\begin{example}\label{ex_2}
Let~$\alpha>0$ and~$r \in \mathbb{N^*}$. And let
$
M = \bigoplus_{i=1}^r M_i \text{ and }
N = \bigoplus_{i=1}^r N_i,
$ be two~$d$-parameter upset-decomposable modules
with supports
\[
\operatorname{supp} (M_i) = \{(x_1,\dots,x_d) \mid (x_1,\dots,x_d) \ge (2i\alpha,\dots,2i\alpha)\}\]
\[
\operatorname{supp} (N_i) = \{(x_1,\dots,x_d) \mid (x_1,\dots,x_d) \ge \left( (2r+1)\alpha,\dots,(2r+1)\alpha \right)\}.
\]
Then one can check that
\[
\operatorname{Pru}_{\alpha}(M) \cong N,
\]
so~$\operatorname{Pru}_{\alpha}(M)$ is an~$\infty$-refinement of~$N$.
Moreover, for every~$\delta>0$,
\[
\operatorname{Pru}_{\alpha+\delta} (M) \cong \operatorname{Pru}_{\delta} (N) \cong N(-\delta),
\]
so we also have~$\operatorname{Pru}_{\alpha+\delta}(M)$ as an~$\infty$-refinement of~$N(-\delta)$.
We can also easily check that~$\forall \varepsilon < \alpha, \operatorname{Pru}_{\varepsilon}(M)$ is not an~$\infty$-refinement of~$N$.
Similarly,~$N$ is an~$\infty$-refinement of~$M$, and for every~$\delta>0$,~$\operatorname{Pru}_{\delta}(N)$ is an~$\infty$-refinement of~$\operatorname{Pru}_{\delta}(M)$.\\
From these observations we obtain that
\[
d_P(M,N) = \alpha.
\]
Furthermore, since all~$N_i$ have the same support, it is easy to check that
\[
d_B(M,N) = d_I(M,N) = (2r-1)\alpha,
\]
showing that the upper bound in Theorem~\ref{theorem_main} can indeed be attained ~$\forall \alpha>0$,~$\forall d \in \mathbb{N}^*$ and~$\forall r \in \mathbb{N}^*$. In particular, the constant~$2r-1$ is optimal, and the inequality cannot be strengthened neither for the bottleneck distance~$d_B$ nor for the interleaving distance~$d_I$.

\end{example}

\markboth{1. Introduction}{}

Developed initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \emph{persistence modules}, which are formally functors from the ordered set~$(\mathbb{R},\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.

The remarkable isometry theorem \cite{chazal_isometry} guarantees that the algebraically defined \emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \cite{Edelsbrunner_Harer}, which can be solved in polynomial time \cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.

When dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \cite{bjerkevik_Botnan_kerber}.

Bjerkevik introduced the \emph{$\varepsilon$-pruning} (Definition~\ref{def_pruning}) of a module in \cite{bjerkevik_stabilizing}. The~$\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.

Importantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,
\[
d_P(M,N) \leq d_I(M,N) \leq 2r\,d_P(M,N).
\]

In this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \subset \mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\ref{sec:pruning-upset}. Our main results, Theorem~\ref{theorem_main} and Theorem~\ref{theorem_3}, show the following

Developed initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \emph{persistence modules}, which are formally functors from the ordered set~$(\mathbb{R},\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.

The remarkable isometry theorem \cite{chazal_isometry} guarantees that the algebraically defined \emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \cite{Edelsbrunner_Harer}, which can be solved in polynomial time \cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.

When dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \cite{bjerkevik_Botnan_kerber}.

Bjerkevik introduced the \emph{$\varepsilon$-pruning} (Definition~\ref{def_pruning}) of a module in \cite{bjerkevik_stabilizing}. The~$\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.

Importantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,
\[
d_P(M,N) \leq d_I(M,N) \leq 2r\,d_P(M,N).
\]

In this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \subset \mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\ref{sec:pruning-upset}. Our main results, Theorem~\ref{theorem_main} and Theorem~\ref{theorem_3}, show the following

\begin{definition}\label{def_pruning}
\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \to M(2\varepsilon)$, we have~$f(I) \subseteq M_{0 \to 2\varepsilon}(I)$. \\
Let~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \to M(2\varepsilon)$, it holds that~$I^{-1}_{0 \to 2\varepsilon}(f(K)) \subseteq K$.\\
We define the \emph{$\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \emph{$\varepsilon$-pruning} of~$M$ to be
\[
\Pru_\varepsilon(M) := (I/K)(-\varepsilon).
\]
\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is
\[
\Pru_0(M) = (M/0)(0) \cong M.
\]
\end{definition}

\begin{definition}
\label{def_pruning_distance}
The \emph{pruning distance}~$d_P$ is defined by
\begin{align*}
d_P(M,N) = \inf\{&\varepsilon\geq 0\mid \forall \delta\geq 0,\\ &\Pru_{\varepsilon+\delta}(M) \text{ is an } \infty\text{-refinement of } \Pru_\delta(N) \text{ and }\\ &\Pru_{\varepsilon+\delta}(N) \text{ is an } \infty\text{-refinement of } \Pru_\delta(M)\}
\end{align*}
for any pfd modules~$M$ and~$N$.
\end{definition}

\begin{theorem}\label{theorem_3}
       Let~$M$ and~$N$ be pfd upset-decomposable modules then
       \[d_P(M,N) \leq d_B(M,N).\]
   \end{theorem}

\begin{theorem}\label{theorem_main}
       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\operatorname{supdim}$~$M < \infty$ then
       \[d_B(M,N) \leq (2r-1)\,d_P(M,N).\]
   \end{theorem}

Importantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,
\[
d_P(M,N) \leq d_I(M,N) \leq 2r\,d_P(M,N).
\]

Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.

\begin{theorem}\label{theorem_3}
 Let~$M$ and~$N$ be pfd upset-decomposable modules then
 \[d_P(M,N) \leq d_B(M,N).\]
 \end{theorem}
\begin{proof}
Let~$r = \operatorname{supdim}$~$M$, if~$\operatorname{supdim}$~$N \neq r$ then both distances~$= \infty$ so nothing to prove.\\
Suppose that~$\operatorname{supdim}$~$N = r$ i.e~$M=\bigoplus_{l=1}^r M_l$ and~$N=\bigoplus_{l=1}^r N_l.$ Let~$\alpha \geq 0$, such that there exists a matching between~$M$ and~$N$. This means that there exists a permutation~$\sigma$ of~$\{1,\dots,r\}$ such that
\[
M_l \subseteq N_{\sigma(l)}(\alpha) \quad \text{and} \quad N_l \subseteq M_{\sigma^{-1}(l)}(\alpha), \quad \forall l \in \{1,\dots,r\}.
\]
Define~$N' = \bigoplus_{l=1}^r N'_l$ where~$N'_l = N_{\sigma(l)}$, then~$N' \cong N$. We get that
 \begin{equation}
M_l \subseteq N'_l(\alpha) \quad \text{and} \quad N'_l \subseteq M_l(\alpha), \quad \forall l \in \{1,\dots,r\}.
 \tag{4}
\label{eq:4}
\end{equation}
From now on, to simplify the notation,~$N$ refers to~$N'$.\\
We want to show that~$\forall \delta\geq 0, \Pru_{\alpha+\delta}(M) \text{ is an } \infty\text{-refinement of } \Pru_\delta(N)$ and~$\Pru_{\alpha+\delta}(N)$ is an~$\infty\text{-refinement of } \Pru_\delta(M).$ We prove only the first statement; the second follows by symmetry. By Lemma~\ref{lem_infinit_refinement} it is sufficient to show that
\[
\forall \delta\geq 0, \, \forall l \in \{1,\dots,r\}, \quad \pi_{l}(\operatorname{Pru}_{\alpha+\delta}(M)) \subseteq \pi_{l}(\operatorname{Pru}_{\delta}(N)).
\]
Fix~$l_1 \in \{1,\dots,r\}$ and~$\delta \geq 0$. Let~$l \in JN_{l_1}$ in the directed graph~$G(N,\delta)$, then there exists a path
\[
P:\; N_{l_1}=N_{l'_0}\to N_{l'_1}\to\cdots\to N_{l'_{c-1}}\to N_{l'_{c}}=N_{l},
\]
for some integer~$c$.
By the construction of~$G(N,\delta)$ each arrow gives a containment shifted by~$2\delta$, hence
\[
N_{l'_0} \subseteq N_{l'_1}(2\delta), N_{l'_1} \subseteq N_{l'_2}(2\delta), \cdots , N_{l'_{c-1}} \subseteq N_{l'_{c}}(2\delta).
\]
By~\eqref{eq:4} we get that
\[
M_{l'_i} \subseteq N_{l'_i}(\alpha) \subseteq N_{l'_{i+1}}(\alpha+2\delta) \subseteq M_{l'_{i+1}}(2\alpha+2\delta), \quad \forall i \in \{0,\dots,c-1\}.
\]
So we get that there exists a path
\[
P':\; M_{l_1}=M_{l'_0}\to M_{l'_1}\to\cdots\to M_{l'_{c-1}}\to M_{l'_{c}}=M_{l},
\]
in the directed graph~$G(M,\alpha+\delta)$. Therefore~$l \in JM_{l_1}$ and
\[
JN_{l_1} \subseteq JM_{l_1}.
\]
Therefore, by Theorem~\ref{theorem_1} and~\eqref{eq:4} we obtain
\[\pi_{l_1}(\operatorname{Pru}_{\alpha+\delta}(M)) = \bigcap_{l \in JM_{l_1}} M_l(-\delta)(-\alpha) \subseteq \bigcap_{l \in JN_{l_1}} N_l(-\delta)=\pi_{l_1}(\operatorname{Pru}_\delta(N)).\]
Since~$l_1$ and~$\delta$ were arbitrary,
\[
\forall \delta \geq 0, \forall l \in \{1,\dots,r\}, \quad \pi_l(\operatorname{Pru}_{\alpha+\delta}(M)) \subseteq \pi_l(\operatorname{Pru}_\delta(N)).
\]
Thus~$\forall \delta\geq 0, \Pru_{\alpha+\delta}(M) \text{ is an } \infty\text{-refinement of } \Pru_\delta(N)$, and we conclude that \[d_P(M,N) \leq d_B(M,N).\]
\end{proof}

Importantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,
\[
d_P(M,N) \leq d_I(M,N) \leq 2r\,d_P(M,N).
\]

\begin{theorem}\label{theorem_3}
 Let~$M$ and~$N$ be pfd upset-decomposable modules then
 \[d_P(M,N) \leq d_B(M,N).\]
 \end{theorem}
\begin{proof}
Let~$r = \operatorname{supdim}$~$M$, if~$\operatorname{supdim}$~$N \neq r$ then both distances~$= \infty$ so nothing to prove.\\
Suppose that~$\operatorname{supdim}$~$N = r$ i.e~$M=\bigoplus_{l=1}^r M_l$ and~$N=\bigoplus_{l=1}^r N_l.$ Let~$\alpha \geq 0$, such that there exists a matching between~$M$ and~$N$. This means that there exists a permutation~$\sigma$ of~$\{1,\dots,r\}$ such that
\[
M_l \subseteq N_{\sigma(l)}(\alpha) \quad \text{and} \quad N_l \subseteq M_{\sigma^{-1}(l)}(\alpha), \quad \forall l \in \{1,\dots,r\}.
\]
Define~$N' = \bigoplus_{l=1}^r N'_l$ where~$N'_l = N_{\sigma(l)}$, then~$N' \cong N$. We get that
 \begin{equation}
M_l \subseteq N'_l(\alpha) \quad \text{and} \quad N'_l \subseteq M_l(\alpha), \quad \forall l \in \{1,\dots,r\}.
 \tag{4}
\label{eq:4}
\end{equation}
From now on, to simplify the notation,~$N$ refers to~$N'$.\\
We want to show that~$\forall \delta\geq 0, \Pru_{\alpha+\delta}(M) \text{ is an } \infty\text{-refinement of } \Pru_\delta(N)$ and~$\Pru_{\alpha+\delta}(N)$ is an~$\infty\text{-refinement of } \Pru_\delta(M).$ We prove only the first statement; the second follows by symmetry. By Lemma~\ref{lem_infinit_refinement} it is sufficient to show that
\[
\forall \delta\geq 0, \, \forall l \in \{1,\dots,r\}, \quad \pi_{l}(\operatorname{Pru}_{\alpha+\delta}(M)) \subseteq \pi_{l}(\operatorname{Pru}_{\delta}(N)).
\]
Fix~$l_1 \in \{1,\dots,r\}$ and~$\delta \geq 0$. Let~$l \in JN_{l_1}$ in the directed graph~$G(N,\delta)$, then there exists a path
\[
P:\; N_{l_1}=N_{l'_0}\to N_{l'_1}\to\cdots\to N_{l'_{c-1}}\to N_{l'_{c}}=N_{l},
\]
for some integer~$c$.
By the construction of~$G(N,\delta)$ each arrow gives a containment shifted by~$2\delta$, hence
\[
N_{l'_0} \subseteq N_{l'_1}(2\delta), N_{l'_1} \subseteq N_{l'_2}(2\delta), \cdots , N_{l'_{c-1}} \subseteq N_{l'_{c}}(2\delta).
\]
By~\eqref{eq:4} we get that
\[
M_{l'_i} \subseteq N_{l'_i}(\alpha) \subseteq N_{l'_{i+1}}(\alpha+2\delta) \subseteq M_{l'_{i+1}}(2\alpha+2\delta), \quad \forall i \in \{0,\dots,c-1\}.
\]
So we get that there exists a path
\[
P':\; M_{l_1}=M_{l'_0}\to M_{l'_1}\to\cdots\to M_{l'_{c-1}}\to M_{l'_{c}}=M_{l},
\]
in the directed graph~$G(M,\alpha+\delta)$. Therefore~$l \in JM_{l_1}$ and
\[
JN_{l_1} \subseteq JM_{l_1}.
\]
Therefore, by Theorem~\ref{theorem_1} and~\eqref{eq:4} we obtain
\[\pi_{l_1}(\operatorname{Pru}_{\alpha+\delta}(M)) = \bigcap_{l \in JM_{l_1}} M_l(-\delta)(-\alpha) \subseteq \bigcap_{l \in JN_{l_1}} N_l(-\delta)=\pi_{l_1}(\operatorname{Pru}_\delta(N)).\]
Since~$l_1$ and~$\delta$ were arbitrary,
\[
\forall \delta \geq 0, \forall l \in \{1,\dots,r\}, \quad \pi_l(\operatorname{Pru}_{\alpha+\delta}(M)) \subseteq \pi_l(\operatorname{Pru}_\delta(N)).
\]
Thus~$\forall \delta\geq 0, \Pru_{\alpha+\delta}(M) \text{ is an } \infty\text{-refinement of } \Pru_\delta(N)$, and we conclude that \[d_P(M,N) \leq d_B(M,N).\]
\end{proof}

This result improves the right-hand inequality in Bjerkevik’s conjecture by reducing the constant from~$2r$ to~$2r-1$, and strengthens it by replacing the interleaving distance with the bottleneck distance, which dominates the interleaving distance. Moreover, in Example~\ref{ex_2} we show that the constant~$2r-1$ is optimal on the class of upset-decomposable persistence modules, for  both the interleaving distance and the bottleneck distance. The left-hand inequality follows from our analysis and is weaker than the conjectured inequality involving the interleaving distance.

Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.

Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in the following theorem. \begin{theorem}\label{theorem_1}
Let~$\alpha \geq 0$ and let~$M = \bigoplus_{l=1}^r M_l$ be an upset-decomposable module. Then
\[
\pi_{l_1}(\operatorname{Pru}_\alpha(M)) = \bigcap_{l \in JM_{l_1}} M_l(-\alpha),
\]
where~$JM_{l_1}$ is as defined above.
\end{theorem}
To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.

If~$M$ and~$N$ are two upset-decomposable modules of pointwise dimension at most~$r$, then
\[
d_P(M,N) \leq d_B(M,N) \leq (2r-1)d_P(M,N).
\],

Question: Let $M$ and $N$ be persistence modules indexed by $(\mathbb{R}^n,\leq)$ with the product order. Assume that both are upset-decomposable, meaning that each can be written as a direct sum of interval persistence modules supported on subsets $U\subseteq \mathbb{R}^n$ that are upward closed under the product order, and assume their pointwise dimensions are bounded by $r$ (i.e. at every parameter value, the corresponding vector space has dimension at most $r$). If $d_P$ denotes the pruning distance (defined via the $\varepsilon$-prunings $\operatorname{Pru}_\varepsilon$) and $d_B$ denotes the bottleneck distance for these decompositions, which quantitative estimate holds?

For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N) \le (2r-1)\, d_P(M,N).
\]

Choices:
B.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N) \le 2r\, d_P(M,N).
\]
C.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N).
\]
D.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_B(M,N) \le d_P(M,N) \le (2r-1)\, d_B(M,N).
\]
E.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N) \le (2^r-1)\, d_P(M,N).
\]

sharp constant from reachable-index/counting argument

dropped the upper Lipschitz bound $d_B\le(2r-1)d_P$

direction of comparison between pruning and bottleneck distances

linear dependence on pointwise dimension $r$ replaced by exponential dependence

The stem does not explicitly reveal the correct option. It states the hypotheses and asks which distance comparison is valid, without embedding the sharp constant or direction of the inequalities.

This is essentially a theorem-recall item: the stem lists the exact hypotheses and asks for the corresponding conclusion. It closely mirrors a standard theorem statement rather than testing transfer or application.

There is some pressure to distinguish a sharp equivalence from weaker, reversed, or mis-constant variants, but the item mainly rewards memory of the theorem rather than substantial derivation or generative reasoning.

The distractors are plausible and mathematically targeted: one has a slightly wrong constant, one gives only a weaker true statement, one incorrectly removes dependence on r, and one reverses the inequality direction. These reflect realistic failure modes.

A technically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, yet it is largely tautological and only moderately tests reasoning.


Question: Let $M$ and $N$ be persistence modules indexed by $(\mathbb{R}^n,\leq)$ with the product order. Assume that both are upset-decomposable, meaning that each can be written as a direct sum of interval persistence modules supported on subsets $U\subseteq \mathbb{R}^n$ that are upward closed under the product order, and assume their pointwise dimensions are bounded by $r$ (i.e. at every parameter value, the corresponding vector space has dimension at most $r$). If $d_P$ denotes the pruning distance (defined via the $\varepsilon$-prunings $\operatorname{Pru}_\varepsilon$) and $d_B$ denotes the bottleneck distance for these decompositions, which quantitative estimate holds?

Choices:
B.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N) \le 2r\, d_P(M,N).
\]
C.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N).
\]
D.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_B(M,N) \le d_P(M,N) \le (2r-1)\, d_B(M,N).
\]
E.
For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has
\[
d_P(M,N) \le d_B(M,N) \le (2^r-1)\, d_P(M,N).
\]

The stem does not explicitly reveal the correct inequality or constant. It specifies the setup and asks which estimate holds, so there is no direct answer leakage beyond the theorem context itself.

This is essentially a direct recall of a specific theorem statement: under the listed hypotheses, identify the exact quantitative bound. It does not substantially transform the underlying result into a new inference task.

There is some reasoning pressure because the student must distinguish the sharp bound from a weaker true statement and several near-miss variants. However, the task is still primarily theorem recognition/recall rather than genuine derivation.

The distractors are strong: one is a close constant perturbation, one is a weaker true statement, one reverses the comparison, and one introduces an implausibly stronger growth rate. These reflect realistic mathematical confusions.

A mathematically well-constructed recall-style MCQ with strong distractors, but it is close to a theorem restatement and only moderately tests generative reasoning.


\begin{mainthm}\label{thm:chordarc}
    Let $3\leq n\leq 5$, \(\Sigma^n \subset \mathbb{R}^{n+1}\) be a two-sided, embedded minimal disk with \(\sup_{\Sigma}|A| < \infty\).
    Then there exists a constant \(c = c\bigl(n,\sup_{\Sigma}|A|\bigr) > 0\) with the following property: for every \(x\in \Sigma\) and every \(R>1\) such that the intrinsic ball \(\mathcal{B}_{R}(x)\subset \Sigma\), the connected component \(\Sigma_{x, R}\) of the Euclidean ball \(B_{R}(x)\cap \Sigma\) containing \(x\) satisfies
    \[
        \Sigma_{x, R}\subset \mathcal{B}_{cR^n}(x).
    \]
    In particular, intrinsic distances are quantitatively controlled by extrinsic distances on \(\Sigma_{x, R}\), with constants depending only on the dimension and the curvature bound.
\end{mainthm}

In 1965, E. Calabi \cite{Ca} proposed the following conjecture:
\begin{con}\label{con:1}
Let $n\geq 2$. A complete minimal hypersurface $\Sigma^{n}\subset \mathbb{R}^{n+1}$ must be unbounded.
\end{con}
When $n=2$, Nadirashvili \cite{Na1} constructed a complete minimal immersion of a disk into the unit ball, thus disproving the conjecture for minimal immersions in $\mathbb{R}^3$. Conjecture \ref{con:1} also appeared in \cite[pg.~212]{chern1966geometry} and was later highlighted in Yau’s survey \cite[pg.~360]{Ya2}, where, referring to Nadirashvili's construction, Yau asked what geometric properties such examples must satisfy. In particular, Yau asked whether any such bounded minimal surface can be embedded. This was addressed in \cite{CM10}, where Colding--Minicozzi proved the Calabi--Yau conjectures by showing that any complete embedded minimal surface in $\mathbb{R}^3$ with finite topology is proper and therefore unbounded. It is therefore natural to ask what remains true in higher dimensions, that is, for $n\ge 3$.

Our first main result proves Conjecture \ref{con:1} under a uniform curvature bound, via a chord--arc estimate for embedded minimal disks:

Let \(\Sigma \subset \bbR^n\) be an embedded minimal hypersurface with bounded curvature, i.e. \(\sup_\Sigma |A| < \infty\), and let \(x \in \Sigma\). For \(R > 0\), we let \(B_R(x)\) denote an extrinsic ball of radius \(R\) in \(\bbR^{n+1}\), and we let \(\calB_R(x)\) denote an intrinsic ball of radius \(R\) in \(\Sigma\). We denote by \(\Sigma_{x, R}\) the component of \(\Sigma \cap B_R(x)\) containing \(x\).
Furthermore, since we assume $\Sigma$ to have bounded curvature, we can find a graphical radius as follows:
\begin{lem}\label{graphical radius and chord arc for graphical radius}
 Let \(\Sigma^{n} \subset \mathbb{R}^{n+1}\) be an embedded minimal surface with bounded curvature \(\sup_{\Sigma}|A| \le C_A\). Then there is a number \(0 < R_{\Sigma} < 1/C_A\) such that for all \(x \in \Sigma\) and \(R \le R_{\Sigma}\), the following properties hold:
 \begin{enumerate}
 \item The intrinsic ball \(\mathcal{B}_{R}(x)\) is graphical.
 \item \(\calB_R(x)\) can be written as a graph of a function \(u: \Omega \subset T_x\Sigma \rightarrow \mathbb{R}^{n + 1}\) such that \(|\nabla u| < 1\) and \(|\nabla^2 u| < c(C_A)\).
 \item There exists a \(\delta_c > 0\) depending only on dimension and the curvature bound \(C_A\) such that
 \begin{equation}\label{eq: graphical weak chord arc}
 \Sigma_{x, \delta_cR} \subset \mathcal{B}_{R/2}(x).
 \end{equation}
 \end{enumerate}
 Importantly, the number \(R_\Sigma\) depends only on the curvature bound; in particular, this means that \(\calB_{R_\Sigma}(x)\) may contain points in \(\partial \Sigma\).

The final ingredient needed is based on the recent work on Stable Bernstein Conjecture by \cite{chodosh2023stable,CL, catino2024two,chodosh2024stable,mazet2024stable} and its extension to the $\delta$-stable case by \cite{hong2024delta,cheng2025complete}.
\begin{thm}\label{thm:delta-stable-flat}
For $3 \leq n \leq 5$ and $\delta\in (0,1-\delta_1(n))$, an $n$-dimensional complete two-sided $\delta$--stable minimal hypersurface $\Sigma^n \subset \mathbb{R}^{n+1}$ is a hyperplane, where $\delta_1(3)=3 / 8$, $\delta_1(4)=2 / 3$ and $\delta_1(5)=21 / 22$.
\end{thm}
By a standard blow up argument, Theorem \ref{thm:delta-stable-flat} implies the following curvature estimate,
\begin{cor}\label{cor:curvature-estimate}
Let $3 \leq n \leq 5$ and \(\delta \in (0, 1-\delta_1(n))\) as in Theorem \ref{thm:delta-stable-flat}. There exists \(C(\delta, n) < \infty\) such that if \(\Sigma^n \subset \mathbb{R}^{n+1}\) is a two-sided, \(\delta\)-stable minimal immersion, then
\[|A_M|d(p, \partial M) \le C(\delta, n).\]
\end{cor}
We now ready to give a proof of Theorem \ref{thm:chordarc}.
\begin{proof}[Proof of Theorem \ref{thm:chordarc}]
For clarity, we will rename \(R\) in the theorem statement to \(R_0\) in this proof. Because \(\Sigma\) is an embedded disk, \(\Sigma_{x, R_0}\) is also an embedded disk by the maximum principle and the monotonicity of topology (cf. \cite{CM1}). Suppose that \(\Sigma_{x, R_0}\) were not contained in an intrinsic ball \(\mathcal{B}_{2NR_{\Sigma}}(x)\) for any integer \(N\), where $R_\Sigma>0$ is defined in Lemma \ref{graphical radius and chord arc for graphical radius}. Then we can choose points \(z_i \in \partial \mathcal{B}_{2R_{\Sigma}i} \cap \Sigma_{x, R_0}\) for every \(i\), as well as a curve \(\sigma \subset \Sigma_{x, R_0}\) which passes through each \(z_i\). This sequence is contained in the ball \(B_{R_0}\), and thus is Cauchy. Choose \(z_{i_1}, z_{i_2} \in \{z_i\}\) with
\[|z_{i_1} - z_{i_2}| < \veps < \veps_0\]
where \(\veps_0\) is produced from Lemma \ref{lem: lemma 2.11 in CM}. For \(r_{\Sigma} < R_\Sigma/2\) the intrinsic balls \(\mathcal{B}_{r_{\Sigma}}(z_{i_j})\) are graphical and disjoint. Reducing \(r_\Sigma\) until it is less than the radius \(r_0\) from Lemma \ref{lem: lemma 2.11 in CM}, the balls \(\mathcal{B}_{r_{\Sigma}}(z_{i_1})\) and \(\mathcal{B}_{r_{\Sigma}}(z_{i_2})\) are both graphs over \(T_{z_{i_1}}\Sigma\), and it is possible to define \(u: \mathcal{B}_{r_{\Sigma}}(z_{i_1}) \rightarrow \mathbb{R}\) such that \(x + u(x)\mathbf{n}_{\Sigma}(x) \in \calB_{r_\Sigma}(z_{i_2})\) for \(x \in \mathcal{B}_{r_{\Sigma}}(z_{i_1})\) with \(|\nabla u| + |u||A| < 1\).

By our original construction, the surfaces \(\Sigma_i\) lie above the plane \(\{x_n = 0\}\), because before the rotation of \(\mathbb{R}^n\) they were in the interior of \(B_{R_0}(0)\). By the maximum principle, the graph of \(u\) must lie entirely in the plane \(\{x_n = 0\}\) or go below the plane \(\{x_n = 0\}\), since the graph is a minimal surface and \(u(0) = 0\). But then it follows that some \(\Sigma_i\) leaves the ball \(B_{R_0}\), a contradiction. This completes the proof.
\end{proof}
Finally, while we cannot prove that a complete embedded minimal hypersurface $\Sigma^n\subset \R^{n+1}$ with bounded curvature is proper, we observe that $\bar{\Sigma}$ is a minimal lamination of $\R^{n+1}$. Recall that a minimal lamination of a smooth manifold $M^{n+1}$ is defined as follows (cf. Appendix B of \cite{CM6})
\begin{defn}[Minimal lamination]\label{def:minimal-lamination}
Let $M^{n+1}$ be an $(n+1)$-dimensional smooth manifold.
A \emph{codimension-one lamination} of $M$ is a collection $\mathcal L$ of smooth, disjoint $n$-dimensional submanifolds of $M$ (called \emph{leaves})
such that:
\begin{enumerate}
 \item The union $\bigcup_{\Lambda\in\mathcal L}\Lambda$ is a closed subset of $M$.
 \item For every $x\in M$ there exists an open neighborhood $U\ni x$ and a coordinate chart
 $\Phi:U\to \Phi(U)\subset \mathbb R^{n+1}$ with the property that, in these coordinates,
 the leaves of $\mathcal L$ pass through $\Phi(U)$ in slices of the form $(\mathbb R^n\times\{0\})\cap \Phi(U)$.
\end{enumerate}
A \emph{minimal lamination} is a lamination whose leaves are minimal. A \emph{limit leaf} is a leaf $\Lambda$ of $\calL$ that is contained in the closure of $\calL\setminus \Lam$ (cf. Definition 4.3.2 in \cite{zhou2024geometry}).
\end{defn}
Furthermore, by the recent resolution of the Stable Bernstein Conjecture in $\R^{n+1}$ for $3\leq n \leq 5$ we can now show that the limit leaves of $\bar{\Sigma}$ are flat, which yields the following trichotomy:
\begin{lem}
Let $3\leq n\leq 5$. Suppose $\Sigma^n\subset \R^{n+1}$ is a complete connected embedded minimal surface in $\mathbb{R}^{n+1}$ with $|A_\Sigma|\leq C_A <\infty$. Then \(\overline{\Sigma}\) is a minimal lamination, and one of the following holds:
\begin{enumerate}
 \item $\Sigma$ is properly embedded in $\mathbb{R}^{n+1}$,
 \item $\Sigma$ is properly embedded in an open halfspace of $\mathbb{R}^{n+1}$ with limit set the boundary plane of this halfspace, or
 \item $\Sigma$ is properly embedded in an open slab of $\mathbb{R}^{n+1}$ with limit set consisting of the boundary planes.
\end{enumerate}
\end{lem}
\begin{proof}
The proof follows from the same argument as in the proof of Lemma 1.1 in \cite{meeks2005uniqueness} where the authors, using the bound on the curvature, first show that $\bar{\Sigma}$ is a minimal lamination of $\R^{n+1}$. Next, by considering the universal cover $\hat{\Lam}$ of any limit leaf $\Lam$ one can show that $\hat{\Lam}$ is stable since any compact domain $\hat{D}$ of $\hat{\Lam}$ is a limit of disjoint minimal domains, which in turn implies that $\hat{D}$ is stable by a contradiction argument. Therefore, $\Lam$ is stable, which by the works of \cite{chodosh2023stable,CL, catino2024two,chodosh2024stable,mazet2024stable} implies that is $\Lam$ is a plane, thus completing the proof of the Lemma.
\end{proof}

The second part of the paper addresses the stronger version of the Calabi--Yau conjecture in higher dimensions, namely properness. While properness holds for complete embedded minimal surfaces in \(\mathbb{R}^3\) by \cite{CM10}, we show that the natural higher-dimensional analogue fails in general:
\begin{mainthm}\label{thm:nonproper}
For every \(n\ge 3\), there exists a complete, embedded, minimal hypersurface \(\Sigma_\infty^{n}\subset \mathbb{R}^{n+1}\) that is not proper.
\end{mainthm}

The restrictions on the dimensions in Theorem \ref{thm:chordarc} arise from curvature estimates coming from the recent resolution of the Stable Bernstein conjecture in dimensions $3\leq n \leq 5$. Since Theorem \ref{thm:chordarc} implies unboundedness for complete embedded minimal hypersurfaces with bounded curvature in dimensions $3\leq n\leq 5$, we also include, for completeness, a proof of unboundedness in all dimensions $n\geq 3$. While this fact seems well known to experts (cf.\ \cite{Carlotto_2016} and the proof of Theorem 2 in \cite{rosenberg2001intersection}), we have not found a detailed argument in the literature that applies uniformly to all dimensions.
\begin{mainthm}\label{thm:unbounded}
Let \(\Sigma^{n} \subset \mathbb{R}^{n+1}\) be a complete, embedded minimal hypersurface and assume that \(\sup_{\Sigma}|A|^2 < \infty\). Then \(\Sigma\) is unbounded.
\end{mainthm}

Extending the above result to show properness for complete embedded minimal hypersurfaces with bounded curvature seems nontrivial, in part due to the failure of the half-space theorem. We expect that with suitable topological assumptions one could get around these issues. However, we can show that for a complete, improperly embedded minimal surface \(\Sigma\), the closure \(\bar \Sigma\) is a minimal lamination, as defined in \cite{CM6} Appendix B with planar limit leaves in ambient dimensions $3\leq n \leq 5.$

At a conceptual level, the proofs of Theorem \ref{thm:chordarc} and Theorem \ref{thm:unbounded} follow the strategy of \cite{CM10} in the bounded-curvature regime: when two embedded minimal sheets are very close but disjoint, their separation function satisfies a uniformly elliptic equation whose coefficients are controlled by \(|A|\); a Harnack inequality then yields quantitative control on how fast the sheets can separate.

Theorem \ref{thm:nonproper} is inspired by the work of Fakhi and Pacard \cite{Fakhi-Pacard} and Kapouleas \cite{MR1601434}. Fahki and Pacard construct complete, immersed minimal surfaces in \(\mathbb{R}^{n+1}\), \(n \ge 3\), with finite total curvature and arbitrarily many ends, and Kapouleas produces complete, embedded minimal surfaces in \(\mathbb{R}^3\) with finite total curvature and arbitrarily many ends. These results have also been generalized by Coutant in \cite{coutant2012deformation} to higher dimensions where, amongst other things, the author constructs examples of complete, embedded minimal hypersurfaces in \(\mathbb{R}^{n+1}\) with finite total curvature and arbitrarily many ends. In our work, we modify Fahki and Pacard's construction to produce embedded examples:
\begin{mainthm}\label{thm:k-ends}
For \(n\ge 3\) and \(k \ge 1\), there exists a complete, embedded, minimal hypersurface \(\Sigma_k^{n}\subset \mathbb{R}^{n+1}\) that has finite total curvature and \(k\) ends.
\end{mainthm}
Notably, both Kapouleas \cite{MR1601434} and Fahki-Pacard \cite{Fakhi-Pacard} produce their examples by gluing catenoidal necks. We proceed similarly by iteratively gluing catenoids to form a sequence of surfaces \(\Sigma_k^n\) with \(k\)-ends. To prove Theorem \ref{thm:nonproper}, we will construct such a sequence \(\Sigma_k^n\) that converges locally smoothly to a hypersurface \(\Sigma_\infty^n\) that is complete, embedded, and improper. To our knowledge, this gives the first example of an improperly embedded minimal hypersurface in $\mathbb{R}^{n+1}$. The constructed hypersurface has unbounded curvature, infinite genus, and infinite ends. In particular, it shows that the properness conjecture fails for embedded minimal hypersurfaces of infinite topology in higher dimensions. By contrast, in the surface case $n=2$, the results of \cite{CM10, meeks2021embedded} indicate that properness for embedded minimal surfaces should hold under the additional assumption of finite genus and therefore it would be interesting to find an analogue of Theorem \ref{thm:nonproper} when $n=2.$

Finally, we direct the reader to the survey \cite{MR4415896} by Breiner, Kapouleas, and Kleene for an overview on the state of gluing constructions for constant mean curvature surfaces.

\subsection{Acknowledgments}
The authors would like to thank their advisors Tobias Colding and William Minicozzi for many helpful discussions and encouragement. We are also grateful for stimulating discussions with Otis Chodosh, David Jersion and Peter McGrath. The first author acknowledges support from the Simons Dissertation Fellowship and the second author acknowledges support from the National Science Foundation.

At a conceptual level, the proofs of Theorem \ref{thm:chordarc} (and Theorem \ref{thm:unbounded}) “follow the strategy of \cite{CM10} in the bounded-curvature regime”: when two embedded minimal sheets are “very close but disjoint,” their “separation function satisfies a uniformly elliptic equation whose coefficients are controlled by \(|A|\)”; then “a Harnack inequality” gives “quantitative control on how fast the sheets can separate.”

At a conceptual level, the proof of the main theorem (and the following theorem) “follow the strategy of \cite{CM10} in the bounded-curvature regime”: when two embedded minimal sheets are “very close but disjoint,” their “separation function satisfies a uniformly elliptic equation whose coefficients are controlled by \(|A|\)”; then “a Harnack inequality” gives “quantitative control on how fast the sheets can separate.” We first prove the following theorem.

\begin{mainthm}\label{thm:unbounded}
Let \(\Sigma^{n} \subset \mathbb{R}^{n+1}\) be a complete, embedded minimal hypersurface and assume that \(\sup_{\Sigma}|A|^2 < \infty\). Then \(\Sigma\) is unbounded.
\end{mainthm}

Existential–Universal
Inequality or Bound

Question: Let $3\le n\le 5$, and let $\Sigma^n\subset \mathbb{R}^{n+1}$ be a two-sided, embedded minimal disk with bounded curvature, i.e. $\sup_{\Sigma}|A|<\infty$. For $x\in \Sigma$, let $\mathcal B_R(x)$ denote the intrinsic ball of radius $R$ in $\Sigma$ (with respect to the induced metric), let $B_R(x)$ denote the Euclidean ball of radius $R$ in $\mathbb{R}^{n+1}$, and let $\Sigma_{x,R}$ be the connected component of $B_R(x)\cap \Sigma$ that contains $x$. Which of the following quantitative estimates holds for every $x\in\Sigma$ and every $R>1$ such that $\mathcal B_R(x)\subset \Sigma$?

There exists a constant $c=c\bigl(n,\sup_{\Sigma}|A|\bigr)>0$ such that for every $x\in\Sigma$ and every $R>1$ with $\mathcal B_R(x)\subset \Sigma$, one has
\[
\Sigma_{x,R}\subset \mathcal B_{cR^n}(x).
\]
Equivalently, on each such component $\Sigma_{x,R}$, intrinsic distance is quantitatively controlled by extrinsic distance, with constants depending only on $n$ and the curvature bound.

Choices:
B.
There exists a constant $c=c\bigl(n,\sup_{\Sigma}|A|\bigr)>0$ such that for every $x\in\Sigma$ and every $R>1$ with $\mathcal B_R(x)\subset \Sigma$, one has
\[
\Sigma_{x,R}\subset \mathcal B_{cR}(x).
\]
Equivalently, on each such component $\Sigma_{x,R}$, intrinsic distance is quantitatively controlled linearly by extrinsic distance, with constants depending only on $n$ and the curvature bound.
C.
There exists a constant $c=c\bigl(n,\sup_{\Sigma}|A|\bigr)>0$ such that for every $x\in\Sigma$ and every $R>1$ with $\mathcal B_R(x)\subset \Sigma$, one has
\[
\Sigma_{x,R}\subset \mathcal B_{cR^{n+1}}(x).
\]
D.
For every $x\in\Sigma$ and every $R>1$ with $\mathcal B_R(x)\subset \Sigma$, there exists a constant $c=c\bigl(x,R,n,\sup_{\Sigma}|A|\bigr)>0$ such that
\[
\Sigma_{x,R}\subset \mathcal B_{cR^n}(x).
\]
Equivalently, intrinsic distance is quantitatively controlled by extrinsic distance on each $\Sigma_{x,R}$, but the controlling constant may depend on the center and scale.
E.
There exists a constant $c=c\bigl(n,\sup_{\Sigma}|A|\bigr)>0$ such that for every $x\in\Sigma$ and every $R>1$ with $\mathcal B_R(x)\subset \Sigma$, the entire intersection with the Euclidean ball satisfies
\[
B_R(x)\cap \Sigma\subset \mathcal B_{cR^n}(x).
\]
Equivalently, every point of $B_R(x)\cap \Sigma$ has intrinsic distance at most $cR^n$ from $x$, with constants depending only on $n$ and the curvature bound.

The stem does not explicitly reveal choice A, nor does it contain a giveaway phrase uniquely matching the correct option. It asks for the valid uniform existence statement, so the answer must be inferred from the options.

This is essentially a theorem-recall item: the correct option states the target conclusion almost verbatim, with the task being to recognize the precise formulation rather than derive a new consequence.

Some reasoning is required to track quantifiers, uniformity of the constant, the radius threshold, and the growth exponent. However, the main burden is careful recognition of the exact theorem statement rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: they vary the radius range, uniformity in the constant, direction of containment, and the exponent. These reflect realistic failure modes in parsing geometric estimates.

A technically well-constructed theorem-identification MCQ with strong distractors and no obvious answer leakage, but it scores poorly on tautology avoidance because it mainly tests recognition of the exact statement rather than genuine derivational reasoning.


\label{MainThm}
		$(T_t^{(2)})_{t\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies
		\[
		\widetilde{c}(\beta) (p - 1) \leq e^{2\tau t_p} \leq \widetilde{C}(\beta) (p - 1), \quad \forall\, 2 < p < \infty,
		\]
		where $\widetilde{c}(\beta)$ and $\widetilde{C}(\beta)$ are positive constants depending only on $\beta$.

\label{Introduction}
	In the celebrated papers \cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.

	Gross discovered the equivalence of log-Sobolev inequalities and hypercontractivity, and also gave a new proof of the hypercontractivity for the classical Ornstein-Uhlenbeck semigroup in the remarkable work \cite{LG1975}. This surprising observation led to the application of hypercontractivity in numerous fields, such as Boolean analysis, concentration inequalities, geometric inequalities, statistical physics, among others. We refer to \cite{LG1993,BGL2013} for more information.

	Quantum Markov semigroups, which are a generalization of classical Markov semigroups, are a fundamental tool to describe open quantum systems. In \cite{CFL2000}, Cipriani \emph{et al.} rigorously proved that a specific unbounded Lindblad-type operator generates a quantum Markov semigroup by virtue of the remarkable quantum semigroup theory established in \cite{C1992,C1997}. Such unbounded Lindblad-type operator has been extensively studied in quantum optics models of masers and lasers \cite{D1976,FRS1994}. The quantum Markov semigroup appearing in \cite{CFL2000}, which is now called quantum Ornstein-Uhlenbeck semigroup, has also been shown to be hypercontractive in the seminal work \cite{CS2008}.

	The need to construct quantum Markov semigroups on von Neumann algebras, which are symmetric with respect to a nontracial state, is clear for various applications to open quantum systems, quantum statistical mechanics and quantum information (see \cite{KP} for more information). In the remarkable paper \cite{KP}, Ko \emph{et al.} constructed a family of quantum Ornstein-Uhlenbeck semigroups, which can be seen as a generalization of the quantum Ornstein-Uhlenbeck semigroup appearing in \cite{CFL2000, D1976,FRS1994}.

	In this paper, we aim to study hypercontractivities of the quantum Ornstein-Uhlenbeck semigroups established in \cite{KP}. At first, we introduce such quantum Ornstein-Uhlenbeck semigroups. To this end, we present Weyl operators and CCR.

	Let $H=\ell_2(\mathbb{N})$, where $\mathbb{N}=\{0,1,2, \cdots\}$ is the set of all natural numbers. Let $\{e_n\}_{n=0}^\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows
	\[
	a^* e_n = \sqrt{n+1} \, e_{n+1}, \quad a e_n = \sqrt{n} \, e_{n-1}, \quad (e_{-1} := 0).
	\]
	It is clear that they satisfy the canonical commutation relation (CCR)
	\[
	[a, a^*] = 1,
	\]
	where $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as
	\[
	N e_n = n \cdot e_n.
	\]
	The position and momentum operators are defined as
	\[
	Q = \frac{1}{\sqrt{2}}(a + a^*), \quad P = \frac{1}{\sqrt{2}i}(a - a^*),
	\]
	and so
	\[
	[Q, P] = i \cdot 1.
	\]
	For any $z \in \mathbb{C}$, define the Weyl operators
	\[
	W(z) = e^{\frac {i}{\sqrt{2}}(z a^* + \bar{z} a)} = e^{i ( \Re z \cdot Q + \Im z \cdot P ) }.
	\]
	These operators satisfy the Weyl relation
	\[
	W(z)^*=W(-z), \quad \text{and} \quad \forall z, w\in\mathbb{C}, \ W(z) W(w) = e^{-\frac{i}{2} \Im (\bar{z} w)} W(z + w).
	\]
	Let $\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\mathcal{A}\subsetneqq B(H)$ and $\mathcal{A}{''}=B(H)$. Let $\omega$ be the normal faithful state on $B(H)$, which is defined by
	$$ \omega(W(z))=e^{-\frac{|z|^2}{4} (1+e^{-\beta})(1-e^{-\beta})^{-1}}, \quad \forall \, z\in \mathbb{C}  $$
	where $\beta>0$ is a fixed inverse temperature. Note that $e^{-\beta N}$ is a trace class operator in $B(H)$. Let
	$$ \rho=\dfrac{e^{-\beta N}}{\text{Tr}(e^{-\beta N})}=(1-e^{-\beta})e^{-\beta N}. $$
	Indeed, $\omega$ is a Gibbs state, and for any $x\in B(H)$,
	$$  \omega(x)=\text{Tr}(\rho x).  $$
	We refer the interested reader to \cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \cite{KP}. Let the parameters $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{R}$ satisfy the following relations
	\begin{align*}
		& \frac{1}{2}\left(1+\alpha_2^2\right) \sinh (\beta / 2)=-\alpha_1 \cosh (\beta / 2), \\
		& \frac{1}{2}\left(\alpha_1^2+\alpha_3^2\right) \sinh (\beta / 2)=\alpha_2 \alpha_3 \cosh (\beta / 2).
	\end{align*}
	Let $G$ be the elliptic operator on $B(H)$ given by
	\begin{equation*}
		\begin{aligned}
			G(A)=\frac{\gamma}{2}\left(1+\alpha_2^2\right)[P,[P, A]]&+\frac{\gamma}{2}\left(\alpha_1^2+\alpha_3^2\right)[Q,[Q, A]]-i \gamma \alpha_1(Q[P, A]+[P, A] Q)\\
			&-i \gamma \alpha_2 \alpha_3(P[Q, A]+[Q, A] P), \quad \quad \forall \, A\in B(H),
		\end{aligned}
	\end{equation*}
	where $\gamma=\left(1+\alpha_2^2\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\mathcal{M}$, the above is a formal expression.

	For $0<p\leq \infty$, let $S_p$ be the Schatten $p$-class of $B(H)$ with the usual Schatten $p$-norm denoted by $\|\cdot\|_p$. We use Kosaki's definition \cite{Ko1984} for noncommutative $L_p$ spaces.
	Let $L_p(\rho)$ be the closure of all elements in $\rho^{1/2}B(H)\rho^{1/2}$ with respect to the following norm
	$$  \| \rho^{1/2} x \rho^{1/2} \|_{L_p(\rho)}:= \| \rho^{1/2p} x \rho^{1/2p}\|_p, \quad  \forall \, x\in B(H). $$
	Define $\mathcal{G}^{(2)}$ on $S_2$ by
	$$  \mathcal{G}^{(2)}(\rho^{1/4}x\rho^{1/4})=\rho^{1/4}G(x)\rho^{1/4}, \quad \forall \, x\in B(H).  $$
	According to \cite{KP}, $P_t= e^{-t\mathcal{G}^{(2)}}$ ($\forall \, t\geq 0$) is a symmetric semigroup on $S_2$. For any $x\in B(H)$, define
	$$  \rho^{1/4} T_t^{(\infty)}(x) \rho^{1/4}= P_t(\rho^{1/4}x\rho^{1/4}).  $$
	Then $T_t^{(\infty)}$ is an ergodic quantum Markov semigroup on $S_\infty$, and $\omega$ is the unique invariant state associated with $T_t^{(\infty)}$. See \cite{KP} for more details.

	For $1\leq p<\infty$, denote by $T_t^{(p)}: L_p(\rho)\rightarrow L_p(\rho)$ the usual restriction of $T_t^{(\infty)}$ to $L_p(\rho)$, by using the same way as in \cite{GL1995}. More explicitly, for $p=2$, if $x\in B(H)$, then
	$$  T_t^{(2)}(\rho^{1/2}x\rho^{1/2})=\rho^{1/4}P_t(\rho^{1/4}x\rho^{1/4})\rho^{1/4}. $$
	Denote by $\tau$ the spectral gap of $T_t^{(2)}$, which is defined in \eqref{gap}.

	We present the definition of hypercontractivity. Let $(T_t)_{t\geq 0}$ be an ergodic quantum Markov semigroup with the invariant state induced by $\rho$. Then $(T_t)_{t\geq 0}$ is said to be hypercontractive if for any $2<p<\infty$, there exists $t_p>0$ such that for any $t \geq t_p$, and for any $ x\in B(H) $ with $\omega(x)=0$,
	\begin{equation}\label{hypercontractivity}
		\| T_t(\rho^{1/2}x\rho^{1/2}) \|_{L_p(\rho)} \leq \|\rho^{1/2}x\rho^{1/2}\|_{L_2(\rho)}.
	\end{equation}
	Moreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.

	Our result is the following theorem.

\label{Introduction}
 In the celebrated papers \cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.

Let $H=\ell_2(\mathbb{N})$, where $\mathbb{N}=\{0,1,2, \cdots\}$ is the set of all natural numbers. Let $\{e_n\}_{n=0}^\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows
 \[
 a^* e_n = \sqrt{n+1} \, e_{n+1}, \quad a e_n = \sqrt{n} \, e_{n-1}, \quad (e_{-1} := 0).
 \]
 It is clear that they satisfy the canonical commutation relation (CCR)
 \[
 [a, a^*] = 1,
 \]
 where $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as
 \[
 N e_n = n \cdot e_n.
 \]
 The position and momentum operators are defined as
 \[
 Q = \frac{1}{\sqrt{2}}(a + a^*), \quad P = \frac{1}{\sqrt{2}i}(a - a^*),
 \]
 and so
 \[
 [Q, P] = i \cdot 1.
 \]
 For any $z \in \mathbb{C}$, define the Weyl operators
 \[
 W(z) = e^{\frac {i}{\sqrt{2}}(z a^* + \bar{z} a)} = e^{i ( \Re z \cdot Q + \Im z \cdot P ) }.
 \]
 These operators satisfy the Weyl relation
 \[
 W(z)^*=W(-z), \quad \text{and} \quad \forall z, w\in\mathbb{C}, \ W(z) W(w) = e^{-\frac{i}{2} \Im (\bar{z} w)} W(z + w).
 \]
 Let $\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\mathcal{A}\subsetneqq B(H)$ and $\mathcal{A}{''}=B(H)$. Let $\omega$ be the normal faithful state on $B(H)$, which is defined by
 $$ \omega(W(z))=e^{-\frac{|z|^2}{4} (1+e^{-\beta})(1-e^{-\beta})^{-1}}, \quad \forall \, z\in \mathbb{C} $$
 where $\beta>0$ is a fixed inverse temperature. Note that $e^{-\beta N}$ is a trace class operator in $B(H)$. Let
 $$ \rho=\dfrac{e^{-\beta N}}{\text{Tr}(e^{-\beta N})}=(1-e^{-\beta})e^{-\beta N}. $$
 Indeed, $\omega$ is a Gibbs state, and for any $x\in B(H)$,
 $$ \omega(x)=\text{Tr}(\rho x). $$
 We refer the interested reader to \cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \cite{KP}. Let the parameters $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{R}$ satisfy the following relations
 \begin{align*}
 & \frac{1}{2}\left(1+\alpha_2^2\right) \sinh (\beta / 2)=-\alpha_1 \cosh (\beta / 2), \\
 & \frac{1}{2}\left(\alpha_1^2+\alpha_3^2\right) \sinh (\beta / 2)=\alpha_2 \alpha_3 \cosh (\beta / 2).
 \end{align*}
 Let $G$ be the elliptic operator on $B(H)$ given by
 \begin{equation*}
 \begin{aligned}
 G(A)=\frac{\gamma}{2}\left(1+\alpha_2^2\right)[P,[P, A]]&+\frac{\gamma}{2}\left(\alpha_1^2+\alpha_3^2\right)[Q,[Q, A]]-i \gamma \alpha_1(Q[P, A]+[P, A] Q)\\
 &-i \gamma \alpha_2 \alpha_3(P[Q, A]+[Q, A] P), \quad \quad \forall \, A\in B(H),
 \end{aligned}
 \end{equation*}
 where $\gamma=\left(1+\alpha_2^2\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\mathcal{M}$, the above is a formal expression.

For $0<p\leq \infty$, let $S_p$ be the Schatten $p$-class of $B(H)$ with the usual Schatten $p$-norm denoted by $\|\cdot\|_p$. We use Kosaki's definition \cite{Ko1984} for noncommutative $L_p$ spaces.
 Let $L_p(\rho)$ be the closure of all elements in $\rho^{1/2}B(H)\rho^{1/2}$ with respect to the following norm
 $$ \| \rho^{1/2} x \rho^{1/2} \|_{L_p(\rho)}:= \| \rho^{1/2p} x \rho^{1/2p}\|_p, \quad \forall \, x\in B(H). $$
 Define $\mathcal{G}^{(2)}$ on $S_2$ by
 $$ \mathcal{G}^{(2)}(\rho^{1/4}x\rho^{1/4})=\rho^{1/4}G(x)\rho^{1/4}, \quad \forall \, x\in B(H). $$
 According to \cite{KP}, $P_t= e^{-t\mathcal{G}^{(2)}}$ ($\forall \, t\geq 0$) is a symmetric semigroup on $S_2$. For any $x\in B(H)$, define
 $$ \rho^{1/4} T_t^{(\infty)}(x) \rho^{1/4}= P_t(\rho^{1/4}x\rho^{1/4}). $$
 Then $T_t^{(\infty)}$ is an ergodic quantum Markov semigroup on $S_\infty$, and $\omega$ is the unique invariant state associated with $T_t^{(\infty)}$. See \cite{KP} for more details.

For $1\leq p<\infty$, denote by $T_t^{(p)}: L_p(\rho)\rightarrow L_p(\rho)$ the usual restriction of $T_t^{(\infty)}$ to $L_p(\rho)$, by using the same way as in \cite{GL1995}. More explicitly, for $p=2$, if $x\in B(H)$, then
 $$ T_t^{(2)}(\rho^{1/2}x\rho^{1/2})=\rho^{1/4}P_t(\rho^{1/4}x\rho^{1/4})\rho^{1/4}. $$
 Denote by $\tau$ the spectral gap of $T_t^{(2)}$, which is defined in \eqref{gap}.

We present the definition of hypercontractivity. Let $(T_t)_{t\geq 0}$ be an ergodic quantum Markov semigroup with the invariant state induced by $\rho$. Then $(T_t)_{t\geq 0}$ is said to be hypercontractive if for any $2<p<\infty$, there exists $t_p>0$ such that for any $t \geq t_p$, and for any $ x\in B(H) $ with $\omega(x)=0$,
 \begin{equation}\label{hypercontractivity}
 \| T_t(\rho^{1/2}x\rho^{1/2}) \|_{L_p(\rho)} \leq \|\rho^{1/2}x\rho^{1/2}\|_{L_2(\rho)}.
 \end{equation}
 Moreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.

Our result is the following theorem.

\begin{equation}\label{gap}
		\tau=\min\{\tau_1, \tau_2\}>0.
	\end{equation}

Our result is the following theorem.

We can also obtain some estimate of the optimal time of $(T_t^{(2)})_{t\geq 0}$ for all $ x\in B(H) $ without the restriction $\omega(x)=0$ with the help of Theorem \ref{MainThm}. For any $2<p<\8$, let $t_p'>0$ be the least constant such that for any $t \geq t_p'$, and for any $ x\in B(H) $,
 \begin{equation*}
 \| T_t^{(2)}(\rho^{1/2}x\rho^{1/2}) \|_{L_p(\rho)} \leq \|\rho^{1/2}x\rho^{1/2}\|_{L_2(\rho)}.
 \end{equation*}

\begin{corollary}\label{MainThm2}
 For $2<p<\8$, the optimal time $t'_p$ satisfies
 \[
 \wt{c}_1(\beta) (p - 1) \leq e^{2\tau t'_p} \leq \wt{C}_1(\beta) (p - 1)^2,
 \]
 where $\wt{c}_1(\beta)$ and $\wt{C}_1(\beta)$ are positive constants depending only on $\beta$.
 \end{corollary}

\begin{proposition}\label{Prop2.1}
 $T_t^{(2)}$ is hypercontractive if for any $2<p<\8$, there exists a constant $c_p>0$ such that for any $x \in L_k$, $k\in \nat$,
 \begin{align}\label{KL}
 \left\| \rho^{1/(2p) - 1/4} x \rho^{1/(2p) - 1/4}\right\|_{p} \leq c_p^k \left\| x \right\|_{2}.
 \end{align}
 Moreover, the optimal time $t_p$ satisfies
 \begin{equation}\label{optimal-1}
 e^{2\tau t_p} \leq 2c_p^2, \quad \forall \, 2<p<\8.
 \end{equation}
 \end{proposition}

\section{Proof of Theorem \ref{Main-Lem}}\label{Section3}
 In this section, we will show Theorem \ref{Main-Lem}, which finishes the proof of Theorem \ref{MainThm}. To begin with, we introduce Meixiner polynomials, which are the main ingredient for our proof of Theorem \ref{Main-Lem}. Recall that $p\geq 2$.
 \subsection{Orthogonal polynomials}
 We present a special family of Meixiner polynomials (or Laguerre polynomials), which is orthogonal with respect to the weight $w(n) = e^{-n\beta}$ ($n\in \nat$). For any $k \geq 0$, define
 \begin{equation}
 L_k(n) = e^{-k\beta}\sum_{j = 0}^k (-1)^{j} (e^{\beta} - 1)^j \binom{k}{j} \binom{n}{j}, \quad \forall \, n\in\nat.
 \end{equation}
 We have the following proposition, see \cite[Chapter 9]{koekoek2010}.
 \begin{proposition}\label{Orthogonal}
 The family $\lk{ L_0, L_1, \cdots, L_n, \cdots }$ is orthogonal with respect to the weight $w(n) = e^{-n \beta}$. More precisely, for any $m, l\in \nat$
 \begin{equation}\label{Orthogonal-eq1}
 \sum_{n = 0}^\infty e^{-n \beta} L_m(n) L_l(n) = \frac{ e^{-m\beta} }{ 1 - e^{-\beta} } \delta_{ml},
 \end{equation}
 where $\delta_{ml}$ is the Kronecker delta function.
 \end{proposition}

We restate Theorem \ref{MainThm} on the estimate of the optimal time $t_p$.
 \begin{thm}\label{Optimal}
 Let $p\geq 2$. There exist positive constants $\wt{c}(\beta)$, $\wt{C}(\beta)$ such that,
 \[
 \wt{c}(\beta)(p-1) \leq e^{2\tau t_p} \leq \wt{C}(\beta) (p - 1).
 \]
 More precisely,
 \[
 \wt{c}(\beta) = \frac{ \min\lk{\beta^{-2}, \beta^{-1}} }{ 36e}, \quad \wt{C}(\beta) = 24 (1 - e^{-\beta})^{-1}C(\beta),
 \]
 where $C(\beta)$ is the constant in Theorem \ref{Main-Lem}.
 \end{thm}

Note that one has
 \begin{equation}\label{Lower-1}
 \e \| e^{-\tau t_p} \rho^{1/(2p) - 1/4} A_1^*\xi_0 \rho^{1/(2p) - 1/4} \|_p = \| T_{t_p}^{(2)}( \rho^{1/2} x \rho^{1/2} ) \|_{L_p(\rho)} \leq \| \rho^{1/2} x \rho^{1/2} \|_{L_2(\rho)} = \e.
 \end{equation}
 By definition, we can write $A_1^*\xi_0$ as a matrix of the form
 \begin{align*}
 A_1^*\xi_0 & = \frac{1}{\sqrt{2}} (D_1^* - D_2^*)\rho^{1/2} \\
 & = \frac{ \sk{ 2 \sinh(\beta/2) }^{-1/2} }{\sqrt{2}} \sk{ e^{\beta/4} - e^{-3\beta/4} } \sk{ a^*\rho^{1/2} + \rho^{1/2}a } \\
 & = \frac{\sk{ 2\sinh(\beta/2) }^{-1/2}}{\sqrt{2}} \sk{ e^{\beta/4} - e^{-3\beta/4} } \sum_{n = 0}^\infty \frac{ e^{-n \beta/2}\sqrt{n + 1} }{ \sk{ 1 - e^{-\beta} }^{1/2} } \sk{ e_n \otimes e_{n + 1} + e_{n + 1} \otimes e_n } .
 \end{align*}
 For convenience, we denote
 \[
 \quad Z = e^{-\tau t_p} \rho^{1/(2p) - 1/4} A_1^*\xi_0 \rho^{1/(2p) - 1/4}.
 \]
 Hence
 $$ Z=e^{-\tau t_p} \frac{ \sk{ 2 \sinh(\beta/2) }^{1/2-1/p} }{\sqrt{2}} \sum_{n = 0}^\infty \sqrt{n + 1} e^{-n\beta/p } \sk{ e_n \otimes e_{n + 1} + e_{n + 1} \otimes e_n }. $$
 However, form \eqref{Lower-1} and by \eqref{Schatten-norm},
 \begin{align}\label{Z-p-norm}
 1 & \geq \| Z \|_p \geq e^{-\tau t_p} \frac{ 2^{1/p}\sk{ 2 \sinh(\beta/2) }^{1/2-1/p} }{3\sqrt{2}} \sk{ \sum_{n = 0}^\infty (n + 1)^{p/2} e^{-n\beta } }^{1/p} \notag\\
 & \geq e^{-\tau t_p} \frac{ 2^{1/p}\sk{ 2 \sinh(\beta/2) }^{1/2-1/p} }{3\sqrt{2}} \beta^{-(1/2 + 1/p)} \sk{ \frac{ \pi p}{4} }^{1/(2p)} \sk{ \frac{p}{2e} }^{1/2} \notag\\
 & \geq e^{-\tau t_p} \frac{ \min\lk{ \beta^{-1}, \beta^{-1/2} } }{6 \sqrt{e}} p^{1/2}.
 \end{align}
 This implies
 $$ e^{\tau t_p} \geq \frac{ \min\lk{ \beta^{-1}, \beta^{-1/2} } }{6 \sqrt{e}} p^{1/2}. $$
 Hence, the proof is completed.

\begin{proof}[Proof of Corollary \ref{MainThm2}]
 It suffices to show
 $$ e^{2\tau t'_p} \leq \wt{C}_1(\beta) (p - 1)^2. $$
 We use the same notation with Proposition \ref{Prop2.1}. Take an arbitrary $y\in B(H)$ where $\omega(y)$ might be non-zero. Now applying the Ball-Carlen-Lieb convexity inequality (see \cite{BCL1994,OZ1999,RX2016}), one has for all $p \geq 2$,
 \[
 \| \rho^{1/2} y \rho^{1/2} \|_{L_p(\rho)}^2 \leq |\omega(y)|^2 + (p - 1) \| \rho^{1/2} (y - \omega(y)) \rho^{1/2} \|_{L_p(\rho)}^2.
 \]
 Then we obtain
 \begin{align*}
 \| T_t^{(2)}( \rho^{1/2} y \rho^{1/2} ) \|_{L_p(\rho)} & \leq \sk{ | \omega(y) |^2 + (p - 1) \| T_t^{(2)}\sk{ \rho^{1/2} (y - \omega(y) ) \rho^{1/2} } \|_{L_p(\rho)}^2 }^{1/2} \\
 & \leq \sk{ | \omega(y) |^2 + (p - 1) \frac{ e^{-2\tau t} c_p^2 }{ 1 - e^{-2\tau t}c_p^2 } \| \rho^{1/2} ( y - \omega(y) ) \rho^{1/2} \|_{L_2(\rho)}^2 }^{1/2} \\
 & \leq \sk{ | \omega(y) |^2 + \| \rho^{1/2} ( y - \omega(y) ) \rho^{1/2} \|_{L_2(\rho)}^2 }^{1/2} = \| \rho^{1/2} y \rho^{1/2} \|_{L_2(\rho)},
 \end{align*}
 where we require
 \[
 (p - 1)\frac{ e^{-2\tau t} c_p^2 }{ 1 - e^{-2\tau t}c_p^2 } \leq 1 \quad \text{i.e.} \quad e^{2\tau t} \geq pc_p^2.
 \]
 Hence, the optimal time $t_p'$ satisfies
 \[
 e^{2\tau t_p'} \leq p c_p^2.
 \]
 We complete the proof.
 \end{proof}

\begin{thm}\label{Main-Lem}
		Let $f_{k, n, m}$ be defined as before. For all $k \geq 1$, $-k \leq m \leq k$, there exists a constant $C(\beta)$ such that
		\begin{equation}
			\left( \sum_{n=0}^\8  e^{-n\beta}| f_{k, n, m} |^p  \right)^{1/p}  \leq e^{m\beta(1/2p-1/4)} \mk{ C(\beta)p }^{ k/2 } \left( \sum_{n=0}^\8 e^{-n\beta} | f_{k, n, m} |^2  \right)^{1/2}.
		\end{equation}
	\end{thm}

\begin{thm}\label{MainThm}
		$(T_t^{(2)})_{t\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies
		\[
		\wt{c}(\beta) (p - 1) \leq e^{2\tau t_p} \leq \wt{C}(\beta) (p - 1), \quad \forall\, 2 < p < \infty,
		\]
		where $\wt{c}(\beta)$ and $\wt{C}(\beta)$ are positive constants depending only on $\beta$.
	\end{thm}

\begin{corollary}\label{MainThm2}
		For $2<p<\8$, the optimal time $t'_p$ satisfies
		\[
		\wt{c}_1(\beta) (p - 1) \leq e^{2\tau t'_p} \leq \wt{C}_1(\beta) (p - 1)^2,
		\]
		where $\wt{c}_1(\beta)$ and $\wt{C}_1(\beta)$ are positive constants depending only on $\beta$.
	\end{corollary}

\begin{proposition}\label{Prop2.1}
		$T_t^{(2)}$ is hypercontractive if for any $2<p<\8$, there exists a constant $c_p>0$ such that for any $x \in L_k$, $k\in \nat$,
		\begin{align}\label{KL}
			\left\| \rho^{1/(2p) - 1/4} x \rho^{1/(2p) - 1/4}\right\|_{p} \leq c_p^k \left\|  x  \right\|_{2}.
		\end{align}
		Moreover, the optimal time $t_p$ satisfies
		\begin{equation}\label{optimal-1}
			e^{2\tau t_p} \leq 2c_p^2, \quad \forall \, 2<p<\8.
		\end{equation}
	\end{proposition}

We can also obtain some estimate of the optimal time of $(T_t^{(2)})_{t\geq 0}$ for all $ x\in B(H) $ without the restriction $\omega(x)=0$ with the help of Theorem \ref{MainThm}. For any $2<p<\infty$, let $t_p'>0$ be the least constant such that for any $t \geq t_p'$, and for any $ x\in B(H) $,
	\begin{equation*}
		\| T_t^{(2)}(\rho^{1/2}x\rho^{1/2}) \|_{L_p(\rho)} \leq \|\rho^{1/2}x\rho^{1/2}\|_{L_2(\rho)}.
	\end{equation*}

	\begin{corollary}\label{MainThm2}
		For $2<p<\infty$, the optimal time $t'_p$ satisfies
		\[
		\widetilde{c}_1(\beta) (p - 1) \leq e^{2\tau t'_p} \leq \widetilde{C}_1(\beta) (p - 1)^2,
		\]
		where $\widetilde{c}_1(\beta)$ and $\widetilde{C}_1(\beta)$ are positive constants depending only on $\beta$.
	\end{corollary}

	Several comments are in order. Theorem \ref{MainThm} implies that $(p-1)^{1/2}$ is the optimal order of $e^{\tau t_p}$ up to a constant depending only on $\beta$. Our proof of Theorem \ref{MainThm} relies heavily on Meixiner polynomials. To the best of our knowledge, this is perhaps the first time that Meixiner polynomials are used to study quantum Ornstein-Uhlenbeck semigroups. Therefore, the proof of Theorem \ref{MainThm} is different from the previous ones. Throughout this paper, we do not care about the constants $\widetilde{c}(\beta)$ and $\widetilde{C}(\beta)$ as we are only concerned with the order $(p-1)^{1/2}$. In addition, we would like to mention that at the time of this writing, we do not know the optimal order of the optimal time $t_p'$. We come up with a conjecture on $t_p'$ at the end of this paper.

	$(T_t^{(2)})_{t\geq 0}$ is called quantum Ornstein-Uhlenbeck semigroups, but it does not behave like the classcial Ornstein-Uhlenbeck semigroup. Indeed, for the classcial Ornstein-Uhlenbeck semigroup \cite{EN1973} or the Ornstein-Uhlenbeck semigroup in the mixed $q$-Gaussian setting \cite{CL1993,BI1997}, or even for the Ornstein-Uhlenbeck semigroup on the \(q\)-Araki--Woods algebras \cite{LR2011}, the optimal time has been already calculated explicitly or estimated. Besides, they are all ultracontractive \cite{BM2021}. This is because the constant $c_p$ appearing in Proposition \ref{Prop2.1} is always a universal constant which does not depend on $p$ for all Ornstein-Uhlenbeck semigroups mentioned above except the quantum Ornstein-Uhlenbeck semigroups considered in this paper. We will see that for quantum Ornstein-Uhlenbeck semigroups, the constant $c_p$ appearing in Proposition \ref{Prop2.1} does depend on $p$, which is also the main difficulty in the estimate of $t_p$ and $t'_p$.

	In particular, when $\alpha_1=-\alpha_2$ and $\alpha_3=1$, the quantum Ornstein-Uhlenbeck semigroup $T_t^{(2)}$ is as same as in that in \cite{CFL2000}. On the one hand, in this special case, it has already been shown in \cite{CS2008} that $T_t^{(2)}$ is hypercontractive. The authors in \cite{CS2008} employed the birth and death processes in an ingenious way to show the hypercontractivity of $T_t^{(2)}$. However, they only showed that $T_t^{(2)}$ is contractive from $L_2(\rho)$ to $L_4(\rho)$ once $t$ is larger than some fixed time. They did not consider directly the contractivity of $T_t^{(2)}$ from $L_2(\rho)$ to $L_p(\rho)$ for $2\leq p\neq 4<\infty$. It seems that their method does not yield the best estimate of optimal time when $p$ is very large. On the other hand, the best constants of $p$-log-Sobolev inequalities of $T_t^{(2)}$ have been calculated explicitly in the remarkable papers by Beigi and Rahimi-Keshari \cite{BK2024} for $1\leq p\leq 2$ and by Carlen and Maas \cite{CJ2017} for $p=1$ respectively. In \cite{BK2024}, Beigi \emph{et al.} also gave an estimate of the optimal time, but their coefficient of $\ln(p-1)$ for the optimal time is larger than ours given in Theorem \ref{MainThm} or Corollary \ref{MainThm2}. This also reveals that in general, the best constants of log-Sobolev inequalities cannot yield the optimal time, which is quite different from the case of the classical Ornstein-Uhlenbeck semigroup. In addition, we would like to emphasize that the quantum Ornstein-Uhlenbeck semigroups which we consider in this paper are much more general than that in \cite{CFL2000}.

	The paper is organized as follows. In Section \ref{Section2}, we present more details on the eigenvalues and corresponding eigenvectors of $T_t^{(2)}$. We also reduce the proof of Theorem \ref{MainThm} to that of Theorem \ref{Main-Lem}. Section \ref{Section3} is devoted to the proof of Theorem \ref{Main-Lem}. In Section \ref{Section4} and Section \ref{Section5}, we will give the estimate of the optimal time $t_p$ and $t'_p$ respectively.

	\bigskip

Our proof of Theorem~\ref{MainThm} relies heavily on Meixiner polynomials; to the best of our knowledge, this is perhaps the first time that Meixiner polynomials are used to study quantum Ornstein--Uhlenbeck semigroups, and therefore the proof of Theorem~\ref{MainThm} is different from the previous ones. In Section~\ref{Section2}, we present more details on the eigenvalues and corresponding eigenvectors of $T_t^{(2)}$, and we also reduce the proof of Theorem~\ref{MainThm} to that of Theorem~\ref{Main-Lem}. Section~\ref{Section3} is devoted to the proof of Theorem~\ref{Main-Lem}. In Section~\ref{Section4} we give the estimate of the optimal time $t_p$ (yielding the order $(p-1)^{1/2}$ for $e^{\tau t_p}$ up to constants depending only on $\beta$).

Our proof of the main theorem relies heavily on Meixiner polynomials; to the best of our knowledge, this is perhaps the first time that Meixiner polynomials are used to study quantum Ornstein--Uhlenbeck semigroups, and therefore the proof of the main theorem is different from the previous ones. Next, we present more details on the eigenvalues and corresponding eigenvectors of $T_t^{(2)}$, and we also reduce the proof of the main theorem to the following theorem.

\begin{thm}\label{Main-Lem}
		Let $f_{k, n, m}$ be defined as before. For all $k \geq 1$, $-k \leq m \leq k$, there exists a constant $C(\beta)$ such that
		\begin{equation}
			\left( \sum_{n=0}^\8  e^{-n\beta}| f_{k, n, m} |^p  \right)^{1/p}  \leq e^{m\beta(1/2p-1/4)} \mk{ C(\beta)p }^{ k/2 } \left( \sum_{n=0}^\8 e^{-n\beta} | f_{k, n, m} |^2  \right)^{1/2}.
		\end{equation}
	\end{thm}

We then prove this theorem. Finally, we give the estimate of the optimal time $t_p$ (yielding the order $(p-1)^{1/2}$ for $e^{\tau t_p}$ up to constants depending only on $\beta$).

Question:
Fix an inverse temperature $\beta>0$. Let $H=\ell_2(\mathbb N)$, let $N=a^*a$ be the number operator, and let
\[
\rho=(1-e^{-\beta})e^{-\beta N},\qquad \omega(x)=\operatorname{Tr}(\rho x).
\]
Let $\alpha_1,\alpha_2,\alpha_3\in\mathbb R$ satisfy
\[
\tfrac12(1+\alpha_2^2)\sinh(\beta/2)=-\alpha_1\cosh(\beta/2),
\]
\[
\tfrac12(\alpha_1^2+\alpha_3^2)\sinh(\beta/2)=\alpha_2\alpha_3\cosh(\beta/2).
\]
Using these parameters, form the corresponding quantum Ornstein-Uhlenbeck semigroup $\bigl(T_t^{(2)}\bigr)_{t\ge 0}$ on $L_2(\rho)$, where $T_t^{(2)}$ is induced from $P_t=e^{-t\mathcal G^{(2)}}$ via
\[
\mathcal G^{(2)}(\rho^{1/4}x\rho^{1/4})=\rho^{1/4}G(x)\rho^{1/4}.
\]
Let $\tau$ denote the spectral gap of $T_t^{(2)}$. For $2<p<\infty$, call $\bigl(T_t^{(2)}\bigr)_{t\ge0}$ hypercontractive if there exists $t_p>0$ such that for all $t\ge t_p$ and all $x\in B(H)$ with $\omega(x)=0$,
\[
\|T_t^{(2)}(\rho^{1/2}x\rho^{1/2})\|_{L_p(\rho)}\le \|\rho^{1/2}x\rho^{1/2}\|_{L_2(\rho)}.
\]
When this holds, let $t_p$ be the optimal such time. Which statement holds for this semigroup for every $2<p<\infty$?

The semigroup $\bigl(T_t^{(2)}\bigr)_{t\ge 0}$ is hypercontractive, and there exist positive constants $\widetilde c(\beta)$ and $\widetilde C(\beta)$, depending only on $\beta$, such that for every $2<p<\infty$,
\[
\widetilde c(\beta)(p-1)\le e^{2\tau t_p}\le \widetilde C(\beta)(p-1).
\]

Choices:
B.
The semigroup $\bigl(T_t^{(2)}\bigr)_{t\ge 0}$ is hypercontractive, and there exist positive constants $\widetilde c(\beta)$ and $\widetilde C(\beta)$, depending only on $\beta$, such that for every $2<p<\infty$,
\[
\widetilde c(\beta)(p-1)^2\le e^{2\tau t_p}\le \widetilde C(\beta)(p-1)^2.
\]
C.
The semigroup $\bigl(T_t^{(2)}\bigr)_{t\ge 0}$ is hypercontractive, and for each fixed $p\in(2,\infty)$ there exist positive constants $c_{p,\beta}$ and $C_{p,\beta}$ such that
\[
c_{p,\beta}\le e^{2\tau t_p}\le C_{p,\beta}.
\]
D.
The semigroup $\bigl(T_t^{(2)}\bigr)_{t\ge 0}$ is hypercontractive, and there exist positive constants $\widetilde c$ and $\widetilde C$, independent of both $p$ and $\beta$, such that for every $2<p<\infty$,
\[
\widetilde c(p-1)\le e^{2\tau t_p}\le \widetilde C(p-1).
\]
E.
The semigroup $\bigl(T_t^{(2)}\bigr)_{t\ge 0}$ is hypercontractive, and there exist positive constants $\widetilde c(\beta)$ and $\widetilde C(\beta)$, depending only on $\beta$, such that for every $2<p<\infty$,
\[
\widetilde c(\beta)(p-1)\le e^{\tau t_p}\le \widetilde C(\beta)(p-1).
\]

uniform linear bounds with constants depending only on \beta

exponentiated quantity is e^{2\tau t_p}, not e^{\tau t_p}

The stem does not reveal the correct estimate explicitly or through obvious phrasing. It sets up the semigroup and asks for the valid quantitative bound without signaling which dependence on p or beta is correct.

The item is close to a theorem-recall question: it asks which quantitative estimate holds for the optimal hypercontractive time. Although the choices introduce meaningful variants, the task is still largely to recognize the theorem’s stated conclusion.

Some reasoning is needed to distinguish p from p-1, one-sided from two-sided bounds, and beta-dependent from beta-independent constants. However, the question mainly tests recognition of the known result rather than generating a conclusion from first principles.

The distractors are plausible and mathematically targeted: they vary asymptotic order, remove a lower bound, or alter parameter dependence. These reflect realistic failure modes rather than arbitrary wrong answers.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it only moderately tests genuine generative reasoning because it is close to direct recall of a specific quantitative result.