# Notes on noncommutative ergodic theorems

Semyon Litvinov  76 University Drive, Pennsylvania State University, Hazleton
18202<EMAIL_ADDRESS>
(Date: July 30, 2023)

###### Abstract.

Given a semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful
normal semifinite trace $\tau$, we prove that the spaces
$L^{0}(\mathcal{M},\tau)$ and $\mathcal{R}_{\tau}$ are complete with respect
to pointwise, almost uniform and bilaterally almost uniform, convergences in
$L^{0}(\mathcal{M},\tau)$. Then we show that the pointwise Cauchy property for
a special class of nets of linear operators in the space
$L^{1}(\mathcal{M},\tau)$ can be extended to pointwise convergence of such
nets in any fully symmetric space $E\subset\mathcal{R}_{\tau}$, in particular,
in any space $L^{p}(\mathcal{M},\tau)$, $1\leq p<\infty$. Some applications of
these results in the noncommutative ergodic theory are discussed.

###### Key words and phrases:

Semifinite von Neumann algebra; almost uniform convergence; bilaterally almost
uniform convergence; completeness; fully symmetric space; individual ergodic
theorem

###### 2010 Mathematics Subject Classification:

47A35(primary), 46L51(secondary)

##  1\. Introduction and Preliminaries

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a faithful
normal semifinite trace $\tau$, and let $\mathcal{P}(\mathcal{M})$ stand for
the complete lattice of projections in $\mathcal{M}$; see [1, 21, 20]. If
$\mathbf{1}$ is the identity of $\mathcal{M}$ and
$e\in\mathcal{P}(\mathcal{M})$, we write $e^{\perp}=\mathbf{1}-e$. In what
follows, we assume that $\tau(\mathbf{1})=\infty$. Denote by
$L^{0}=L^{0}(\mathcal{M},\tau)$ the algebra of $\tau$-measurable operators
affiliated with $\mathcal{M}$. Let $L^{p}=L^{p}(\mathcal{M},\tau)$ if $1\leq
p<\infty$ and $L^{\infty}(\mathcal{M},\tau)=\mathcal{M}$ be the noncommutative
$L^{p}$\- spaces associated with $(\mathcal{M},\tau)$. For detailed accounts
on the spaces $L^{p}(\mathcal{M},\tau)$, $p\in\\{0\\}\cup[1,\infty)$, see [19,
15, 22, 17].

Let $(\Lambda,\leq)$ be a directed set, and let $\|\cdot\|_{\infty}$ be the
uniform norm in $\mathcal{M}$. The following definition presents two
noncommutative counterparts of the classical pointwise (almost everywhere)
convergence.

######  Definition 1.1.

A net $\\{x_{\alpha}\\}_{\alpha\in\Lambda}\subset L^{0}$ is said to converge
almost uniformly (a.u.) (bilaterally almost uniformly (b.a.u.)) to
$\widehat{x}\in L^{0}$ if for any given $\varepsilon>0$ there exists a
projection $e\in\mathcal{P}(\mathcal{M})$ such that
$\tau(e^{\perp})\leq\varepsilon$ and

$\|(\widehat{x}-x_{\alpha})e\|_{\infty}\to 0\text{ \ (\,respectively,\
}\|e(\widehat{x}-x_{\alpha})e\|_{\infty}\to 0\,).$

######  Remark 1.1.

(1) A.u. convergence clearly implies b.a.u. convergence, while the converse in
not true in general - see [6, Example 3.1].

(2) Pointwise convergence in the space $L^{0}$ is not topological in general,
that is, it may not agree with any topology in $L^{0}$; see, for example, the
note [16].

First ergodic theorem in the space $L^{1}$ equipped with b.a.u. convergence
appeared in the groundbreaking paper [23]. Ergodic theorems in the spaces
$L^{p}$, $1\leq p<\infty$, endowed with b.a.u. or a.u. for $p\geq 2$
convergence were comprehensively studied in another fundamental paper [11]. An
alternative approach, based directly on [23], to deriving ergodic theorems in
the spaces $L^{p}$, $1<p<\infty$, was suggested in [14].

Given $x\in L^{0}$ and $t>0$, denote

$\mu_{t}(x)=\inf\big{\\{}\|xe\|_{\infty}:\,e\in\mathcal{P}(\mathcal{M}),\,\tau(e^{\perp})\leq
t\big{\\}},$

the $t$-th generalized singular number of $x$ \- see [10]. The function
$\mu_{t}(x)$, $t>0$, is called the non-increasing rearrangement of $x$. It is
worth mentioning that, as it follows from the proof of [10, Proposition 2.2],

$\mu_{t}(x)=\inf\big{\\{}\|xe\|_{\infty}:\,e\in\mathcal{P}(\mathcal{A}),\,\tau(e^{\perp})\leq
t\big{\\}},$

where $\mathcal{A}$ is any subalgebra of $\mathcal{M}$ containing the spectral
family $\big{\\{}e_{\lambda}(|x|)\big{\\}}_{\lambda\geq 0}$ of the absolute
value $|x|=(x^{*}x)^{1/2}$ of an operator $x\in L^{0}$. Thus, $\mu_{t}(x)$ can
be alternatively defined as

$\mu_{t}(x)=\inf\big{\\{}\lambda>0:\,\tau(e_{\lambda}(|x|)^{\perp})\leq
t\big{\\}}.$

Now, let us denote

$\mathcal{R}_{\tau}=\big{\\{}x\in L^{1}+\mathcal{M}:\ \mu_{t}(x)\to 0\text{\,\
as\ }t\to\infty\big{\\}}.$

In view of [10, Lemma 2.5], $\mathcal{R}_{\tau}$ is a $\ast$-algebra. Besides,
it is clear that

$\mathcal{R}_{\tau}=\big{\\{}x\in L^{1}+\mathcal{M}:\
\tau(e_{\lambda}(|x|)^{\perp})<\infty\,\ \forall\ \lambda>0\big{\\}},$

so $\mathcal{R}_{\tau}$ is the noncommutative analogue of Fava’s space
$\mathcal{R}_{0}$ \- see [9, Section (2.2.15)].

Sequential completeness of the space $L^{0}$ with respect to a.u. and b.a.u.
convergences was proved in [7, Theorem 2.3]. In Section 2, we show that the
space $L^{0}$ and - more importantly for this article - the space
$\mathcal{R}_{\tau}$ are complete with respect to these convergences.

Section 3 is devoted to the description how the a.u. (b.a.u.) convergence of a
generic class of nets of linear maps in any fully symmetric space
$E\subset\mathcal{R}_{\tau}$ \- in particular, in any space $L^{p}$, $1\leq
p<\infty$ \- can be deduced from the a.u. (respectively, b.a.u.) Cauchy
property for such nets in the space $L^{1}$.

In Section 4 we proceed to outline some possible applications of these results
to the nocommutative ergodic theory.

##  2\. Pointwise completeness in $L^{0}(\mathcal{M},\tau)$

Here we aim to show that any almost uniform or bilaterally almost uniform
Cauchy net converges in $L^{0}$ and to derive the same for the space
$\mathcal{R}_{\tau}$. We will employ the notion of measure topology in $L^{0}$
\- in which $L^{0}$ is a complete metric space - and use the fact that a
Cauchy net in a complete metric space is convergent.

The measure topology $t_{\tau}$ in $L^{0}$ is given by the system

$\mathcal{N}(\varepsilon,\delta)=\big{\\{}x\in L^{0}:\ \exists\
e\in\mathcal{P}(\mathcal{M})\text{\,\ such that\,\
}\tau(e^{\perp})\leq\varepsilon\text{\,\ and\
}\|xe\|_{\infty}\leq\delta\big{\\}}\,;$

$\varepsilon>0$, $\delta>0$, of (closed) neighborhoods of zero. The following
is an immediate corollary of [15, Theorem 1]; note that it is understood that
the space $L^{0}$ is the completion of $\mathcal{M}$ with respect to
$t_{\tau}$.

######  Theorem 2.1.

$(L^{0},t_{\tau})$ is a complete metrizable topological $\ast$-algebra.

######  Definition 2.1.

A net $\\{x_{\alpha}\\}_{\alpha\in\Lambda}\subset L^{0}$ is said to converge
to $\widehat{x}\in L^{0}$ in measure (bilaterally in measure) if for every
$\varepsilon>0$ and $\delta>0$ there is $\gamma=\gamma(\varepsilon,\delta)\in
I$ such that, given $\alpha\in\Lambda$ with $\alpha\geq\gamma$, there exists a
projection $e_{\alpha}\in\mathcal{P}(\mathcal{M})$ satisfying conditions
$\tau(e^{\perp}_{\alpha})\leq\varepsilon$ and

$\|(\widehat{x}-x_{\alpha})e_{\alpha}\|_{\infty}\leq\delta\,\
(\text{respectively},\,\|e_{\alpha}(\widehat{x}-x_{\alpha})e_{\alpha}\|_{\infty}\leq\delta).$

Proof of the following lemma - which idea is attributed to E. Nelson (see [18,
Appendix, Theorem B]); note also the paragraph before [23, Theorem 1] \- can
be found in [7, Lemma 2.1, Theorem 2.2].

######  Lemma 2.1.

Let $x\in L^{0}$ and $e\in\mathcal{P}(\mathcal{M})$ be such that
$exe\in\mathcal{M}$. Then there exists $f\in\mathcal{P}(\mathcal{M})$ such
that

$\tau(f^{\perp})\leq 2\tau(e^{\perp})\text{ \ and \
}\|xf\|_{\infty}\leq\|exe\|_{\infty}.$

The following statement is a version of [7, Theorem 2.2] for nets. We provide
a proof for the sake of the reader’s convenience.

######  Proposition 2.1.

A net $\\{x_{\alpha}\\}_{\alpha\in\Lambda}\subset L^{0}$ is Cauchy in measure
if and only if it is Cauchy bilaterally in measure.

###### Proof.

Clearly, if is sufficient to show the ”if” implication. So, let
$\\{x_{\alpha}\\}_{\alpha\in\Lambda}\subset L^{0}$ be Cauchy bilaterally in
measure and fix $\varepsilon>0$, $\delta>0$. Then there is
$\gamma=\gamma(\varepsilon,\delta)$ such that for any $\alpha,\beta\in I$ with
$\alpha,\beta\geq\gamma$ there exists
$e_{\alpha,\beta}\in\mathcal{P}(\mathcal{M})$ for which

$\tau(e_{\alpha,\beta}^{\perp})\leq\frac{\varepsilon}{2}\text{ \ and \
}\|e_{\alpha,\beta}(x_{\alpha}-x_{\beta})e_{\alpha,\beta}\|_{\infty}\leq\delta.$

Then, by Lemma 2.1, for every $\alpha,\beta\geq\gamma$ there exists
$f_{\alpha,\beta}\in\mathcal{P}(\mathcal{M})$ such that

$\tau(f_{\alpha,\beta}^{\perp})\leq
2\tau(e_{\alpha,\beta}^{\perp})\leq\varepsilon\text{ \ and \
}\|(x_{\alpha}-x_{\beta})f_{\alpha,\beta}\|_{\infty}\leq\|e_{\alpha,\beta}(x_{\alpha}-x_{\beta})e_{\alpha,\beta}\|_{\infty}\leq\delta,$

implying that the net $\\{x_{\alpha}\\}_{\alpha\in\Lambda}\subset L^{0}$ is
Cauchy in measure. ∎

Now we are ready to prove main results of this section:

######  Theorem 2.2.

The space $L^{0}$ is complete with respect to a.u. and b.a.u. convergences.

###### Proof.

We will show that $L^{0}$ is b.a.u. complete. Proof in the a.u. case is
similar. Let $\\{x_{\alpha}\\}_{\alpha\in\Lambda}\subset L^{0}$ be b.a.u.
Cauchy. Then $\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ is clearly Cauchy
bilaterally in measure, which, by Proposition 2.1, implies that it is Cauchy
in measure.

Let $d$ be a metric in $L^{0}$ compatible with the measure topology
$t_{\tau}$. Then for every $\delta>0$ there is $\gamma\in I$ such that
$d(x_{\alpha},x_{\beta})\leq\delta$ for all $\alpha,\beta\geq\gamma$.

So, pick $\gamma_{1}\in I$ such that $d(x_{\alpha},x_{\beta})\leq 1$ whenever
$\alpha,\beta\geq\gamma_{1}$. Further, given $\gamma_{n}\in I$ such that
$\displaystyle d(x_{\alpha},x_{\beta})\leq\frac{1}{n}$ for all
$\alpha,\beta\geq\gamma_{n}$, choose $\gamma_{n+1}\in I$ such that

$\gamma_{n+1}\geq\gamma_{n}\text{\, \ and\, \
}d(x_{\alpha},x_{\beta})\leq\frac{1}{n+1}\ \ \forall\
\alpha,\beta\geq\gamma_{n+1}.$

It follows that the sequence $\\{x_{\gamma_{n}}\\}$ is Cauchy in measure,
hence, in view of Theorem 2.1, there exists $\widehat{x}\in L^{0}$ such that
$d(x_{\gamma_{n}},\widehat{x})\to 0$ as $n\to\infty$.

Given $\delta>0$, there is $n_{1}$ such that
$d(x_{\gamma_{n}},\widehat{x})\leq\delta/2$ whenever $n\geq n_{1}$. In
addition, there is $n_{2}$ such that $\displaystyle
d(x_{\alpha},x_{\beta})\leq\frac{\delta}{2}$ whenever
$\alpha,\beta\geq\gamma_{n_{2}}$. If $n_{0}=\max\\{n_{1},n_{2}\\}$, then

$d(x_{\gamma_{n_{0}}},\widehat{x})\leq\frac{\delta}{2}\text{\ \ \ and, as\
}\gamma_{n_{0}}\geq\gamma_{n_{1}},\
d(x_{\alpha},x_{\gamma_{n_{0}}})\leq\frac{\delta}{2}\ \ \forall\
\alpha\geq\gamma_{n_{0}}.$

Therefore $d(x_{\alpha},\widehat{x})\leq\delta$ whenever
$\alpha\geq\gamma_{n_{0}}$, implying that $x_{\alpha}\to\widehat{x}$ in
measure.

Finally, fix $\varepsilon>0$. Since $\\{x_{\alpha}\\}_{\alpha\in\Lambda}$ is
b.a.u. Cauchy, there exists $e\in\mathcal{P}(\mathcal{M})$ with
$\tau(e^{\perp})\leq\varepsilon$ such that the net
$\\{ex_{\alpha}e\\}_{\alpha\in\Lambda}$ is $\|\cdot\|_{\infty}$ \- Cauchy.
Then, as $(\mathcal{M},\|\cdot\|_{\infty})$ is a complete metric space, we
show as above that there is $x_{e}\in\mathcal{M}$ such that
$\|ex_{\alpha}e-x_{e}\|_{\infty}\to 0$, implying that $ex_{\alpha}e\to x_{e}$
in measure. Also, as $x_{\alpha}\to\widehat{x}$ in measure, it follows from
Theorem 2.1 that $ex_{\alpha}e\to e\widehat{x}e$ in measure, hence
$x_{e}=e\widehat{x}e$, and we obtain

$\|e(\widehat{x}-x_{\alpha})e\|_{\infty}=\|x_{e}-ex_{\alpha}e\|_{\infty}\to
0.$

Therefore, it follows that $L^{0}$ is b.a.u. complete. ∎

######  Theorem 2.3.

The space $\mathcal{R}_{\tau}$ is complete with respect to a.u. and b.a.u.
convergences.

###### Proof.

In view of Theorem 2.2 and, since $\mathcal{R}_{\tau}\subset L^{0}$ and a.u.
convergence implies b.a.u. convergence, it is sufficient to show that
$\mathcal{R}_{\tau}$ is b.a.u. closed.

Let a net $\\{x_{\alpha}\\}_{\alpha\in I}\subset\mathcal{R}_{\tau}$ be such
that $x_{\alpha}\to\widehat{x}\in L^{0}$ b.a.u. Since $\mu_{t}(x)$ is a non-
increasing function of $t$, in order to show that
$\widehat{x}\in\mathcal{R}_{\tau}$, that is, that $\mu_{t}(x)\to 0$ as
$t\to\infty$, it is sufficient to verify that for every $\delta>0$ there is
$t>0$ such that $\mu_{t}(x)<\delta$. Since $\|xe\|_{\infty}<\delta$ for some
$e\in\mathcal{P}(\mathcal{M})$ with $\tau(e^{\perp})\leq t$ implies that
$\mu_{t}(x)<\delta$, this in turn will follow if we show that there exists
$f\in\mathcal{P}(\mathcal{M})$ such that

$\tau(f^{\perp})<\infty\text{ \ and \ }\|\widehat{x}f\|_{\infty}<\delta.$

Since $x_{\alpha}\to\widehat{x}$ b.a.u., given $n\in\mathbb{N}$, there exits
$e_{n}\in\mathcal{P}(\mathcal{M})$ such that

$\tau(e_{n}^{\perp})\leq 2^{-n}\text{ \ and \
}\|e_{n}(\widehat{x}-x_{\alpha})e_{n}\|_{\infty}\xrightarrow[\alpha]{\ }0.$

Therefore, for each $n$ there is $x_{n}\in\\{x_{\alpha}\\}_{\alpha\in I}$ for
which
$\|e_{n}(\widehat{x}-x_{n})e_{n}\|_{\infty}\leq\displaystyle\frac{1}{n}$.

Fix $\varepsilon>0$, and let $N$ be such that
$\sum\limits_{n=N}^{\infty}2^{-n}\leq\varepsilon$. Then, with
$e=\bigwedge\limits_{n\geq N}e_{n}$, we have

$\tau(e^{\perp})\leq\varepsilon\text{ \ and \
}\|e(\widehat{x}-x_{n})e\|_{\infty}\leq\frac{1}{n}\ \ \forall\ n\geq N,$

implying that $\mathcal{R}_{\tau}\ni x_{n}\to\widehat{x}$ b.a.u.

Now, let $\delta>0$. Since $x_{1}\in\mathcal{R}_{\tau}$, there is
$e_{1}\in\mathcal{P}(\mathcal{M})$ with $\tau(e_{1}^{\perp})<\infty$ such that
$\|x_{1}e_{1}\|_{\infty}<\delta$. As $x_{2}\in L^{0}$, there exists
$\mathcal{P}(\mathcal{M})\ni h_{1}\leq e_{1}$ such that
$\tau(h_{1}^{\perp})<\infty$ and $x_{2}h_{1}\in\mathcal{M}$. Next,
$x_{2}\in\mathcal{R}_{\tau}$ entails that there is
$\mathcal{P}(\mathcal{M})\ni e_{2}\leq h_{1}$ such that
$\tau(e_{2}^{\perp})<\infty$ and $\|x_{2}e_{2}\|_{\infty}<\delta$. Repeating
this process, one can construct a sequence
$\\{e_{n}\\}\subset\mathcal{P}(\mathcal{M})$ satisfying conditions

$e_{n}\geq e_{n+1},\,\ \tau(e_{n}^{\perp})<\infty,\text{\ and \
}\|x_{n}e_{n}\|_{\infty}<\delta,\text{\,hence\
}\|e_{n}x_{n}e_{n}\|_{\infty}<\delta,\,\ \forall\ n.$

Further, $x_{n}\to\widehat{x}$ b.a.u. implies that there exists
$\mathcal{P}(\mathcal{M})\ni e\leq e_{1}$ such that $\tau(e-e_{1})<\infty$,
hence $\tau(e^{\perp})<\infty$, and $\|e(\widehat{x}-x_{n})e\|_{\infty}\to 0$,
implying that

$\|(e_{n}\land e)\widehat{x}(e_{n}\land e)\|_{\infty}-\|(e_{n}\land
e)x_{n}(e_{n}\land e)\|_{\infty}\to 0.$

Therefore, since $\|(e_{n}\land e)x_{n}(e_{n}\land e)\|_{\infty}<\delta$ for
every $n$, it follows that there exists $n_{0}$ such that $\|(e_{n_{0}}\land
e)\widehat{x}(e_{n_{0}}\land e)\|_{\infty}<\delta$. Then, by Lemma 2.1, it is
possible to find $f\in\mathcal{P}(\mathcal{M})$ such that

$\tau(f^{\perp})\leq 2\tau((e_{n_{0}}\land e)^{\perp})\leq
2\tau(e_{n_{0}}^{\perp})+2\tau(e^{\perp})<\infty\text{ \ and \
}\|\widehat{x}f\|_{\infty}<\delta,$

which completes the argument. ∎

##  3\. Pointwise convergence in fully symmetric spaces
$E\subset\mathcal{R}_{\tau}$

A non-zero Banach space $(E,\|\cdot\|_{E})\subset L^{0}$ is called fully
symmetric if conditions

$x\in E,\ y\in L^{0},\text{\ and\
}\int_{0}^{s}\mu_{t}(y)dt\leq\int_{0}^{s}\mu_{t}(x)dt\ \ \forall\ s>0\text{\
(denoted\ }y\prec\prec x)$

imply that $y\in E$ and $\|y\|_{E}\leq\|x\|_{E}$. It is well-known that the
space $L^{p}=(L^{p}(\mathcal{M},\tau),\|\cdot\|_{p})$ is fully symmetric and
$L^{p}\subset\mathcal{R}_{\tau}$ for each $1\leq p<\infty$. Besides, if
$E\subset L^{0}$ is a fully symmetric space, then $E\subset\mathcal{R}_{\tau}$
if and only if $\mathbf{1}\notin E$ [6, Proposition 2.2], implying that the
noncommutative Orlicz, Lorentz and Marcinkiewicz spaces $E$ on
$(\mathcal{M},\tau)$ with $\mathbf{1}\notin E$ also lie in
$\mathcal{R}_{\tau}$; see, for example, [4, Section 7].

######  Remark 3.1.

In contrast with Theorem 2.3, even in the commutative case, a fully symmetric
space may not be complete with respect to almost uniform convergence. Indeed,
denote by $\nu$ Lebesgue measure on the interval $(0,\infty)$ and define

$\mathcal{R}_{\nu}=\big{\\{}\,f\in L^{1}(\nu)+L^{\infty}(\nu):\
\nu\\{|f|>\lambda\\}<\infty\ \ \forall\ \lambda>0\,\big{\\}},$

the commutative precursor of $\mathcal{R}_{\tau}$, known as Fava’s space
$\mathcal{R}_{0}$; see [3] for details.

Given $t\in(0,\infty)$, set

$f_{n}(t)=\sum_{k=1}^{n}2^{k}\cdot\chi_{(2^{-k},\,2^{-k+1}]}(t),\
\,n=1,2,\dots$

It is easy to see that $\\{f_{n}\\}\subset L^{1}(\nu)$ and that $\\{f_{n}\\}$
converges almost uniformly to a function $f\in\mathcal{R}_{\nu}$ such that
$f\notin L^{1}(\nu)$. Therefore, the fully symmetric space
$L^{1}(\nu)\subset\mathcal{R}_{\nu}$ is not complete with respect to almost
uniform convergence. However, for a special class of pointwise Cauchy nets in
a fully symmetric space $E$ the limits belong to $E$ \- see Theorem 3.1 below.

A linear operator $A:L^{1}+\mathcal{M}\to L^{1}+\mathcal{M}$ is called a
Dunford-Schwartz operator (denoting $A\in DS$) if $A(L^{1})\subset L^{1}$,
$A(\mathcal{M})\subset\mathcal{M}$ and

$\|A(x)\|_{1}\leq\|x\|_{1}\,\ \forall\ x\in L^{1}\text{ \ and \
}\|A(x)\|_{\infty}\leq\|x\|_{\infty}\,\ \forall\ x\in\mathcal{M}.$

The following fact is well-known in the commutative case; see, for example,
[12, Chapter II, Section 4].

######  Proposition 3.1.

If $A\in DS$, then $A(x)\prec\prec x$ for all $x\in L^{1}+\mathcal{M}$.

###### Proof.

As it is proven right after [10, Theorem 4.4] \- see also [8, Proposition 2.5]
\- given $x\in L^{1}+\mathcal{M}$ and $s>0$, we have

$\int_{0}^{s}\mu_{t}(x)ds=\inf\big{\\{}\|y\|_{1}+s\|z\|_{\infty}:\ x=y+z,\
x\in L^{1},\ y\in\mathcal{M}\big{\\}}.$

Let $x\in L^{1}+\mathcal{M}$ and $s>0$. As $x=y+z$, with $y\in L^{1}$ and
$z\in\mathcal{M}$, implies that $A(x)=A(y)+A(z)$, where $A(y)\in L^{1}$ and
$A(z)\in\mathcal{M}$, we can write

$\begin{split}\int_{0}^{s}\mu_{t}(A(x))ds&=\inf\big{\\{}\|y\|_{1}+s\|z\|_{\infty}:\
A(x)=y+z,\ x\in L^{1},\ y\in\mathcal{M}\big{\\}}\\\
&\leq\inf\big{\\{}\|A(y)\|_{1}+s\|A(z)\|_{\infty}:\ A(x)=A(y)+A(z),\ x\in
L^{1},\ y\in\mathcal{M}\big{\\}}\\\
&\leq\inf\big{\\{}\|y\|_{1}+s\|z\|_{\infty}:\ A(x)=A(y)+A(z),\ x\in L^{1},\
y\in\mathcal{M}\big{\\}}\\\ &\leq\inf\big{\\{}\|y\|_{1}+s\|z\|_{\infty}:\
x=y+z,\ x\in L^{1},\ y\in\mathcal{M}\big{\\}}\\\
&=\int_{0}^{s}\mu_{t}(x)ds,\end{split}$

so the result follows. ∎

The following statement provides a useful characterization of the $*$-algebra
$\mathcal{R}_{\tau}$.

######  Proposition 3.2.

Let $x^{*}=x\in L^{0}$. Then $x\in\mathcal{R}_{\tau}$ if and only if for any
$\delta>0$ there exist $y_{\delta}\in L^{1}$ and $z_{\delta}\in\mathcal{M}$
such that

(1) $\|z_{\delta}\|_{\infty}\leq\delta\text{ \ and \
}x=y_{\delta}+z_{\delta}.$

###### Proof.

$"\Rightarrow":\ $ Fix $\delta>0$. Since $|x|\in\mathcal{R}_{\tau}$, we have
$0\leq x_{+}=\displaystyle\frac{|x|+x}{2}\in\mathcal{R}_{\tau}$ and $0\leq
x_{-}=\displaystyle\frac{|x|-x}{2}\in\mathcal{R}_{\tau}$. If
$x_{+}=\int_{0}^{\infty}\lambda de_{\lambda}$ is the spectral decomposition of
$x_{+}$, define $y_{\delta}^{+}=\int_{\delta/2}^{\infty}\lambda de_{\lambda}$
and $z_{\delta}^{+}=\int_{0}^{\delta/2}\lambda de_{\lambda}$. Then we have

$x_{+}=y_{\delta}^{+}+z_{\delta}^{+}\text{ \ and \
}\|z_{\delta}^{+}\|_{\infty}\leq\frac{\delta}{2}.$

Next, $x_{+}\in\mathcal{R}_{\tau}$ implies that
$\tau(e_{\delta/2}^{\perp})<\infty$ and $x_{+}=y^{+}+z^{+}$ for some $y^{+}\in
L^{1}$ and $z^{+}\in\mathcal{M}$, hence

$y_{\delta}^{+}=y_{\delta}^{+}e_{\delta/2}^{\perp}=y^{+}e_{\delta/2}^{\perp}+(z^{+}-z_{\delta}^{+})e_{\delta/2}^{\perp}\in
L^{1}.$

Similarly, $x_{-}=y_{\delta}^{-}+z_{\delta}^{-}$, where $y_{\delta}^{-}\in
L^{1}$ and $\|z_{\delta}^{-}\|_{\infty}\leq\delta/2$, so it follows that

$x=(y_{\delta}^{+}-y_{\delta}^{-})+(z_{\delta}^{+}-z_{\delta}^{-}),\text{ \
where \ }y_{\delta}^{+}-y_{\delta}^{-}\in L^{1}\text{ \ and \
}\|z_{\delta}^{+}-z_{\delta}^{-}\|_{\infty}\leq\delta.$

$"\Leftarrow":\ $ Let $x^{*}=x\in L^{0}$ be such that for any $\delta>0$ there
exist $y_{\delta}\in L^{1}$ and $z_{\delta}\in\mathcal{M}$ for which
conditions (1) hold. Fix $\delta>0$. There exist $y_{\delta}\in L^{1}$ and
$z_{\delta}\in\mathcal{M}$ such that $\|z_{\delta}\|_{\infty}\leq\delta/2$ and
$x=y_{\delta}+z_{\delta}$. Then, by [10, Lemma 2.5], we have

$\mu_{t}(x)\leq\mu_{t/2}(y_{\delta})+\mu_{t/2}(z_{\delta})\leq\mu_{t/2}(y_{\delta})+\frac{\delta}{2}$

whenever $t>\delta$. Also, since $y_{\delta}\in L^{1}$, it follows that
$\mu_{t}(y_{\delta})\to 0$ as $t\to\infty$, so there is $t_{0}>0$ such that
$t\geq t_{0}$ entails $\mu_{t}(y_{\delta})\leq\delta/2$. Thus, we conclude
that $\mu_{t}(x)\leq\delta$ for all $t>\max\\{\delta,t_{0}\\}$, implying that
$\mu_{t}(x)\to 0$ as $t\to\infty$, so $x\in\mathcal{R}_{\tau}$. ∎

We will call a map $A:L^{0}\to L^{0}$ selfadjoint if $A(x)^{*}=A(x)$ whenever
$x^{*}=x$. Here is an immediate consequence of Proposition 3.2:

######  Corollary 3.1.

If a linear selfadjoint map $A:L^{1}+\mathcal{M}\to L^{1}+\mathcal{M}$ is such
that $A(L^{1})\subset L^{1}$, $A(\mathcal{M})\subset\mathcal{M}$, and
$\|A(x)\|_{\infty}\leq c\|x\|_{\infty}$ for some $c>0$ and any
$x\in\mathcal{M}$, then $A(\mathcal{R}_{\tau})\subset\mathcal{R}_{\tau}$.

###### Proof.

Let $x\in\mathcal{R}_{\tau}$ and fix $\delta>0$. By Proposition 3.2, there
exist $y_{\delta}\in L^{1}$ and $z_{\delta}\in\mathcal{M}$ such that
$\|z_{\delta}\|_{\infty}\leq\delta/c$ and $x=y_{\delta}+z_{\delta}$. Then
$A(x)=A(y_{\delta})+A(z_{\delta})$, where $A(y_{\delta})\in L^{1}$ and
$\|A(z_{\delta})\|_{\infty}\leq\delta$, and Proposition 3.2 implies that
$A(x)\in\mathcal{R}_{\tau}$. ∎

######  Theorem 3.1.

Let a net of linear selfadjoint maps $A_{\alpha}:L^{1}+\mathcal{M}\to
L^{1}+\mathcal{M}$, $\alpha\in\Lambda$, be such that

  1. (1)

$A_{\alpha}(L^{1})\subset L^{1}$ for every $\alpha\in\Lambda$, and the net
$\\{A_{\alpha}(x)\\}_{\alpha\in\Lambda}$ is a.u. (b.a.u.) Cauchy for each
$x\in L^{1}$;

  2. (2)

$A_{\alpha}(\mathcal{M})\subset\mathcal{M}$ for every $\alpha\in\Lambda$, and
there exists $c_{\infty}>0$ such that

$\sup_{\alpha\in\Lambda}\|A_{\alpha}(x)\|_{\infty}\leq
c_{\infty}\|x\|_{\infty}\ \ \forall\ x\in\mathcal{M}.$

Then $A_{\alpha}(x)\to\widehat{x}\in\mathcal{R}_{\tau}$ a.u. (respectively,
b.a.u.) for every $x\in\mathcal{R}_{\tau}$. If, in addition,
$\|A_{\alpha}(x)\|_{1}\leq c_{1}\|x\|_{1}$ for some $c_{1}>0$ and all $x\in
L^{1}$, then, given a fully symmetric space $E\subset\mathcal{R}_{\tau}$, we
have $A_{\alpha}(E)\subset E$ for every $\alpha\in\Lambda$ and
$A_{\alpha}(x)\to\widetilde{x}\in E$ a.u. (respectively, b.a.u.) for every
$x\in E$.

###### Proof.

We will prove the statement for a.u. convergence. Proof for b.a.u. convergence
is similar. Assume, without loss of generality, that $x^{*}=x$ and fix
$\varepsilon>0$. By Proposition 3.2, given $m\in\mathbb{N}$, there exist
$y_{m}\in L^{1}$ and $z_{m}\in\mathcal{M}$ such that
$\displaystyle\|z_{m}\|_{\infty}\leq\frac{1}{m}$ and $x=y_{m}+z_{m}$. Since
the net $\\{A_{\alpha}(y_{m})\\}_{\alpha\in\Lambda}$ is a.u. Cauchy, there
exist $e_{m}\in\mathcal{P}(\mathcal{M})$ and $\gamma_{m}\in I$ such that

$\tau(e_{m}^{\perp})\leq\frac{\varepsilon}{2^{m}}\text{ \ \ and \ \
}\|(A_{\alpha}(y_{m})-A_{\beta}(y_{m}))e_{m}\|_{\infty}\leq\frac{1}{m}\,\ \
\forall\ \alpha,\beta\geq\gamma_{m}.$

Thus, letting $e=\bigwedge_{m}e_{m}$, we obtain

$\tau(e^{\perp})\leq\varepsilon\text{ \ \ and \ \
}\|(A_{\alpha}(y_{m})-A_{\beta}(y_{m}))e\|_{\infty}\leq\frac{1}{m}\,\ \
\forall\ \alpha,\beta\geq\gamma_{m}.$

Therefore, given $\alpha,\beta\geq\gamma_{m}$, it follows that

$\begin{split}\|(A_{\alpha}(x)-A_{\beta}(x))e\|_{\infty}&=\|(A_{\alpha}(y_{m})-A_{\beta}(y_{m}))e\|_{\infty}\\\
&+\|(A_{\alpha}(z_{m})-A_{\beta}(z_{m}))e\|_{\infty}\leq\frac{1+2c_{\infty}}{m},\end{split}$

so the net $\\{A_{\alpha}(x)\\}_{\alpha\in\Lambda}$ is a.u. Cauchy. Since, by
Corollary 3.1, $\\{A_{\alpha}(x)\\}_{\alpha\in I}\subset\mathcal{R}_{\tau}$,
and, by Theorem 2.3, the space $\mathcal{R}_{\tau}$ is a.u. complete, we
conclude that there exists $\widehat{x}\in\mathcal{R}_{\tau}$ such that
$A_{\alpha}(x)\to\widehat{x}\in\mathcal{R}_{\tau}$ a.u.

Let now $E\subset\mathcal{R}_{\tau}$ be a fully symmetric space, and assume,
in addition, that $\|A_{\alpha}(x)\|_{1}\leq c_{1}\|x\|_{1}$ for some
$c_{1}>0$ and all $x\in L^{1}$. Put $c=\max\\{c_{\infty},c_{1}\\}$ and denote
$\widetilde{A}_{\alpha}=c^{-1}A_{\alpha}(x)$, $\alpha\in\Lambda$. As
$\\{\widetilde{A}_{\alpha}\\}_{\alpha\in I}\subset DS$, we conclude, in view
of Proposition 3.1, that $\widetilde{A}_{\alpha}(x)\prec\prec x$ for all $x\in
L^{1}+\mathcal{M}$, hence $\widetilde{A}_{\alpha}(E)\subset E$, and so
$A_{\alpha}(E)\subset E$ for all $\alpha\in\Lambda$.

Next, let $x\in E$. We have
$\widetilde{A}_{\alpha}(x)\to\widehat{x}\in\mathcal{R}_{\tau}$ a.u. To
simplify the argument, choose, as in the proof of Theorem 2.3, a sequence
$\\{\widetilde{A}_{n}(x)\\}\subset\\{\widetilde{A}_{\alpha}(x)\\}$ with
$\widetilde{A}_{n}(x)\to\widehat{x}$ a.u. Then it is clear that
$\widetilde{A}_{n}(x)\to\widehat{x}$ in measure, which, by [10, Lemma 3.4],
implies that $\mu_{t}(\widetilde{A}_{n}(x))\to\mu_{t}(\widehat{x})$ almost
everywhere on $((0,\infty),\nu)$. Further, as $\widetilde{A}_{n}(x)\prec\prec
x$, we have

$\int_{0}^{s}\mu_{t}(\widetilde{A}_{n}(x))dt\leq\int_{0}^{s}\mu_{t}(x)dt\ \ \
\forall\ s>0,\,n\in\mathbb{N}.$

Therefore, as $\mu_{t}(\widetilde{A}_{n}(x))\to\mu_{t}(\widehat{x})$ almost
everywhere on $((0,s),\nu)$, it follows from Fatou’s lemma that

$\int_{0}^{s}\mu_{t}(\widehat{x})dt\leq\int_{0}^{s}\mu_{t}(x)dt\ \ \ \forall\
s>0,$

that is $\widehat{x}\prec\prec x$. Since $E$ is a fully symmetric space, we
conclude that $\widehat{x}\in E$, hence $A_{\alpha}(x)\to c\,\widehat{x}\in E$
a.u., completing the argument. ∎

##  4\. Applications

For the sake of convenience, let us state the following immediate corollary of
Theorem 3.1.

######  Proposition 4.1.

Let $A_{\alpha}:L^{1}+\mathcal{M}\to L^{1}+\mathcal{M}$, $\alpha\in\Lambda$,
be a net of linear selfadjoint maps such that

  1. (1)

there exist $c_{1}>0$ and $c_{\infty}>0$ satisfying conditions

$\|A_{\alpha}(x)\|_{1}\leq c_{1}\|x\|_{1}\ \ \forall\ x\in L^{1}\text{\, \
and\, \ }\|A_{\alpha}(x)\|_{\infty}\leq c_{\infty}\|x\|_{\infty}\ \ \forall\
x\in\mathcal{M};$

  2. (2)

the net $\\{A_{\alpha}(x)\\}_{\alpha\in\Lambda}$ is a.u. (b.a.u.) Cauchy for
each $x\in L^{1}$.

Then, given a fully symmetric space $E\subset L^{0}$ with $\mathbf{1}\notin E$
and $x\in E$, there exists $\widehat{x}\in E$ such that
$A_{\alpha}(x)\to\widehat{x}$ a.u. (respectively, b.a.u.).

We will now present some applications of Proposition 4.1. Given
$d\in\mathbb{N}$, denote

$\mathbb{N}^{d}_{0}=\\{\,{\bf
n}=(n_{1},\dots,n_{d}):\,n_{1},\dots,n_{d}\in\mathbb{N}\cup{0}\\}.$

Let $\big{\\{}{\bf
n}_{\alpha}=(n_{1}(\alpha),\dots,n_{d}(\alpha))\big{\\}}_{\alpha\in\Lambda}$
be a net in $\mathbb{N}_{0}^{d}$. We write ${\bf n}_{\alpha}\to\infty$ if
$n_{i}(\alpha)\to\infty$ for each $1\leq i\leq d$. Denote $|{\bf
n}_{\alpha}|^{\prime}=n_{1}(\alpha)\cdot\ldots\cdot n_{d}(\alpha)$, and let
$|{\bf n}_{\alpha}|$ stand for $|{\bf n}_{\alpha}|^{\prime}$, with
$n_{i}(\alpha)$ replaced by $1$ whenever $n_{i}(\alpha)=0$.

Assume that $\\{T_{1},\dots,T_{d}\\}\subset DS^{+}$ and let $x\in
L^{1}+\mathcal{M}$. If $\alpha\in\Lambda$, set

(2) $A_{\alpha}(x)=\frac{1}{|{\bf n}_{\alpha}|}\sum_{k_{1}\leq
n_{1}(\alpha),\dots,k_{d}\leq n_{d}(\alpha)}T_{1}^{k_{1}}\cdot\ldots\cdot
T_{d}^{k_{d}}(x).$

Then $\\{A_{\alpha}\\}_{\alpha\in\Lambda}$ is a net of linear selfadjoint
(positive) maps in $L^{1}+\mathcal{M}$ that naturally satisfies condition (1)
of Proposition 4.1. To ensure that this net also satisfies condition (2) of
Proposition 4.1, we need an additional assumption that the net ${\bf
n}_{\alpha}\to\infty$ remains in a sector of $\mathbb{N}_{0}^{d}$ (see [13,
Ch. 6, § 6.2, 6.3] for the sequential, commutative case):

######  Definition 4.1.

A net ${\bf n}_{\alpha}\to\infty$ is said to remain in a sector of
$\mathbb{N}_{0}^{d}$ if there exists $0<c_{0}<\infty$ such that

$\displaystyle\frac{n_{i}(\alpha)}{n_{j}(\alpha)}\leq c_{0}\ \ \ \forall\
1\leq i,j\leq d\text{\,\ and\,\ }\alpha\in\Lambda,$

assuming that $n_{j}(\alpha)\neq 0$.

Now, straightforwardly adjusting to the net setting the argument in [6] that
yields [6, Theorem 5.3], we derive from Yeadon’s weak-type (1,1) maximal
inequality [23, Theorem 1] a weak-type $(1,1)$ maximal inequality for the net
$\\{A_{\alpha}\\}_{\alpha\in\Lambda}$ in (2) and use it to show that it is
b.a.u. Cauchy for every $x\in L^{1}$. Thus, as a corollary of Proposition 4.1,
we arrive at the following (cf. [6, Theorem 5.6]).

######  Theorem 4.1.

Let $\\{T_{1},\dots,T_{d}\\}\subset DS^{+}$ be a commuting family, and let
$E\subset\mathcal{R}_{\tau}$ be a fully symmetric space, for example, any
space $L^{p}$, $1\leq p<\infty$. Then, given $x\in E$, the net
$\\{A_{\alpha}(x)\\}_{\alpha\in\Lambda}$ given by (2) converges b.a.u. to some
$\widehat{x}\in E$, provided the net ${\bf n}_{\alpha}\to\infty$ remains in a
sector of $\mathbb{N}_{0}^{d}$.

Theorem 4.1 for $d=1$ yields the following version of [23, Theorem 2] (recall
that the class of contractions $\alpha$ considered in [23] coincides with the
class $DS^{+}$ modulo a unique extension - see [2, Proposition 1.1]).

######  Theorem 4.2.

Let $T\in DS^{+}$, and let
$\\{n_{\alpha}\\}_{\alpha\in\Lambda}\subset\mathbb{N}$ be a net such that
$n_{\alpha}\to\infty$. Then, given $x\in L^{1}$, the net

$A_{\alpha}(x)=\frac{1}{n_{\alpha}}\sum_{k=0}^{n_{\alpha}-1}T^{k}(x),\
\alpha\in\Lambda,$

converges b.a.u. to some $\widehat{x}\in L^{1}$.

Next, let $\mathbb{C}_{1}=\\{z\in\mathbb{C}:|z|=1\\}$. A function
$p:\mathbb{R}_{+}\to\mathbb{C}$ is called a trigonomertic polynomial if
$p(t)=\sum\limits_{j=1}^{n}w_{j}\lambda_{j}^{t}$, where $n\in\mathbb{N}$,
$\\{w_{j}\\}_{1}^{n}\subset\mathbb{C}$, and
$\\{\lambda_{j}\\}_{1}^{n}\subset\mathbb{C}_{1}$. A Lebesgue measurable
function $\beta:\mathbb{R}_{+}\to\mathbb{C}$ will be called a bounded
Besicovitch function if $\|\beta\|_{\infty}<\infty$, and for every
$\varepsilon>0$ there is a trigonometric polynomial $p_{\varepsilon}$ such
that

$\limsup_{t\to\infty}\frac{1}{t}\int_{0}^{t}|\beta(s)-p_{\varepsilon}(s)|ds<\varepsilon.$

Let a semigroup $\\{T_{t}\\}_{t\geq 0}\subset DS$ be strongly continuous in
$L^{1}$, that is,

$\|T_{t}(x)-T_{t_{0}}(x)\|_{1}\to 0\text{ \ \ whenever\ \ }t\to t_{0}\text{ \
\ for all\ \ }x\in L^{1}.$

Then, as it is shown in [5], given a Lebesgue measurable function
$\beta:\mathbb{R}_{+}\to\mathbb{C}$ with $\|\beta\|_{\infty}<\infty$, for any
$x\in L^{1}$ and $t>0$ there exists the Bochner integral

$B_{t}(x)=\frac{1}{t}\int_{0}^{t}\beta(s)T_{s}(x)ds\in L^{1}.$

Note that if $\|\beta\|_{\infty}\neq 0$, then for each $t>0$ the operator

$\|\beta\|_{\infty}^{-1}B_{t}:\,L^{1}\to L^{1}$

can be uniquely extended to a Dunford-Schwartz operator - see [2].

The following theorem was proved in [5]. Here we present an alternative,
straightforward, argument.

######  Theorem 4.3.

Let $\\{T_{t}\\}_{t\geq 0}\subset DS^{+}$ be a strongly continuous semigroup
in $L^{1}$, and let $\beta(t)$ be a bounded Besicovitch function. If
$\mathbf{1}\notin E\subset L^{0}$ is a fully symmetric space, then, given
$x\in E$, the flow $\\{B_{t}(x)\\}_{t>0}$ converges b.a.u. to some
$\widehat{x}\in E$ as $t\to\infty$. In particular, $B_{t}(x)\to\widehat{x}\in
L^{p}$ b.a.u. for every $x\in L^{p}$, $1\leq p<\infty$.

###### Proof.

It is clear that the net $\\{B_{t}\\}_{t>0}$ satisfies condition (1) of
Proposition 4.1. As the proof of [5, Theorem 3.1] shows that
$\\{B_{t}(x)\\}_{t>0}$ is b.a.u. Cauchy for all $x\in L^{1}$, that is,
condition (2) of Proposition 4.1 is also satisfied, the assertion follows. ∎

######  Remark 4.1.

One can verify, using the approach in [11], or in [14, 6], that for $p\geq 2$
the convergence in Theorems 4.1 and 4.3 is a.u.

Finally, we notice that the results of this article can be applied to simplify
and streamline some arguments in the noncommutative ergodic theory; for
example, some presented in [4, 6].

Acknowledgment. The author would like to express his gratitude to Dr. Vladimir
Chilin for his comments that helped to improve the article’s presentation. The
author is partially supported by the Pennsylvania State University traveling
program.

## References

  * [1] O. Bratelli, N. D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer-Verlag Berlin - Heidelberg - New York, 1979.
  * [2] V. Chilin, S. Litvinov, Ergodic theorems in fully symmetric spaces of $\tau$-measurable operators, Studia Math. 288(2) (2015), 177-195.
  * [3] V. Chilin, S. Litvinov, The validity space of Dunford–Schwartz pointwise ergodic theorem, J. Math. Anal. Appl. 461 (2018), 234–247.
  * [4] V. Chilin, S. Litvinov, Local ergodic theorems in symmetric spaces of measurable operators, Integr. Equ. Oper. Theory (2019) 91:15.
  * [5] V. Chilin, S. Litvinov, Noncommutative weighted individual ergodic theorems with continuous time, Inf. Dim. Anal. Quantum Prob. Rel. Topics 23 (2) (2020).
  * [6] V. Chilin, S. Litvinov, On individual ergodic theorems for semifinite von Neumann algebras, J. Math. Anal. Appl. 495 (2021).
  * [7] V. Chilin, S. Litvinov, A. Skalski, A few remarks in noncommutative ergodic theory, J. Oper. Theory 53(2) (2005), 331-350.
  * [8] P. G. Dodds, T. K. Dodds, and B. Pagter, Noncommutative Köthe duality, Trans. Amer. Math. Soc. 339(2) (1993), 717-750.
  * [9] G. A. Edgar, L. Sucheston, Stopping Times and Directed Processes, Cambridge University Press, 1992.
  * [10] T. Fack, H. Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific. J. Math. 123 (1986), 269-300.
  * [11] M. Junge, Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20(2) (2007), 385-439.
  * [12] S. G. Krein, Ju. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators, Translations of Mathematical Monographs, Amer. Math. Soc., 54, 1982.
  * [13] U. Krengel, Ergodic Theorems, Walter de Gruyer, Berlin-New York, 1985.
  * [14] S. Litvinov, Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems, Proc. Amer. Math. Soc. 140(7) (2012), 2401-2409.
  * [15] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116.
  * [16] E. T. Ordman, Convergence almost everywhere is not topological, Amer. Math. Monthly 73 (2) (1966).
  * [17] G. Pisier, Q. Xu, Noncommutative $L^{p}$-spaces, in: Handbook of the Geometry of Banach Spaces, 2, 2003, 1459-1517.
  * [18] C. Radin, Automorphisms of von Neumann algebras as point transformations, Proc. Amer. Math. Soc. 57 (1973), 343-346.
  * [19] I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. 57 (1953), 401-457.
  * [20] S. Strătilă, L. Zsidó, Lectures on von Neumann Algebras, Editura Academiei Române, Abacus Press, Bucureşti, 1979.
  * [21] M. Takesaki, Theory of Operator Algebras I, II, III, Springer-Verlag New York, 1979.
  * [22] F. J. Yeadon, Non-commutative $L^{p}$-spaces, Math. Proc. Camb. Phil. Soc. 77 (1975), 91-102.
  * [23] F. J. Yeadon, Ergodic theorems for semifinite von Neumann algebras. I, J. London Math. Soc. 16(2) (1977), 326-332.


# Ising Meson Spectroscopy on a Noisy Digital Quantum Simulator

Christopher Lamb<EMAIL_ADDRESS>Department of Physics and
Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA  Yicheng Tang
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ
08854-8019 USA  Robert Davis  Department of Physics and Astronomy, Rutgers
University, Piscataway, NJ 08854-8019 USA  Ananda Roy
<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, NJ 08854-8019 USA

###### Abstract

Quantum simulation has the potential to be an indispensable technique for the
investigation of non-perturbative phenomena in strongly-interacting quantum
field theories (QFTs). In the modern quantum era, with Noisy Intermediate
Scale Quantum (NISQ) simulators widely available and larger-scale quantum
machines on the horizon, it is natural to ask: what non-perturbative QFT
problems can be solved with the existing quantum hardware? We show that
existing noisy quantum machines can be used to analyze the energy spectrum of
a large family of strongly-interacting 1+1D QFTs. The latter exhibit a wide-
range of non-perturbative effects like ‘quark confinement’ and ‘false vacuum
decay’ which are typically associated with higher-dimensional QFTs of
elementary particles. We perform quench experiments on IBM’s ibmq_mumbai
quantum simulator to compute the energy spectrum of 1+1D quantum Ising model
with a longitudinal field. The latter model is particularly interesting due to
the formation of mesonic bound states arising from a confining potential for
the Ising domain-walls, reminiscent of t’Hooft’s model of two-dimensional
quantum chromodynamics. Our results demonstrate that digital quantum
simulation in the NISQ era has the potential to be a viable alternative to
numerical techniques such as density matrix renormalization group or the
truncated conformal space methods for analyzing QFTs.

Investigation of non-perturbative phenomena in strongly interacting quantum
field theories (QFTs) remains one of the outstanding challenges of modern
physics. Despite impressive progress, ab-initio lattice computations of
properties of arbitrary QFTs will likely remain beyond the computational
capabilities of the most powerful classical computer. This is due to the
exponentially-growing Hilbert space which poses an insurmountable challenge
for exact simulation of these lattice models. Nevertheless, quasi-exact or
approximate classical computing techniques have been successful in
investigating a large family of quantum many-body systems that realize QFTs in
the scaling limit. For example, states with ‘relatively low entanglement’ can
be efficiently represented using tensor networks [1, 2, 3, 4]. This has
allowed high-precision investigation of low-energy states of one-dimensional,
local, gapped and gapless Hamiltonians using matrix product states [5] as well
as their two-dimensional generalizations [6, 7, 8].

Despite the success of classical computing methods, simulation of high-energy
states or non-equilibrium dynamics remains challenging even for generic,
strongly-interacting 1+1D quantum systems. This is due to the rapid growth of
entanglement between different subsystems [9, 10, 11, 12]. Quantum simulation,
both analog and digital, have the potential to become a viable alternative to
the aforementioned approaches for tackling these questions [13, 14]. Analog
quantum simulation involves tailoring a given quantum system to emulate a
specific target model [15, 16, 17, 18, 19, 20] and has been remarkably
successful in analyzing a wide range of physical problems. However, it is
gate-based, error-corrected, digital quantum simulation that is the long-term,
universal solution for simulation of QFTs [21]. In this approach, lattice
regularizations of QFTs can be encoded into registers of error-corrected
qubits with suitable initial state preparation, time evolution and measurement
protocols. To that end, protocols for digitization of scalar fields [22, 23],
quantum algorithms for computation of scattering processes in scalar [24] and
fermionic [25] QFTs as well as simulation protocols for models of quantum
chromodynamics [26, 27], the Schwinger model [28] and the SU(3) Yang-Mills
theory [29] have already been proposed.

Figure 1:  (a) Schematic of the periodic Ising chain. The coral circles
indicate spin-1/2 sites. The black, green and blue bonds respectively
correspond to the ferromagnetic interaction, transverse and longitudinal
fields respectively. The quench protocol considered in this work involves
turning on the longitudinal and transverse field strengths: $g\leq 1,h<1$ in
the Hamiltonian of Eq. (1) after preparing the system in the ground state of
$H$ with $g=h=0$: $|\rightarrow,\ldots,\rightarrow\rangle$. (b) Schematic of
the qubit layout in IBM’s ibmq_mumbai quantum simulator. Of the available 27
qubits (in gray), a periodic chain of 12 qubits (in coral) and 20 qubits (in
yellow) were chosen for the quench experiments. (c) Schematic of the
trotterized unitary time-evolution generated by the Hamiltonian $H$. The
action of the full unitary operator is decomposed into the unitary operators:
$U_{e}$ and $U_{o}$. At the end of the time-evolution, single qubit
measurements are performed in the $\sigma_{y}$ basis. (d) The decompositions
of $U_{e}$ and $U_{o}$ in terms of single and two qubit gates are shown.
Figure 2:  Ising meson spectroscopy simulation and experimental results for
quench from $g=h=0$. The Qiskit twirled readout error extinction and dynamical
decoupling schemes were used to mitigate the effects of noise present in the
IBM Mumbai simulator. (Top panels) Results for
$\sigma^{y}(t)=$$\sum_{j}\langle\sigma_{j}^{y}(t)\rangle/L$ for different
quench parameters are shown. A periodic chain with length $L=12~{}(20)$ qubits
[in pink (yellow) in Fig. 1] was used for quenches to $g<1(=1)$ with $h=0.3$.
The results obtained using exact simulation of the time evolution of $L$
qubits with $\Delta t=0.01$ are shown with green solid lines. The noiseless
numerical Qiskit simulation results are performed for $\Delta t=0.4$ (orange
dashed line). The expectation values for the noiseless simulations are
computed by trotterized time-evolution and making measurements in the
$\sigma^{y}$ basis for each qubit after every time-step. The data was averaged
over $10^{5}$ and 8192 shots for the 12 and 20 qubit systems respectively. The
close agreement of the noiseless simulation and the exact results confirm the
relatively low trotterization error for $\sigma^{y}(t)$ even for the quench to
$g=1$. The experimental data from the quantum hardware (IBM Mumbai) are shown
as blue diamonds (squares) for $g<1(=1)$. (Bottom panels) The squared absolute
values of the Fourier transforms $\sigma^{y}(\omega)$ are shown. The peaks in
the latter correspond to the rest energies ($e_{n}$), of the mesonic
excitations labeled by $n$, and their differences ($e_{mn}$). The gray dashed
lines correspond to the exact diagonalization results obtained in the zero
momentum sector.

However, implementing a quantum simulation protocol for general QFTs on
existing quantum hardware remains a daunting challenge. In spite of the recent
progress in the fabrication and manipulation photonic and solid-state systems,
the number of available qubits across the different quantum computing
platforms remains around $10^{2}$ with modest coherence times. Given the large
overhead in the number of required qubits to implement quantum error
correcting codes [30], it is an open problem to implement error-corrected
digital simulation of QFTs in the near term. This raises the question: what
non-perturbative QFT problems can be tackled with current Noisy Intermediate
Scale Quantum (NISQ) machines?

In this work, a superconducting NISQ simulator is used to compute the energy
spectra of certain strongly-interacting QFTs that arise as scaling limits of
one-dimensional quantum spin chains. The latter QFTs offer a valuable
playground for investigation of a wide range of non-perturbative phenomena
such as ‘quark confinement’ [31] and ‘false vacuum decay’ [32] that are
typically associated with QFTs in higher-dimensions. Importantly, there are
two practical advantages: i) these spin-chains can be directly mapped to
arrays of qubits without additional overhead involved in discretizing the
local lattice degree of freedom, ii) several properties of these spin-chains
can be investigated with existing classical numerical techniques (tensor
networks) and analytical methods (Bethe ansatz and bosonization). This enables
benchmarking NISQ simulators before using them for investigation of problems
that truly lie beyond the reach of the classical computers.

The goal of this work is to demonstrate the aforementioned using the
paradigmatic example of the one-dimensional quantum Ising model in the
presence of a longitudinal field [33]. The Hamiltonian is:

$H=-\sum_{j=1}^{L}\left(\sigma^{x}_{j}\sigma^{x}_{j+1}+g\sigma^{z}_{j}+h\sigma^{x}_{j}\right),$
(1)

where $g$, $h$ are the strengths of the transverse and longitudinal felds and
we have chosen periodic boundary conditions. We consider the case $g\leq
1,h<1$. The presence of the longitudinal field introduces a confining
potential between the domain-wall excitations of the Ising model. This leads
to formation of ‘mesonic’ bound states, analogous to hadron-formation in
quantum chromodynamics [34]. The energy spectrum has been analyzed using semi-
classical [35, 36], truncated conformal space [37, 38, 39, 40] and exact
diagonalization [41] approaches. While signatures of confinement have been
observed for this model in a NISQ simulator [42] (see also Refs. [43, 44]),
digital quantum simulation of the energy spectrum has remained elusive. This
is performed in this work using a quench experiment on IBM’s ibmq_mumbai
(hereafter referred to as IBM Mumbai) simulator which is one of the IBM
Quantum Falcon Processors. The quantum simulation protocol is first tested
using the Qiskit software package [45] and benchmarked against exact
computations. Subsequently, it is implemented on the noisy quantum hardware.

The quantum simulation protocol is as follows (see Fig. 1). First, the qubits
are prepared in the state: $|\rightarrow,\ldots,\rightarrow\rangle$, which is
one of the two degenerate ground state of the Hamiltonian $H$ with $g=h=0$
111The protocol could be implemented also with the initial state being the
ground state of the Hamiltonian with $g<1,h=0$. However, preparing such an
initial state on a quantum simulator is nontrivial [58, 59]. For simplicity,
we restrict ourselves to initial product states.. This is done by applying a
Hadamard gate on each qubit after initializing them in the ground state. Next,
a global quench is performed to $0<g\leq 1,h<1$. The quench experiments with
$g<1~{}(=1)$ were performed on a 12 (20) qubit loop [in pink (yellow) in Fig.
1] within the IBM Mumbai simulator. The quench is implemented in the NISQ
hardware by the application of single and two-qubit gates (see Fig. 1(c,d) for
the gate decomposition) [47] that together give rise to the unitary operator
$U={\rm e}^{-{\rm i}H\Delta t}$. Here, $\Delta t$ is the size of the time-
step. The rest energy spectrum of the mesonic excitations formed due to the
confinement of fermionic Ising domain-walls manifests itself in the
observables that involve only single-qubit operators [41]. This is
particularly suitable for implementation in a NISQ machine where single qubit
measurements are performed with relatively low error. In particular, in this
work, the single-qubit observable $\langle\sigma_{j}^{y}(t)\rangle$, $j$ being
the site-index, is measured during the course of this evolution 222The meson
rest energy spectrum can be also computed by measuring
$\langle\sigma_{j}^{x}(t)\rangle$ or $\langle\sigma_{j}^{z}(t)\rangle$. On a
noisy simulator, the best noise-resilience for the energy computation was
obtained for $\langle\sigma_{j}^{y}(t)\rangle$. The same choice was found to
be optimal with regards to the errors arising from trotterization of the
continuous time-evolution..

Fig. 2 shows the results for the quenches from $g=h=0$ to $g\leq 1,h=0.3$. The
top panels show the measured variation of
$\sigma^{y}(t)=$$\sum_{j}\langle\sigma_{j}^{y}(t)\rangle/L$ with time. The
solid green curves act as reference and are obtained from exact computation of
the time-evolution of the chain of qubits with $\Delta t=0.01$ 333Exact
simulation of $g=1$ with 20 qubits was challenging for the classical
computational resources available for this work. This case was computed with
Qutip [60, 61]. The dashed orange curves are obtained from numerical
simulations on a noiseless Qiskit simulator for $\Delta t=0.4$. In this case,
the expectation value $\langle\sigma_{j}^{y}(t)\rangle$ was computed for each
qubit at each time-step by measuring it in the corresponding basis and
averaging over 8192 shots. The close agreement of the noiseless simulation
results for the two choices of $\Delta t$ indicates a small trotterization
error for this observable. The quantum simulation experiment on IBM Mumbai was
performed with the same parameters as the noiseless simulations. The
experimental data is shown as blue diamonds (squares) for $g<1(=1)$. To
mitigate the effects of errors and noise on the quantum hardware, the in-built
Qiskit twirled readout error extinction and dynamical decoupling schemes were
chosen. The bottom panels show the absolute values of the corresponding
Fourier transforms. The rest energies of the mesons are inferred from the
location of the peaks, with the dashed vertical lines being results obtained
using exact diagonalization in the zero momentum sector. Note that the quality
of the data from the quantum hardware for $g=1$ is worse than the $g<1$ cases.
This is due to the lower gap between the ground and first excited states of
the energy spectrum. This makes the experiments more susceptible to the noise
arising due to trotterization and noise in the IBM Mumbai simulator.
Nevertheless, the peak corresponding to the difference between the lowest two
meson energies is clearly discernible for the $g=1$ case.

Figure 3:  Variation of the meson energies obtained from quantum simulation
experiments performed on the IBM Mumbai simulator. The results (in diamonds)
are shown for a quench from $g=h=0$ to $g\in[0.25,0.75]$ and $h=0.3$. The
corresponding predictions from exact diagonalization of the 12-site qubit
chain keeping only the translation-invariant states are shown as dotted lines.
The meson energies are shown as a function of the dimensionless parameter
$\eta=2\pi(1-g)/h^{8/15}$. The peaks for the different meson energies were
identified by computing the $\sigma^{y}(\omega)$ for quenching to different
choices of $g$ [see also Fig. 2]. Notice that the obtained quantum data did
not always permit unambiguous identification of the second and third meson
energies. Nevertheless, the agreement between the experimentally obtained
lowest meson energy and the exact diagonalization results is reasonable.

Fig. 3 shows the variation of the meson energies obtained from different
quench experiments performed on the IBM Mumbai simulator. In every experiment,
the system was initialized in the ground state of $g=h=0$. Subsequently, a
quench was performed to $h=0.3$ and $g$ varying between 0.25 and 0.75 in steps
of 0.05. The obtained meson energies are shown as diamonds. The corresponding
predictions from the exact diagonalization of the qubit chain in the zero
momentum sector are shown as dotted lines. Notice that the noisy data did not
always permit unique identification of the second and third meson energies.
However, the lowest meson energy could always be unambiguously obtained. The
agreement between the experimentally-obtained lowest meson energies and the
exact diagonalization results is reasonable.

Fig. 4 shows the results for the connected correlation function
$G_{x}(r,t)=\langle\sigma_{i}^{x}(t)\sigma_{i+r}^{x}(t)\rangle-\langle\sigma_{i}^{x}(t)\rangle\langle\sigma_{i+r}^{x}(t)\rangle$.
The latter contains signatures of the confinement of the domain-walls into
mesonic excitations [41]. The results of the noiseless Qiskit simulations are
shown in the left panels. The simulation parameters are kept as in Fig. 2. For
$h=0$, the correlation function exhibits a characteristic broadening due to
the propagation of the energy-carrying domain wall excitations [9] until the
separation between the latter is of the order of the system size (top left
panel). For $h\neq 0$, as the domain-walls get separated, they feel the
confining force that leads to the formation of the mesonic bound states. This
results in oscillatory behavior of the correlation function [41] (bottom left
panel). The corresponding expectation value
$\sigma^{x}(t)=\sum_{j}\langle\sigma_{j}^{x}(t)\rangle/L$ for the noiseless
simulations are shown with the dashed orange line in the right panels. The
data from the quantum hardware is shown in the middle panels. With the current
noise levels in the gates and modest lifetimes of the qubits of IBM’s Mumbai
simulator, signatures of confinement could not be reliably discerned for the
noisy experimental data for the correlation function. This is consistent with
the measured rapid decay of $\sigma^{x}(t)$ shown with blue diamonds in the
right panel (compare the oscillations obtained for
$\langle\sigma^{y}(t)\rangle$ shown in Fig. 2). Additional error mitigation
techniques and improved quantum hardware could be used to obtain signatures of
confinement in the correlation functions.

Figure 4:  Results for $G_{x}(r,t)$ and the $\langle\sigma^{x}(t)\rangle$ for
a quench to $g=0.25,h=0$ and $g=0.25,h=0.2$. The noiseless simulation results
for $G_{x}(r,t)$ are shown in the left panels. The corresponding results for
$\sigma^{x}(t)$ are shown with orange dashed lines in the right panels. The
simulation parameters are the same as in Fig. 2. The results of the quantum
simulation experiments on the IBM Mumbai simulator are shown in the middle
panel and with blue diamonds in the right panel. With the current noisy gates
and qubits on the quantum hardware, a distinct signature of confinement could
not be discerned. This is consistent with the rapid decay of
$\langle\sigma^{x}(t)\rangle$ (blue diamonds in the right panels). Additional
error mitigation techniques and improved quantum hardware can be used to
obtain more reliable signatures of the confinement in the correlation
functions.

In summary, this work highlights the potential of NISQ simulators for
investigation of non-perturbative problems in strongly-interacting QFTs. In
particular, a noisy IBM simulator is used to compute the energy-spectrum of
the mesonic excitations occurring in the one-dimensional Ising model with a
longitudinal field. These mesonic excitations have masses that are larger than
twice the mass of the Ising domain walls [33]. Crucially, the confinement of
the Ising domain walls into mesonic excitations leads to a small-enough
length-scale making this model particularly suitable for realization on a NISQ
simulator.

In contrast to traditional approaches based on discretization of local degrees
of freedom of the QFT using a register of qubits, a quantum spin chain
provides a natural regularization of the QFT under consideration with minimal
overhead. As such, this demonstrated quantum simulation scheme can be
straightforwardly generalized to analyze a wide family of 1+1D QFTs admitting
similar spin chain regularizations. An example is the quantum sine-Gordon
model perturbed by a cosine potential with twice the periodicity. In the
latter model, the free Ising domain walls are replaced by interacting sine-
Gordon solitons [51, 52, 53] and this model can be realized as the scaling
limit of the XYZ spin chain [50] in the presence of a longitudinal field as
well as in a recently-proposed analog quantum electronic circuit array [54].
The details of the simulation protocol and noiseless simulation results at the
free-fermion point of the latter model are given in [55]. With improved noise
properties, larger sizes, and more sophisticated error-mitigation techniques
that are likely to be available in the coming years, it is conceivable that a
wider range of QFTs will become amenable to quantum simulation on quantum
hardware in the imminent future.

The authors are grateful to Martin Savage for pointing out the recent
promising results on simulation of SU(2) hadrons [56] and 1+1D models of
quantum chromodynamics [57]. The authors acknowledge discussions with Nicholas
Bronn, Robert Konik and Olivia Lanes and the use of IBM Quantum services for
this work. The views expressed are those of the authors, and do not reflect
the official policy or position of IBM or the IBM Quantum team. AR was
supported from a grant from the Simons Foundation (825876, TDN).

## References

  * Hastings [2007] M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. 0708, P08024 (2007), arXiv:0705.2024 [quant-ph] .
  * Vidal [2008] G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101, 110501 (2008).
  * Schuch  _et al._ [2008] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Entropy scaling and simulability by matrix product states, Phys. Rev. Letters 100, 10.1103/physrevlett.100.030504 (2008).
  * Verstraete  _et al._ [2008] F. Verstraete, V. Murg, and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Advances in Physics 57, 143 (2008).
  * Schollwöck [2011] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue.
  * Verstraete and Cirac [2004] F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions (2004).
  * Schuch  _et al._ [2007] N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).
  * Zaletel and Pollmann [2020] M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  * Calabrese and Cardy [2005] P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, Journal of Statistical Mechanics: Theory and Experiment 2005, P04010 (2005).
  * Vidal [2007] G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, 070201 (2007).
  * Paeckel  _et al._ [2019] S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig, Time-evolution methods for matrix-product states, Annals of Physics 411, 167998 (2019).
  * Lin  _et al._ [2022] S.-H. Lin, M. P. Zaletel, and F. Pollmann, Efficient simulation of dynamics in two-dimensional quantum spin systems with isometric tensor networks, Phys. Rev. B 106, 10.1103/physrevb.106.245102 (2022).
  * Feynman [1982] R. P. Feynman, Simulating Physics with Quantum Computers, Inernational Journal of Theoretical Physics 21, 467 (1982).
  * Lloyd [1996] S. Lloyd, Universal Quantum Simulators, Science 273, 1073 (1996).
  * Douçot  _et al._ [2004] B. Douçot, L. B. Ioffe, and J. Vidal, Discrete non-abelian gauge theories in josephson-junction arrays and quantum computation, Phys. Rev. B 69, 214501 (2004).
  * Cirac  _et al._ [2010] J. I. Cirac, P. Maraner, and J. K. Pachos, Cold atom simulation of interacting relativistic quantum field theories, Phys. Rev. Lett. 105, 190403 (2010).
  * Büchler  _et al._ [2005] H. P. Büchler, M. Hermele, S. D. Huber, M. P. A. Fisher, and P. Zoller, Atomic quantum simulator for lattice gauge theories and ring exchange models, Phys. Rev. Lett. 95, 040402 (2005).
  * Casanova  _et al._ [2011] J. Casanova, L. Lamata, I. L. Egusquiza, R. Gerritsma, C. F. Roos, J. J. García-Ripoll, and E. Solano, Quantum simulation of quantum field theories in trapped ions, Phys. Rev. Lett. 107, 260501 (2011).
  * Gross and Bloch [2017] C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017), https://science.sciencemag.org/content/357/6355/995.full.pdf .
  * Roy and Saleur [2019] A. Roy and H. Saleur, Quantum electronic circuit simulation of generalized sine-gordon models, Phys. Rev. B 100, 155425 (2019).
  * Jordan  _et al._ [2012] S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum Algorithms for Quantum Field Theories, Science 336, 1130 (2012), arXiv:1111.3633 [quant-ph] .
  * Macridin  _et al._ [2018] A. Macridin, P. Spentzouris, J. Amundson, and R. Harnik, Digital quantum computation of fermion-boson interacting systems, Phys. Rev. A 98, 10.1103/physreva.98.042312 (2018).
  * Klco and Savage [2019] N. Klco and M. J. Savage, Digitization of scalar fields for quantum computing, Phys. Rev. A 99, 10.1103/physreva.99.052335 (2019).
  * Jordan  _et al._ [2014a] S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum Computation of Scattering in Scalar Quantum Field Theories, Quant. Inf. Comput. 14 (2014a), arXiv:1112.4833 [hep-th] .
  * Jordan  _et al._ [2014b] S. P. Jordan, K. S. M. Lee, and J. Preskill, Quantum Algorithms for Fermionic Quantum Field Theories, (2014b), arXiv:1404.7115 [hep-th] .
  * Farrell  _et al._ [2022a] R. C. Farrell, I. A. Chernyshev, S. J. M. Powell, N. A. Zemlevskiy, M. Illa, and M. J. Savage, Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (I) Axial Gauge, (2022a), arXiv:2207.01731 [quant-ph] .
  * Farrell  _et al._ [2022b] R. C. Farrell, I. A. Chernyshev, S. J. M. Powell, N. A. Zemlevskiy, M. Illa, and M. J. Savage, Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (II) Single-Baryon $\beta$-Decay in Real Time, (2022b), arXiv:2209.10781 [quant-ph] .
  * Klco  _et al._ [2018] N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, and M. J. Savage, Quantum-classical computation of Schwinger model dynamics using quantum computers, Phys. Rev. A 98, 032331 (2018), arXiv:1803.03326 [quant-ph] .
  * Ciavarella  _et al._ [2021] A. Ciavarella, N. Klco, and M. J. Savage, Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis, Phys. Rev. D 103, 094501 (2021), arXiv:2101.10227 [quant-ph] .
  * Nielsen and Chuang [2000] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
  * Greensite [2011] J. Greensite,  _An introduction to the confinement problem_, Vol. 821 (2011).
  * Coleman [1988] S. Coleman,  _Aspects of Symmetry: Selected Erice Lectures_ (Cambridge University Press, 1988).
  * McCoy and Wu [1978] B. M. McCoy and T. T. Wu, Two-dimensional ising field theory in a magnetic field: Breakup of the cut in the two-point function, Phys. Rev. D 18, 1259 (1978).
  * ’t Hooft [1974] G. ’t Hooft, A two-dimensional model for mesons, Nuclear Physics B 75, 461 (1974).
  * Rutkevich [2005] S. B. Rutkevich, Large-$n$ excitations in the ferromagnetic ising field theory in a weak magnetic field: Mass spectrum and decay widths, Phys. Rev. Lett. 95, 250601 (2005).
  * Rutkevich [2008] S. B. Rutkevich, Energy spectrum of bound-spinons in the quantum ising spin-chain ferromagnet, J. Stat. Phys. 131, 917 (2008).
  * Fonseca and Zamolodchikov [2003] P. Fonseca and A. Zamolodchikov, Ising field theory in a magnetic field: Analytic properties of the free energy, J. Stat. Phys. 110, 527 (2003).
  * Fonseca and Zamolodchikov [2006] P. Fonseca and A. Zamolodchikov, Ising spectroscopy. I. Mesons at $T<T_{c}$, (2006), arXiv:hep-th/0612304 .
  * James  _et al._ [2019] A. J. A. James, R. M. Konik, and N. J. Robinson, Nonthermal states arising from confinement in one and two dimensions, Phys. Rev. Lett. 122, 130603 (2019).
  * Robinson  _et al._ [2019] N. J. Robinson, A. J. A. James, and R. M. Konik, Signatures of rare states and thermalization in a theory with confinement, Phys. Rev. B 99, 195108 (2019).
  * Kormos  _et al._ [2016] M. Kormos, M. Collura, G. Takács, and P. Calabrese, Real-time confinement following a quantum quench to a non-integrable model, Nature Physics 13, 246 (2016).
  * Vovrosh and Knolle [2021] J. Vovrosh and J. Knolle, Confinement and entanglement dynamics on a digital quantum computer, Scientific Reports 11, 11577 (2021).
  * Knaute and Hauke [2022] J. Knaute and P. Hauke, Relativistic meson spectra on ion-trap quantum simulators, Phys. Rev. A 105, 022616 (2022), arXiv:2107.09071 [cond-mat.str-el] .
  * Tan  _et al._ [2021] W. L. Tan, P. Becker, F. Liu, G. Pagano, K. S. Collins, A. De, L. Feng, H. B. Kaplan, A. Kyprianidis, R. Lundgren, W. Morong, S. Whitsitt, A. V. Gorshkov, and C. Monroe, Domain-wall confinement and dynamics in a quantum simulator, Nature Physics 17, 742 (2021).
  * tA-v et al [2021] A. tA-v et al, Qiskit: An open-source framework for quantum computing (2021).
  * Note [1] The protocol could be implemented also with the initial state being the ground state of the Hamiltonian with $g<1,h=0$. However, preparing such an initial state on a quantum simulator is nontrivial [58, 59]. For simplicity, we restrict ourselves to initial product states.
  * Smith  _et al._ [2019] A. Smith, M. S. Kim, F. Pollmann, and J. Knolle, Simulating quantum many-body dynamics on a current digital quantum computer, npj Quantum Inf. 5, 106 (2019), arXiv:1906.06343 [quant-ph] .
  * Note [2] The meson rest energy spectrum can be also computed by measuring $\langle\sigma_{j}^{x}(t)\rangle$ or $\langle\sigma_{j}^{z}(t)\rangle$. On a noisy simulator, the best noise-resilience for the energy computation was obtained for $\langle\sigma_{j}^{y}(t)\rangle$. The same choice was found to be optimal with regards to the errors arising from trotterization of the continuous time-evolution.
  * Note [3] Exact simulation of $g=1$ with 20 qubits was challenging for the classical computational resources available for this work. This case was computed with Qutip [60, 61].
  * Baxter [2013] R. Baxter,  _Exactly Solved Models in Statistical Mechanics_, Dover Books on Physics (Dover Publications, 2013).
  * Delfino and Mussardo [1998] G. Delfino and G. Mussardo, Non-integrable aspects of the multi-frequency sine-gordon model, Nuclear Physics B 516, 675–703 (1998).
  * Bajnok  _et al._ [2001] Z. Bajnok, L. Palla, G. Takacs, and F. Wagner, Nonperturbative study of the two frequency sine-Gordon model, Nucl. Phys. B 601, 503 (2001), arXiv:hep-th/0008066 .
  * Mussardo  _et al._ [2004] G. Mussardo, V. Riva, and G. Sotkov, Semiclassical particle spectrum of double sine-Gordon model, Nucl. Phys. B 687, 189 (2004), arXiv:hep-th/0402179 .
  * Roy and Lukyanov [2023] A. Roy and S. Lukyanov, Soliton Confinement in a Quantum Circuit, arXiv (2023), arXiv:2302.06289 [quant-ph] .
  * [55] See supplementary material for details.
  * Atas  _et al._ [2021] Y. Y. Atas, J. Zhang, R. Lewis, A. Jahanpour, J. F. Haase, and C. A. Muschik, SU(2) hadrons on a quantum computer via a variational approach, Nature Commun. 12, 6499 (2021), arXiv:2102.08920 [quant-ph] .
  * Atas  _et al._ [2022] Y. Y. Atas, J. F. Haase, J. Zhang, V. Wei, S. M. L. Pfaendler, R. Lewis, and C. A. Muschik, Simulating one-dimensional quantum chromodynamics on a quantum computer: Real-time evolutions of tetra- and pentaquarks, (2022), arXiv:2207.03473 [quant-ph] .
  * Verstraete  _et al._ [2009] F. Verstraete, J. I. Cirac, and J. I. Latorre, Quantum circuits for strongly correlated quantum systems, Phys. Rev. A 79, 032316 (2009).
  * Ho and Hsieh [2019] W. W. Ho and T. H. Hsieh, Efficient variational simulation of non-trivial quantum states, SciPost Phys. 6, 029 (2019).
  * Johansson  _et al._ [2012] J. Johansson, P. Nation, and F. Nori, Qutip: An open-source python framework for the dynamics of open quantum systems, Computer Physics Communications 183, 1760 (2012).
  * Johansson  _et al._ [2013] J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Computer Physics Communications 184, 1234 (2013).