# On the motivic homotopy type of algebraic stacks

Neeraj Deshmukh  Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
<EMAIL_ADDRESS>
###### Abstract.

We construct smooth presentations of algebraic stacks that are local
epimorphisms in the Morel-Voevodsky ${\mathbb{A}}^{1}$-homotopy category. As a
consequence we show that the motive of a smooth stack (in Voevodsky’s
triangulated category of motives) has many of the same properties as the
motive of a smooth scheme.

##  1\. Introduction

An important technique in motivic homotopy theory of algebraic stacks is
reduction to the scheme case by means of homotopical descent. This is
possible, for instance, when the stacks in question are Nisnevich locally
quotient stacks. The results in [AHR] (and further generalisation in [AHLHR])
show that stacks with linearly reductive stabilisers are Nisnevich locally
quotient stacks.

In this note we establish a certain homotopy descent result for any quasi-
separated algebraic stack. This will allows us to conclude that the various
formalisms of motives and motivic homotopy theory111In the case of stacks one
has to distinguish between genuine vs Kan-extended motivic spectra. For
instance $K$-theory is genuine but Chow groups are Kan-extended. In this note
we will be concerned with the latter kind of objects. produce the correct
results for algebraic stacks. To wit, we will show that the motive of a smooth
algebraic stack has similar properties as the motive of a smooth scheme. This
generalises the results in [CDH] that were established for Nisnevich locally
quotient stacks. Another improvement, albeit minor, that we can make is to
show that for algebraic stacks with spearated diagonal the existing notions of
stable homotopy category coincide: in [Cho], it is shown that the stable
motivic homotopy category of [Cho] and the lisse-extended category of [KR] are
equivalent when the stack admits a smooth presentation with a Nisnevich local
section; we will show that all quasi-separated stacks admit such a
presentation. Following [Pir], we will call such coverings _smooth-Nisnevich
coverings_.

######  Definition 1.1.

A _smooth-Nisnevich morphism_ of algebraic stacks is a morphism
$f:{\mathcal{Y}}\rightarrow{\mathcal{X}}$ such that $f$ is smooth, surjective
and any morphism ${\rm Spec\,}k\rightarrow{\mathcal{X}}$ from the spectrum of
a field $k$ lifts to ${\mathcal{Y}}$. We say that $f$ is _smooth-Nisnevich
covering_ when ${\mathcal{Y}}$ is an algebraic space.

When $f$ is étale such covering are called Nisnevich coverings in literature.

Smooth-Nisnevich coverings were first discussed in [Pir], where it is shown
that such coverings exist for finite type stacks with affine stabilisers that
are defined over an infinite field. The aim of this note is extend this result
to all finite type stacks with affine stabilisers. The following is our main
geometric result.

######  Theorem 1.2.

Let ${\mathcal{X}}$ be a quasi-separated algebraic stack with separated
diagonal over a scheme $S$.

  1. (1)

There exists a smooth-Nisnevich covering $X\rightarrow{\mathcal{X}}$ with $X$
a scheme.

  2. (2)

Assume $S$ is Noetherian and that ${\mathcal{X}}$ is of finite type over $S$
with affine stabilisers, then $X$ can also be chosen to be of finite type over
$S$.

Note that the argument for Part (1) of the above theorem is essentially
contained in [LMB, §6.7]. Part (2) is new in the mixed charactersitic setting
and is built from a modification of Pirisi’s argument [Pir, Proposition 3.6]
which works over an infinite field.

The existence of such covers has implications for motivic homotopy theory of
algebraic stacks. More specifically, it shows that the Nisnevich homotopy type
of an algebraic stack can be described by a simplicial scheme (or simplicial
algebraic space). This makes many homotopical descent arguments accessible for
algebraic stacks.

######  Theorem 1.3.

Let $p:X\rightarrow{\mathcal{X}}$ be a smooth-Nisnevich covering over a field
$k$. Let $X_{\bullet}$ denote the Čech nerve of $p$. Then the morphism
$p_{\bullet}:X_{\bullet}\rightarrow{\mathcal{X}}$ induces an equivalence in
the Morel-Voevodsky ${\mathbb{A}}^{1}$-homotopy category, ${\mathcal{H}}(k)$.

A consequence of the above result is that the motive of a smooth algebraic
stack continues to enjoy the same properties as the motive of a smooth scheme
(see Section 3). So far this was only known for stacks which satisfied local
structure theorems in the sense of [AHR, AHLHR] or in a different direction
for motives of smooth stacks in the étale topology. We describe various
results of this kind in the text.

######  Remark 1.4.

Another application of Theorem 1.2 is the following: In [KM], the authors use
Theorem 1.2(2) to apply Elkik’s approximation technique to the Picard stack
and study the following question of Grothendieck: when is the map

$H^{2}(X,{\mathbb{G}}_{m})\rightarrow\mathop{\varprojlim}\limits
H^{2}(X_{n},{\mathbb{G}}_{m})$

injective for a proper morphism $X\rightarrow{\rm Spec\,}A$, where
$(A,\mathfrak{m})$ be a complete Noetherian ring and $X_{n}:=X\times_{A}{\rm
Spec\,}A/\mathfrak{m}^{n+1}$ is the $n$-th infinitesimal thickening.

Conventions. We work with algebraic stacks in the sense of [LMB], i.e, all
stacks are assumed to be quasi-separated with separated diagonal.

Acknowledgements. I would like to thank Roberto Pirisi and Tuomas Tajakka for
helpful discussions. I also thank Amit Hogadi, Siddharth Mathur and Andrew
Kresch for their comments on this note and Utsav Choudhury for teaching me
about motives with compact support.
The author was supported by the Swiss National Science Foundation (SNF),
project 200020_178729 during the course of this work. He also acknowledges the
support of the University of Zurich under the UZH Postdoc Grant, Verfügung Nr.
FK-22-111.

##  2\. Smooth-Nisnevich coverings

In order to prove Theorem 1.2, we will use the following results from [LMB,
§6]

###  2.1. Moduli of finite étale covers

Fix a base scheme $S$. For a separated and representable $S$-morphism
$\pi:{\mathcal{X}}\rightarrow{\mathcal{Y}}$ of algebraic stacks, we recall the
construction and basic properties of the stacks
$SEC_{d}({\mathcal{X}}/{\mathcal{Y}})$ and
$ET_{d}({\mathcal{X}}/{\mathcal{Y}})$.

Consider the $d$-fold fibre product
$({\mathcal{X}}/{\mathcal{Y}})^{d}:={\mathcal{X}}\times_{\mathcal{Y}}{\mathcal{X}}\times_{\mathcal{Y}}\ldots\times_{\mathcal{Y}}{\mathcal{X}}$.
For any $S$-scheme $U$, an object
$u:U\rightarrow({\mathcal{X}}/{\mathcal{Y}})^{d}$ is equivalent to the data of
a morphism $U\rightarrow{\mathcal{Y}}$ together with $d$ sections
$x_{1},\ldots,x_{d}$ of the morphism
${\mathcal{X}}\times_{\mathcal{Y}}U\rightarrow U$. Since $\pi$ is separated,
the locus where any two such section agree is closed. Thus, the complement of
this locus is an open substack which is denoted by
$SEC_{d}({\mathcal{X}}/{\mathcal{Y}})$.

The symmetric group $S_{d}$ acts naturally on
$({\mathcal{X}}/{\mathcal{Y}})^{d}$ and the restriction of this action to
$SEC_{d}({\mathcal{X}}/{\mathcal{Y}})$ is free. We denote the quotient stack
by

$ET_{d}({\mathcal{X}}/{\mathcal{Y}}):=[SEC_{d}({\mathcal{X}}/{\mathcal{Y}})/S_{d}].$

Almost by construction $SEC_{d}$ and $ET_{d}$ satisfy the following
properties:

######  Proposition 2.1 ([LMB, Propositions 6.6.2, 6.6.3]).

Let $\pi:{\mathcal{X}}\rightarrow{\mathcal{Y}}$ be a representable and
separated morphism of algebraic stacks over $S$. Then

  1. (1)

The constructions $SEC_{d}({\mathcal{X}}/{\mathcal{Y}})$ and
$ET_{d}({\mathcal{X}}/{\mathcal{Y}})$ behave well with respect to base change.
That is, given any morphism $U\rightarrow{\mathcal{Y}}$, we have cononical
isomorphisms $SEC_{d}({\mathcal{X}}/{\mathcal{Y}})\times_{\mathcal{Y}}U\simeq
SEC_{d}({\mathcal{X}}\times_{\mathcal{Y}}U/U)$ and
$ET_{d}({\mathcal{X}}/{\mathcal{Y}})\times_{\mathcal{Y}}U\simeq
ET_{d}({\mathcal{X}}\times_{\mathcal{Y}}U/U)$.

  2. (2)

The natural morphisms
$SEC_{d}({\mathcal{X}}/{\mathcal{Y}})\rightarrow{\mathcal{Y}}$ and
$ET_{d}({\mathcal{X}}/{\mathcal{Y}})\rightarrow{\mathcal{Y}}$ are
representable and separated. Moreover, they are smooth (respectively, étale)
if $\pi$ is.

  3. (3)

If ${\mathcal{X}}$ is an algebraic space, then
$SEC_{d}({\mathcal{X}}/{\mathcal{Y}})$ is an algebraic space and, therefore,
$ET_{d}({\mathcal{X}}/{\mathcal{Y}})$ is a Deligne-Mumford stack.

  4. (4)

For any $S$-scheme $U$, a $U$-point of $ET_{d}({\mathcal{X}}/{\mathcal{Y}})$
is equivalent to a morphism $U\rightarrow{\mathcal{Y}}$ together with a closed
subscheme $Z\subset U\times_{\mathcal{Y}}{\mathcal{X}}$ such that
$Z\rightarrow U$ is finite étale of degree $d$.

  5. (5)

If $\pi$ is finite étale of degree $d$, then
$ET_{d}({\mathcal{X}}/{\mathcal{Y}})\simeq{\mathcal{Y}}$.

###  2.2. Proof of Theorem 1.2

For part (2) of the theorem, we will need the following result about stacks
admitting stratification by global quotient stacks as found in [Kre].

######  Definition 2.2.

Let ${\mathcal{X}}$ be an algebraic stack over $S$. We say that
${\mathcal{X}}$ admits a stratification by global quotient stacks if there
exists a finite chain
${\mathcal{X}}\supset{\mathcal{X}}_{1}\supset\ldots{\mathcal{X}}_{n}$ of
closed substacks such that for each $i$,
${\mathcal{X}}_{i}\setminus{\mathcal{X}}_{i-1}$ is a global quotient stack
over $S$.

######  Theorem 2.3.

[Kre, Theorem 3.5.9] Let ${\mathcal{X}}$ be an algebraic stack of finite type
over a Noetherian scheme $S$. Assume that ${\mathcal{X}}$ has affine
stabilizers at geometric points. Then ${\mathcal{X}}$ admits a stratification
global quotient stacks over $S$.

We also need the following result whose proof can also be found in [LMB,
Théorème 6.1]

######  Proposition 2.4.

Let ${\mathcal{Y}}$ be a Deligne-Mumford stack. Assume that ${\mathcal{Y}}$ is
of the form $[Y/G]$ with $Y$ an algebraic space and $G$ a finite group. Then
there exists a smooth covering $X\rightarrow{\mathcal{Y}}$ such that given any
semi-local ring ${\mathcal{O}}$ any ${\mathcal{O}}$-valued point of
${\mathcal{Y}}$ lifts to $X$.

We will now prove Theorem 1.2.

###### Proof of 1.2.

For both part (a) and part (b), we will show that we can find such a covering
with $X$ an algebraic space. As any algebraic space admits a Nisnevich
covering by a scheme (see [Knu, Theorem 6.4]),we conclude by replacing $X$
with a Nisnevich covering by a scheme.

Proof of part (1). The argument for (1) is essentially in [LMB, §6.7] and we
reproduce it here for completeness. It suffices to consider the case when
${\mathcal{X}}$ is quasi-compact. Let $Y\rightarrow{\mathcal{X}}$ be a smooth
presentation with $Y$ affine. Let $K$ be a field and choose a $K$-valued point
${\rm Spec\,}K\rightarrow{\mathcal{X}}$. Then there exists a finite separable
extension $L\supset K$ and a closed immersion $f:{\rm Spec\,}L\rightarrow Y$.
Let $d:=[L:K]$. Denote by $ET_{d}(Y/{\mathcal{X}})$ the moduli of degree $d$
étale covers. Then the closed immersion $f:{\rm Spec\,}L\rightarrow X$ defines
a $K$-point of $ET_{d}(XY/{\mathcal{X}})$ giving us a commutative diagram,

${ET_{d}(Y/{\mathcal{X}})}$${{\rm Spec\,}K}$${{\mathcal{X}}}$

As $ET_{d}(Y/{\mathcal{X}}):=[SEC_{d}(Y/{\mathcal{X}})/S_{d}]$ and $Y$ is
affine, $SEC_{d}(Y/{\mathcal{X}})$ is an algebraic space. Applying the
previous proposition, we see that ${\rm Spec\,}K$ lifts to
$X_{d}:=SEC_{d}(Y/{\mathcal{X}})\times^{S_{d}}GL_{n}$. Then the countable
collection $\\{X_{n}\\}_{n}$ gives a smooth-Nisnevich covering
$\cup_{n}X_{n}\rightarrow{\mathcal{X}}$. As every algebraic space admits a
Nisnevich covering by a scheme we can also assume that the family
$\\{X_{n}\\}$ is a countable collection of schemes.

Proof of part (2). Assume ${\mathcal{X}}$ is finite type with affine
stabilisers and that $S$ is Noetherian. Let ${\mathcal{A}}:=\\{X_{n}\\}_{n}$
be the countable family constructed in part (1), and
$p:\cup_{n}X_{n}\rightarrow{\mathcal{X}}$ be the smooth-Nisnevich covering. By
the Theorem 2.3, ${\mathcal{X}}$ admits a stratification by global quotient
stacks. Thus, it suffices to show that for each stratum
${\mathcal{Y}}_{i}:={\mathcal{X}}_{i}\setminus{\mathcal{X}}_{i-1}$, there is a
finite collection $X_{i_{1}},X_{i_{2}},\ldots,X_{i_{k}}\in{\mathcal{A}}$ such
that
$\cup_{k}(X_{i_{k}}\times_{\mathcal{X}}{\mathcal{Y}}_{i})\rightarrow{\mathcal{Y}}_{i}$
is a smooth-Nisnevich covering. The union of all such $X_{i_{k}}$ is a finite
type scheme that is a smooth-Nisnevich covering of ${\mathcal{X}}$. Thus, it
suffices to consider the case of global quotient stacks.

Let $[Y/GL_{n}]$ be a global quotient stack. Let
$\cup_{n}X_{n}\rightarrow[Y/GL_{n}]$ be the countable family of part (1). We
have the following cartesian diagram where each arrow is a smooth-Nisnevich
covering:

${\cup_{n}(X_{n}\times_{[Y/GL_{n}]}Y)}$${Y}$${\cup_{n}X_{n}}$${{[Y/GL_{n}]}}$

Denote $\cup_{n}(X_{n}\times_{[Y/GL_{n}]}Y):=\cup_{n}Z_{n}$. Since
$\cup_{n}Z_{n}\rightarrow Y$ is smooth-Nisnevich, the generic point of $Y$
admits a lift to some $Z_{k}$. Thus, there is an open subset $U\subset Y$ that
lifts to $Z_{k}$. We may now consider the complement of $U$, and apply the
same procedure. This gives a descending sequence $Y\subset Y_{1}\subset\ldots$
whose generic points lift to $\cup_{n}Z_{n}$. Since $Y$ is Noetherian, there
exists an $N$ such that $\cup_{n\leq N}Z_{n}$ stabilises.

We will show that the map $\cup_{n\leq N}Z_{n}\rightarrow Y$ is a smooth-
Nisnevich covering. Let $x:{\rm Spec\,}k\rightarrow Y$ be a morphism. By
construction, there exists an $i$ such that the morphism $x$ factors as ${\rm
Spec\,}k\rightarrow Y_{i}\setminus Y_{i-1}\subset Y$. Then the lifting
$Y_{i}\setminus Y_{i-1}\subset Z_{i}\subset\cup_{n\leq N}Z_{n}$, gives a lift
of $x:{\rm Spec\,}k\rightarrow Y$. Further, as $Y\rightarrow[Y/GL_{n}]$ is a
smooth-Nisnevich covering, it is easy to see that $\cup_{n\leq
N}X_{n}\rightarrow[Y/GL_{n}]$ is also a smooth-Nisnevich covering. ∎

######  Remark 2.5.

We note that the proof of part (2) above is essentially the same argument as
in [Pir, Proposition 3.6]. A careful choice of presentations for the strata
allows us to to relax the characteristic zero assumption in [Pir, Proposition
3.6] and also deal with the mixed characteristic situation.

######  Remark 2.6.

The argument for part (1) is extracted from the proof of [LMB, Theorem 6.7]
which states for any field $k$ and a $k$-point of ${\mathcal{X}}$ there exists
smooth affine neighbourhood that lifts the given $k$-point. While it is
tempting to take union over all such neighbourhood to get a smooth-Nisnevich
covering, it is unclear to me how to do this if the stack is not defined over
a field. This is due the fact that there is no notion of a residue field for
algebraic stacks, hence there is no concept of “minimal” fields through which
any field valued point factors.

In fact smooth-Nisnevich coverings also lift Hensel local points by the
following corollary.

######  Corollary 2.7.

Let $X\rightarrow{\mathcal{X}}$ be a smooth-Nisnevich covering. Let ${\rm
Spec\,}{\mathcal{O}}$ be the spectrum of a Henselian local ring
${\mathcal{O}}$. Then any morphism ${\rm
Spec\,}{\mathcal{O}}\rightarrow{\mathcal{X}}$ lifts to a morphism ${\rm
Spec\,}{\mathcal{O}}\rightarrow X$.

###### Proof.

Let $k$ be the residue field of ${\mathcal{O}}$ and
$X_{\mathcal{O}}:=X\times_{\mathcal{X}}{\rm Spec\,}{\mathcal{O}}$ denote the
base change of $X$ to ${\rm Spec\,}{\mathcal{O}}$. Then the composite ${\rm
Spec\,}k\rightarrow{\rm Spec\,}{\mathcal{O}}\rightarrow{\mathcal{X}}$ lifts to
a morphism ${\rm Spec\,}k\rightarrow X$. This also gives us a morphism ${\rm
Spec\,}k\rightarrow X_{\mathcal{O}}$. Now, as $X_{\mathcal{O}}\rightarrow{\rm
Spec\,}{\mathcal{O}}$ is smooth, we have a surjection of sets
$X_{\mathcal{O}}({\mathcal{O}})\twoheadrightarrow X_{\mathcal{O}}(k)$. Thus,
we have a section ${\rm Spec\,}{\mathcal{O}}\rightarrow X_{\mathcal{O}}$.
Composing with the projection map $X_{\mathcal{O}}\rightarrow X$ gives us the
desire lift. ∎

##  3\. Applications to Motivic homotopy theory of algebraic stacks

We will now mention some applications of Theorem 2.3 to the motivic homotopy
theory of algebraic stacks. The proof of all the statements are the same as in
[CDH] using the smooth-Nisnevich covering constructed in Theorem 1.2.

We will now recall the notion of the (unstable) ${\mathbb{A}}^{1}$-homotopy
category over a field.

Fix a base field $k$. Let $Sm/k$ denote the category of smooth schemes over
$k$. Let $\Delta^{op}PSh(Sm/k)$ be the category of simplicial presheaves on
$Sm/k$. $\Delta^{op}PSh(Sm/k)$ has a local model structure with respect to the
Nisnevich topology (see [Jar]). A morphism $f:X\rightarrow Y$ in
$\Delta^{op}PSh(Sm/k)$ is a weak equivalence if the induced morphisms on
stalks (for the Nisnevich topology) are weak equivalences of simplicial sets.
Cofibrations are monomorphisms, and fibrations are characterised by the right
lifting property.

We Bousfield localise this model structure with respect to the class of maps
$X\times{\mathbb{A}}^{1}\rightarrow X$ (see [MV, 3.2]). The resulting model
structure is called the Nisnevich motivic model structure. Denote by
$\mathcal{H}(k)$ the resulting homotopy category. This is the (unstable)
${\mathbb{A}}^{1}$-homotopy category for smooth schemes over $k$.

Let us also rapidly recall the definition of Voevodsky’s triangulated category
of motive $\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})$. Let $Cor_{k}$ denote
the category of finite correspondences whose objects are smooth separated
schemes over $k$. For any two $X,Y$, the morphisms of $Cor_{k}$ are given by
$Cor(X,Y)$ which is the free abelian group generated by irreducible closed
subschemes $W\subset X\times Y$ that are finite and surjective over $X$. An
additive functor $F:Cor_{k}^{op}\rightarrow\mathbf{Ab}$ is called a presheaf
with transfers. Let $PST(k,{\mathbb{Z}})$ denote the category presheaves with
transfers. For any smooth scheme $X$, let ${\mathbb{Z}}_{tr}(X)$ be the
presheaf with tranfers which on any smooth scheme $Y$ is defined as

${\mathbb{Z}}_{tr}(X)(Y):=Cor(X,Y)$

Let $K(PST(k,{\mathbb{Z}}))$ denote the category of complexes of presheaves
with transfers. The category $K(PST(k,{\mathbb{Z}}))$ also has a Nisnevich
motivic model structure which is defined analogously as in the case of
$\Delta^{op}PSh(Sm/k)$. We denote the associated homotopy category by
$\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})$. This is Voevodsky’s triangulated
category of mixed motives in the Nisnevich topology (for details, see [MVW]).

######  Proposition 3.1.

Let ${\mathcal{X}}$ be a quasi-separated stack. Let
$X\rightarrow{\mathcal{X}}$ be a smooth-Nisnevich covering. Then the
associated Čech hypercover $X_{\bullet}$ is weakly equivalent to
${\mathcal{X}}$ in the Nisnevich local model structure on the category
$\Delta^{op}PSh(Sm/k)$ of simplicial presheaves.

###### Proof.

By Corollary 2.7, we know that hensel local points lift along smooth-Nisnevich
coverings. Adapting the proof of [CDH, Theorem 1.2] gives the result. ∎

The proof of Corollary 1.3 follows from ${\mathbb{A}}^{1}$-localising the
above proposition.

###### Proof of Corollary 1.3.

As ${\mathbb{A}}^{1}$-localisation preserves simplicial equivalences, the
result follows from the previous proposition. ∎

As a consequence, we have the following result.

######  Corollary 3.2.

The weak equivalence above induces an equivalence of motives
$M(X_{\bullet})\simeq M({\mathcal{X}})$ in
$\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})$.

###### Proof.

Use the functor
$M:{\mathcal{H}}_{\bullet}(k)\rightarrow\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})$
(see [CDH] for details). ∎

We will now state some results about motives and motivic cohomology of smooth
stacks.

The proofs are exactly as in [CDH] after replacing $GL_{n}$-presentations with
smooth-Nisnevich presentations.

######  Theorem 3.3.

Let ${\mathcal{X}}$ be a smooth algebraic stack. Then its motive
$M({\mathcal{X}})\in\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})$ satisfies the
following properties:

  1. (1)

$M({\mathcal{X}})$ satisfies Nisnevich descent.

  2. (2)

(Projective bundle formula) For any vector bundle ${\mathcal{E}}$ of rank
$n+1$ over ${\mathcal{X}}$, we have a canonical isomorphism

$M({{\mathbb{P}}roj}({\mathcal{E}}))\simeq\bigoplus_{i=0}^{n}M({\mathcal{X}})(i)[2i]$

  3. (3)

(Blow-up formula) For ${\mathcal{Z}}\subset{\mathcal{X}}$ a smooth closed
substack of pure codimension $c$ we have

$M(Bl_{{\mathcal{Z}}}({\mathcal{X}}))\simeq
M({\mathcal{X}})\oplus_{i=0}^{c-1}M({\mathcal{Z}})(i)[2i]$

  4. (4)

(Gysin triangle) For ${\mathcal{Z}}\subset{\mathcal{X}}$ a smooth closed
substack of codimension c, we have a Gysin triangle:

$M({\mathcal{X}}\setminus{\mathcal{Z}})\rightarrow M({\mathcal{X}})\rightarrow
M({\mathcal{Z}})(c)[2c]\rightarrow M({\mathcal{X}}\setminus{\mathcal{Z}})[1].$

###### Proof.

The proofs in [CDH, §4] go through using a smooth-Nisnevich covering
$X\rightarrow{\mathcal{X}}$. ∎

######  Remark 3.4.

Note that the transfer functor
$M:{\mathcal{H}}_{\bullet}(k)\rightarrow\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})$
allows us to define the motive of any smooth stack. What is apriori unclear is
whether this motive has any properties (good or bad). This is why in [CDH],
the authors work with stacks that are Nisnevich locally quotient stacks in
order to use homotopical descent to prove the above formulae for the motive.
With Corollary 1.3, those arguments now work for all smooth stacks.

###  3.1. Motivic cohomology as hypercohomology

Recall the definitions of smooth-Nisnevich and smooth-Zariski sites of an
algebraic stack

######  Definition 3.5.

Let ${\mathcal{X}}$ be an algebraic stack. The smooth-Nisnevich (resp. smooth-
Zariski) site of ${\mathcal{X}}$, denoted by ${\mathcal{X}}_{\text{lis-nis}}$
(resp. ${\mathcal{X}}_{\text{lis-zar}}$) is the category whose objects consist
of pairs $(U,p)$ where $p$ is a smooth morphism $p:U\rightarrow{\mathcal{X}}$
from an algebraic space $U$ and coverings in ${\mathcal{X}}_{\text{lis-nis}}$
are given by Nisnevich coverings (resp. Zariski coverings) of algebraic
spaces.

The following result states that motivic cohomology can be computed as
hypercohomology of the motivic complexes ${\mathbb{Z}}(i)$ on the smooth-
Nisnevich site of the stack.

######  Corollary 3.6.

Let ${\mathcal{X}}$ be a smooth stack over a field $k$. The motivic cohomology
of ${\mathcal{X}}$ agrees with the hypercohomology of the motivic complexes
${\mathbb{Z}}(j)$ on ${\mathcal{X}}_{\text{lis-nis}}$. That is,

$Ext^{i}_{D({\mathcal{X}})}({\mathbb{Z}},{\mathbb{Z}}(j)|_{\mathcal{X}})\simeq
Ext^{i}_{DA({\mathcal{X}})}({\mathbb{Z}},{\mathbb{Z}}(j)|_{\mathcal{X}})\simeq
Hom_{\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})}(M({\mathcal{X}}),{\mathbb{Z}}(j)[i]),$

where ${\mathbb{Z}}$ denotes the constant sheaf ${\mathbb{Z}}$ on the
${\mathcal{X}}_{\text{lis-nis}}$.

###### Proof.

See [CDH, Theorem 5.2]. ∎

In fact, for smooth separated Deligne-Mumford stacks with schematic coarse
space, we can also use the smooth-Zariski site for the above computation. This
follows from the following obvious corollary of [KV]:

######  Corollary 3.7.

Let ${\mathcal{X}}$ be a smooth separated Deligne-Mumford stack with schematic
coarse space. Then ${\mathcal{X}}$ is Zariski locally a quotient stack.

###### Proof.

By [KV], it is immediate that ${\mathcal{X}}$ admits finite flat covers Zarski
locally on the coarse space. As resolution property descends along finite flat
maps, we are done. ∎

The above propostion show that for any such stack one can find a
$GL_{n}$-presentation which is a _Zariski local_ equivalence.

We note the following refinement of Corollary 3.6 which is true for smooth
separated Deligne-Mumford stacks with schematic coarse space. The proof is
that same using a Zariki local presentation (instead of a Nisnevich local
presentation).

######  Corollary 3.8.

Let ${\mathcal{X}}$ be a smooth separated Deligne Mumford stack over a field
$k$. Assume that the coarse moduli space of ${\mathcal{X}}$ is a scheme. The
motivic cohomology of ${\mathcal{X}}$ agrees with the hypercohomology of the
motivic complexes ${\mathbb{Z}}(j)$ on ${\mathcal{X}}_{\text{lis-zar}}$. That
is,

$Ext^{i}_{D({\mathcal{X}})}({\mathbb{Z}},{\mathbb{Z}}(j)|_{\mathcal{X}})\simeq
Ext^{i}_{DA({\mathcal{X}})}({\mathbb{Z}},{\mathbb{Z}}(j)|_{\mathcal{X}})\simeq
Hom_{\mathbf{DM}^{\mathit{eff}}(k,{\mathbb{Z}})}(M({\mathcal{X}}),{\mathbb{Z}}(j)[i]),$

where ${\mathbb{Z}}$ denotes the constant sheaf ${\mathbb{Z}}$ on the
${\mathcal{X}}_{\text{lis-nis}}$.

###  3.2. Motive cohomology with finite coefficients

We also have the following comparison theorem relating motivic cohomology with
${\mathbb{Z}}/n{\mathbb{Z}}$ to étale cohomology with $\mu_{n}$-coefficients.

######  Corollary 3.9.

Let ${\mathcal{X}}$ be a smooth stack over a field $k$. Then the homomorphisms

$H^{p,q}_{M}({\mathcal{X}},{\mathbb{Z}}/n{\mathbb{Z}})\rightarrow
H^{p}_{\acute{e}t}({\mathcal{X}},\mu_{n}^{\otimes q}),$

are isomorphisms for $p\leq q$ and monomorphisms for $p=q+1$.

###### Proof.

See [CDH, Corollary 5.3]. ∎

###  3.3. Motive with compact support

In this subsection, we will define the notion of a motive with compact support
for smooth algebraic stacks. The definition is very similar to Totaro’s
construction for quotient stacks using approximation by vector bundles [Tot],
except that since we cannot approximate non-quotient stacks by algebraic
spaces, we are forced to work with homotopy limits.

######  Definition 3.10.

Let ${\mathcal{X}}$ be an algebraic stack locally of finite type. Consider a
smooth-Nisnevich covering $p:X\rightarrow{\mathcal{X}}$ and let $n$ denote the
relative dimension of $p:X\rightarrow{\mathcal{X}}$. Let
$p_{\bullet}:X_{\bullet}\rightarrow{\mathcal{X}}$ be the Čech nerve of $p$.
Then we define the compactly supported motive of ${\mathcal{X}}$ as

$M^{c}({\mathcal{X}}):={\rm
holim\,}\\!_{i}M^{c}(X_{i})(-(i+1)n^{2})[-2n^{2}i].$

This definition is a bit weird looking, but it has the following pleasant
feature, which also shows that the definition is independent of choices for
smooth algebraic stacks.

######  Theorem 3.11 (Poincaré Duality).

Let ${\mathcal{X}}$ be a smooth algebraic stack of dimension $d$. Then we have
an isomorphism,

$M^{c}({\mathcal{X}})=M({\mathcal{X}})^{\vee}(d)[2d].$

###### Proof.

The proof is a straightforward manipulation of the definition. Tensoring by
$\otimes{\mathbb{Z}}(-d)[-2d]$ we get,

$\displaystyle M^{c}({\mathcal{X}})\otimes{\mathbb{Z}}(-d)[-2d]$
$\displaystyle=({\rm
holim\,}\\!_{i}M^{c}(X_{i})(-(i+1)n^{2})[-2n^{2}i])\otimes{\mathbb{Z}}(-d)[-2d]$
$\displaystyle={\rm holim\,}\\!_{i}M^{c}(X_{i})(-(i+1)n^{2}-d)[-2n^{2}i-2d]$
$\displaystyle={\rm holim\,}\\!_{i}M(X_{i})^{\vee}.$

where the last step follows from the fact that each $X_{i}$ is smooth.
Further, since $M({\mathcal{X}})\simeq{\rm hocolim\,}\\!_{i}(X_{i})$, taking
dual we get that $M({\mathcal{X}})^{\vee}={\rm
holim\,}\\!_{i}M(X_{i})^{\vee}$. Thus, we have proved that

$M^{c}({\mathcal{X}})(-d)[2d]\simeq M({\mathcal{X}})^{\vee}.$

Tensoring by $\otimes{\mathbb{Z}}(d)[2d]$, gives the required expression. ∎

In the following subsection, we note an application to stable motivic homotopy
theory of stacks. We will not explain any details to prevent drowning the
reader in $\infty$-categories. Our intention is to simply indicate how the
geometric content of Theorem 1.2 can be applied in the world of motivic
homotopy theory. The interested reader may consult [Cho] or [KR] for further
details.

###  3.4. Stable motivic category of a stack

Recently, two different definitions of motivic homotopy category for stacks
have appeared in literature. The limit-extended category
$SH_{\lhd}({\mathcal{X}})$ in [KR] and the category
$SH_{ext}^{\otimes}({\mathcal{X}})$ in [Cho] that is extended from schemes to
stacks having Nisnevich local sections. In [Cho] it is proved that the two
definition are equivalent whenever the stack admits a smooth-Nisnevich
covering. As coverings such always exist for quasi-separated stacks by Theorem
1.2, we have a strengthening of [Cho, Corollary 2.5.4].

######  Corollary 3.12.

Let ${\mathcal{X}}$ be a quasi-separated algebraic stack. Then
$SH_{\lhd}({\mathcal{X}})\simeq SH_{ext}^{\otimes}({\mathcal{X}})$, whenever
they are defined.

###### Proof.

The proof in [Cho, Corollary 2.5.4] works verbatim using a smooth-Nisnevich
covering $X\rightarrow{\mathcal{X}}$. ∎

######  Remark 3.13.

Theorem 1.2 also improves [KR, Corollary 12.28] in a similar fashion.

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# Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render

Ana-Maria Acu  Heiner Gonska  Margareta Heilmann  Dedicated to professor Ioan
Rasa on the occasion of his 70th birthday Lucian Blaga University of Sibiu,
Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No.5-7,
RO-550012 Sibiu, Romania, e-mail<EMAIL_ADDRESS>University of
Duisburg-Essen, Faculty of Mathematics, Bismarckstr. 90, D-47057 Duisburg,
Germany, e-mail<EMAIL_ADDRESS>University of Wuppertal, School of
Mathematics and Natural Sciences, Gaußstraße 20, D-42119 Wuppertal, Germany,
e-mail<EMAIL_ADDRESS>
###### Abstract

The Bernstein-type operator of Aldaz, Kounchev and Render (2009) is discussed.
New direct results in terms of the classical second order modulus as well as
in a modification following Marsden and Schoenberg are given.

###### keywords:

Bernstein-type operator; King operator; second order modulus of continuity;
Marsden-Schoenberg; modulus of order $j$.

###### MSC:

[2020] 41A25, 41A36.

††journal: …

##  1 Introduction

This brief note is dedicated to our dear friend and long term collaborator
Ioan Rasa on the occasion of his 70th birthday. Over the years he did a lot,
most fruitful work on quite a number of Bernstein-type and related operators,
among others. Here we deal with a then surprising, 12-year old object of this
type offering a lot of challenges.

Starting from the classical Bernstein operators $B_{n}$ defined for $f\in
C[0,1]$ as

$B_{n}(f;x)=\displaystyle\sum_{k=0}^{n}f\left(\dfrac{k}{n}\right)p_{n,k}(x),\,\,\text{
where}$ $p_{n,k}(x)={n\choose k}x^{k}(1-x)^{n-k},$

during recent years many, probably much too many, modifications have been
considered. One of the present hypes follows a 2003 paper written by J.P. King
[16] in order to obtain linear positive operators which preserve two functions
different from the classical test functions $e_{0}$ and $e_{1}$. Here we used
the convention $e_{j}(x)=x^{j},\,\,j=0,1,\dots$

King modified the classical Bernstein operators as follows:

$f\to\left(B_{n}f\right)\circ r_{n},\,\,f\in C[0,1],$ (1.1)

where

$r_{n}(x)=\left\\{\begin{array}[]{ll}x^{2},&n=1,\\\
\frac{1}{2(n-1)}\left(-1+\sqrt{1+4n(n-1)x^{2}}\right),&n=2,3,\dots\\\
\end{array}\right.$

These operators preserve the functions $e_{i}(x)=x^{i}$ for $i=0,2$. A slight
extension was considered by Cárdenas et al. in [6] where a sequence of
operators $B_{n,\alpha}$ that preserve $e_{0}$ and $e_{2}+\alpha e_{1}$,
$\alpha\in[0,+\infty)$ was introduced. It is clear that the operator of King
does not produce polynomials in the general case. Therefore, in their 2005
remarks on the article of King, Gonska and Piţul (see [15]) asked the question
if there exist linear and positive polynomial operators that reproduce
$e_{2}$.

Using a continuous strictly increasing function $\tau$ defined on $[0,1]$ with
$\tau(0)=0$ and $\tau(1)=1$, $\tau^{\prime}(x)>0,x\in[0,1]$, Cárdenas-Morales
et al. (see [6]) introduced a modification of the Bernstein operator as
follows:

$B_{n}^{\tau}(f;x):=\displaystyle\sum_{k=0}^{n}{n\choose
k}\tau(x)^{k}(1-\tau(x))^{n-k}(f\circ\tau^{-1})\left(k/n\right),\,\,f\in
C[0,1],\,\,x\in[0,1].$ (1.2)

Note that $\tau^{\prime}(0)>0$ is essential for their results. A predecessor
can be found in Cottin et al. [9], but $\tau(x)=\sqrt{x}$ there.

These operators preserve the functions $e_{0}$ and $\tau$. So, for
$\tau(x)=x^{j}$, one has

$B_{n}^{\tau}(e_{0};x)=e_{0}\textrm{ and }B_{n}^{\tau}(e_{j};x)=e_{j}.$

In this case $B_{n}^{\tau}$ produces polynomials of degree $jn.$ If the
function $\tau$ is not a polynomial, then the modified operator (1.2) is not a
polynomial operator.

Fix $j>1$, $j\in{\mathbb{N}}$. Let $\Pi_{n}$ be the space of polynomials over
$[0,1]$ of degree less than or equal to $n$. For every $n\geq j$, Aldaz,
Kounchev and Render [3] introduced a polynomial Bernstein operator
$B_{n,0,j}:C[0,1]\to\Pi_{n}$ that fixes $e_{0}$ and $e_{j}$, and converges in
the strong operator topology to the identity as $n\to\infty$. The operator is
a linear combination of the classical Bernstein basis
$\\{p_{n,k}\\}_{k=0,\dots,n}$, thus produces polynomials of degree $n$ and is
explicitly given by

$B_{n,0,j}(f;x)=\displaystyle\sum_{k=0}^{n}f\left(t_{n,k}^{j}\right)p_{n,k}(x),$
(1.3)

where

$t_{n,k}^{j}=\left(\dfrac{k(k-1)\dots(k-j+1)}{n(n-1)\dots(n-j+1)}\right)^{1/j}.$

######  Remark 1.1.

  * (i)

For $j=1$ this is the classical Bernstein operator (reproducing $e_{0}$ and
$e_{1}$).

  * (ii)

$B_{n,0,j}$ is linear, positive and of the form

$\displaystyle\sum_{k=0}^{n}l_{n,k}(f)p_{n,k}(x),$ (1.4)

where $l_{n,k}$ are positive linear functionals with $l_{n,k}(e_{0})=1$.
Obviously all the $B_{n,0,j}$s map into $\Pi_{n}[0,1]$ (in contrast to the
$B_{n}^{\tau}$s from above, case $\tau=e_{j}$).

  * (iii)

For $j$ fixed and $n=j$ we have $(t_{n,k}^{j})_{k=0,\dots,n}=(0,\dots,0,1)$
with $n$ zeros preceding the $1$.

For $n=j+1$ one obtains
$(t_{n,k}^{j})_{k=0,\dots,n}=\left(0,\dots,0,n^{\frac{1}{j}},1\right)$, and so
on. That is, only for $n$ big enough all the nodes will be distinct.

  * (IV)

The notation $B_{n,0,j}$ is motivated by the fact that the operator reproduces
$e_{0}$ and $e_{j}$. As shown by Finta in [10], see Theorem 2.1 there, no
sequence $(L_{n})$ of type (1.4) can reproduce $e_{i}$ and $e_{j}$,
$i,j\in\\{1,2,\dots\\},\,\,i<j$. So reproduction of $e_{0}$ is a must!

Moreover, there exist infinitely many sequences of operators $L_{n}$ of type
(1.4) which approximate each continuous function on $[0,1]$, and have the
functions $e_{0}$ and $e_{j}$ as fixed points, where $j\in\\{1,2,\dots\\}$ is
given (see [10, Th. 2.4]).

##  2 On direct estimates for $B_{n,0,j}$

We start with some brief history. For the case $j=2$, $n\geq 2,$ an inequality
involving $\omega_{1}$ (following Shisha $\&$ Mond) was given by Cardenas-
Morales et al. in [8, Proposition 3.1] as follows

$|B_{n,0,2}f(x)-f(x)|\leq\omega_{1}(f,\delta)\left(1+\dfrac{1}{\delta}\sqrt{2x(1-x)\dfrac{1-(1-x)^{n-1}}{n-1}}\right),$

$f\in C[0,1],\,x\in[0,1]\textrm{ and }\delta>0.$

Moreover, in 2014 Finta showed (see [11])

$|B_{n,0,j}(f;x)-f(x)|\leq
2\omega_{1}\left(f;\left(\dfrac{2j(3+4j)}{\sqrt{n}}\right)^{\frac{1}{2j}}\right),$

$f\in C[0,1],\,x\in[0,1]\textrm{ and }n\geq j\geq 2.$

In 2018 Aldaz and Render considered a generalization of the classical
Bernstein operators on the polynomial spaces, which reproduce ${\bf 1}$ and a
polynomial $f_{1}$, strictly increasing on $(0,1)$. Denote by $B_{n}^{f_{1}}$
this Bernstein type operator. For $f\in C[0,1]$, $x\in[0,1]$ (see [4, Remark
following Th. 6.5])

$|B_{n}^{f_{1}}(f;x)-f(x)|\leq(c_{S}+1)\omega_{1}\left(f,n^{-\frac{1}{2}}\right),$

where $c_{S}=\dfrac{4306+837\sqrt{6}}{5832}$, Sikkema’s constant. The drawback
here is that the latter is shown to be valid only for $n$ sufficiently large.

We will improve the estimates known so far in three different ways.

######  Proposition 2.1.

Let $f\in C[0,1]$, $x\in[0,1]$, $n\geq j\geq 1$. Then

$|f(x)-B_{n,0,j}(f;x)|\leq
1\cdot\omega_{2}\left(f;\dfrac{1}{\sqrt{n}}\right)+\omega_{1}\left(f;\dfrac{j-1}{n}\right).$
(2.1)

###### Proof.

We have

$\displaystyle|f(x)-B_{n,0,j}(f;x)|$
$\displaystyle\leq|f(x)-B_{n}(f;x)|+|B_{n}(f;x)-B_{n,0,j}(f;x)|$
$\displaystyle\leq
1\cdot\omega_{2}\left(f;\dfrac{1}{\sqrt{n}}\right)+\omega_{1}\left(f;\dfrac{j-1}{n}\right).$

The first summand is taken from Păltănea [18, 19], while the second was given
by Acu and Raşa [1]. For $j=1$ this reduces to a best possible result for
classical Bernstein operators. ∎

There is a second way to prove a similar inequality. To this end we use
Păltănea’s general theorem for positive linear operators reproducing constant
functions (see [19, Cor.2.2.1, p.31]).

$\displaystyle|B_{n,0,j}(f;x)-f(x)|\leq\dfrac{1}{h}|B_{n,0,j}(e_{1}-x;x)|\cdot\omega_{1}(f;h)$
$\displaystyle+\left(1+\dfrac{1}{2h^{2}}B_{n,0,j}\left((e_{1}-x)^{2};x\right)\right)\cdot\omega_{2}(f;h),\,\,0<h\leq\frac{1}{2}.$

######  Proposition 2.2.

Let $f\in C[0,1]$, $x\in[0,1]$, $1\leq j\leq n$. Then

  * a)

$|B_{n,0,2}(f;x)\\!-\\!f(x)|\\!\leq\\!\dfrac{1}{\sqrt{n-1}}d_{n}(x)\omega_{1}\left(f;\dfrac{1}{\sqrt{n\\!-\\!1}}\right)\\!+\\!\left(1\\!+\\!xd_{n}(x)\right)\omega_{2}\left(f;\dfrac{1}{\sqrt{n\\!-\\!1}}\right)$,
$n\geq 5$, where $d_{n}(x):=(1-x)\left[1-(1-x)^{n-1}\right]$;

  * b)

$|B_{n,0,j}(f;x)-f(x)|\leq(j-1)\dfrac{1}{\sqrt{n}}\omega_{1}\left(f;\dfrac{1}{\sqrt{n}}\right)+\left(\dfrac{1}{8}+j\right)\omega_{2}\left(f;\dfrac{1}{\sqrt{n}}\right),\,n\geq
4.$

###### Proof.

a) In order to use Păltănea’s result we estimate $B_{n,0,2}(e_{1}-x;x)$
and$B_{n,0,2}((e_{1}-x)^{2};x)$.

According to [8, (3.7)] there holds

$\displaystyle|B_{n,0,2}(e_{1}-x;x)|$ $\displaystyle=|B_{n,0,2}(e_{1};x)-x|$
$\displaystyle\leq\dfrac{1-x}{n-1}\left(1-(1-x)^{n-1}\right)=\dfrac{d_{n}(x)}{n-1}.$

Furthermore,

$\displaystyle 0\leq B_{n,0,2}\left((e_{1}-x)^{2};x\right)$
$\displaystyle=2x\left(x-B_{n,0,2}(e_{1};x)\right)$
$\displaystyle\leq\dfrac{2xd_{n}(x)}{n-1}.$

Taking $h=\dfrac{1}{\sqrt{n-1}}$, we get the estimation a).

In order to illustrate the nonsymmetry of the situation, in Fig.1 are the
graphs of the function $d_{n}(x)$ for $n=5$ and $n=10$, $x\in[0,1]$.

Figure 1: $d_{n}(x)$ for $n=5$ and $n=10$, $x\in[0,1]$

b) One has to determine global estimates for $|B_{n,0,j}(e_{1}-x;x)|$ and
$B_{n,0,j}\left((e_{1}-x)^{2};x\right)$, using again $B_{n,0,j}(e_{0};x)=1$.
We have

$\displaystyle
B_{n,0,j}(e_{1}-x;x)=\displaystyle\sum_{k=0}^{n}\left[\left(\dfrac{k}{n}\dots\dfrac{k-j+1}{n-j+1}\right)^{1/j}-\dfrac{k}{n}\right]p_{n,k}(x).$

The result from [1, Section 4.7], namely

$-\dfrac{j-1}{n}\leq\left(\dfrac{k}{n}\dots\dfrac{k-j+1}{n-j+1}\right)^{1/j}-\dfrac{k}{n}\leq
0,$

yields

$|B_{n,0,j}(e_{1}-x;x)|\leq\dfrac{j-1}{n}.$ (2.2)

The second moment can be written as

$\displaystyle 0$ $\displaystyle\leq B_{n,0,j}\left((e_{1}-x)^{2};x\right)$
$\displaystyle=B_{n,0,j}(e_{2};x)-x^{2}-2x\left[B_{n,0,j}(e_{1};x)-x\right]$
$\displaystyle\leq B_{n,0,j}(e_{2};x)-x^{2}+2x\dfrac{j-1}{n},$ (2.3)

where we used (2.2).

We have

$\displaystyle B_{n,0,j}(e_{2};x)-x^{2}$
$\displaystyle=B_{n,0,j}(e_{2};x)-B_{n}(e_{2};x)+B_{n}(e_{2};x)-x^{2}$
$\displaystyle=\displaystyle\sum_{k=0}^{n}\left[\left(\dfrac{k}{n}\cdots\dfrac{k-j+1}{n-j+1}\right)^{\frac{2}{j}}-\left(\dfrac{k}{n}\right)^{2}\right]p_{n,k}(x)+\dfrac{x(1-x)}{n}$
$\displaystyle=\displaystyle\sum_{k=0}^{n}\left[\left(\dfrac{k}{n}\cdots\dfrac{k-j+1}{n-j+1}\right)^{\frac{1}{j}}-\dfrac{k}{n}\right]\left[\left(\dfrac{k}{n}\cdots\dfrac{k-j+1}{n-j+1}\right)^{\frac{1}{j}}+\dfrac{k}{n}\right]p_{n,k}(x)+\dfrac{x(1-x)}{n}$
$\displaystyle\leq\dfrac{x(1-x)}{n}.$

Together with (2) this yields

$0\leq
B_{n,0,j}\left((e_{1}-x)^{2};x\right)\leq\dfrac{x(1-x)}{n}+2x\dfrac{j-1}{n}.$

So we arrive at

$\displaystyle|B_{n,0,j}(f;x)-f(x)|$
$\displaystyle\leq\dfrac{j-1}{n}\dfrac{1}{h}\omega_{1}(f;h)+\left\\{1+\dfrac{1}{2h^{2}}\left[\dfrac{x(1-x)}{n}+2x\dfrac{j-1}{n}\right]\right\\}\omega_{2}(f;h)$
$\displaystyle\leq\dfrac{j-1}{n}\dfrac{1}{h}\omega_{1}(f;h)+\left\\{1+\dfrac{1}{2nh^{2}}\left(\dfrac{1}{4}+2j\right)\right\\}\omega_{2}(f;h).$
(2.4)

Let $h=\dfrac{1}{\sqrt{n}}$, $n\geq 4$. This leads to our proposition.

∎

######  Remark 2.1.

a) The right hand side of inequality (2) is not a pointwise one and so does
not express interpolation at the endpoints. There is room for improvement.

b) If one compares the estimate from Proposition 2.2 b) and the estimate from
Proposition 2.1 it is clear that
$\omega_{2}\left(f;\dfrac{1}{\sqrt{n}}\right)$ is always the worse (i.e.,
dominant) term. In case $f\in C^{2}[0,1]$ both inequalities give
${\mathcal{O}}\left(\dfrac{1}{n}\right)$. This cannot be reached by any of the
previous inequalities in terms of $\omega_{1}$ only .

The inequalities (2.1) and those from Proposition 2.2 do not imply that the
function $e_{j}$ is reproduced by $B_{n,0,j}$ if $j\geq 2$. In order to
achieve this we follow an idea of Marsden and Schoenberg [17, p.82] (see also
[14, Corollary 3.4]). Below we will use a variation of an analogue of their
$\omega_{1}^{*}(f;\delta)$, namely $\omega_{j}^{*}(f;\delta)$ given by

$\omega_{j}^{*}(f;\delta):=\displaystyle\inf_{l}\left\\{\omega_{1}(f-l;\delta)\right\\},\textrm{
where }l(x)=ae_{j},\,\,a\in\mathbb{R}.$

Note that we won’t loose anything in doing so!

######  Proposition 2.3.

For $f\in C[0,1]$, $x\in[0,1]$ and $n\geq j$ we have

$\displaystyle|f(x)-B_{n,0,j}(f;x)|\leq\displaystyle\inf_{l}\left\\{\omega_{2}\left(f-l;\dfrac{1}{\sqrt{n}}\right)+\omega_{1}\left(f-l;\dfrac{j-1}{n}\right)\right\\}.$

###### Proof.

Since $B_{n,0,j}$ reproduces $e_{j}$ we have

$\displaystyle|f(x)-B_{n,0,j}(f;x)|$
$\displaystyle=\left|(f-l)(x)-B_{n,0,j}(f-l;x)\right|$ $\displaystyle\leq
1\cdot\omega_{2}\left(f-l;\dfrac{1}{\sqrt{n}}\right)+\omega_{1}\left(f-l;\dfrac{j-1}{n}\right).$

Since $l=ae_{j}$ was arbitrary, we pass to the $inf$ and get the result.

Moreover, if $f$ is of the form
$f(x)=a_{0}+a_{j}x^{j},\,a_{0},a_{j}\in{\mathbb{R}}$, we arrive at

$|f(x)-B_{n,0,j}(f;x)|=0.$

∎

Analogous statements follow from the inequalities in Proposition 2.2.

######  Remark 2.2.

Except for the papers previously cited we did not find any reference
mentioning or using $\omega_{l}^{*}$ or modifications thereof. The similarity
to K-functionals is obvious.

In [14] it is mentioned that

$\omega_{1}^{*}(f;\delta)\leq c\cdot\omega_{2}(f;\sqrt{\delta}).$

##  3 Iterates of $B_{n,0,j}$

Using the approach of Agratini and Rus (see [2]) the following result for the
sequence of iterates $(B_{n,0,j}^{m})_{m\geq 1}$ is obtained.

######  Theorem 3.1.

For $n\geq j$ fixed and $m\geq 1$, the iterates sequence
$(B_{n,0,j}^{m})_{m\geq 1}$ satifies

$\displaystyle\lim_{m\to\infty}B_{n,0,j}^{m}(f;x)=f(0)+\left[f(1)-f(0)\right]e_{j}(x).$

###### Proof.

Denote

$X_{\alpha,\beta}:=\left\\{f\in
C[0,1]\,\,\left|\,\,f(0)=\alpha,\,f(1)=\beta\right.\right\\},\,\,(\alpha,\beta)\in{\mathbb{R}}\times{\mathbb{\mathbb{R}}}.$

The operators $B_{n,0,j}$ interpolate $0$ and $1$:

$\displaystyle B_{n,0,j}(f;0)=f(t_{n,0}^{j})p_{n,0}(x)=f(0),$ $\displaystyle
B_{n,0,j}(f;1)=f(t_{n,n}^{j})p_{n,0}(x)=f(1).$

Let $f,g\in X_{\alpha,\beta}$. Then, for $x\in[0,1]$,

$\displaystyle|B_{n,0,j}(f;x)-B_{n,0,j}(g;x)|$
$\displaystyle=\left|\displaystyle\sum_{k=1}^{n-1}[f(t_{n,k})-g(t_{n,k})]p_{n,k}(x)\right|$
$\displaystyle\leq\left(1-p_{n,0}(x)-p_{n,n}(x)\right)\cdot\|f-g\|_{\infty}$
$\displaystyle\leq\left(1-\dfrac{1}{2^{n-1}}\right)\|f-g\|_{\infty}.\ $

Therefore, $B_{n,0,j}|_{X_{\alpha,\beta}}:X_{\alpha,\beta}\to
X_{\alpha,\beta}$ is a contraction for every
$(\alpha,\beta)\in{\mathbb{R}}\times{\mathbb{R}}$ and $n\geq j$ fixed.

The function $p_{\alpha,\beta}^{*}:=f(0)+\left(f(1)-f(0)\right)e_{j}$ belongs
to $X_{f(0),f(1)}$ and is a fixed point of $B_{n,0,j}$.

By contradiction we obtain the claim.

∎

The above result was also obtained by Gavrea and Ivan as an example for a
general theorem concerning the limit of the iterates of positive linear
operators (see [13]). No corresponding quantitative version is known to us.

##  4 Voronovskaya result for the Bernstein-type operators $B_{n,0,j}$

The asymptotic formula of the operator $B_{n,0,j}$ was stated as a conjecture
by Cárdenas-Morales et al. [8] and proved by Birou in [5].

######  Theorem 4.1.

(see [8],[5] ) For all $f\in C[0,1]$, $x\in(0,1)$ and $j\geq 1$, whenever
$f^{\prime\prime}(x)$ exists,

$\displaystyle\lim_{n\to\infty}n(B_{n,0,j}(f;x)-f(x))=\dfrac{x(1-x)}{2}f^{\prime\prime}(x)-\dfrac{(j-1)(1-x)}{2}f^{\prime}(x).$
(4.1)

A quantitative pre-Voronovskaya theorem for the Bernstein type operator
$B_{n,0,j}$ was proved by Finta [12] using the first order Ditzian-Totik
modulus of smoothness defined by

$\omega_{\varphi}^{1}(f;\delta)=\displaystyle\sup_{0<h\leq\delta}\sup_{x\pm\frac{1}{2}h\varphi(x)}\left|f\left(x+\frac{1}{2}h\varphi(x)\right)-f\left(x-\frac{1}{2}h\varphi(x)\right)\right|,$

where $\varphi(x)=\sqrt{x(1-x)}$, $x\in[0,1]$. We call this ”pre-Voronovskaya”
since it appears to point into a good direction but the limit is not
explicitly given.

######  Theorem 4.2.

[12] There exists $C>0$ depending only on $j$ such that

$\left|n(B_{n,0,j}(f;x)-f(x))+f^{\prime}(x)nB_{n,0,j}(xe_{0}-e_{1};x)-\frac{1}{2}f^{\prime\prime}(x)nB_{n,0,j}((e_{1}-xe_{0})^{2};x)\right|$
$\leq C\omega_{\varphi}^{1}\left(f^{\prime\prime};\dfrac{1}{\sqrt{n}}\right),$

for all $x\in[0,1]$, $f\in C^{2}[0,1]$ and $n\geq j\geq 1$.

No quantitative version of a Voronovskaya-type theorem for the operators of
Aldaz et al. is known to us.

## References

  * [1] A.M. Acu, I. Raşa, New estimates for the differences of positive linear operators, Numerical Algorithms, 73(3), 775-789, 2016.
  * [2] O. Agratini, I.A. Rus, Iterates of a class of discrete linear operator via contraction principle, Comment. Math. Univ. Carol. 44(3), 2003, 555-563.
  * [3] J.M. Aldaz, O. Kounchev, H. Render, Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 2009, 114(1), 1–25.
  * [4] J.M. Aldaz, H. Render, Generalized Bernstein operators on the classical polynomial spaces. Mediterr. J. Math. 15, 222 (2018).
  * [5] M. Birou, A proof of a conjecture about the asymptotic formula of a Bernstein type operator, Results Math. 72 (2017), 1129–1138.
  * [6] D. Cárdenas-Morales, P. Garrancho, F.J. Muñoz-Delgado, Shape preserving approximation by Bernstein-type operators which fix polynomials, Appl. Math. Comput. 182 (2006), 1615–1622.
  * [7] D. Cárdenas-Morales, P. Garrancho, I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62 (2011), 158–163.
  * [8] D. Cárdenas-Morales, P. Garrancho, I. Raşa, Asymptotic formulae via a Korovkin-type result, Abstract and Applied Analysis Volume 2012, Article ID 217464, 12 pages.
  * [9] C. Cottin, I. Gavrea, H. Gonska, D. Kacsó. Ding-xuan Zhou, Global smoothness preservation and the variation-diminishing property. J. Inequal. Appl. 4 (1999), no. 2, 91–114.
  * [10] Z. Finta, Bernstein type operators having $1$ and $x^{j}$ as fixed points, Cent. Eur. J. Math. 11 (2013), 2257–2261
  * [11] Z. Finta, Note on a Korovkin-type theorem, J. Math. Anal. Appl. 415(2014), 750–759.
  * [12] Z. Finta, A quantitative variant of Voronovskaja’s theorem for King-type operators, Constructive Mathematical Analysis, 2 (2019), No. 3, 124-129.
  * [13] I. Gavrea, M. Ivan, Asymptotic behaviour of the iterates of positive linear operators, Abstract and Applied Analysis Volume 2011, Article ID 670509, 11 pages.
  * [14] H. Gonska, On approximation by linear operators: improved estimates, Anal. Numér. Théor. Approx. 14 (1985), 7-32.
  * [15] H. Gonska, P. Piţul, Remarks on an article of J.P. King, Comment. Math. Univ. Carol. 46 (2005), 645–652.
  * [16] J.P. King, Positive linear operators which preserve $x^{2}$, Acta Math. Hungar. 99 (3) (2003), 203–208.
  * [17] M. Marsden and I.J. Schoenberg, On variation diminishing spline approximation methods, Mathematica 8 (31), no.1, 1966, 61-82.
  * [18] R. Păltănea, Optimal constant in approximation by Bernstein operators. J. Comput. Anal. Appl. 5 (2003), 195–235.
  * [19] R. Păltănea, Approximation Theory using Positive Linear Operators, Birkhäuser, Boston, 2004.


# Spatiotemporal effects in heralded state preparation

Filippus S. Roux  University of Kwazulu-Natal, Private Bag X54001, Durban
4000, South Africa<EMAIL_ADDRESS>
###### Abstract

Heralding, which is often used for preparing quantum optical states, is
studied to determine the effects of the spatiotemporal properties of the
process. Incorporating all the spatiotemporal degrees of freedom, we follow a
Wigner functional approach to consider cases where these states are prepared
to have Wigner functionals with negative regions, being suitable resources for
quantum information technologies. General expressions are derived for single-
photon-subtracted and single-photon-added states. As examples, we consider the
photon-subtracted squeezed vacuum state, the photon-added coherent state, and
the photon-added thermal state. The Wigner functional approach reveals the
importance of the spatiotemporal transformations imposed by the experimental
conditions.

##  I Introduction

Quantum information technology involves exotic measurements performed on the
outputs from quantum processing systems, applied to exotic quantum states
serving as quantum resources. In photonic quantum information systems, such
quantum resources often require sophisticated preparation techniques, such as
heralded quantum state preparation Dakna et al. (1998); Lvovsky et al. (2020).
The quality of such quantum information systems depends on the quality of the
quantum states used as resources.

A heralded quantum optical state is produced conditioned on a suitable
detection made on a portion of an input state. The portion is separated off
from the an input state by some means and then subjected to a heralding
measurement. A successful detection registered by the heralding measurement
system signals that the remainder of the state has attained the required form
or property. Such heralding processes have been used to prepare Fock states
Lvovsky et al. (2001); Zavatta et al. (2004a); Ourjoumtsev et al. (2006a),
photon subtracted states Biswas and Agarwal (2007); Walschaers et al. (2020),
and photon added states Agarwal and Tara (1991); Zavatta et al. (2007). Such
states are considered to be suitable resources for quantum information
process. Their quantum nature is associated with the fact that their Wigner
functions have negative regions. Photon subtraction is also used to produce
states with Wigner functions that resemble those of Schrödinger cat states. In
the limit of a small number of photons, a photon subtracted state is
equivalent to a Schrödinger cat state Dakna et al. (1997); Ourjoumtsev et al.
(2006b); Ourjoumtsev et al. (2007).

The basic mechanism of heralded state preparation involves only the particle-
number degrees of freedom Roux (2021a). Although this mechanism provides some
insight into the quantum properties of the states that are produced, the other
degrees of freedom also play significant roles in such experiments Lvovsky and
Raymer (2009). The effects of various experimental parameters can only be
revealed through an analysis that incorporates the spatiotemporal degrees of
freedom.

Here, the effects of the spatiotemporal degrees of freedom in heralded state
preparation is investigated by analyzing the process with a Wigner functional
formalism, which combines the spatiotemporal degrees of freedom with the
particle-number degrees of freedom Mrowczynski and Mueller (1994); Roux
(2020a). For this purpose, we focus on the applications of heralded state
preparation for the experimental implementation of photon subtraction and
photon addition. As an example of the former, we consider the heralded
preparation of a photon-subtracted squeezed vacuum state Biswas and Agarwal
(2007). Examples of the latter that are considered here are photon-added
coherent states Zavatta et al. (2004b) and photon-added thermal states Zavatta
et al. (2007). Inevitably, the incorporation of the spatiotemporal degrees of
freedom in the analysis implies a significant increase in the complexity of
such an analysis. Fortunately, the complexity can be significantly alleviated
with the aid of generating functions Roux (2021a). In addition, we introduce
some further simplifications that are relevant under general experimental
conditions.

##  II Heralding

First, we provide a general model for the processes whereby states are
prepared with the aid of heralding. The idea is that the required state is
only produced when a specific measurement is performed successfully. In that
case, the measurement heralds the existence of the required state. The latter
is then processed and subsequently measured. The final measurements are done
in coincidence with the heralding measurement, ensuring the existance of the
heralded state during the final measurements. The requirement for coincidence
implies that the heralding measurement and the final measurements combine into
a measurement protocol performed on the initial state.

###  II.1 Generic process

In terms of quantum mechanics, we can express the state that is produced by
the heralding process as

$\hat{\rho}_{\text{her}}=\text{tr}_{B}\\{\hat{P}_{B}\hat{U}_{\text{bs}}\left(\hat{\rho}_{\text{in}}\otimes\hat{\rho}_{\text{vac}}\right)\hat{U}_{\text{bs}}^{{\dagger}}\\},$
(1)

where $\hat{P}_{B}$ is the projection operator for the heralding measurement,
$\hat{U}_{\text{bs}}$ is a unitary operator for the (homogeneous or
inhomogeneous) beamsplitter, and
$\hat{\rho}_{\text{in}}\otimes\hat{\rho}_{\text{vac}}$ is the tensor product
of the initial state and the vacuum state, respectively entering the two input
ports of the beamsplitter. A partial trace is evaluated over the part of the
state that is received from the output port of the beamsplitter on which the
projection operator is applied. The difference between a homogeneous
beamsplitter and an inhomogeneous beamsplitter, is that the former applies the
splitting ratio indiscriminately on all photons, and the latter uses the
spatiotemporal properties of the photons to determine the splitting ratio.
Inhomogeneous beamsplitters come in handy when modeling loss processes
introduced by structures such as apertures.

To analize the effects of the physical system, incorporating any physical
restrictions associated with the spatiotemporal degrees of freedom of the
system, we use Wigner functionals. The effect of the beamsplitter on such
Wigner functionals is represented by transformations of the field variables on
which they depend.

The heralding measurement is modeled with generalized projection operations
expressed in terms of a generating function for the Wigner functionals of such
projection operators. It is given by

$\mathcal{P}(J)=\mathcal{N}^{\text{tr}\\{D\\}}\exp\left(-2\mathcal{J}\alpha^{*}\diamond
D\diamond\alpha\right),$ (2)

where $\alpha$ and $\alpha^{*}$ represent the field variables. We define

$\mathcal{N}=\frac{2}{1+J}~{}~{}~{}\text{and}~{}~{}~{}\mathcal{J}=\frac{1-J}{1+J},$
(3)

in terms of $J$, the auxilary parameter used to generate the individual Wigner
functionals of the projection operators, and use $\diamond$-contractions to
represent

$\alpha^{*}\diamond
D\diamond\alpha=\int\alpha^{*}(\mathbf{k}_{1})D(\mathbf{k}_{1},\mathbf{k}_{2})\alpha(\mathbf{k}_{2})\frac{d^{3}k_{1}}{(2\pi)^{3}}\
\frac{d^{3}k_{2}}{(2\pi)^{3}},$ (4)

with $D(\mathbf{k}_{1},\mathbf{k}_{2})$ denoting a detector kernel.

Heralding is a post-selection process, which is not trace-preserving. As a
result, the raw heralded state is not normalized and thus needs to be
normalized by dividing it by its trace.

Figure 1: Diagrammatic representation of the experimental setup for heralded
photon subtraction.

##  III Photon subtracted states

One way to produce a Wigner functional with a negative region is to subtract a
photon from a squeezed state; the latter having a Wigner functional that is
positive everywhere in the functional phase space. Formally, a photon
subtraction is done by applying an annihilation operator to an initial state.
In a practical experiment, photon subtraction is performed with the aid of
heralding. A small portion of the state is separated off by an unbalanced
homogeneous beamsplitter, having a small reflectivity, and sent to a photon-
number resolving detector, as shown in Fig. 1. The detection of $n$ photons by
this detector then heralds the formation of an $n$-photon subtracted state.

The unitary transformation imposed by the homogeneous beamsplitter on the
input state is implement by the transformations of the field variables of the
Wigner functional on which it operates. For the unbalanced beamsplitters with
a small reflectivity, we represent the amplitude reflectivity by $\zeta$; the
intensity reflectivity is $\zeta^{2}$. The transformation of the field
variables is given by

$\displaystyle\begin{split}\alpha\rightarrow&\sqrt{1-\zeta^{2}}\alpha+i\zeta\beta,\\\
\beta\rightarrow&\sqrt{1-\zeta^{2}}\beta+i\zeta\alpha.\end{split}$ (5)

where $\alpha$ and $\beta$ represent the field variables of the respective
input ports or the respective output ports. This transformation is applied to
the product of the Wigner functionals of the input state and a vacuum state.
The Wigner functional of the vacuum state is

$\displaystyle W_{\text{vac}}[\beta]=$
$\displaystyle\mathcal{N}_{0}\exp\left(-2\beta^{*}\diamond\beta\right),$ (6)

where the field variables are $\beta$ and $\beta^{*}$, and $\mathcal{N}_{0}$
is a (divergent) normalization constant. After applying the beamsplitter
transformation, the combined state becomes

$\displaystyle W_{\text{bs}}[\alpha,\beta]=$
$\displaystyle\mathcal{N}_{0}\exp\left[-2\left(\sqrt{1-\zeta^{2}}\beta^{*}-i\zeta\alpha^{*}\right)\right.$
$\displaystyle\left.\diamond\left(\sqrt{1-\zeta^{2}}\beta+i\zeta\alpha\right)\right]$
$\displaystyle\times
W_{\text{in}}\left[\sqrt{1-\zeta^{2}}\alpha+i\zeta\beta\right].$ (7)

Assuming a small reflectivity, we expand the Wigner functional of the state
after the beamsplitter as a power series in $\zeta$. Then we use Eq. (2) as a
functional of $\beta$ to apply measurements on the $\beta$-degrees of freedom,
tracing over $\beta$. It produces a generating function for the Wigner
functionals of the heralded state associated with the detection of a certain
number of photons. To detect $n$ photons, we need to expand the transformed
Wigner functional to $\zeta^{2n}$. Here, we consider single-photon subtraction
only, for which we need to make the expansion up the $\zeta^{2}$. The
procedure can be readily generalized for larger numbers of photon
subtractions.

All the uneven order terms in the expansion are removed by the trace over
$\beta$. The zeroth order term represents the vacuum, which does not
contributed to the measurement. So, for single-photon subtraction, we only
need to compute the second order term in the expansion. The two derivatives
with respect to $\zeta$ produces functional derivatives of the Wigner
functional $W_{\text{in}}$. The result for the second order term is

$\displaystyle\tfrac{1}{2}\zeta^{2}\left.\partial_{\zeta}^{2}W_{\text{bs}}[\alpha,\beta]\right|_{\zeta=0}=$
$\displaystyle\tfrac{1}{2}\zeta^{2}\left[2\beta^{*}\diamond\left(\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha}+2\alpha\frac{\delta
W_{\text{in}}}{\delta\alpha}+2\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\alpha^{*}+2W_{\text{in}}\mathbf{1}+4\alpha
W_{\text{in}}\alpha^{*}\right)\diamond\beta-\beta^{*}\diamond\left(\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha^{*}}\right.\right.$
$\displaystyle\left.+4\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\alpha+4\alpha
W_{\text{in}}\alpha\right)\diamond\beta^{*}-\beta\diamond\left(\frac{\delta^{2}W_{\text{in}}}{\delta\alpha\delta\alpha}+4\alpha^{*}\frac{\delta
W_{\text{in}}}{\delta\alpha}+4\alpha^{*}W_{\text{in}}\alpha^{*}\right)\diamond\beta$
$\displaystyle\left.-\frac{\delta
W_{\text{in}}}{\delta\alpha}\diamond\alpha-\alpha^{*}\diamond\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}-4\alpha^{*}\diamond\alpha
W_{\text{in}}\right]\mathcal{N}_{0}\exp\left(-2\beta^{*}\diamond\beta\right),$
(8)

where $\mathbf{1}$ is the identity, defined such that
$\mathbf{1}\diamond\alpha=\alpha$.

For the heralding measurements on the $\beta$-degrees of freedom, we multiply
Eq. (8) by Eq. (2), expressed as a functional of $\beta$, and perform a
functional integration over $\beta$. The latter produces a superposition of
moment integrals of the form

$M_{m,n}=\int\beta^{*m}\beta^{n}\mathcal{V}[\beta]\ {\cal D}^{\circ}[\beta],$
(9)

where

$\mathcal{V}[\beta]=\mathcal{N}^{\text{tr}\\{D\\}}\mathcal{N}_{0}\exp\left(-2\beta^{*}\diamond\beta-2\mathcal{J}\beta^{*}\diamond
D\diamond\beta\right).$ (10)

Such moment integrals can be computed with the aid of a generating function,
given by

$\displaystyle\mathcal{M}[\mu^{*},\nu]=$
$\displaystyle\mathcal{N}^{\text{tr}\\{D\\}}\mathcal{N}_{0}\int\exp\left(-2\beta^{*}\diamond\beta\right.$
$\displaystyle\left.-2\mathcal{J}\alpha^{*}\diamond
D\diamond\alpha+\beta^{*}\diamond\nu+\mu^{*}\diamond\beta\right)\ {\cal
D}^{\circ}[\beta]$ $\displaystyle=$
$\displaystyle\exp\left[\tfrac{1}{2}\mu^{*}\diamond\nu-\tfrac{1}{4}(1-J)\mu^{*}\diamond
D\diamond\nu\right].$ (11)

The moments are produced by functional derivatives with respect to the
auxiliary field variables $\mu^{*}$ and $\nu$ producing factors of $\beta$ and
$\beta^{*}$ respectively, after which the auxiliary field variables are set
equal to zero. For example

$\displaystyle M_{1,1}=$
$\displaystyle\int\beta^{*}(\mathbf{k})\beta(\mathbf{k}^{\prime})\mathcal{V}[\beta]\
{\cal D}^{\circ}[\beta]$ $\displaystyle=$
$\displaystyle\left.\frac{\delta^{2}\mathcal{M}[\mu^{*},\nu]}{\delta\nu(\mathbf{k})\delta\mu^{*}(\mathbf{k}^{\prime})}\right|_{\mu^{*}=\nu=0}$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}\mathbf{1}(\mathbf{k}^{\prime},\mathbf{k})-\tfrac{1}{4}(1-J)D(\mathbf{k}^{\prime},\mathbf{k}).$
(12)

Similarly, we obtain $M_{2,0}=M_{0,2}=0$ and $M_{0,0}=1$. With these moments,
the generating function of the heralded single-photon subtracted state becomes

$\displaystyle\mathcal{W}_{1\text{ps}}[\alpha]=$
$\displaystyle\tfrac{1}{2}\text{tr}\left\\{\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha}\right\\}+\tfrac{1}{2}\frac{\delta
W_{\text{in}}}{\delta\alpha}\diamond\alpha+\tfrac{1}{2}\alpha^{*}\diamond\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}$ $\displaystyle+\Omega
W_{\text{in}}-\tfrac{1}{4}(1-J)\left(\text{tr}\left\\{D\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha}\right\\}\right.$
$\displaystyle+2\text{tr}\\{D\\}W_{\text{in}}+4\alpha^{*}\diamond
D\diamond\alpha W_{\text{in}}$ $\displaystyle\left.+2\frac{\delta
W_{\text{in}}}{\delta\alpha}\diamond D\diamond\alpha+2\alpha^{*}\diamond
D\diamond\frac{\delta W_{\text{in}}}{\delta\alpha^{*}}\right),$ (13)

where we discarded the factor of $\zeta^{2}$, and
$\Omega=\text{tr}\\{\mathbf{1}\\}$. Note that the resulting generating
function is linear in the auxiliary parameter $J$. It can therefore only
produce the single-photon subtracted state. The result obtained from the
generating function still needs to be normalized. Assuming that the detector
can be modeled by a single-mode kernel
$D(\mathbf{k}_{1},\mathbf{k}_{2})=M(\mathbf{k}_{1})M^{*}(\mathbf{k}_{2})$,
which implies that $\text{tr}\\{D\\}=1$, we obtain the single-photon
subtracted state as

$\displaystyle W_{1\text{ps}}[\alpha]=$ $\displaystyle
N\left\\{\left.\partial_{J}\mathcal{W}_{1\text{ps}}\right|_{J=0}\right\\}$
$\displaystyle=$
$\displaystyle\mathcal{N}\left(M^{*}\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha}\diamond
M+2W_{\text{in}}\right.$ $\displaystyle+4\alpha^{*}\diamond
MM^{*}\diamond\alpha W_{\text{in}}+2\frac{\delta
W_{\text{in}}}{\delta\alpha}\diamond MM^{*}\diamond\alpha$
$\displaystyle\left.+2\alpha^{*}\diamond MM^{*}\diamond\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\right),$ (14)

for an arbitrary initial state $W_{\text{in}}$, where $N\\{\cdot\\}$ is a
normalization process, and $\mathcal{N}$ is the normalization constant.

The normalization can be computed with the generating function that is
obtained by integrating Eq. (13) over $\alpha$. For this purpose, we note
that, since the inital Wigner functional is normalized,

$\int W_{\text{in}}\ {\cal D}^{\circ}[\alpha]=1.$ (15)

If the integrant is a total derivative, the result is 0 because the Wigner
functional tend to 0 at infinity. Hence,

$\int\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha}\ {\cal
D}^{\circ}[\alpha]=0.$ (16)

With the aid of partial functional integration, we get

$\displaystyle\int\alpha(\mathbf{k})\frac{\delta
W_{\text{in}}}{\delta\alpha(\mathbf{k}^{\prime})}\ {\cal D}^{\circ}[\alpha]=$
$\displaystyle\int\alpha^{*}(\mathbf{k})\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}(\mathbf{k}^{\prime})}\ {\cal
D}^{\circ}[\alpha]$ $\displaystyle=$
$\displaystyle-\delta(\mathbf{k}-\mathbf{k}^{\prime}).$ (17)

Using these identities, we obtain

$\displaystyle\mathcal{N}^{-1}=$
$\displaystyle\left.\partial_{J}\int\mathcal{W}_{1\text{ps}}\ {\cal
D}^{\circ}[\alpha]\right|_{J=0}$ $\displaystyle=$
$\displaystyle\int\alpha\diamond D\diamond\alpha^{*}W_{\text{in}}\ {\cal
D}^{\circ}[\alpha]-\tfrac{1}{2}\text{tr}\\{D\\}.$ (18)

We are left with one integral to evaluate, for which we need to know the
initial Wigner functional.

Considering different input states, we note that coherent states remain
unaffected by photon subtraction processes. Thermal states are affected, but
their Wigner functionals remain positive everywhere. To have negative regions,
we need squeezed states as input states.

###  III.1 Photon subtracted squeezed vacuum state

The Wigner functional of a squeezed vacuum state is

$\displaystyle W_{\text{sv}}=$
$\displaystyle\mathcal{N}_{0}\exp\left(-2\alpha^{*}\diamond
A\diamond\alpha-\alpha^{*}\diamond B\diamond\alpha^{*}\right.$
$\displaystyle\left.-\alpha\diamond B^{*}\diamond\alpha\right),$ (19)

where $A$ and $B$ are a Hermitian kernel and a symmetric kernel, respectively.
The purity of the state ensures that

$A-B\diamond A^{*-1}\diamond B^{*}=A^{-1}.$ (20)

Substituting Eq. (19) into Eq. (14), we obtain a Wigner functional for a
single-photon subtracted squeezed vacuum state in the form of a polynomial
Gaussian state that becomes negative at the origin.

To compute the normalization factor, we produce a generating functional for
the moments in the same way we did in Eq. (11). For the squeezed vacuum state,
this generating functional is given by

$\displaystyle\mathcal{W}_{\mathcal{N}}[\mu^{*},\nu]=$
$\displaystyle\int\mathcal{N}_{0}\exp\left(-2\alpha^{*}\diamond
A\diamond\alpha-\alpha^{*}\diamond B\diamond\alpha^{*}\right.$
$\displaystyle\left.-\alpha\diamond
B^{*}\diamond\alpha+\alpha^{*}\diamond\nu+\mu^{*}\diamond\alpha\right)\ {\cal
D}^{\circ}[\alpha]$ $\displaystyle=$
$\displaystyle\exp\left(\tfrac{1}{2}\mu^{*}\diamond
A\diamond\nu-\tfrac{1}{4}\nu\diamond A^{*}\diamond B^{*}\diamond
A^{-1}\diamond\nu\right.$ $\displaystyle\left.-\tfrac{1}{4}\mu^{*}\diamond
A^{-1}\diamond B\diamond A^{*}\diamond\mu^{*}\right).$ (21)

The inverse normalization constant thus becomes

$\displaystyle\mathcal{N}^{-1}=$
$\displaystyle\left.\text{tr}\left\\{D\diamond\frac{\delta^{2}\mathcal{W}_{\mathcal{N}}}{\delta\nu\delta\mu^{*}}\right\\}\right|_{\mu^{*}=\nu=0}-\tfrac{1}{2}\text{tr}\\{D\\}$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}\text{tr}\left\\{D\diamond(A-\mathbf{1})\right\\}=\tfrac{1}{2}M^{*}\diamond
E\diamond M,$ (22)

for a single-mode detector, where $E=A-\mathbf{1}$.

The normalized heralded state is now given by

$\displaystyle W_{1\text{ps}}^{(\text{sv})}=$ $\displaystyle
2\left(\frac{\left|\alpha^{*}\diamond
M_{E}+M_{B}^{*}\diamond\alpha\right|^{2}}{\eta}-1\right)W_{\text{sq}},$ (23)

where we defined

$\displaystyle\begin{split}M_{E}=&E\diamond M=(A-\mathbf{1})\diamond M,\\\
M_{B}^{*}=&B^{*}\diamond M,\\\ \eta=&M^{*}\diamond E\diamond M.\end{split}$
(24)

At the origin ($\alpha=0$), we get $W_{1\text{ps}}^{(\text{sv})}[0]=-2$,
showing that there is a region where the Wigner functional is negative. For
larger numbers of subtracted photons, the number of regions where the Wigner
functional is negative increases, but for even numbers of subtracted photons
the origin does not lie in a negative region.

Although the functional phase space on which the Wigner functional is defined
is infinite dimensional, the $\diamond$-contractions in the polynomial
prefactor in Eq. (23) reduces it to a four-dimensional subspace of the
functional phase space defined by the two transformed modes $M_{E}$ and
$M_{B}$. When the Wigner functional is measured with a homodyne tomography
process Leonhardt and Paul (1993); Lvovsky and Raymer (2009); Roux (2022), the
best results are obtained when the mode of the local oscillator lies within
this subspace. Knowledge of the detector mode $M$ and the expressions of the
$E$ and $B$ kernels Roux (2020b) make it possible to compute these transformed
modes.

##  IV Photon added states

The addition of photons to states is more powerful in producing Wigner
functionals with negative regions than photon subtraction. Photon addition is
formally represented by the application of a creation operator to an initial
state. Experimentally, it is done by using the state as the seed in stimulated
parametric down-conversion Roux (2021b) and then heralding the photon-added
state (in the signal beam) by detecting a certain number of photons in the
difference-frequency (idler) beam, as shown in Fig. 2.

In terms of quantum mechanics, the process of heralded photon addition is
represented by

$\hat{\rho}_{\text{hpa}}=\text{tr}_{B}\left\\{\hat{P}_{B}\hat{U}_{\text{ibs}}\left[\left(\hat{U}_{\text{sq}}\hat{\rho}_{\text{in}}\hat{U}_{\text{sq}}^{{\dagger}}\right)\otimes\hat{\rho}_{\text{vac}}\right]\hat{U}_{\text{ibs}}^{{\dagger}}\right\\}.$
(25)

The input state is squeezed by stimulated parametric down-conversion and,
together with a vacuum state, sent through an inhomogeneous beamsplitter to
separate the squeezed state into two beams, one of which is subjected to a
measurement by a number-resolving detector, represented by the projection
operator $\hat{P}_{B}$.

Figure 2: Diagrammatic representation of the experimental setup for heralded
photon addition with stimulated parametric down-conversion (PDC).

In terms of Wigner functionals, the stimulated parametric down-conversion
process, together with the inhomogenous beamsplitter, can be performed with a
twin-beam Bogoliubov transformation, which separates the signal and idler
beams into two different phase spaces. To suppress the production of multiple
photon pairs, the squeezing (i.e., the efficiency of the stimulated parametric
down-conversion) is designed to be weak. Therefore, the squeezing parameter
can be used as an expansion parameter, similar to the way we used the
reflectivity as an expansion parameter in Section III. The Bogoliubov kernels
Roux (2021b) can then be expressed as

$U\rightarrow\mathbf{1}+\xi^{2}F~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}V\rightarrow\xi
V,$ (26)

where we display the squeezing parameter $\xi$ explicitly, and $F$ represents
the subleading part of $U$. The twin-beam Bogoliubov transformation is thus
given by

$\displaystyle\begin{split}\alpha\rightarrow&(\mathbf{1}+\xi^{2}F)\diamond\alpha+\xi
V\diamond\beta^{*},\\\ \beta\rightarrow&(\mathbf{1}+\xi^{2}F)\diamond\beta+\xi
V\diamond\alpha^{*}.\end{split}$ (27)

After applying this transformation to the combination of the Wigner
functionals for the seed state and the vacuum state, we have

$\displaystyle W_{\text{ibs}}[\alpha,\beta]=$ $\displaystyle
W_{\text{in}}\left[\alpha+\xi^{2}F\diamond\alpha+\xi
V\diamond\beta^{*}\right]$
$\displaystyle\times\mathcal{N}_{0}\exp\left[-2\left(\beta^{*}+\xi^{2}\beta^{*}\diamond
F+\xi\alpha\diamond V^{*}\right)\right.$
$\displaystyle\left.\diamond\left(\beta+\xi^{2}F\diamond\beta+\xi
V\diamond\alpha^{*}\right)\right].$ (28)

Similar to what we had with photon subtraction, we only need to compute the
second order term for single-photon addition. It reads

$\displaystyle\tfrac{1}{2}\xi^{2}\left.\partial_{\xi}^{2}W_{\text{ibs}}[\alpha,\beta]\right|_{\xi=0}=$
$\displaystyle\xi^{2}\left[\beta^{*}\diamond
V\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha\delta\alpha^{*}}\diamond
V^{*}\diamond\beta+\tfrac{1}{2}\beta^{*}\diamond
V\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha\delta\alpha}\diamond
V\diamond\beta^{*}+\tfrac{1}{2}\beta\diamond
V^{*}\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha^{*}}\diamond
V^{*}\diamond\beta\right.$ $\displaystyle-2\left(\alpha\diamond
V^{*}\diamond\beta+\beta^{*}\diamond
V\diamond\alpha^{*}\right)\left(\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\diamond V^{*}\diamond\beta+\beta^{*}\diamond
V\diamond\frac{\delta W_{\text{in}}}{\delta\alpha}\right)$
$\displaystyle-2\left(2\beta^{*}\diamond F\diamond\beta+\alpha\diamond
V^{*}\diamond V\diamond\alpha^{*}\right)W_{\text{in}}+2\left(\alpha\diamond
V^{*}\diamond\beta+\beta^{*}\diamond
V\diamond\alpha^{*}\right)^{2}W_{\text{in}}$ $\displaystyle\left.+\frac{\delta
W_{\text{in}}}{\delta\alpha}\diamond F\diamond\alpha+\alpha^{*}\diamond
F\diamond\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\right]\mathcal{N}_{0}\exp\left(-2\beta^{*}\diamond\beta\right).$
(29)

We again use Eq. (2) for the photon measurements. As in Section III, the
functional integration over $\beta$ produces a superposition of moment
integrals, which can be computed with the iad of Eq. (11). It leads to

$\displaystyle\mathcal{W}_{1\text{pa}}=$
$\displaystyle\text{tr}\left\\{F\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha^{*}\delta\alpha}\right\\}-\frac{\delta
W_{\text{in}}}{\delta\alpha}\diamond F\diamond\alpha-\alpha^{*}\diamond
F\diamond\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}-2\text{tr}\left\\{F\right\\}W_{\text{in}}$
$\displaystyle-\tfrac{1}{4}(1-J)\left(\text{tr}\left\\{V^{*}\diamond D\diamond
V\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha\delta\alpha^{*}}\right\\}-4\text{tr}\left\\{D\diamond
F\right\\}W_{\text{in}}+4\alpha\diamond V^{*}\diamond D\diamond
V\diamond\alpha^{*}W_{\text{in}}\right.$ $\displaystyle\left.-2\alpha\diamond
V^{*}\diamond D\diamond V\diamond\frac{\delta
W_{\text{in}}}{\delta\alpha}-2\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\diamond V^{*}\diamond D\diamond
V\diamond\alpha^{*}\right).$ (30)

Here we dropped the factor of $\xi^{2}$ and we replaced $V\diamond
V^{*}\rightarrow 2F$. This identity can be derived under weak squeezing
conditions from Eq. (20) and the relationships between the Bogoliubov kernels
and the squeezed state kernels given by

$A=U\diamond U+V\diamond V^{*}~{}~{}~{}\text{and}~{}~{}~{}B=U\diamond
V+V\diamond U^{*}.$ (31)

Being linear in the generating parameter $J$, the generating function only
allows single photon addition. Higher order expansions are required for the
addition of more photons. The expression for the single-photon added state
with a single-mode detector kernel reads

$\displaystyle W_{1\text{pa}}=$
$\displaystyle\mathcal{N}\left[\tfrac{1}{4}M^{*}\diamond
V\diamond\frac{\delta^{2}W_{\text{in}}}{\delta\alpha\delta\alpha^{*}}\diamond
V^{*}\diamond M\right.$ $\displaystyle+\left(\alpha\diamond V^{*}\diamond
MM^{*}\diamond V\diamond\alpha^{*}-M^{*}\diamond F\diamond
M\right)W_{\text{in}}$ $\displaystyle-\tfrac{1}{2}\alpha\diamond V^{*}\diamond
MM^{*}\diamond V\diamond\frac{\delta W_{\text{in}}}{\delta\alpha}$
$\displaystyle\left.-\tfrac{1}{2}\frac{\delta
W_{\text{in}}}{\delta\alpha^{*}}\diamond V^{*}\diamond MM^{*}\diamond
V\diamond\alpha^{*}\right],$ (32)

where $\mathcal{N}$ is the normalization constant, which is required because
heralded photon addition is not a trace-preserving process. With the aid of
the same calculation used in Section III to compute the normalization
constant, we obtain

$\displaystyle\mathcal{N}^{-1}=$ $\displaystyle\int\alpha\diamond
V^{*}\diamond D\diamond V\diamond\alpha^{*}W_{\text{in}}\ {\cal
D}^{\circ}[\alpha]$ $\displaystyle+\text{tr}\left\\{D\diamond F\right\\}.$
(33)

It again requires knowledge of the initial Wigner functional to evaluate the
functional integral.

###  IV.1 Single-photon added coherent state

Contrary to the situation with photon subtraction, photon addition applied to
most states produces Wigner functionals with negative regions. We substitute
the coherent state Wigner functional

$W_{\text{coh}}[\alpha]=\mathcal{N}_{0}\det\\{T\\}\exp\left[-2(\alpha^{*}-\xi^{*})\diamond(\alpha-\xi)\right],$
(34)

with parameter function $\xi$, into Eq. (32). Then we compute the
normalization constant, with a generating functional for the moments, produced
in the same way we did in Eq. (11). For the coherent state, this generating
functional is given by

$\displaystyle\mathcal{W}_{\mathcal{N}}[\mu^{*},\nu]=$
$\displaystyle\int\mathcal{N}_{0}\exp\left(-2(\alpha^{*}-\xi^{*})\diamond(\alpha-\xi)\right.$
$\displaystyle\left.+\alpha^{*}\diamond\nu+\mu^{*}\diamond\alpha\right)\ {\cal
D}^{\circ}[\alpha]$ $\displaystyle=$
$\displaystyle\exp\left(\xi^{*}\diamond\nu+\mu^{*}\diamond\xi+\tfrac{1}{2}\mu^{*}\diamond\nu\right).$
(35)

leading to

$\left.\frac{\delta^{2}\mathcal{W}_{\mathcal{N}}}{\delta\nu(\mathbf{k})\delta\mu^{*}(\mathbf{k}^{\prime})}\right|_{0}=\xi^{*}(\mathbf{k})\xi(\mathbf{k}^{\prime})+\tfrac{1}{2}\mathbf{1}(\mathbf{k}^{\prime},\mathbf{k}).$
(36)

So, the normalization constant comes out to be

$\mathcal{N}^{-1}=M_{V}^{*}\diamond\xi\xi^{*}\diamond M_{V}+\|M_{V}\|^{2},$
(37)

where the transformed detector mode is $M_{V}=V\diamond M^{*}$. The resulting
expression for the Wigner functional of the normalized photon-added coherent
state is

$\displaystyle W_{1\text{pa}}^{(\text{coh})}=$
$\displaystyle\frac{M_{V}^{*}\diamond(2\alpha-\xi)(2\alpha^{*}-\xi^{*})\diamond
M_{V}-\|M_{V}\|^{2}}{M_{V}^{*}\diamond\xi\xi^{*}\diamond M_{V}+\|M_{V}\|^{2}}$
$\displaystyle\times W_{\text{coh}},$ (38)

This Wigner functional is negative at $\alpha=\tfrac{1}{2}\xi$. However, for
$\|\xi\|\gg 1$, the negative amplitude of the Wigner functional at
$\alpha=\tfrac{1}{2}\xi$ is severely suppressed by the Gaussian shape of the
coherent state Wigner functional Zavatta et al. (2004b).

Figure 3: Wigner function of photon-added coherent state.

The polynomial factor in Eq. (38) exists in a subspace of the functional phase
space associated with a single mode, namely $M_{V}$. Therefore, only the parts
of the field variable $\alpha$ and the parameter function $\xi$ that are
proportional to $M_{V}$ play a role in the shape of the Wigner functional. We
can substitute $M_{V}\rightarrow m_{0}G$, $\alpha\rightarrow\alpha_{0}G$ and
$\xi\rightarrow\xi_{0}G$, where $G$ is a normalized mode, $\alpha_{0}$ is a
complex variable and $\xi_{0}$ and $m_{0}$ are constants. After tracing over
the part of the functional phase space that is orthogonal to $G$, we end up
with a two-dimensional Wigner function (not a functional anymore), given by

$\displaystyle W_{1\text{pa}}^{(\text{coh})}(\alpha_{0})=$ $\displaystyle
2\frac{|2\alpha_{0}-\xi_{0}|^{2}-1}{|\xi_{0}|^{2}+1}\exp\left(-2|\alpha_{0}-\xi_{0}|^{2}\right).$
(39)

The magnitude of the transformed mode $|m_{0}|=\|M_{V}\|$ cancels out
everywhere. If we assume that the coherent state is weak so that
$|\xi_{0}|\approx 1$, we can plot the two-dimensional Wigner function of the
single-photon added coherent state. It is shown in Fig. 3, as a function of
$q$ and positive $p$ to reveal the negative region lying between the origin
and maximum of the Wigner function.

###  IV.2 Single-photon added thermal state

We also consider a thermal state as initial state for the photon-addition
process Zavatta et al. (2007). Its Wigner functional is

$W_{\text{th}}[\alpha]=\mathcal{N}_{0}\det\\{T\\}\exp(-2\alpha^{*}\diamond
T\diamond\alpha),$ (40)

where $T$ is the thermal state kernel. Substituted into Eq. (32), with a
single-mode detector kernel, it gives

$\displaystyle W_{1\text{pa}}^{(\text{th})}=$
$\displaystyle\mathcal{N}\left[M_{V}^{*}\diamond(T+\mathbf{1})\diamond\alpha\alpha^{*}\diamond(T+\mathbf{1})\diamond
M_{V}\right.$
$\displaystyle\left.-\tfrac{1}{2}M_{V}^{*}\diamond(T+\mathbf{1})\diamond
M_{V}\right]W_{\text{th}}.$ (41)

Following the same steps as before, we produce a generating functional for the
moments, given by

$\displaystyle\mathcal{W}_{\mathcal{N}}[\mu^{*},\nu]=$
$\displaystyle\int\det\\{T\\}\exp\left(-2\alpha^{*}\diamond
T\diamond\alpha\right.$
$\displaystyle\left.+\alpha^{*}\diamond\nu+\mu^{*}\diamond\alpha\right)\ {\cal
D}^{\circ}[\alpha]$ $\displaystyle=$
$\displaystyle\exp\left(\tfrac{1}{2}\mu^{*}\diamond T^{-1}\diamond\nu\right),$
(42)

so that

$\left.\frac{\delta^{2}\mathcal{W}_{\mathcal{N}}}{\delta\nu(\mathbf{k})\delta\mu^{*}(\mathbf{k}^{\prime})}\right|_{0}=\tfrac{1}{2}T^{-1}(\mathbf{k}^{\prime},\mathbf{k}).$
(43)

The normalization constant is then given by

$\mathcal{N}^{-1}=\tfrac{1}{2}M_{V}^{*}\diamond(T^{-1}+\mathbf{1})\diamond
M_{V}.$ (44)

The normalized photon-added thermal state is

$\displaystyle W_{1\text{pa}}^{(\text{th})}=$
$\displaystyle\frac{M_{V}^{*}\diamond(T+\mathbf{1})\diamond\alpha\alpha^{*}\diamond(T+\mathbf{1})\diamond
M_{V}-\Lambda_{0}}{\Lambda_{1}}$ $\displaystyle\times W_{\text{th}}.$ (45)

where

$\displaystyle\begin{split}\Lambda_{0}=&\tfrac{1}{2}M_{V}^{*}\diamond(T+\mathbf{1})\diamond
M_{V},\\\
\Lambda_{1}=&\tfrac{1}{2}M_{V}^{*}\diamond(T^{-1}+\mathbf{1})\diamond
M_{V}.\end{split}$ (46)

For simplicity, we consider a single-mode thermal state, which is often
encountered in experiments where the thermal source is represented by a beam
of thermal light with a specific spatiotemporal mode. The kernel of such a
single-mode thermal state can be represented by

$T=\left(\mathbf{1}+\tau\Theta\Theta^{*}\right)^{-1}=\mathbf{1}-\frac{\tau}{1+\tau}\Theta\Theta^{*},$
(47)

where $\tau$ is the average number of photons in the thermal state and
$\Theta$ is a normalized mode. The determinant becomes
$\det\\{T\\}=(1+\tau)^{-1}$. Assuming that the transformed detector mode is
proportional to the mode of the thermal state, we replace $M_{V}\rightarrow
m_{0}\Theta$. With the part orthogonal to $\Theta$ traced out, the photon-
added thermal state’s Wigner function becomes

$\displaystyle W_{1\text{pa}}^{(\text{th})}(\alpha_{0})=$
$\displaystyle\frac{2}{(1+\tau)^{2}}\left(2\frac{2+\tau}{1+\tau}|\alpha_{0}|^{2}-1\right)$
$\displaystyle\times\exp\left(-\frac{2|\alpha_{0}|^{2}}{1+\tau}\right),$ (48)

where $\alpha_{0}=\Theta^{*}\diamond\alpha$. In Fig. 4, the Wigner function of
the photon-added thermal state is shown for $\tau=5$, plotted as a function of
$q$ and positive $p$. It reveals the negative region at the origin.

Figure 4: Wigner function of photon-added thermal state.

##  V Conclusions

The effect of the spatiotemporal degrees of freedom in the experimental
implementations of heralded quantum state preparation is investigated. The
Wigner functional formalism is used to derive general expressions for the
states produced by heralded photon subtraction and heralded photon addition.
While the initial states are parameterized in terms of kernels and parameter
functions, the heralding measurement is represented by a detector kernel,
which can be parameterized in terms of a single spatiotemporal detector mode.

As examples, the heralded preparation of photon-subtracted squeezed vacuum
states, photon-added coherent states, and photon-added thermal states are
considered. In all these cases, the single-mode measurements reduce the
dimensionality of the Wigner functional of the heralded state. The Wigner
functional of the photon-subtracted squeezed vacuum states is represented
within a four-dimensional subspace of the functional phase space spanned by
two complex-valued transformed detector modes, because the initial squeezed
vacuum states is parameterized by two kernels. Both the examples of photon-
added states are rendered in two-dimensional subspaces of the functional phase
space, represented as a complex plane associated with the complex-valued
transformed detector mode. The reason is that the heralded photon addition
process is based on a measurement of the difference-frequency component of the
Bogoliubov transformed input state. It thus involves only one Bogoliubov
kernel. The two-dimensional Wigner functions of these photon-added states can
therefore be shown in plots as provided.

Although the domains of these heralded states are significantly reduced by the
single-mode heralding measurements, there are still enough variability in the
properties of the detector modes and the initial states to provide much
diversity for the properties for the heralded states. With the aid of this
analysis, one can investigate these properties for optimal design of the
quantum information systems involved.

One of the important aspects resulting from this analysis is that the finite-
dimensional domain on which the heralded state’s Wigner function is
represented, is defined by transformed detector modes. When such a Wigner
function is sent into a subsequent system for further processing (for example,
being measured with the aid of homodyne tomography), it is important that the
spatiotemproal degrees of freedom of the modes used in such a subsequent
process (such as the local oscillator in the homodyne tomography system) match
the transformed detector mode of the heralding detector, and not the original
heralding detector mode. Any mismatch between these modes and the transformed
detector mode would lead to diminished or distorted results Roux (2022).

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