varphi Y,Z,V)+g(\varphi
Y,V)\,g((h-\varphi)\varphi X,\varphi Z)-g(\varphi Y,\varphi
Z)\,g(\varphi(h-\varphi)V,X)$ $\displaystyle+g(\varphi Y,\varphi
V)\,g((h{-}\varphi)\varphi X,Z)+g(\varphi Y,Z)\,g(\varphi
X,(h{-}\varphi)\varphi V)-2\,g(\varphi^{2}X,Y)\,g(Z,\widetilde{Q}V)$
$\displaystyle+(1/2)\,g(\varphi^{2}X,Z)\,g(Y,\varphi(h-\varphi)V)+(1/2)\,g(\varphi
X,Z)\,g(\varphi Y,(h-\varphi)\varphi V)=0.$ (33)

Replacing $V$ by $\xi$ in (5), we obtain

$\displaystyle
2\,g(R_{\,QX,QY}Z,\xi)-2\,\eta(X)\,g(R_{\,\xi,QY}Z,\xi)+2\,\eta(Y)\,g(R_{\,\xi,QX}Z,\xi)$
$\displaystyle+\,g(R_{\,\xi,Z}\,\varphi X,\varphi Y)+\delta(\varphi X,\varphi
Y,Z,\,\xi)=0.$ (34)

Replacing $X$ by $\varphi X$ and also $Y$ by $\varphi Y$ in (5), we obtain

$\displaystyle 4\,g(R_{\,Q\,\varphi X,\,Q\,\varphi
Y}Z,\xi)+2\,\delta(\varphi^{2}X,\varphi^{2}Y,Z,\,\xi)+2\,g(R_{\,\xi,Z}\,QX,QY)$
$\displaystyle-2\,\eta(X)\,g(R_{\,\xi,Z}\,\xi,QY)+2\,\eta(Y)\,g(R_{\,\xi,Z}\,\xi,QX)=0.$
(35)

Adding (5) and (5) and applying $Q={\rm id}_{\,TM}+\widetilde{Q}$, we get

$\displaystyle 3\,g(R_{\,\xi,Z}\,\varphi X,\varphi
Y)=-4\,g(R_{\,\xi,Z}\,\widetilde{Q}\varphi X,\varphi
Y)-4\,g(R_{\,\xi,Z}\,\varphi X,\widetilde{Q}\varphi Y)$
$\displaystyle-4\,g(R_{\,\xi,Z}\,\widetilde{Q}\varphi X,\widetilde{Q}\varphi
Y)+\delta(\varphi X,\varphi
Y,Z,\,\xi)+2\,\delta(\varphi^{2}X,\varphi^{2}Y,Z,\,\xi).$ (36)

Using the condition (6), we get

$g(R_{\,\xi,Z}\,\widetilde{Q}\varphi X,\varphi Y)=g(R_{\,\xi,Z}\,\varphi
X,\widetilde{Q}\varphi Y)=g(R_{\,\xi,Z}\,\widetilde{Q}\varphi
X,\widetilde{Q}\varphi Y)=0$

and $\delta(\varphi X,\varphi
Y,Z,\,\xi)=\delta(\varphi^{2}X,\varphi^{2}Y,Z,\,\xi)=0$. Therefore, from (5)
we obtain (17). Since the restriction $\varphi\,|_{\,\ker\eta}$ is non-
degenerate, the distribution $\ker\eta$ is curvature invariant, see (9).
$\square$

## Conclusions

We have shown that the weak nearly Sasakian structure is useful for studying
contact metric structures and totally geodesic foliations. Some well known
results on nearly Sasakian manifolds were extended to weak nearly Sasakian
manifolds with the conditions (5) and (6), and our main result is that such
weak nearly Sasakian manifolds have a foliated structure with two types of
totally geodesic foliations. Motivated by a criterion for an almost contact
metric manifold to be Sasakian in [6], we pose the following question: are
weak nearly Sasakian manifolds of dimension higher than five, satisfying the
conditions (5) and (6), Sasakian manifolds?

Based on the numerous applications of the nearly Sasakian structure, we expect
that certain weak structure will also be useful for geometry and physics,
e.g., in twistor string theory.

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