# Essential normality of quotient modules vs. Hilbert-Schmidtness of
submodules in $H^{2}(\mathbb{D}^{2})$

Penghui Wang Chong Zhao Zeyou Zhu

School of Mathematics, Shandong University
Jinan, Shandong 250100, The People’s Republic of China
Partially supported by NSFC(No. 11871308,12271298), E-mail<EMAIL_ADDRESS>Partially supported by NSFC(No. 12071253), and the Young Scholars Program of
Shandong University, E-mail<EMAIL_ADDRESS>Partially supported by
NSFC(No. 11871308,12271298), E-mail<EMAIL_ADDRESS>
###### Abstract

In the present paper, we prove that all the quotient modules in
$H^{2}(\mathbb{D}^{2})$, associated to the finitely generated submodules
containing a distinguished homogenous polynomial, are essentially normal,
which is the first result on the essential normality of non-algebraic quotient
modules in $H^{2}(\mathbb{D}^{2})$. Moreover, we obtain the equivalence of the
essential normality of a quotient module and the Hilbert-Schmidtness of its
associated submodule in $H^{2}(\mathbb{D}^{2})$, in the case that the
submodule contains a distinguished homogenous polynomial. As an application,
we prove that each finitely generated submodule containing a polynomial is
Hilbert-Schmidt, which partially gives an affirmative answer to the conjecture
of Yang [39].

2000 MSC. 47A13, 46H25

Key Words. Essentially normal quotient module, Hilbert-Schmidt submodule,
distinguished variety, Hardy space over the polydisc

##  1 Intoduction

The Hardy space $H^{2}(\mathbb{D}^{d})$ on the unit polydisc can be viewed as
a Hilbert module [5, 15] on the polynomial ring
$\mathbb{C}[z_{1},\ldots,z_{d}]$, equipped with a natural module action
defined by the obvious multiplication by polynomials. A closed subspace
$\mathcal{M}$ of $H^{2}(\mathbb{D}^{d})$ that is invariant under the module
actions is called a submodule, and $\mathcal{N}={\mathcal{M}}^{\perp}$ is
called the associated quotient module. The module actions on submodules and
quotient modules are given by

$p\cdot
h=R_{p,\mathcal{M}}h,~{}p\in\mathbb{C}[z_{1},\ldots,z_{d}],~{}h\in\mathcal{M},$

and

$p\cdot
g=S_{p,\mathcal{N}}g,~{}p\in\mathbb{C}[z_{1},\ldots,z_{d}],~{}g\in\mathcal{N},$

where $R_{p,\mathcal{M}}=M_{p}|_{\mathcal{M}}$ and
$S_{p,\mathcal{N}}=P_{\mathcal{N}}M_{p}|_{\mathcal{N}}$. For brevity, we write
$R_{p}=R_{p,\mathcal{M}}$ and $S_{p}=S_{p,\mathcal{N}},$ if there is no
confusion on the modules referred to. A quotient module $\mathcal{N}$ is
called essentially normal if all the commutators $[S_{p}^{*},S_{p}]$ for
$p\in\mathbb{C}[z_{1},\ldots,z_{d}]$ are compact. Let
$\mathcal{K}(\mathcal{N})$ be the ideal of compact operators on $\mathcal{N}$,
which is written as $\mathcal{K}$ for short.

In [2], Arveson conjectured that all the homogeneous submodules of the
$d$-shift module over the complex unit ball are essentially normal. Much work
has been done along this line, such as [2, 10, 11, 12, 13, 14, 17, 18, 19, 26,
30, 34] and references therein, and there is a good survey on this problem
provided in [23]. A polydisc version of Arveson’s conjecture is to
characterize the essential normality of homogeneous quotient modules in
$H^{2}(\mathbb{D}^{2})$, which was solved by the first two authors in [37].
Other efforts on the polydisc version of Arveson’s conjecture can be found in
[6, 9, 21, 22, 33, 35, 36, 38] and references therein.

It is known that the essential normality of quotient modules of analytic
Hilbert modules over the polydisc is closely related to distinguished
varieties, which were introduced by Agler and $\operatorname{M^{c}Carthy}$ [1]
to deal with Ando’s inequality and Pick interpolation problem, etc. Recall
that an ideal $I$ in $\mathbb{C}[z_{1},\ldots,z_{d}]$ is distinguished if the
zero variety of $I$ satisfies

$\partial\big{(}Z(I)\cap\mathbb{D}^{d}\big{)}\subset\mathbb{T}^{d}.$

In this case, the submodule $[I],$ which is the closure of $I$, and the
associated quotient module $[I]^{\bot}$ are called distinguished. Recently, in
[24], by introducing the Grassmannian structure for the distinguished quotient
modules, Guo and the first two authors proved the following result.

######  Theorem 1.1.

([24, Theorem 1.3]) Let $V$ be an algebraic subvariety of $\mathbb{D}^{2}$,
and $I_{V}$ be the radical idea of $\mathbb{C}[z,w]$ determined by $V$. Then
$V$ is distinguished if and only if, as a quotient module of some weighted
Bergman space $A_{\alpha}^{2}(\mathbb{D}^{2})$, $[I_{V}^{2}]^{\perp}$ is
essentially normal.

In this theorem, the choice of weighted Bergman space depends on the algebraic
rank of the variety. Therefore, it is worth to find a fixed analytic Hilbert
module over the bidisk, such as the Hardy space $H^{2}(\mathbb{D}^{2})$, on
which all the distinguished quotient modules are essentially normal. There are
two important results along this line. Firstly, it was proved in [36] that all
the distinguished homogeneous quotient modules in $H^{2}(\mathbb{D}^{d})$ are
essentially normal. Secondly, Clark [6] proved that for nonconstant finite
Blaschke products $B_{i}(z_{i})$, $i=1,\ldots,d$, the quotient module
$[B_{1}(z_{1})-B_{2}(z_{2}),\cdots,B_{d-1}(z_{d-1})-B_{d}(z_{d})]^{\perp}$ in
$H^{2}(\mathbb{D}^{d})$ is a kind of weighted Bergman space over the
distinguished variety
$\bigcap\limits_{i=1}^{d-1}Z\big{(}B_{i}(z_{i})-B_{i+1}(z_{i+1})\big{)}$,
whence essentially normal. Inspired by these observations, we introduce the
concept of $\emph{Clark-type quotient modules}.$

In what follows, for $q\in\mathbb{C}[z_{1},\ldots,z_{d}]$ and a $d$-tuple of
inner functions
$\eta(z)=\big{(}\eta_{1}(z_{1}),\ldots,\eta_{d}(z_{d})\big{)},$ we write

$p_{\eta}(z)=p\big{(}\eta_{1}(z_{1}),\ldots,\eta_{d}(z_{d})\big{)},$

and $J_{\eta}=\\{q\circ\eta:q\in J\\}$ for subsets
$J\subset\mathbb{C}[z_{1},\ldots,z_{d}]$.

To continue, we introduce the terminology of quasi-homogeneity. By the term
weight we meant a given tuple $\mathbf{K}=(K_{1},\ldots,K_{d})$ of positive
integers. For a monomial $z^{\alpha}$, the integer
$\alpha\cdot\mathbf{K}=\sum_{i=1}^{d}\alpha_{i}K_{i}$ is called its
$\mathbf{K}$-degree. A linear combination of monomials with the same
$\mathbf{K}$-degree is called $\mathbf{K}$-quasi-homogeneous. Ideals generated
by $\mathbf{K}$-quasi-homogeneous polynomials are called $\mathbf{K}$-quasi-
homogeneous, and so do the associated submodules and quotient modules. The
essential normality of $\mathbf{K}$-quasi-homogeneous quotient modules of
$H^{2}(\mathbb{D}^{d})$ was completely characterized in [38].

######  Definition 1.2.

Let $I\subset\mathbb{C}[z_{1},\ldots,z_{d}]$ be an ideal. If there exists a
$d$-tuple of nonconstant finite Blaschke products
$B(z)=\big{(}B_{1}(z_{1}),\ldots,B_{d}(z_{d})\big{)}$ and a distinguished
quasi-homogeneous ideal $J\subset\mathbb{C}[z_{1},\ldots,z_{d}],$ such that

$I=\\{f\in\mathbb{C}[z,w]:f=qp_{B},~{}p\in
J,~{}q\in\mathbb{C}[z_{1},\ldots,z_{d}]\\},$

then we call $I$ a Clark-type ideal, $Z(I)\cap\mathbb{D}^{d}$ a Clark-type
variety, and $[I]^{\perp}$ a Clark-type quotient module.

Evidently the Clark-type varieties are distinguished. Since the denominators
of finite Blaschke products are invertible outer functions, the submodule
$[I]$ and the submodule generated by $J_{B}$ coincide. Note that an ideal with
Clark-type variety is not necessarily Clark-type.

In the case $d=2$, every distinguished K-quasi-homogeneous ideal $J$ has a set
of generators $\\{p_{i}(z^{K_{2}},w^{K_{1}}):1\leq i\leq n\\}$, where each
$p_{i}$ is a distinguished homogeneous polynomial. Temporarily write
$J^{\prime}$ the ideal generated by $\\{p_{1},\ldots,p_{n}\\}$, and let $I$ be
the Clark-type ideal defined by $J$ and $B=(B_{1},B_{2})$, then

$\displaystyle I$ $\displaystyle=$
$\displaystyle\\{f\in\mathbb{C}[z,w]:f=qp_{B},~{}p\in
J,~{}q\in\mathbb{C}[z,w]\\}$ $\displaystyle=$
$\displaystyle\\{f\in\mathbb{C}[z,w]:f=q\cdot
p\big{(}B_{1}(z)^{K_{2}},B_{2}(z)^{K_{1}}\big{)},~{}p\in
J^{\prime},~{}q\in\mathbb{C}[z,w]\\}$ $\displaystyle=$
$\displaystyle\\{f\in\mathbb{C}[z,w]:f=qp_{(B_{1}^{K_{2}},B_{2}^{K_{1}})},~{}p\in
J^{\prime},~{}q\in\mathbb{C}[z,w]\\}$

is just the Clark-type ideal defined by the homogeneous ideal $J^{\prime}$ and
Blaschke products $B_{1}^{K_{2}}$ and $B_{2}^{K_{1}}$. For this reason, when
$d=2$, in Definition 1.2 we assume that $J$ is homogeneous. Moreover, if
$p\in\mathbb{C}[z,w]$ is a distinguished homogeneous polynomial, then we call
$p_{B}$ a Clark-type function.

The following theorem is one of the main results in this paper.

######  Theorem 1.3.

Let $p_{B}$ be a Clark-type function, and $\mathcal{M}$ be a finitely
generated submodule containing $p_{B}$, then the quotient module
$\mathcal{M}^{\perp}$ of $H^{2}(\mathbb{D}^{2})$ is essentially normal.

It is worth pointing out that Theorem 1.3 is the first result on the essential
normality of non-algebraic submodules over the polydisc, where a submodule is
called algebraic if it is generated by polynomials. Even the essential
normality of a typical Clark-type quotient module, such as
$[(B_{1}(z)-B_{2}(w))^{n}]^{\bot},$ is unknown before. More surprisingly, in
the case that a submodule $[I]$ contains a Clark-type function, we find that
the essential normality of $[I]^{\perp}$ is equivalent to the Hilbert-
Schmidtness of $[I]$.

To state this clearly, we need some notations. For a submodule $\mathcal{M}$
in $H^{2}(\mathbb{D}^{2})$, the core operator

$C_{\mathcal{M}}=I-R_{z}R_{z}^{*}-R_{w}R_{w}^{*}+R_{zw}R_{zw}^{*},$

introduced by Guo and Yang [25] plays an important role in the study of the
submodule. A submodule ${\mathcal{M}}$ is called Hilbert-Schmidt if
$C_{\mathcal{M}}$ is a Hilbert-Schmidt operator. It was conjectured by Yang
[39, Page 403] that

######  Conjecture 1.4.

All the finitely generated submodules of $H^{2}(\mathbb{D}^{2})$ are Hilbert-
Schmidt.

Some positive results on this conjecture have been obtained. In [39, 40], it
was proved that all the algebraic submodules of $H^{2}(\mathbb{D}^{2})$ are
Hibert-Schmidt. Luo, Izuchi and Yang [27] proved that a submodule of
$H^{2}(\mathbb{D}^{2})$ containing $z-B(w)$ is Hilbert-Schmidt if and only if
it is finitely generated, where $B$ is a nonconstant finite Blaschke product.
This result was then generalized by Zu, Yang and Lu [42] to the submodules
containing $B_{1}(z)-B_{2}(w)$ for finite Blaschke products $B_{1}$ and
$B_{2}$. The following result allows us to build up the equivalence between
the essential normality of the quotient modules and the Hilbert-Schmidtness of
the associated submodules in $H^{2}(\mathbb{D}^{2})$.

######  Theorem 1.5.

Let $\mathcal{M}$ be a submodule of $H^{2}(\mathbb{D}^{2})$ containing a
Clark-type function, then $\mathcal{M}$ is Hilbert-Schmidt if and only if
$\mathcal{M}^{\perp}$ is essentially normal.

Without the assumption that the submodule contains a Clark-type function,
Theorem 1.5 fails. For example, it is obvious that the submodule $[z^{2}]$ of
$H^{2}(\mathbb{D}^{2})$ is Hilbert-Schmidt, while its quotient
$[z^{2}]^{\perp}$ is not essentially normal. The following result attacks
Conjecture 1.4.

######  Theorem 1.6.

Let $\mathcal{M}$ be a finitely generated submodule of $H^{2}(\mathbb{D}^{2})$
containing a polynomial, then $\mathcal{M}$ is Hilbert-Schmidt.

This paper is organized as follows. In Section 2, we obtain the essential
normality of Clark-type quotient modules in $H^{2}(\mathbb{D}^{d})$. In
Section 3, the K-homology elements of Clark-type quotient modules are
considered, and the non-triviality of the K-homology elements induced by the
associated $C^{*}$-extensions is obtained. In Section 4, for submodules
$\mathcal{M}$ containing a Clark-type ideal, the equivalence between Hilbert-
Schmidtness of $\mathcal{M}$ and the essential normality of
$\mathcal{M}^{\perp}$ is established. In Section 5, finitely generated non-
algebraic submodules of $H^{2}(\mathbb{D}^{2})$ containing a polynomial are
proven to be Hilbert-Schmidt. And finally in Section 6, several examples on
non-algebraic submodules and the associated quotient modules are investigated.

##  2 $(1,\infty)$-essentially normal quotient modules

To characterize the essential normality of quotient modules over
$\mathbb{D}^{d}$ precisely, we introduce the concept of Macaev ideals. Let
$\lambda_{1},\lambda_{2},\ldots$ be the list of singular values of a compact
operator $A\in B(H)$ in the descending order, multiplicity being counted. If
the sequence
$\Big{\\{}\frac{1}{\ln(n+1)}{\sum\limits_{k=1}^{n}\lambda_{k}}\Big{\\}}$ is
bounded, then $A$ is said to be in the Macaev ideal
$\mathfrak{L}^{(1,\infty)}$. Readers are referred to [7] for more details of
Macaev ideals. As in [36], a quotient module $N$ in $H^{2}(\mathbb{D}^{d})$ is
called $(1,\infty)$-essentially normal if
$[S_{p}^{*},S_{q}]\in\mathfrak{L}^{(1,\infty)}$ for all polynomials $p$ and
$q$.

For $A\in B(H)$, if it happens that
$\mathrm{Id}_{H}-A^{*}A\in\mathfrak{L}^{(1,\infty)}$, then $A$ is called
$(1,\infty)$-essentially isometric. The adjoint operators of
$(1,\infty)$-essentially isometries are called $(1,\infty)$-essentially co-
isometries. If an operator is simultaneously $(1,\infty)$-essentially
isometric and $(1,\infty)$-essentially co-isometric, then it is called
$(1,\infty)$-essentially unitary.

######  Lemma 2.1.

Let $\mathcal{N}$ be a quotient module of $H^{2}(\mathbb{D}^{d})$, and $\xi$
be an inner function on $\mathbb{D}^{d}$ such that $S_{\xi}$ is
$(1,\infty)$-essentially unitary, and
$\varphi_{a}:z\mapsto\frac{a-z}{1-\bar{a}z}$ be a Mobius transform. Then
$S_{\varphi_{a}(\xi)}$ is also $(1,\infty)$-essentially unitary.

###### Proof.

Because $S_{\xi}$ is essentially unitary, it holds that
$\sigma_{e}(S_{\xi})\subset\mathbb{T}$. By the spectral mapping theorem, the
essential spectrum

$\sigma_{e}(S_{\varphi_{a}(\xi)})=\varphi_{a}\big{(}\sigma_{e}(S_{\xi})\big{)}\subset\mathbb{T},$

which implies the Fredholmness of $S_{\varphi_{a}(\xi)}$.

Since $S_{\xi}$ is $(1,\infty)$-essentially unitary, we have

$P_{\mathcal{N}}M_{\xi}^{*}P_{\mathcal{N}^{\perp}}M_{\xi}P_{\mathcal{N}}=(\mathrm{Id}_{\mathcal{N}}-S_{\xi}^{*}S_{\xi})P_{\mathcal{N}}\in\mathcal{L}^{(1,\infty)},$

and therefore
$P_{\mathcal{N}^{\perp}}M_{\xi}P_{\mathcal{N}}\in\mathcal{L}^{(2,\infty)}$.
Temporarily we write
$c=\|P_{\mathcal{N}^{\perp}}M_{\xi}P_{\mathcal{N}}\|_{(2,\infty)}$. Observe
that

$\displaystyle\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k+1}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle=$
$\displaystyle\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k}(P_{\mathcal{N}^{\perp}}+P_{\mathcal{N}})M_{\xi}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle\leq$
$\displaystyle\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k}P_{\mathcal{N}^{\perp}}M_{\xi}P_{\mathcal{N}}\|_{(2,\infty)}+\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k}P_{\mathcal{N}}M_{\xi}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle\leq$ $\displaystyle
c+\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k}P_{\mathcal{N}}\|_{(2,\infty)},$

then by induction we get

$\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k}P_{\mathcal{N}}\|_{(2,\infty)}\leq
kc,~{}\forall k\in\mathbb{N}.$

Consequently

$\displaystyle\|P_{\mathcal{N}^{\perp}}M_{\varphi_{a}(\xi)}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle=$
$\displaystyle\|P_{\mathcal{N}^{\perp}}(a-M_{\xi})(1-\bar{a}M_{\xi})^{-1}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle=$
$\displaystyle\|\sum_{k=0}^{\infty}a\bar{a}^{k}P_{\mathcal{N}^{\perp}}M_{\xi}^{k}P_{\mathcal{N}}-\sum_{k=0}^{\infty}\bar{a}^{k}P_{\mathcal{N}^{\perp}}M_{\xi}^{k+1}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}|a|^{k+1}\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k}P_{\mathcal{N}}\|_{(2,\infty)}+\sum_{k=0}^{\infty}|a|^{k}\|P_{\mathcal{N}^{\perp}}M_{\xi}^{k+1}P_{\mathcal{N}}\|_{(2,\infty)}$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}kc|a|^{k+1}+\sum_{k=0}^{\infty}(k+1)c|a|^{k}$
$\displaystyle<$ $\displaystyle\infty,$

which ensures
$P_{\mathcal{N}^{\perp}}M_{\varphi_{a}(\xi)}P_{\mathcal{N}}\in\mathcal{L}^{(2,\infty)}$,
and moreover

$\mathrm{Id}_{\mathcal{N}}-S_{\varphi_{a}(\xi)}^{*}S_{\varphi_{a}(\xi)}=P_{\mathcal{N}}M_{\varphi_{a}(\xi)}^{*}P_{\mathcal{N}^{\perp}}M_{\varphi_{a}(\xi)}P_{\mathcal{N}}\mid_{\mathcal{N}}\in\mathcal{L}^{(1,\infty)}.$
(2.1)

Let $S_{\varphi_{a}(\xi)}=W|S_{\varphi_{a}(\xi)}|$ be the polar decomposition,
where
$|S_{\varphi_{a}(\xi)}|=(S_{\varphi_{a}(\xi)}^{*}S_{\varphi_{a}(\xi)})^{1/2}$
and $W$ is the partial isometry that maps $\mathcal{N}$ unitarily onto
$\mathrm{ran}~{}S_{\varphi_{a}(\xi)}$. Since $S_{\varphi_{a}(\xi)}$ is
Fredholm, the subspace
$\mathrm{ran~{}}(\mathrm{Id}_{\mathcal{N}}-WW^{*})=\mathrm{coker~{}}S_{\varphi_{a}(\xi)}$
is finite dimensional. Then it follows that

$\mathrm{Id}_{\mathcal{N}}-S_{\varphi_{a}(\xi)}S_{\varphi_{a}(\xi)}^{*}=W(\mathrm{Id}_{\mathcal{N}}-S_{\varphi_{a}(\xi)}^{*}S_{\varphi_{a}(\xi)})W^{*}+(\mathrm{Id}_{\mathcal{N}}-WW^{*})\in\mathcal{L}^{(1,\infty)},$

which completes the proof of the lemma. ∎

For a tuple $z\in\mathbb{C}^{d}$, and multi-index $\alpha\in\mathbb{N}^{d}$,
we write $z^{\alpha}=\prod_{i=1}^{d}z_{i}^{\alpha_{i}}$. The following result
gives the essential normality of a large class of inhomogeneous quotient Hardy
modules.

######  Theorem 2.2.

Let $I\subset\mathbb{C}[z_{1},\ldots,z_{d}]$ be a Clark-type ideal, then the
quotient module $[I]^{\perp}$ of $H^{2}(\mathbb{D}^{d})$ is
$(1,\infty)$-essentially normal. Moreover, $S_{z_{i}}$ is
$(1,\infty)$-essentially unitary for $i=1,\ldots,d$.

###### Proof.

Let $J,B_{1},\ldots,B_{d}$ be as in Definition 1.2. Write
$H_{0}=\bigotimes_{i=1}^{d}\ker M_{B_{i}(z_{i})}^{*}$ which is finite
dimensional, then by the Wold decomposition we have

$H^{2}(\mathbb{D}^{d})=\bigoplus_{\alpha\in\mathbb{N}^{d}}M_{B^{\alpha}}H_{0}.$

For $p\in\mathbb{C}[z_{1},\ldots,z_{d}]$ and $f\in H^{2}(\mathbb{D}^{d})$ we
write $A_{p}f=p(B)f$, then $A$ defines a
$\mathbb{C}[z_{1},\ldots,z_{d}]$-module structure on $H^{2}(\mathbb{D}^{d})$.
Define

$\displaystyle U:H^{2}(\mathbb{D}^{d})\otimes H_{0}$ $\displaystyle\to$
$\displaystyle H^{2}(\mathbb{D}^{d})$
$\displaystyle\sum_{\alpha\in\mathbb{N}^{d}}z^{\alpha}\otimes f_{\alpha}$
$\displaystyle\mapsto$
$\displaystyle\sum_{\alpha\in\mathbb{N}^{d}}B^{\alpha}f_{\alpha},$

then $U$ is unitary and $U(M_{p}\otimes\mathrm{Id}_{H_{0}})=A_{p}U$. For $f\in
H^{2}(\mathbb{D}^{d})\otimes H_{0}$, we have

$\displaystyle Uf\in[I]^{\perp}$ $\displaystyle\Leftrightarrow$ $\displaystyle
M_{p}^{*}Uf=0,~{}\forall p\in I$ $\displaystyle\Leftrightarrow$ $\displaystyle
M_{q(B)}^{*}Uf=0,~{}\forall q\in J$ $\displaystyle\Leftrightarrow$
$\displaystyle A_{q}^{*}Uf=0,~{}\forall q\in J$ $\displaystyle\Leftrightarrow$
$\displaystyle U(M_{q}\otimes\mathrm{Id}_{H_{0}})^{*}f=0,~{}\forall q\in J$
$\displaystyle\Leftrightarrow$
$\displaystyle(M_{q}\otimes\mathrm{Id}_{H_{0}})^{*}f=0,~{}\forall q\in J$
$\displaystyle\Leftrightarrow$ $\displaystyle f\in[J]^{\perp}\otimes H_{0},$

and therefore $[I]^{\perp}=U([J]^{\perp}\otimes H_{0})$. By [36, 38] it holds
that

$P_{[J]}M_{z_{i}}P_{[J]^{\perp}}\in\mathcal{L}^{(2,\infty)},$

and

$P_{[J]^{\perp}}-P_{[J]^{\perp}}M_{z_{i}}M_{z_{i}}^{*}P_{[J]^{\perp}}\in\mathcal{L}^{(1,\infty)}.$

Write $B_{i}(z_{i})=\varphi_{a}(z_{i})\tilde{B}_{i}(z_{i})$ where
$\varphi_{a}$ is a Mobius transform and $\tilde{B}_{i}$ is a Blaschke product.
It follows that

$P_{[I]}M_{B_{i}(z_{i})}P_{[I]^{\perp}}=U\big{(}(P_{[J]}M_{z_{i}}P_{[J]^{\perp}})\otimes\mathrm{Id}_{H_{0}}\big{)}U^{*}\in\mathcal{L}^{(2,\infty)},$

and

$\displaystyle
P_{[I]^{\perp}}-S_{\varphi_{a}(z_{i})}S_{\varphi_{a}(z_{i})}^{*}$
$\displaystyle=$ $\displaystyle
P_{[I]^{\perp}}-P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}M_{\varphi_{a}(z_{i})}^{*}P_{[I]^{\perp}}$
(2.2) $\displaystyle=$ $\displaystyle P_{[I]^{\perp}}P_{\ker
M_{\varphi_{a}(z_{i})}^{*}}P_{[I]^{\perp}}$ $\displaystyle\leq$ $\displaystyle
P_{[I]^{\perp}}P_{\ker M_{B_{i}(z_{i})}^{*}}P_{[I]^{\perp}}$ $\displaystyle=$
$\displaystyle
P_{[I]^{\perp}}-P_{[I]^{\perp}}M_{B_{i}(z_{i})}M_{B_{i}(z_{i})}^{*}P_{[I]^{\perp}}$
$\displaystyle=$ $\displaystyle
U(P_{[J]^{\perp}}-P_{[J]^{\perp}}M_{z_{i}}M_{z_{i}}^{*}P_{[J]^{\perp}})\otimes\mathrm{Id}_{H_{0}})U^{*}$
$\displaystyle\in$ $\displaystyle\mathcal{L}^{(1,\infty)}.$

Similarly, from

$\displaystyle
P_{[I]^{\perp}}M_{B_{i}(z_{i})}^{*}P_{[I]}M_{B_{i}(z_{i})}P_{[I]^{\perp}}$
$\displaystyle=$ $\displaystyle
P_{[I]^{\perp}}-P_{[I]^{\perp}}M_{B_{i}(z_{i})}^{*}P_{[I]^{\perp}}M_{B_{i}(z_{i})}P_{[I]^{\perp}}$
$\displaystyle=$ $\displaystyle
P_{[I]^{\perp}}-P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}^{*}P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}P_{[I]^{\perp}}+P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}^{*}(P_{[I]^{\perp}}-M_{\tilde{B}_{1}(z)}^{*}P_{[I]^{\perp}}M_{\tilde{B}_{1}(z)})M_{\varphi_{a}(z_{i})}P_{[I]^{\perp}}$
$\displaystyle\geq$ $\displaystyle
P_{[I]^{\perp}}-P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}^{*}P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}P_{[I]^{\perp}}$
$\displaystyle=$ $\displaystyle
P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}^{*}P_{[I]}M_{\varphi_{a}(z_{i})}P_{[I]^{\perp}},$

we obtain

$\mathrm{Id}_{[I]^{\perp}}-S_{\varphi_{a}(z_{i})}^{*}S_{\varphi_{a}(z_{i})}=P_{[I]^{\perp}}M_{\varphi_{a}(z_{i})}^{*}P_{[I]}M_{\varphi_{a}(z_{i})}P_{[I]^{\perp}}\mid_{[I]^{\perp}}\in\mathcal{L}^{(1,\infty)}.$
(2.3)

By (2.2) and (2.3), $S_{\varphi_{a}(z_{i})}$ is $(1,\infty)$-essentially
unitary. Then it follows from Lemma 2.1 that
$S_{z_{i}}=S_{\varphi_{a}\circ\varphi_{a}(z_{i})}$ is $(1,\infty)$-essential
unitary. ∎

In the proof of Proposition 2.2, if one of the inner functions
$B_{1},\ldots,B_{d}$ is not a finite Blaschke products, namely some $B_{i}$ is
a Blaschke product with infinite order, or even has a nontrivial singular
part, then $H_{0}$ is infinite dimensional. The unitary $U$ builds up the
equivalence between the quotient modules $[I]^{\perp}$ and $[J]^{\perp}\otimes
H_{0}$, which are not essentially normal. This derives the following result.

######  Corollary 2.3.

Let $\eta=\big{(}\eta_{1}(z_{1}),\ldots,\eta_{d}(z_{d})\big{)}$ be a $d$-tuple
of nonconstant inner functions, and $J\subset\mathbb{C}[z_{1},\ldots,z_{d}]$
be a distinguished quasi-homogeneous ideal. Then $[J_{\eta}]^{\perp}$ is
essentially normal if and only if all the $\eta_{i}$’s are finite Blaschke
products.

By the similar technique in [38, Section 5], Proposition 2.2 can be
generalized to the case of integer-weighted Bergman modules. For simplicity,
let $[I]_{\alpha}^{\perp}$ be the quotient module of
$A_{\alpha}^{2}(\mathbb{D}^{2})$ determined by the ideal $I.$ Without
confusion, in the weighted Bergman space we also denote
$S_{f,[I]_{\alpha}^{\perp}}=P_{[I]_{\alpha}^{\perp}}M_{f}|_{[I]_{\alpha}^{\perp}}.$

######  Corollary 2.4.

Let $I\subset\mathbb{C}[z,w]$ be a Clark-type ideal, then for every integer
$\alpha\geq-1$, the quotient module $[I]_{\alpha}^{\perp}$ is
$(1,\infty)$-essentially normal.

###### Proof.

The case $\alpha=-1$ is included in Proposition 2.2, so in the remaining we
assume $\alpha\geq 0$.

By the argument following Definition 1.2, there is a distinguished homogeneous
ideal $J\in\mathbb{C}[z,w]$ and nonconstant finite Blaschke products
$B_{1},B_{2}$, such that $I$ is generated by the numerators of functions in
$J_{B}$. Let $J_{1}$ be the homogeneous ideal of
$\mathbb{C}[z_{1},\ldots,z_{\alpha+2},w_{1},\ldots,w_{\alpha+2}]$ generated by

$\\{q(z_{i},w_{i}),~{}z_{i}-z_{j},~{}w_{i}-w_{j}:q\in J,~{}1\leq
i,j\leq\alpha+2\\},$

and
$I_{1}\subset\mathbb{C}[z_{1},\ldots,z_{\alpha+2},w_{1},\ldots,w_{\alpha+2}]$
be the ideal generated by the numerators of functions in

$\\{q\big{(}B_{1}(z_{1}),\ldots,B_{1}(z_{\alpha+2}),B_{2}(w_{1}),\ldots,B_{2}(w_{\alpha+2})\big{)}:q\in
J_{1}\\}.$

By Proposition 2.2, on the quotient Hardy module $[I_{1}]^{\perp}\subset
H^{2}(\mathbb{D}^{2\alpha+4})$, the operators $S_{z_{i},[I_{1}]^{\perp}}$ and
$S_{w_{i},[I_{1}]^{\perp}}$ are $(1,\infty)$-essentially co-isometric.

Let
$I_{2}\subset\mathbb{C}[z_{1},\ldots,z_{\alpha+2},w_{1},\ldots,w_{\alpha+2}]$
be the ideal generated by the numerators of functions in

$\\{p(z_{i},w_{i}),~{}z_{i}-z_{j},~{}w_{i}-w_{j}:1\leq i,j\leq\alpha+2,~{}p\in
I\\},$

then from $I_{1}\subset I_{2}$ we deduce
$[I_{2}]^{\perp}\subset[I_{1}]^{\perp}$. Consequently, both
$S_{z_{i},[I_{2}]^{\perp}}$ and $S_{w_{i},[I_{2}]^{\perp}}$ are
$(1,\infty)$-essentially co-isometric. By [6, 36], for the ideal $I_{3}$
generated by $\\{z_{i}-z_{j},~{}w_{i}-w_{j}:1\leq i\leq\alpha+2\\}$, the
quotient Hardy module $[I_{3}]^{\perp}$ is unitarily equivalent to the
weighted Bergman module $\mathcal{A}_{\alpha}^{2}(\mathbb{D}^{2})$, via the
mapping

$f(z_{1},\ldots,z_{\alpha+2},w_{1},\ldots,w_{\alpha+2})\mapsto
f(z,\ldots,z,w,\ldots,w).$

Hence $[I_{2}]^{\perp}$ is unitarily equivalent to $[I]_{\alpha}^{\perp}$.
Therefore both $S_{z,[I]_{\alpha}^{\perp}}$ and $S_{w,[I]_{\alpha}^{\perp}}$
are $(1,\infty)$-essentially co-isometric. Finally, by [24, Proposition 2.5],
$[I]_{\alpha}^{\perp}$ is $(1,\infty)$-essentially normal. ∎

######  Theorem 2.5.

If $I\subset\mathbb{C}[z,w]$ is a distinguished ideal with Clark-type variety,
then for every integer $\alpha\geq-1$, the quotient module
$[I]_{\alpha}^{\perp}$ is $(1,\infty)$-essentially normal.

###### Proof.

By the argument following Definition 1.2, there exists a homogeneous ideal
$J\subset\mathbb{C}[z,w]$ and finite Blaschke products $B_{1},B_{2}$, such
that

$Z(I)\cap\mathbb{D}^{d}=Z(I_{0})\cap\mathbb{D}^{d},$

where $I_{0}$ is the ideal generated by the numerators of functions in
$J_{B}$. Since $I$ is distinguished, $I_{0}$ and a fortiori $J$ is
distinguished.

The Hilbert’s Nullstellensatz ensures the existence of an integer $N$ such
that $I\supset I_{0}^{N}$, where $I_{0}^{N}$ is the ideal generated by the
numerators of $q\circ B,~{}q\in J^{N}$. Then $I_{0}^{N}$ is of Clark-type.
Proposition 2.2 and Corollary 2.4 guarantee the $(1,\infty)$-essential
normality of $[I_{0}^{N}]_{\alpha}^{\perp}$, hence

$\mathrm{Id}_{[I]_{\alpha}^{\perp}}-S_{z,[I]_{\alpha}^{\perp}}S_{z,[I]_{\alpha}^{\perp}}^{*}=P_{[I]_{\alpha}^{\perp}}\left(\mathrm{Id}_{[I_{0}^{N}]_{\alpha}^{\perp}}-S_{z,[I_{0}^{N}]_{\alpha}^{\perp}}S_{z,[I_{0}^{N}]_{\alpha}^{\perp}}^{*}\right)\mid_{[I]_{\alpha}^{\perp}}\in\mathcal{L}^{(1,\infty)},$

and symmetrically

$\mathrm{Id}_{[I]_{\alpha}^{\perp}}-S_{w,[I]_{\alpha}^{\perp}}S_{w,[I]_{\alpha}^{\perp}}^{*}\in\mathcal{L}^{(1,\infty)}.$

Then the proof of the corollary is finished by [24, Proposition 2.5]. ∎

Combining Theorem 2.5 and [24, Corollary 6.4], we immediately get the
following result.

######  Corollary 2.6.

Let $I\subset\mathbb{C}[z,w]$ be an ideal which is the intersection of ideals
$I_{1},\ldots,I_{n}$ with Clark-type varieties, such that $Z(I_{i})\cap
Z(I_{j})\cap\mathbb{T}^{d}=\emptyset$ whenever $i\neq j$. Then $[I]^{\perp}$
is essentially normal.

##  3 K-homology for Clark-type quotient modules

In this section, we will consider the K-homology defined by Clark-type
quotient modules. Recall that, by the BDF-theory [4], when a quotient module
$[I]^{\perp}$ is essentially normal, we have the following short exact
sequence

$\displaystyle 0\to{\mathcal{K}}\to C^{*}([I]^{\perp})\to
C(\sigma_{e}(S_{z_{1}},\cdots,S_{z_{d}}))\to 0,$ (3.1)

where $C^{*}([I]^{\perp})$ is the $C^{*}$-algebra generated by
$\\{\mathrm{Id}_{[I]^{\perp}},S_{z_{1}},\cdots,S_{z_{d}}\\}$, and
$\sigma_{e}(S_{z_{1}},\cdots,S_{z_{d}})$ is the Taylor’s joint essential
spectrum [8, 31] of the tuple $(S_{z_{1}},\cdots,S_{z_{d}})$. It will be shown
that the short exact sequence (3.1) is not splitting, which gives a nontrivial
element $e_{I}$ in the K-homology group
$K\big{(}\sigma_{e}(S_{z_{1}},\cdots,S_{z_{d}})\big{)}$. To continue, for
Clark-type quotient modules $[I]^{\perp}$ we will prove that

$\sigma_{e}(S_{z_{1}},\cdots,S_{z_{d}})=Z(I)\cap\partial D^{d}.$

For $z\in\mathbb{C}^{d}$ and a subset $S$ of $\\{1,\ldots,d\\}$, we write
$z_{S}=\prod_{i\in S}z_{i}$. For a Hilbert
$\mathbb{C}[z_{1},\ldots,z_{d}]$-module $\mathcal{H}$, we set

$C_{\mathcal{H}}=\sum_{S\subset\\{1,\ldots,d\\}}(-1)^{\mathrm{card~{}}S}M_{z_{S}}M_{z_{S}}^{*},$

then for a submodule $\mathcal{M}$ of $H^{2}(\mathbb{D}^{d})$,
$C_{\mathcal{M}}$ is just the core operator defined by Guo and Yang [25]. It
is routine to verify that $C_{H^{2}(\mathbb{D}^{d})}=1\otimes 1$ is compact.
For the weighted Bergman module $\mathcal{A}_{\alpha}^{2}(\mathbb{D}^{d})$
where $\alpha\in(-1,\infty)$, an orthonormal basis is

$\left\\{\sqrt{\prod_{i=1}^{d}\frac{\Gamma(\beta_{i}+\alpha+2)}{\Gamma(\beta_{i}+1)\Gamma(\alpha+2)}}z^{\beta}:\beta\in\mathbb{N}^{d}\right\\},$

and it is routine to verify that

$C_{\mathcal{A}_{\alpha}^{2}(\mathbb{D}^{d})}z^{\beta}=\frac{(\alpha+1)^{d}}{\prod_{i=1}^{d}(\beta_{i}+\alpha+1)}z^{\beta},~{}\forall\beta\in\mathbb{N}^{d}.$

Hence $C_{\mathcal{A}_{\alpha}^{2}(\mathbb{D}^{d})}$ is compact for all
$\alpha\in[-1,\infty)$.

######  Proposition 3.1.

Let $I\subset\mathbb{C}[z_{1},\ldots,z_{d}]$ be an ideal such that

$\partial\big{(}Z(I)\cap\mathbb{D}^{d}\big{)}=Z(I)\cap\partial\mathbb{D}^{d},$
(3.2)

and $\alpha\geq-1$ be a real number, such that the quotient module
$[I]^{\perp}$ of $\mathcal{A}_{\alpha}^{2}(\mathbb{D}^{d})$ is essentially
normal, then the Taylor essential spectrum [31, 32]

$\sigma_{e}(S_{z_{1}},\ldots,S_{z_{d}})=Z(I)\cap\partial\mathbb{D}^{d}.$

###### Proof.

Since $p(S_{z_{1}},\ldots,S_{z_{d}})=0$ for all polynomials $p\in I$, the
spectral mapping theorem [32] gives

$\sigma_{e}(S_{z_{1}},\ldots,S_{z_{d}})\subset\sigma(S_{z_{1}},\ldots,S_{z_{d}})\subset
Z(I).$

From

$C_{[I]_{\alpha}^{\perp}}=P_{[I]_{\alpha}^{\perp}}C_{\mathcal{A}_{\alpha}^{2}(\mathbb{D}^{d})}\mid_{[I]_{\alpha}^{\perp}}$
(3.3)

we obtain the compactness of $C_{[I]_{\alpha}^{\perp}}$. Then since
$C^{*}([I]_{\alpha}^{\perp})$ is essentially commutative, it follows from the
definition of $C_{[I]_{\alpha}^{\perp}}$ that
$\sigma_{e}([I]_{\alpha}^{\perp})\subset\partial\mathbb{D}^{d}$. Therefore we
have proved

$\sigma_{e}(S_{z_{1}},\ldots,S_{z_{d}})\subset
Z(I)\cap\partial\mathbb{D}^{d}.$

For the inverse inclusion, let $\lambda\in Z(I)\cap\partial\mathbb{D}^{d}$. By
[8, Corollary 3.9], we need to deny the Fredholmness of
$T=\sum\limits_{i=1}^{d}(\lambda_{i}-S_{z_{i}})(\lambda_{i}-S_{z_{i}})^{*}$.
If $T$ is Fredholm, then by the positivity of $T$, we can find an invertible
positive operator $B$ and a compact $K$ such that $T=B+K$. Take a sequence
$\\{\mu_{n}\\}$ in $Z(I)\cap\mathbb{D}^{d}$ that converges to $\lambda$. Let
$k_{\mu_{n}}=\frac{K_{\mu_{n}}}{\|K_{\mu_{n}}\|}$ be the normalized
reproducing kernel, then $\\{k_{\mu_{n}}\\}$ converges to $0$ weakly. Hence
there is a number $c>0$ making

$\displaystyle\liminf_{n\to\infty}\langle
Tk_{\mu_{n}},k_{\mu_{n}}\rangle=\liminf_{n\to\infty}\langle(B+K)k_{\mu_{n}},k_{\mu_{n}}\rangle=\liminf_{n\to\infty}\langle
Bk_{\mu_{n}},k_{\mu_{n}}\rangle\geq c.$

However, since $\mu_{n}\in V_{u}$, it holds that $k_{\mu_{n}}\in[I]^{\perp}$,
and consequently

$\displaystyle\lim\limits_{n\to\infty}\langle
Tk_{\mu_{n}},k_{\mu_{n}}\rangle=\lim\limits_{n\to\infty}|\lambda-\mu_{n}|^{2}=0,$
(3.4)

contradicting to the previous inequality. Therefore $T$ is not Fredholm and
$\lambda\in\sigma_{e}(S_{z_{1}},\ldots,S_{z_{d}})$, completing the proof of
the proposition. ∎

For a Clark-type variety $V=Z(p_{B})\cap\mathbb{D}^{d}$, it holds

$\partial V=Z(p_{B})\cap\mathbb{T}^{2}=Z(p_{B})\cap\partial\mathbb{D}^{2},$

therefore the condition (3.2) in Proposition 3.1 is satisfied. This together
with Theorem 2.2 and Theorem 2.5 derives the following corollaries on the
Taylor essential spectrums of the quotient modules.

######  Corollary 3.2.

Let $I\subset\mathbb{C}[z_{1},\ldots,z_{d}]$ be a Clark-type ideal, then

$\sigma_{e}(S_{z_{1}},\ldots,S_{z_{d}})=Z(I)\cap\partial\mathbb{D}^{d}.$

And in the case $d=2,$ the following corollary holds.

######  Corollary 3.3.

Let $I\subset\mathbb{C}[z,w]$ be an ideal with Clark-type variety, and
$\alpha\geq-1$ be an integer, then

$\sigma_{e}(S_{z},S_{w})=Z(I)\cap\partial\mathbb{D}^{2}.$

By Theorem 2.5 and [24, Theorem 5.2], we can prove the non-triviality of the
$C^{*}$-extensions associated to the quotient modules with Clark-type variety.

######  Theorem 3.4.

Let $I\subset\mathbb{C}\ddot{}[z,w]$ be an ideal with Clark-type variety, and
$\alpha\geq-1$ be an integer, then the short exact sequence

$0\to\mathcal{K}\hookrightarrow C^{*}({[I]_{\alpha}^{\perp}})\to
C(Z(I)\cap\partial\mathbb{D}^{2})\to 0$

is not splitting.

To prove the non-triviality of the $C^{*}$-extensions corresponding to the
Clark-type quotient modules in the high dimensional cases, the following
technical lemma is needed.

######  Lemma 3.5.

Let $J\subset\mathbb{C}[z_{1},\ldots,z_{d}]$ be a distinguished
$\mathbf{K}$-quasi-homogeneous ideal, then $S_{z_{1}}$ is a Fredholm operator
with nonzero index.

###### Proof.

By [38], $S_{z_{1}}$ is essentially unitary and therefore Fredholm. Evidently
$1\in\ker S_{z_{1}}^{*}$, hence $\dim\ker S_{z_{1}}^{*}>0$. It remains to
prove the injectivity of $S_{z_{1}}$. Let $h\in\ker S_{z_{1}}$ be
$\mathbf{K}$-quasi-homogeneous, and $J=\bigcap_{k=1}^{n}J_{k}$ be the primary
decomposition of $J$, where each $Z(J_{k})$ is $1$-dimensional. Then

$z_{1}h\in[J]\cap\mathbb{C}[z_{1},\ldots,z_{d}]=\bigcap_{k=1}^{n}J_{k}.$

For each $k$, the variety $Z(J_{k})\subset Z(J)$ is distinguished, hence
$z_{1}^{n}\notin J_{k},~{}\forall n\in\mathbb{N}$. Therefore
$h\in\bigcap_{k=1}^{n}J_{k}=J$, which ensures $\ker S_{z_{1}}=0$. ∎

######  Theorem 3.6.

Let $I\subset\mathbb{C}[z_{1},\ldots,z_{d}]$ be a Clark-type ideal, then the
short exact sequence

$0\to\mathcal{K}\hookrightarrow C^{*}({[I]^{\perp}})\to
C(Z(I)\cap\partial\mathbb{D}^{d})\to 0$

is not splitting.

###### Proof.

By [19, Lemma 5.5], it suffices to prove $S_{B_{1}(z_{1}),[I]^{\perp}}$ is a
Fredholm operator with nonzero index.

Let $J,B_{1},\ldots,B_{d}$ be as in Definition 1.2. By the proof of
Proposition 2.2, $S_{B_{1}(z_{1}),[I]^{\perp}}$ is unitary equivalent to
$S_{z_{1},[J]^{\perp}}\otimes\mathrm{Id}_{H_{0}}$, where
$H_{0}=\bigotimes_{i=1}^{d}\ker M_{B_{i}(z_{i})}^{*}$ is finite dimensional.
By Proposition 3.5, $S_{z_{1},[J]^{\perp}}$ is Fredholm with nonzero index,
then so are $S_{z_{1},[J]^{\perp}}\otimes\mathrm{Id}_{H_{0}}$ and
$S_{B_{1}(z_{1}),[I]^{\perp}}$. ∎

##  4 Hilbert-Schmidt submodules vs. Essentially normal quotient modules

In this section, we establish the connection between Hilbert-Schmidt
submodules and essentially normal quotient modules.

Let $p$ be a polynomial such that

$Z(p)\cap Z(z)\cap\partial\mathbb{D}^{2}=\emptyset.$

Suppose that $\mathcal{M}$ is a finitely generated submodule containing $p$,
i.e. $\mathcal{M}=[p,f_{1},\cdots,f_{n}]$ where $f_{i}\in
H^{2}(\mathbb{D}^{2})$, and set

$Q=\mathcal{M}\ominus[p]=[p]^{\perp}\ominus\mathcal{M}^{\perp}.$

It is not hard to verify that $Q$ is invariant for $S_{z,[p]^{\perp}}$. In
fact, for any $f\in Q$, $g\in\mathcal{M}^{\perp}$ and $q\in\mathbb{C}[z,w]$,
it holds that

$\langle S_{q,[p]^{\perp}}f,g\rangle=\langle
P_{[p]^{\perp}}M_{q}f,g\rangle=\langle qf,P_{[p]^{\perp}}g\rangle=0.$

For $q\in\mathbb{C}[z,w]$, we write

$\displaystyle T_{q,Q}=S_{q,[p]^{\perp}}\mid_{Q}=P_{Q}M_{q}\mid_{Q}.$ (4.1)

The following lemma is inspired by [27].

######  Lemma 4.1.

Under the above assumptions, the subspace $\ker T_{z,Q}^{*}$ is of finite
dimension.

###### Proof.

Please note that

$\displaystyle\ker T_{z,Q}^{*}$ $\displaystyle=$ $\displaystyle Q\ominus
S_{z,[p]^{\perp}}Q$ $\displaystyle=$ $\displaystyle\\{f\in Q:\langle
f,P_{[p]^{\perp}}(zg)\rangle=0,~{}\forall g\in Q\\}$ $\displaystyle=$
$\displaystyle\\{f\in Q:\langle f,zg\rangle=0,~{}\forall g\in Q\\}$
$\displaystyle=$ $\displaystyle\\{f\in Q:\langle f,h\rangle=0,~{}\forall h\in
z\mathcal{M}\ominus z[p]\\}$ $\displaystyle=$
$\displaystyle\mathcal{M}\cap[p]^{\perp}\cap\left(z\mathcal{M}\ominus
z[p]\right)^{\perp}$ $\displaystyle=$
$\displaystyle\mathcal{M}\ominus[p,z\mathcal{M}]$ $\displaystyle\subseteq$
$\displaystyle\mathcal{M}\ominus[z\mathcal{M}+p\mathcal{M}].$

Next, we claim that $\mathcal{M}\ominus[z\mathcal{M}+p\mathcal{M}]$ is of
finite dimension. Since

$Z(p)\cap Z(z)\cap\partial\mathbb{D}^{2}=\emptyset,$

by [20], $zH^{2}(\mathbb{D}^{2})+pH^{2}(\mathbb{D}^{2})$ is closed and

$\dim
H^{2}(\mathbb{D}^{2})\ominus\big{(}zH^{2}(\mathbb{D}^{2})+pH^{2}(\mathbb{D}^{2})\big{)}<\infty.$

It is not hard to select a basis $\\{e_{i}:1\leq i\leq k\\}$ of
$H^{2}(\mathbb{D}^{2})\ominus\big{(}zH^{2}(\mathbb{D}^{2})+pH^{2}(\mathbb{D}^{2})\big{)},$
consisting of bounded functions. We claim that

$\mathcal{M}\ominus[z\mathcal{M}+p\mathcal{M}]\subset\mathrm{span}\left\\{P_{[p]}(e_{i}f_{j}):i=1,\cdots,k,~{}j=1,\cdots,n+1\right\\},$

where for convenience, we denote $f_{n+1}=p$. To prove this, suppose
$g\in\mathcal{M}\ominus[z\mathcal{M}+p\mathcal{M}]$ satisfies

$g\perp\mathrm{span}\left\\{P_{[p]}(e_{i}f_{j}):i=1,\cdots,k,~{}j=1,\cdots,n+1\right\\}.$

Now, for any $h\in H^{2}(\mathbb{D}^{2})$, there exists $h_{1}\in
H^{2}(\mathbb{D}^{2})\ominus[z,p]$ and $h_{2}\in[z,p]$, such that
$h=h_{1}+h_{2}$, then

$P_{[p]}(h_{1}f_{j})\in\mathrm{span}\\{P_{[p]}(e_{i}f_{j}):~{}i=1,\ldots,k\\}.$

Furthermore, it is easy to see $h_{2}f_{j}\in[z\mathcal{M}+p\mathcal{M}]$.
Therefore

$\displaystyle\langle g,hf_{j}\rangle=\langle g,h_{1}f_{j}\rangle+\langle
g,h_{2}f_{j}\rangle=0,$

It follows that $g\perp\mathcal{M}$, and the claim is proved. Consequently,
$\ker T_{z,Q}^{*}$ is of finite dimension. ∎

The following lemma originates from Arveson [2, 3], and we omit its proof
which is almost trivial.

######  Lemma 4.2.

Let $A$ be an essentially normal operator on a Hilbert space, and
$\mathcal{M}$ be a closed invariant subspace for $A$. Then the following
statements are equivalent.

  * (1)

the restriction $A\mid_{\mathcal{M}}$ is $(1,\infty)$-essentially normal
(resp. essentially normal);

  * (2)

the operator $P_{\mathcal{M}}A|_{\mathcal{M}^{\perp}}$ belongs to
$\mathcal{L}^{(2,\infty)}$ (resp. $\mathcal{K}$);

  * (3)

the compression $P_{\mathcal{M}^{\perp}}A|_{\mathcal{M}^{\perp}}$ is
$(1,\infty)$-essentially normal (resp. essentially normal).

######  Theorem 4.3.

Let $\mathcal{M}$ be a submodule in $H^{2}(\mathbb{D}^{2})$ containing a
Clark-type ideal $I,$ then the following statements are equivalent.

  * (1)

$\mathcal{M}$ is Hilbert-Schmidt;

  * (2)

$\mathcal{M}^{\perp}$ is $(1,\infty)$-essentially normal;

  * (3)

$\mathcal{M}_{1}=[I]^{\perp}\ominus\mathcal{M}^{\perp}$ is
$(1,\infty)$-essentially normal.

###### Proof.

The equivalence between (2) and (3) follows immediately from Theorem 2.2 and
Lemma 4.2.

(1)$\Rightarrow$(2): Suppose $\mathcal{M}$ is Hilbert-Schmidt, i.e.

$C_{\mathcal{M}}=\mathrm{Id}_{\mathcal{M}}-R_{z}R_{z}^{*}-R_{w}R_{w}^{*}+R_{z}R_{w}R_{z}^{*}R_{w}^{*}\in\mathcal{L}^{2}.$

By [41, Theorem 4.3], it holds
$\sigma_{e}(S_{z},S_{w})\subset\partial\mathbb{D}^{2}$. On the other hand, for
each $f\in I$ we have $f(S_{z},S_{w})=S_{f}=0$, which forces

$f\big{(}\sigma(S_{z},S_{w})\big{)}=\sigma\big{(}f(S_{z},S_{w})\big{)}\subset\\{0\\}.$

Consequently $\sigma_{e}(S_{z},S_{w})\subset
Z(I)\cap\partial\mathbb{D}^{2}\subset\mathbb{T}^{2}$. Applying the spectral
mapping theorem, we find $\sigma_{e}(S_{z})\subset\mathbb{T}$ which ensures
the Fredholmness of $S_{z}$. It follows from Theorem 2.2 that

$\mathrm{Id}_{[I]^{\perp}}-S_{z,[I]^{\perp}}S_{z,[I]^{\perp}}^{*}\in\mathcal{L}^{(1,\infty)},$

and consequently

$\mathrm{Id}_{\mathcal{M}^{\perp}}-S_{z}S_{z}^{*}\in\mathcal{L}^{(1,\infty)},$

which derives the $(1,\infty)$-essential normality of $\mathcal{M}^{\perp}$.

(2)$\Rightarrow$(1): This follows from the direct computation

$\displaystyle C_{\mathcal{M}}$ $\displaystyle=$ $\displaystyle
P_{\mathcal{M}}(\mathrm{Id}-M_{z}P_{\mathcal{M}}M_{z}^{*}-M_{w}P_{\mathcal{M}}M_{w}^{*}+M_{zw}P_{\mathcal{M}}M_{zw}^{*})\mid_{\mathcal{M}}$
$\displaystyle=$ $\displaystyle
P_{\mathcal{M}}(\mathrm{Id}-M_{z}M_{z}^{*}-M_{w}M_{w}^{*}+M_{zw}M_{zw}^{*})\mid_{\mathcal{M}}$
$\displaystyle+P_{\mathcal{M}}(M_{z}P_{\mathcal{M}^{\perp}}M_{z}^{*}+M_{w}P_{\mathcal{M}^{\perp}}M_{w}^{*}-M_{zw}P_{\mathcal{M}^{\perp}}M_{zw}^{*})\mid_{\mathcal{M}}$
$\displaystyle=$ $\displaystyle P_{\mathcal{M}}(1\otimes
1)\mid_{\mathcal{M}}+D_{z}D_{z}^{*}+D_{w}D_{w}^{*}-D_{zw}D_{zw}^{*}$
$\displaystyle\in$ $\displaystyle\mathcal{L}^{(1,\infty)},$

where $D_{p}=P_{\mathcal{M}}M_{p}P_{\mathcal{M}^{\perp}}$ for
$p\in\mathbb{C}[z,w]$, and the last “$\in$” follows from Lemma 4.2. ∎

######  Theorem 4.4.

Let $p$ be a distinguished homogenous polynomial, and $B_{1},B_{2}$ be
nonconstant finite Blaschke products. If $\mathcal{M}$ is a finitely generated
submodule containing $p_{B}$, then

  * (1)

both $S_{z,\mathcal{M}^{\perp}}$ and $S_{w,\mathcal{M}^{\perp}}$ are
$(1,\infty)$-essentially unitary, and hence $\mathcal{M}$ is Hilbert-Schmidt,

  * (2)

the quotient module $\mathcal{M}^{\perp}$ is $(1,\infty)$-essentially normal.

###### Proof.

By Theorem 4.3, we only need to prove (2). For
$\mathcal{M}=[p_{B},f_{1},\cdots,f_{n}],$ we decompose the quotient module
$[p_{B}]^{\perp}$ into

$[p_{B}]^{\perp}=Q\oplus\mathcal{M}^{\perp},$

then accordingly $S_{z,[p_{B}]^{\perp}}$ has a matrix representation

$\displaystyle S_{z,[p_{B}]^{\perp}}=\left(\begin{array}[]{cc}T_{z,Q}&D_{z}\\\
0&S_{z,{\mathcal{M}^{\perp}}}\end{array}\right),$ (4.4)

where $D_{z}=P_{[Q]^{\perp}}S_{z}|_{\mathcal{M}^{\perp}}$. Now, for the Clark-
type function $p_{B}$, $S_{z,[p_{B}]^{\perp}}$ is $(1,\infty)$-essentially
unitary by Theorem 2.2. Since $Q$ is invariant for $S_{z,[p_{B}]^{\perp}}$, we
conclude that $T_{z,Q}$ is $(1,\infty)$-essentially isometric. Moreover, $\ker
T_{z,Q}^{*}$ is finite dimensional by Lemma 4.1, therefore $T_{z,Q}$ is
Fredholm, whence $(1,\infty)$-essentially unitary. Then by Lemma 4.2,
$S_{z,\mathcal{M}^{\perp}}=P_{\mathcal{M}^{\perp}}S_{z,[p_{B}]^{\perp}}\mid_{\mathcal{M}^{\perp}}$
is $(1,\infty)$-essentially normal. Similarly, $S_{w,\mathcal{M}^{\perp}}$ is
also $(1,\infty)$-essentially unitary, and hence $\mathcal{M}^{\perp}$ is
$(1,\infty)$-essentially normal. ∎

##  5 Hilbert-Schmidtness of the submodule containing a polynomial

In this section, we study the Hilbert-Schmidtness of finitely generated
submodules containing a polynomial.

######  Theorem 5.1.

Let $p$ be a polynomial in $\mathbb{C}[z,w]$, and $\mathcal{M}$ be a finitely
generated submodule containing a polynomial $p$, then
$S_{z,\mathcal{M}^{\perp}}$ is Fredholm. Furthermore, $\mathcal{M}$ is
Hilbert-Schmidt.

###### Proof.

The theorem will be proved in two steps.

Step 1. We first consider the case $Z(p)\cap
Z(z)\cap\partial\mathbb{D}^{2}=\emptyset$.

At first, we claim that $S_{z,{[p]^{\perp}}}$ is bounded below. In fact, by
[40], the submodule $[p]$ is Hilbert-Schmidt. Then [41, Theorem 4.3] implies
$\sigma_{e}(S_{z,[p]^{\perp}},S_{w,[p]^{\perp}})\subset\partial\mathbb{D}^{2}$.
On the other hand, we have

$\sigma_{e}(S_{z,[p]^{\perp}},S_{w,[p]^{\perp}})\subset\sigma(S_{z,[p]^{\perp}},S_{w,[p]^{\perp}})\subset
Z(p),$

and it follows that $\sigma_{e}(S_{z,[p]^{\perp}},S_{w,[p]^{\perp}})\subset
Z(p)\cap\partial\mathbb{D}^{2}$. Since $Z(p)\cap
Z(z)\cap\partial\mathbb{D}^{2}=\emptyset$, the spectral mapping theorem gives
$0\notin\sigma_{e}(S_{z,{[p]^{\perp}}})$, hence $S_{z,{[p]^{\perp}}}$ is
Fredholm. According to the proof of [24, Lemma 2.3], $S_{z,[p]^{\perp}}$ is
injective, and hence it is bounded below, which proves the claim.

With respect to $[p]^{\perp}=Q\oplus\mathcal{M}^{\perp}$,
$S_{z,{[p]^{\perp}}}$ can be decomposed as

$S_{z,{[p]^{\perp}}}=\left(\begin{array}[]{cc}T_{z,Q}&D_{z}\\\
O&S_{z,\mathcal{M}^{\perp}}\end{array}\right).$

Therefore $T_{z,Q}$ is bounded below. As a consequence of Lemma 4.1, $T_{z,Q}$
is Fredholm. Hence there exists a compact operator $K$ and $A_{z}\in B(Q)$
such that

$A_{z}T_{z,Q}=I+K.$

Obviously, $\left(\begin{array}[]{cc}A_{z}&O\\\ O&I\end{array}\right)$ is
Fredholm and so is

$\left(\begin{array}[]{cc}A_{z}&O\\\
O&I\end{array}\right)\left(\begin{array}[]{cc}T_{z,Q}&D_{z}\\\
O&S_{z,\mathcal{M}^{\perp}}\end{array}\right)=\left(\begin{array}[]{cc}I+K&D\\\
O&S_{z,\mathcal{M}^{\perp}}\end{array}\right),$

where $D=A_{z}D_{z}$ is a bounded operator. It follows that
$\left(\begin{array}[]{cc}I&D\\\
O&S_{z,\mathcal{M}^{\perp}}\end{array}\right)$ is Fredholm. Now, from

$\left(\begin{array}[]{cc}I&D\\\
O&I\end{array}\right)\left(\begin{array}[]{cc}I&-D\\\
O&I\end{array}\right)=\left(\begin{array}[]{cc}I&-D\\\
O&I\end{array}\right)\left(\begin{array}[]{cc}I&D\\\
O&I\end{array}\right)=\left(\begin{array}[]{cc}I&O\\\ O&I\end{array}\right),$

we conclude that $\left(\begin{array}[]{cc}I&-D\\\ O&I\end{array}\right)$ is
invertible, and hence

$\left(\begin{array}[]{cc}I&O\\\
O&S_{z,\mathcal{M}^{\perp}}\end{array}\right)=\left(\begin{array}[]{cc}I&D\\\
O&S_{z,\mathcal{M}^{\perp}}\end{array}\right)\left(\begin{array}[]{cc}I&-D\\\
O&I\end{array}\right)$

is Fredholm, which ensures the Fredholmness of $S_{z,\mathcal{M}^{\perp}}$,
i.e. $0\not\in\sigma_{e}(S_{z,\mathcal{M}^{\perp}})$. Then by [40, Theorem
2.3], $\mathcal{M}$ is Hilbert-Schmidt.

Step 2. For the case $Z(p)\cap Z(z)\cap\partial\mathbb{D}^{2}\neq\emptyset$,
by [39, Corollary 3.5], we choose an $a\in\mathbb{D}$ such that $p$ has no
zeroes in $\\{a\\}\times\mathbb{T}$. Denote the Mobius transformation by

$\varphi_{a}(z)=\frac{a-z}{1-\bar{a}z},\quad\Phi_{a}(z,w)=(\varphi_{a}(z),w).$

It is easy to see that there exists an $a\in\mathbb{D}$ such that

$Z(z)\cap Z(p_{\Phi_{a}})\cap\partial\mathbb{D}^{2}=\emptyset.$

Let $\mathcal{M}_{\Phi_{a}}=\\{f\circ\Phi_{a},f\in\mathcal{M}\\}$, then by
Step 1 the core operator $C_{\mathcal{M}_{\Phi_{a}}}$ is Hilbert-Schmidt. Then
the Hilbert-Schmidtness of $C_{\mathcal{M}}$ follows from [41, Proposition
2.1]. ∎

##  6 Examples

In this section, we consider several examples. At first, we show that in
Theorem 4.4, without the assumption that the submodule is finitely generated,
the associated quotient module may fail to be essentially normal.

######  Example 6.1.

Consider the quotient module $[z-w]^{\perp}$ of $H^{2}(\mathbb{D}^{2})$, which
can be seen as the Bergman space over the unit disc. Take an invariant
subspace $\mathcal{N}$ of $S_{z}$ such that $\dim(\mathcal{N}\ominus
S_{z}\mathcal{N})=\infty$, then the submodule
$\mathcal{M}=[z-w]\oplus\mathcal{N}$ is infinitely generated. By [27],
$\mathcal{M}$ is not Hilbert-Schmidt, and hence by Proposition 4.3,
$\mathcal{M}^{\perp}$ is not essentially normal.

$\square$

We conjecture that all the quotient modules corresponding to infinitely
generated submodules are not essentially normal. In the following example, we
prove that this is the case for Rudin’s quotient module.

######  Example 6.2.

In [28, Page71], Rudin constructed an invariant subspace of
$H^{2}(\mathbb{D}^{2})$ of infinite rank as follows. It is the submodule
$\mathcal{M}$ consisting of all the elements in $H^{2}(\mathbb{D}^{2})$ with
zero order at least $n$ at $\left(\alpha_{n},0\right)=\left(1-n^{-3},0\right)$
for each positive integer $n$. By [28, Page 71], Rudin${}^{\prime}s$ submodule
is not finitely generated. Let

$\left\\{\begin{array}[]{l}b_{n}(z)=\left(z-\alpha_{n}\right)/\left(1-\alpha_{n}z\right),\\\
q_{0}(z)=\prod_{n=1}^{\infty}b_{n}^{n}(z),\\\
q_{j}(z)=q_{j-1}(z)/\prod_{n=j}^{\infty}b_{n}(z),~{}j\geq
1.\end{array}\right.$

In [29], Seto and Yang obtained the decompositon

$\mathcal{M}^{\bot}=\bigoplus_{j=0}^{\infty}\big{(}H^{2}(z)\ominus
q_{j}(z)H^{2}(z)\big{)}w^{j}.$

In the case $n\geq 2,$ since $\alpha_{n}\in Z(q_{0})\cap Z(q_{1}),$ it is easy
to see that the normalized reproducing kernel

$k_{\alpha_{n}}(z)\in H^{2}(z)\ominus q_{1}(z)H^{2}(z)\subseteq
H^{2}(z)\ominus q_{0}(z)H^{2}(z).$

Routine calculation gives

$(S_{w}^{*}S_{w}-S_{w}S_{w}^{*})k_{\alpha_{n}}(z)=k_{\alpha_{n}}(z),$

therefore $S_{w}^{*}S_{w}-S_{w}S_{w}^{*}$ is not compact, and consequently
$\mathcal{M}^{\bot}$ is not essentially normal.

$\square$

As usual, for a quotient module ${\mathcal{M}}^{\perp}$ in
$H^{2}(\mathbb{D}^{2})$, we briefly write
$\sigma_{e}(\mathcal{M}^{\perp})=\sigma_{e}(S_{z,{\mathcal{M}}^{\perp}},S_{w,{\mathcal{M}}^{\perp}})$.

######  Example 6.3.

In this example, we construct a non-algebraic submodule $\mathcal{M}$
containing $p_{B}$ such that $\mathcal{M}/[p_{B}]$ is infinite dimensional,
while $\sigma_{e}(\mathcal{M}^{\perp})=\sigma_{e}([p_{B}]^{\perp}).$

Let $p=\prod_{i=1}^{n}\left(z-\alpha_{i}\omega\right)^{n_{i}}$ be a
distinguished homogeneous polynomial where each $|\alpha_{i}|=1,$ and
$B_{1},B_{2}$ be two nonconstant finite Blaschke products. Let $B$ be an
infinite Blaschke product satisfying $\mathbb{T}\subseteq\overline{Z(B)}.$ Set
$D=Z(B\circ B_{1})$, and

$E=\bigcup_{i=1}^{n}Z\big{(}B\circ(\alpha_{i}B_{2})\big{)}=Z\big{(}\prod_{i=1}^{n}B\circ(\alpha_{i}B_{2})\big{)}.$

Let $B_{3}=B\circ B_{1}$ and $B_{4}=\prod_{i=1}^{n}B\circ(\alpha_{i}B_{2})$,
then by Theorem 4.4 and [41, Theorem 4.3], the submodule
$\mathcal{M}=[B_{3}(z)B_{4}(w),p_{B}]$ is Hilbert-Schmidt, and moreover,
$\sigma_{e}(\mathcal{M}^{\perp})\subseteq\partial\mathbb{D}^{2}$. Then the
spectral mapping theorem ensures

$\sigma_{e}(\mathcal{M}^{\perp})\subseteq
Z(p_{B})\cap\partial\mathbb{D}^{2}=\sigma_{e}([p_{B}]^{\perp}).$

Conversely, for $(\lambda_{1},\lambda_{2})\in\sigma_{e}([p_{B}]^{\perp}),$
since $B_{1}(\lambda_{1})=\alpha_{i}B_{2}(\lambda_{2})\in\mathbb{T}$ for some
$i,$ there exists a sequence $\\{\mu_{n}\\}$ in $Z(B)$ such that
$(\mu_{n},\mu_{n})\rightarrow(B_{1}(\lambda),\alpha_{i}B_{2}(\mu)).$ Then by
[16, Lemma 2.2], we can find $t_{n}\in\\{B_{1}^{-1}(\mu_{n})\\}\subset
Z(B_{3})$ and $t_{n}^{\prime}\in\\{(\alpha_{i}B_{2})^{-1}(\mu_{n})\\}\subset
Z(B_{4})$ such that
$(t_{n},t_{n}^{\prime})\rightarrow(\lambda_{1},\lambda_{2}).$ Observing that

$p_{B}(t_{n},t_{n}^{\prime})=p\big{(}B_{1}(t_{n}),B_{2}(t_{n}^{\prime})\big{)}=p(\eta_{n},\eta_{n})=0,$

we conclude

$(t_{n},t_{n}^{\prime})\in Z\big{(}B_{3}(z)B_{4}(w)\big{)}\cap Z(p_{B}).$

Then by [16, Lemma 2.2] and [21, Theorem 6.1], it is not difficult to verify

$\sigma_{e}([p_{B}]^{\perp})\subseteq\overline{Z\big{(}B_{3}(z)B_{4}(w)\big{)}\cap
Z(p_{B})}\cap\partial\mathbb{D}^{2}\subseteq\sigma_{e}(\mathcal{M}^{\perp}).$

$\square$

By Theorem 4.4, both
$\mathcal{M}^{\bot}=\left[B_{3}(z)B_{4}(w),p_{B}\right]^{\bot}$ and
$\left[p_{B}\right]^{\bot}$ are essentially normal, hence there are two
$C^{*}$-extensions

$\displaystyle 0\rightarrow\mathcal{K}\rightarrow C^{*}(M^{\bot})\rightarrow
C\big{(}\sigma_{e}([p_{B}]^{\bot})\big{)}\rightarrow 0,$ (6.1)

and

$\displaystyle 0\rightarrow\mathcal{K}\rightarrow
C^{*}([p_{B}]^{\bot})\rightarrow
C\big{(}\sigma_{e}([p_{B}]^{\bot})\big{)}\rightarrow 0.$ (6.2)

From Theorem 3.6, the short exact sequence (6.2) gives a nontrivial element in
the K-homology group $K\big{(}\sigma_{e}([p_{B}]^{\bot})\big{)}$. We end this
section by the following question.

######  Question 6.4.

Are these two $C^{*}$-extensions essentially unitarily equivalent?

Acknowledgement. The authors sincerely thank K. Guo and R. Yang for their
encouragements and interests. The authors also thank R. Yang for valuable
discussions on this topic.

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# Precision Quantum Parameter Inference with Continuous Observation

Bijita Sarma<EMAIL_ADDRESS>Friedrich-Alexander-Universität Erlangen-
Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany  Junxin Chen  LIGO,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Sangkha Borah
<EMAIL_ADDRESS>Max Planck Institute for the Science of Light,
Staudtstraße 2, 91058 Erlangen, Germany  Friedrich-Alexander-Universität
Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany

###### Abstract

Quantum Parameter Estimation (QPE) is important from the perspective of both
fundamental quantum research and various practical applications of quantum
technologies such as for developing optimal quantum control strategies.
Standard and traditional methods for QPE involve projective measurements on
thousands of identically prepared quantum systems. However, these methods face
limitations, particularly in terms of the required number of samples and the
associated experimental resources. In this work, we present a novel method for
precise QPE that diverges from conventional techniques, employs continuous
measurements, and enables accurate QPE with a single quantum trajectory. In an
application, we demonstrate the use of the method for the task of parameter
estimation and force sensing of a levitated nanoparticle.

An experiment of a quantum system typically involves three major steps: (a)
preparation and initialization, (b) measurement, and (c) analysis. During the
preparation and initialization step, the measurement system is isolated and
prepared (in a given state) and the measurement apparatus is initialized
(reset) [1, 2]. Measurement involves coupling the system to the apparatus and
recording the measurement results. In the analysis step, the measurement
records are processed and analyzed to extract relevant information about the
quantum system under study. This process can involve statistical analysis,
comparison with theoretical predictions, and error estimation [3, 4, 5]. The
preparation procedure can be described by a set of classical parameters or
settings of a physical device, and the measurement results encode information
correlated with the preparation and initialization procedure. Often, these
parameters are not known in advance or are only partially known, and the task
of quantum parameter estimation (QPE) is simply to reliably estimate them as
accurately as possible, often with post-processing of the measured data in the
analysis phase[6, 7, 8]. The term “quantum” in QPE comes from the fact that it
is a quantum system that facilitates the transfer of information from the
classical parameters of the preparation process to the classical measurement
results [1, 9, 10].
As quantum technology continues to advance, QPE is expected to play a crucial
role in harnessing the power of quantum systems for practical applications. In
the context of quantum computing, accurate estimation of system parameters can
help optimizing the accuracy of various control protocols, including quantum
error correction (QEC) and error mitigation, as well as for optimizing the
performance and efficiency of various quantum algorithms [11, 12, 13, 14, 15].
Accurate QPE enables a precise understanding of the model of a quantum system,
which can be used to develop accurate QEC techniques and other quantum control
methods [16, 17]. This becomes particularly relevant in light of recent
advances in applying machine learning (ML) to quantum physics problems, in
which model-based ML techniques can be utilized to discover optimal control
and feedback strategies, despite the presence of measurement noise [18, 16].
Additionally, QPE plays a crucial role in quantum sensing and metrology, where
precise parameter estimation is necessary for high-precision measurements and
the detection of quantum signals [1, 9, 10, 3, 4, 5], which empowers both
applications and test of fundamental physics.
While the preparation procedure may be well understood, the measurement
results often exhibit statistical behavior because of the fundamental
principles of quantum mechanics. As a result, a single preparation and
measurement step is often insufficient to accurately estimate the parameters
[19]. To address this, it is common practice to repeat the preparation and
measurement process in multiple identically prepared systems to accumulate
statistical data and make more reliable estimations of the parameters of
interest. However, this approach can be time-consuming and resource-intensive
and introduces the potential for errors and inconsistencies while preparing
and initializing the ensemble of quantum systems and measurement devices.
Therefore, researchers are constantly seeking new techniques and technologies
to improve the efficiency and accuracy of quantum measurements [9, 10, 5, 20,
6, 7, 8]. One such technique is QPE with continuous measurement [1, 21, 2].
While most previous studies have focused on QPE with projective measurement,
there have been some studies that employed continuous measurement of quantum
systems [22, 23, 24, 25]. Continuous measurement has the advantage of not
fully collapsing a quantum state, but it is limited by measurement noise and
measurement-induced backaction. Measurement noise makes it difficult to
accurately separate the signal from noise, so continuous measurement QPE
methods rely on sampling hundreds of trajectories for estimation. In this
work, we propose a method that can accurately determine the precise parameter
from a single trajectory in an experiment. We also demonstrate the usefulness
of this approach in parameter estimation and force sensing of a levitated
nanoparticle.

Figure 1: Schematic of the proposed scheme for QPE of a unknown parameter
$\lambda_{0}$, encoded in the Hamiltonian of the system, $H(\lambda_{0})$.
Continuous measurement of the system is performed on the experimental setup
(left) to collect a single trajectory of duration $\tau$ with a total of $N$
noisy measurement outcomes. The parameter to be predicted, $\lambda_{0}$ is
obtained by simulating offline trajectory with varying $\lambda$. The protocol
guarantees that for $\lambda=\lambda_{0}$, the experimental trajectory is
perfectly matched by the simulated trajectory, leading to the loss,
$|\Delta\mathcal{M}|^{2}$ [Eq. 3] to be zero (minimum). The same model can
also be used to predict other system and detector properties that are not
directly encoded into the Hamiltonian, such as the unknown decoherence rates,
measurement efficiency, etc.

The schematic of the QPE scheme is shown in Fig. 1. In this approach, a
trajectory is collected via continuous measurement in an experiment (left) and
given to post-processing (right). The trajectory consists of time series
dataset given by
${D}=\\{(t_{0},\mathcal{I}_{0}),...,(t_{j},\mathcal{I}_{j}),...(t_{N-1},\mathcal{I}_{N-1})\\}$,
where $\mathcal{I}_{j}$ denotes the noisy measurement data of the measurement
operator $\mathcal{A}$ at time $t_{j}$. The post-processing task is to
estimate the unknown system parameter $\lambda$ at its optimally accurate
value $\lambda_{\rm optimal}\sim\lambda_{0}$, where $\lambda_{0}$ is the
accurate value of the parameter. With the procedure discussed in the
following, $\lambda_{\rm optimal}$ can be accurately determined by running a
simulation with the quantum master equation (QME) by varying $\lambda$ values
and comparing them with the reference trajectory obtained from the experiment.
The minimum error is to perfect zero for $\Delta\lambda=0$, where
$\Delta\lambda=|\lambda-\lambda_{0}|$. The detail of the method is discussed
below, with a brief introduction to the continuous measurement process.
Continuous quantum measurement involves weak monitoring of a quantum system
and is essentially achieved through weak coupling of the measurement device
with the system. Suppose that the laboratory quantum system, with Hamiltonian
$\hat{H}$, is continuously measured with a weak probe for the measurement
operator $\hat{\mathcal{A}}$ (suitably scaled to make it dimensionless). Such
a continuous measurement process leads to conditional stochastic dynamics of
the system density matrix in time $\rho_{c}(t)$ and is described by the so-
called quantum stochastic master equation (SME),

$\displaystyle{d\rho_{c}(t)}=-i[\hat{H},\leavevmode\nobreak\
\rho_{c}(t)]dt+\kappa\mathcal{D}[\hat{A}]\rho_{c}(t)dt$
$\displaystyle+\sqrt{\kappa\eta}\mathcal{H}[\hat{A}]\rho_{c}(t)d\mathcal{W}(t).$
(1)

Here, $\kappa$ is the measurement rate (the rate at which information is
extracted from the detector), $\eta$ is the measurement efficiency of the
detector and $d\mathcal{W}(t)$ represents an instantaneous random Wiener noise
increment (white noise model with zero mean and variance ${dt}$).
$\mathcal{D}[\hat{A}]$ and $\mathcal{H}[\hat{A}]$ are the superoperators that
describe, respectively, the backaction and diffusion terms in the SME [1].
Probing the system with a weakly coupled meter that, in effect, has a broad
probability distribution of the quantum state leads to noisy measurement as a
time series dataset ${D}=\\{(t,\mathcal{I}_{\rm expt}(t))\\}$, which is used
for the post-processing step given below.
The QPE estimator set up a simulation with unknown parameters with the
modified SME

$\displaystyle d\rho_{c}(t)=$
$\displaystyle-i[\hat{H}(\lambda),\leavevmode\nobreak\
\rho_{c}(t)]dt+\kappa\mathcal{D}[\hat{A}]\rho_{c}(t)dt$ $\displaystyle+$
$\displaystyle{2\kappa\eta}\left[{\cal I}_{\rm
expt}(t)-\langle\hat{A}(t)\rangle_{c}\right]\mathcal{H}[\hat{A}]\rho_{c}(t)dt,$
(2)

where $H(\lambda)$ is the Hamiltonian of the system in terms of the unknown
parameter $\lambda$. $\mathcal{I}_{\rm expt}(t)$ is the noisy measurement
current from $\mathcal{I}_{\rm expt}$ the trajectory collected from the
experiment, and $\langle\hat{A}(t)\rangle_{c}=\mathrm{Tr}[\rho_{c}\hat{A}]$ is
the conditional mean calculated locally at time $t$ during simulation. Given
an estimator modeled according to Eq. 2 and initialized with a random quantum
state $\rho(0)$, it is guaranteed that it converges to the true quantum state
$\rho_{\rm expt}(t)$ after a sufficiently long time $t$ [of the order of
$t>\kappa^{-1}$] [1, 26, 27, 21]. The time of convergence $t$ is reduced as a
function of the increasing fidelity between the initial states of the
estimator $\rho(0)$ and the real system $\rho_{\rm expt}(0)$, i.e.
$\mathcal{F}(\rho_{\rm expt}(0),\rho(0))$. In fact, the estimator would
immediately start following the correct dynamics of the real system if
$\mathcal{F}(\rho_{\rm expt}(0),\rho(0))=1$. This fact can be utilized to form
a powerful and useful QPE protocol. Considering the task of estimating an
unknown parameter $\lambda$ encoded in the Hamiltonian
$\hat{H}=\hat{H}(\lambda)$, the method would include the use of this
simulation with varying $\lambda$ [Eq. 2], providing a straightforward but
innovative approach to the QPE of the parameter $\lambda$. This approach can
likewise be applied to calculate other pertinent systems and detector
parameters, such as the decoherence rate and measurement efficiency, even
though these parameters are not directly integrated into the system’s
Hamiltonian (see discussions below).
Essentially, the task at hand is to calculate the following loss function

${\Delta\mathcal{M}^{2}}=\sum_{t}|\mathcal{I}_{\rm expt}(t)-\mathcal{I}_{\rm
QPE}(t)|^{2},$ (3)

where $\mathcal{I}_{\rm QPE}(t)$ is the measurement current at timestamp $t$
as obtained by the simulator and $\mathcal{I}_{\rm expt}(t)$ is the same
collected from the experiment earlier.

Figure 2: The proposed QPE scheme is applied for force sensing of a levitated
nanoparticle. (a) A nanoparticle of mass $m$ is trapped in a potential $V(x)$.
An unknown force $\vec{F}=\hbar\omega_{0}f\hat{x}$ is applied to it. The task
is to estimate the unknown force component $f$. Here, we have re-scaled $f$ so
that the correct value of the force is $f_{0}=1$. (b) A single trajectory
$\mathcal{I}_{\rm expt}(t)$ generated by an experiment with the setup of the
levitated particle. (c) The loss [Eq. 3] calculated as a post-processing step
by modeling the system on a computer and using the experimental trajectory as
a target. The minimum of the loss should ideally be zero, which makes each
point of the experimental trajectory [in (b)] reproduced, yielding ideally
zero loss. In effect, the conditional means $\langle\hat{x}(t)\rangle$ also
perfectly overlap as shown in (b). (d) The derivative
$df/|\Delta\mathcal{M}|^{2}$ with respect to $f$ is plotted. The sharp peak at
$f=1$ shows that the estimation is rather very accurate, which can be enhanced
by choosing finer grids for $f$. (e) QPE for determining the frequency
($\omega$) and decoherence rate ($\gamma$), with accurate values
$\omega=\gamma=1$.

In the following, we will use the method for accurate force sensing of a
levitated nanoparticle. Consider the center-of-mass motion of the nanoparticle
along the $x$-axis, with its mass $m$ and frequency $\omega_{0}$, with the
bare Hamiltonian given by

$\displaystyle\hat{H}_{0}=\frac{\hat{P}^{2}}{2m}+\frac{1}{2}m\omega_{0}^{2}\hat{X}^{2}=\frac{\hbar\omega_{0}}{4}(\hat{p}^{2}+\hat{x}^{2}),$
(4)

where $\hat{X}=x_{0}\hat{x}$ and $\hat{P}=p_{0}\hat{p}$, with
$[\hat{X},\hat{P}]=i\hbar$ are the position and momentum operators,
respectively. $\hat{x}$ and $\hat{p}$ are dimensionless operators, expressed
in terms of zero point position $x_{0}$ and zero-point momentum
$p_{0}=\hbar/(2x_{0})$. Following earlier studies [28, 29], we consider two
types of decoherence channels: (a) statistical displacement of the center of
mass of the levitated particle due to fluctuations in the levitation
potential, caused by external field fluctuations; and (b) fluctuations in the
spring constant resulting from fluctuations in the alternating current
component of the applied potential to the trap potential [30]. This
effectively introduces displacement and frequency noise channels, namely
${\gamma_{1}}\mathcal{D}[\hat{x}]$ and ${\gamma_{2}}\mathcal{D}[\hat{x}^{2}]$,
where $\gamma_{1}$ and $\gamma_{2}$ are the corresponding decoherence rates.
The Hamiltonian in the presence of a static force ${F}$ is given by

$\hat{H}=\hat{H}_{0}+{F}\hat{X}=\hat{H}_{0}+\hbar\omega_{0}{f}\hat{x},$ (5)

where $f=Fx_{0}/(\hbar\omega_{0})$ is the dimensionless static force to be
detected. With this, the SME is given by

$\displaystyle{d\rho_{c}(t)}$ $\displaystyle=-i[\hat{H},\leavevmode\nobreak\
\rho_{c}(t)]dt+\left[\gamma_{1}\mathcal{D}[\hat{x}]+\gamma_{2}\mathcal{D}[\hat{x}^{2}]\right]\rho_{c}(t)dt$
$\displaystyle+\kappa\mathcal{D}[\hat{a}]\rho_{c}(t)dt+\sqrt{\kappa\eta}\mathcal{H}[\hat{a}]\rho_{c}(t)d\mathcal{W}(t),$
(6)

where $\hat{a}$ is the measurement operator in the homodyne measurement
scheme, that results in the position
$\hat{x}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$ as the output signal of the
measurement. Nevertheless, the measurement current will be masked by the
measurement noise that yields only the noisy measurement output
$\mathcal{I}(t)=\langle\hat{x}(t)\rangle_{c}+{(4\kappa\eta)}^{-1/2}d\mathcal{W}/dt$.
Alternative measurement operators ${\hat{A}}$, which may be more advantageous
for the specific experimental configuration used, are possible. In such a
context, too, the protocol remains equally effective.

Figure 3: (a) Estimation error (in percentage) due to inaccuracy in parameter
(in percentage). We consider the inaccuracy as we deviate from the exact
initial state $\rho_{0}$ and the exact frequency $\omega_{0}$. $\rho_{0}$ are
considered coherent states defined by the complex amplitude $\alpha$. Although
the estimation error for the accurate force $f$ is within 0.5% upto 5% of
error $\rho_{0}$ (defined by the error in $\alpha$), it is about 10% for 5%
change in the value of $\omega_{0}$. (b) Simultaneous estimation of
$\omega_{0}$ and $f$. Each shaded region defines a value of $\omega_{0}$ for
which $f$ is varied. The accurate pair of values, that is,
$(\omega_{0},f)=(2\pi,1.0)$, yields the minimum loss.

Now, as per the procedure, the first step is to collect a homodyne trajectory
$\\{\mathcal{I}_{\rm expt}(t)\\}$ starting from a known initial state
$\rho_{0}$. This trajectory will act as our reference dataset $D$, which in
our case is generated by solving the SME given by Eq. 6. The simulator is
constructed for this system based on the known system model: $\omega_{0}$,
$\gamma_{1}$, $\gamma_{2}$, $\kappa$ and $\eta$. If they are not known a
priori, the same procedure can be applied; for example, we can estimate
$\omega_{0}$ and $\gamma$ beforehand [see Fig. 2(e)]. Therefore, the unknown
part of the Hamiltonian is $\Delta H=\hat{H}-\hat{H}_{0}$, which embodies the
force to be detected. The next step is to solve the problem as a post-
processing step, using Eq. 2 and Eq. 3. The minimum of the loss should ideally
be zero, which would correspond to the accurate parameter $f\sim f_{0}$ to be
estimated.

The results of the force estimation are shown in Fig. 2. A typical
experimental trajectory is shown in Fig. 2(b), which acts as the reference
data set for post-processing. The calculated loss is shown as a function of
$f$ in Fig. 2(c). The minimum of loss [$|\Delta\mathcal{M}|^{2}=0$] makes each
point of the experimental trajectory exactly reproduced, as shown in Fig.
2(b). If we analyze the conditional means $\langle\hat{x}(t)\rangle$, they are
also perfectly converged (shown in bold lines). To quantify the sensitivity of
the estimation with respect to the parameter to be estimated, we have
calculated the derivative of the force with respect to the loss,
$df/|\Delta\mathcal{M}|^{2}$, shown in Fig. 2(d). Fig. 2(e) provides an
example of the QPE scheme, demonstrating the process of determining the
frequency $\omega$ [depicted in blue] of a quantum harmonic oscillator
Hamiltonian, $\hat{H}=\hbar\omega\hat{a}^{\dagger}\hat{a}$, where
$\hat{a}(\hat{a}^{\dagger})$ represents the annihilation (creation) operator
of the oscillator. Furthermore, if the system possesses an unknown damping at
a rate of $\gamma$, the same procedure can be utilized to estimate it
[illustrated in orange]. To simplify matters, we rescale the parameters
$\lambda=(\omega,\gamma)$ such that $\lambda_{\rm accurate}=1$. The inset
shows a zoomed-in view around $\lambda=1$ demonstrating the fact that the loss
$\Delta\mathcal{M}\to 0$ as $\omega\to\omega_{\rm accurate}$ and
$\gamma\to\gamma_{\rm accurate}$.

We show in Fig. 3(a), the estimation error resulting from the imprecision in
the initial parameter selections of the simulator relative to the experimental
parameters. Here, we consider the inaccuracy as we deviate from the exact
initial state $\rho_{0}$ and the exact frequency $\omega_{0}$. For this
exercise, we considered $\rho_{0}$ as coherent states defined by the complex
amplitude $\alpha$, and consider the errors as $\alpha$ deviates from the true
value $\alpha_{0}$. Although the estimation error for the accurate force $f$
is within 0.5% up to 5% of the error $\rho_{0}$ (defined by the error in
$\alpha$), it is about 10% for the change of 5% in the value of $\omega_{0}$.
This is rather expected, since $\omega_{0}$ appears in both terms in the
Hamiltonian in Eq. 5, and therefore it is expected to be estimated very well
in advance following the method [as in Fig. 2(e)]. In Fig. 3(b), we show the
case where $\omega_{0}$ and $f$ are estimated simultaneously. Each shaded
region defines a value of $\omega_{0}$ for which $f$ is varied. The accurate
pair of values, that is, $(\omega_{0},f)=(2\pi,1.0)$, yields the minimum loss.

Figure 4: The loss as a function of the error in estimation of force, $f_{\rm
error}=f-f_{0}$. $f_{\rm error}=0$ is not shown for which
$|\Delta\mathcal{M}|^{2}=0$. On the twin-axis, the corresponding derivative
$df/d|\Delta\mathcal{M}|^{2}$ is shown. The $y$-axes are in logarithmic scale.

In Fig. 4, we quantify the accuracy in estimating $f$. For this, we plot the
loss $|\Delta\mathcal{M}|^{2}$ [Eq. 3] as a function of the error in
predicting $f$, i.e. $f_{\rm error}=f-f_{0}$. We see that the loss decreases
linearly with the decrease in $f_{\rm error}$ with $|\Delta\mathcal{M}|^{2}\to
0$ as $f_{\rm error}\to 0$. Therefore, the method outlined above is perfectly
accurate, provided that we have perfect accuracy in the model parameters.

The Quantum Cramér-Rao Bound (QCRB) is the fundamental bound for parameter
estimation. Here, we show that our estimation of a constant force obeys the
QCRB, even though the accuracy is perfect. The QCRB can be expressed as a
spectral uncertainty principle in our case [31]:

$\displaystyle
S_{f_{\mathrm{error}}}(\omega)\geq\left(\frac{4}{\hbar^{2}}|\chi_{\mathrm{m}}(\omega)|^{2}S_{f_{\mathrm{ba}}}(\omega)+\frac{1}{S_{f}(\omega)}\right)^{-1},$
(7)

where $S_{f_{\mathrm{error}}}(\omega)$ is the power spectral density of the
estimation error of the force $f$, $\chi_{\mathrm{m}}(\omega)$ the mechanical
susceptibility of the levitated particle, $S_{f_{\mathrm{ba}}}(\omega)$ the
spectrum of the backaction force, and $S_{f}(\omega)$ the prior spectrum of
the force $f$. As $f$ is a static force, $S_{f}(\omega)=f^{2}\delta(0)$. One
can also write the bound in terms of variance,

$\displaystyle
V_{f_{\mathrm{error}}}\geq\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left(\frac{4}{\hbar^{2}}|\chi_{\mathrm{m}}(\omega)|^{2}S_{f_{\mathrm{ba}}}(\omega)+\frac{1}{S_{f}(\omega)}\right)^{-1},$
(8)

where $V_{f_{\mathrm{error}}}$ is the variance of $f_{\mathrm{error}}$.

For given measurement noise, the estimation accuracy is limited by a weaker
bound. The measurement current in the frequency domain
$\mathcal{I}_{\mathrm{expt}}(\omega)$ can be written in displacement unit as
$\hat{y}(\omega)=\chi_{\mathrm{m}}(\omega)(f(\omega)+\hat{z}(\omega))$, where
$\hat{z}(\omega)$ contains all force noises, including quantum backaction
noise and imprecision noise expressed in force unit. Using the smoothing
technique, one can get the minimum achievable error for a given force noise
spectrum $S_{z}(\omega)$[32, 31]:

$\displaystyle
V_{f_{\mathrm{error}}}^{\mathrm{min}}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\left(\frac{1}{S_{z}(\omega)}+\frac{1}{S_{f}(\omega)}\right)^{-1},$
(9)

which is known to obey the QCRB.

Our estimation method is based on the vanishing loss function in Eq. 3, which
can be written as

$\displaystyle{\Delta\mathcal{M}^{2}(\omega)}=|\hat{y}_{\rm
expt}(\omega)-\hat{y}_{\rm
QPE}(\omega)|^{2}=|f_{\mathrm{error}}(\omega)+\hat{z}_{\mathrm{error}}(\omega)|^{2},$
(10)

where $f_{\mathrm{error}}=f_{\rm QPE}-f_{0}$, similarly
$\hat{z}_{\mathrm{error}}=\hat{z}_{\rm QPE}-\hat{z}_{\rm expt}$.
$\Delta\mathcal{M}^{2}(\omega)=0$ infers perfectly anticorrelated
$f_{\mathrm{error}}$ and $\hat{z}_{\mathrm{error}}$, that is,
$f_{\mathrm{error}}=-\hat{z}_{\mathrm{error}}$. Then the spectrum of the
estimation error $S_{f_{\mathrm{error}}}=S_{\hat{z}_{\mathrm{error}}}$. As
$z_{\mathrm{expt}}$ and $z_{\mathrm{QPE}}$ are in the experiment and
simulation, respectively, they are not correlated. Therefore,
$S_{f_{\mathrm{error}}}(\omega)=S_{\hat{z}_{\mathrm{error}}}(\omega)=2S_{z}(\omega)$.
Notice that the integrand of Eq. 9
$\left(1/S_{z}(\omega)+1/S_{f}(\omega)\right)^{-1}<S_{z}(\omega)<S_{f_{\mathrm{error}}}(\omega)$,
our method does not violate the QCRB. This also implies the following
expression of our estimation error variance:

$\displaystyle
V_{f_{\mathrm{error}}}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}2S_{z}(\omega)B(\omega),$
(11)

where $B(\omega)$ defines the effective bandwidth of the force to be
estimated. This effective bandwidth is encoded in the $S_{f}(\omega)$ in the
expression of QCRB. Outside the bandwidth, $S_{f}(\omega)=0$, and the
integrand vanishes. Intuitively, applying $B(\omega)$ reduces the bandwidth of
the integrated noise, which is favorable for parameter estimation.

Obeying the QCRB still allows conceptually perfect accuracy of estimation in
the sense of variance, due to the zero-bandwidth of the static force to be
estimated, i.e. $B(\omega)=0$. In the vicinity of 0, as $S_{z}(0)$ is finite,
the integration of $S_{f_{\mathrm{error}}}=2S_{z}(\omega)$ around 0 vanishes.
Therefore, $V_{f_{\mathrm{error}}}=0$, i.e. the accuracy of the estimate is
perfect. In practice, the experimental spectrum does not have infinite
resolution, and the integration bandwidth is limited by $1/\tau$, where $\tau$
is the length of the trajectory in time. This gives a finite estimation
accuracy, but scales favorably with integration time.

In conclusion, accurate estimation of parameters in experiments in quantum
systems is important from the perspective of a host of applications. In this
work, we have introduced a method that can be useful for estimating and
sensing parameters from a single quantum trajectory from an experiment that
undergoes continuous measurement. As a useful and intuitive demonstration, we
have shown its usefulness for the application of parameter estimation and
force sensing of a levitated nanoparticle, a field that has become of immense
importance in recent years because of the possibilities of realizing them in
laboratory experiments on different platforms, such as optical/ion traps and
magnetic levitation. However, the same method can also be applied for
estimating parameters in other systems and quantum platforms.

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