[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Preprint · April 2024

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/379872315

The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7, Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if \( P \) is a polynomial with no constant term then the equation \( x + P(y - x) = z + w + zw \), where \( y \neq x \) is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-Čech Compactifications of discrete semigroups.

Throughout our article by \( \mathbb{N} \), we denote the set of all positive integers. For any \( r \in \mathbb{N} \), a \( r \)-coloring of a set \( X \) is a partition of \( X \) into \( r \) disjoint sets. By finite coloring we mean any \( r \)-coloring, for any \( r \in \mathbb{N} \). Let \( (S, \cdot) \) be a semigroup, and \( \mathcal{F} \) be a family of subsets of \( S \). We say \( \mathcal{F} \) is partition regular (P.R.) if, for any finite coloring of \( S \), there exists a member of \( F \in \mathcal{F} \) in the same color, i.e. \( F \) is monochromatic. Given a function \( F : \mathbb{Z}^d \rightarrow \mathbb{Z} \) is called P.R. if for every finite coloring of \( \mathbb{Z} \), there exists monochromatic \( x_1, x_2, ..., x_d \) such that \( F(x_1, x_2, ..., x_d) = 0 \). A core problem in arithmetic Ramsey theory is to characterize the partition regular families and functions.

A cornerstone result in arithmetic Ramsey theory is the van der Waerden's theorem [19], which states that for any \( l, r \in \mathbb{N} \), and for any \( r \) coloring of \( \mathbb{Z} \), there exists a monochromatic arithmetic progression (A.P.) of length \( l \), in other words, the collection of \( l \) length A.P.'s is P.R. Another important theorem was due to Schur [18] which states that the collection \( \{ \{x, y, x+y\} : x, y \in \mathbb{Z} \} \) is P.R. Passing to the map \( n \to 2^n \), one can immediately prove the multiplicative version of the van der Waerden's (which says the collection of \( l \) length geometric progressions (G.P.) is P.R.), and the Schur theorem (which says \( \{ \{x, y, x \cdot y\} : x, y \in \mathbb{Z} \} \) is partition regular). However, the combined extension of additive and multiplicative Schur's theorem is one of the hardest and most investigated conjectures in arithmetic Ramsey theory. The following question is a long-standing open problem and motivates us to study combined additive and multiplicative patterns.

**Question 1.1.** [14, Question 3] Does the collection \( \{ \{x, y, x + y, x \cdot y\} : x, y \in \mathbb{N} \} \) P.R.?

In 1979 by N. Hindman [12, Section 4] and R. Graham [9, pages 68-69] proved this pattern is monochromatic for two colorings. In [17], J. Moreira proved the collection \( \{x, x+y, xy\} \) is P.R. Later in [1], J.M. Barrett, M. Lupini, and J. Moreira proved the collection \( \{x, x+y+xy, xy\} \) is P.R. In [7], M. Di Nasso proved the collection \( \{x, y, x+y+xy\} \) is P.R. and then he extended this result in several directions. In this article, he introduces a new operation to study "symmetric structure".

**Mathematics subject classification 2020:** 05D10, 05C55, 22A15, 54D35.

**Keywords:** Symmetric patterns, Partition regularity of equations, Central sets, Algebra of the Stone-Čech Compactifications of discrete semigroups.

Arkabrata Ghosh
arka2686@gmail.com\*

Sayan Goswami
sayan92m@gmail.com†

Sourav Kanti Patra
spatra3@gitam.edu‡

---
\*Ashoka University, plot no 2, Rajiv Gandhi Education city, Sonipat, Haryana-131029, India.

†Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah, West Bengal-711202, India.

‡Department of Mathematics, GITAM University Hyderabad Campus, Hyderabad, Telangana-502329, India.

Definition 1.2 (Symmetric structure). For any \( n \in \mathbb{N} \), and any \( (x_1, x_2, \ldots, x_n) \in \mathbb{N}^n \) a structure \( P(x_1, x_2, \ldots, x_n) \) is called symmetric structure if \( P(x_1, x_2, \ldots, x_n) = P(x_{i_1}, x_{i_2}, \ldots, x_{i_n}) \) for any \( (i_1, i_2, \ldots, i_n) \in S_n \), where \( S_n \) is the set of all permutation of \( \{1, 2, \ldots, n\} \).

If \( P(x_1, x_2, \ldots, x_n) \) is a polynomial, then this structure is called a symmetric polynomial.

For example, \( P(x, y) = x + y + xy \) is a symmetric polynomial. In [7], M. Di Nasso found monochromatic patterns arising from symmetric polynomials. Later in [6], A. Chakraborty and the second author found polynomial extensions of several symmetric patterns.

In this article, we analyze the algebraic structure of the new symmetric operation introduced by M. Di Nasso [7]. In section 2, we recall the algebraic structure of the ultrafilters over discrete semigroups, and some basic notions from [15]. Then in section 3, we study the interplay between the additive, multiplicative, and symmetric large set. This is an analogous study in the direction of [3]. Then we deduce several new combinatorial applications of our results. Also in Corollary 4.12, we find new partition regular families which gives an answer to [7, Final remarks (4)].

In this section, we recall the algebraic structure of the Stone-Čech compactification of discrete semigroups that we use in our work. Then we recall the newly introduced symmetric operation by M. Di Nasso in [7], which will help us to study symmetric patterns.

### 2 Preliminaries

In this section we recall some basic prerequisites from the ultrafilter theory. For details we refer the book [15] to the readers.

A filter \( \mathcal{F} \) over any nonempty set \( X \) is a collection of subsets of \( X \) such that

1. \( \emptyset \notin \mathcal{F} \), and \( X \in \mathcal{F} \),
2. \( A \in \mathcal{F} \), and \( A \subseteq B \) implies \( B \in \mathcal{F} \),
3. \( A, B \in \mathcal{F} \) implies \( A \cap B \in \mathcal{F} \).

Using Zorn's lemma we can guarantee the existence of maximal filters which are called ultrafilters. Any ultrafilter \( p \) has the following property:

• if \( X = \bigcup_{i=1}^r A_i \) is any finite partition of \( X \), then there exists \( i \in \{1, 2, \ldots, r\} \) such that \( A_i \in p \).

Let \( (S, \cdot) \) be any discrete semigroup. Let \( \beta S \) be the collection of all ultrafilters. For every \( A \subseteq S \), define \( \overline{A} = \{ p \in \beta S : A \in p \} \). Now one can check that the collection \( \{\overline{A} : A \subseteq S\} \) forms a basis for a topology. This basis generates a topology over \( \beta S \). We can extend the operation “\(\cdot\)” of \( S \) over \( \beta S \) as: for any \( p, q \in \beta S \), \( A \in p \cdot q \) if and only if \( \{x : x^{-1}A \in q \} \in p \). With this operation “\(\cdot\)”, \( (\beta S, \cdot) \) becomes a compact Hausdorff right topological semigroup. One can show that \( \beta S \) is nothing but the Stone-Čech compactification of \( S \). Hence Ellis’s theorem guarantees that there exist idempotents in \( (\beta S, \cdot) \).

Before we proceed, let us recall the notions of IP sets. For any set \( X \), let \( P_f(X) \) be the collection of all nonempty finite subsets of \( X \).

**Definition 2.1** (IP Sets). Let \( (S, +) \) be a commutative semigroup. A set \( A \subseteq S \) is said to be an IP set if there exists an injective sequence \( \langle x_n \rangle_{n \in \mathbb{N}} \) in \( S \) such that
\[
A = FS(\langle x_n \rangle_{n \in \mathbb{N}}) = \left\{ \sum_{n \in \alpha} x_n : \alpha \in P_f(\mathbb{N}) \right\}.
\]
Abbreviately for \( \alpha \in P_f(\mathbb{N}) \), we write \( x_\alpha = \sum_{n \in \alpha} x_n \).

The following theorem is a cornerstone theorem in Ramsey theory, often known as the Hindman theorem [11].

**Theorem 2.2** (Hindman Theorem). If \( (S, +) \) is a commutative semigroup, then for every finite coloring of \( S \), there exists a monochromatic IP set.

It can be shown that every member of the idempotents of \((\beta S, \cdot)\) contains an \(IP\) set, which means every idempotent witnesses Hindman's theorem. Using Zorn's lemma one can show that \((\beta S, \cdot)\) contains minimal left ideals (minimal w.r.t. the inclusion). A well-known fact is that the union of such minimal left ideals is a minimal two-sided ideal, denoted by \(K(\beta S, \cdot)\). Here we recall a few well-known classes of sets that are relevant to our work.

**Definition 2.3.** Let \((S, \cdot)\) be a semigroup, let \(n \in \mathbb{N}\) and let \(A \subseteq S\). We say that

- \(A\) is a thick set if for any finite subset \(F \subset S\), there exists an element \(x \in S\) such that \(Fx = \{fx : f \in F\} \subset A\);
- \(A\) is a syndetic set if there exists a finite set \(F \subset S\) such that \(S = \bigcup_{x \in F} x^{-1}A\), where \(x^{-1}A = \{y : xy \in A\}\);
- \(A\) is piecewise syndetic if there exists a finite set \(F \subset S\) such that \(\bigcup_{x \in F} x^{-1}A\) is a thick set. It is well known that \(A\) is piecewise syndetic if and only if there exists \(p \in K(\beta S, \cdot)\) such that \(A \in p\).
- \(A\) is central if it belongs to a minimal idempotent in \(\beta S\).

It can be proved that a set \(A\) is thick iff there exists a left ideal \(L\) such that \(L \subseteq A\). And a set \(A\) is syndetic iff for every left ideal \(L\), \(L \cap A \ne \emptyset\). Note that if \(f : (S, \cdot) \to (T, \cdot)\) be any map between discrete semigroups. Then \(\tilde{f} : \beta S \to \beta T\) be the continuous extension of \(f\) defined by \(\tilde{f}(p) = \tilde{f}(p-\lim x) = p-\lim f(x)\), where \(p-\lim f(x) = y\) if and only if for every neighbourhood \(U\) of \(y\), \(\{x : f(x) \in U\} \in p\). For details about the theory of ultrafilters, we refer to the book [15].

## 2.2 A brief review over the symmetric operation

After lifting the multiplicative operation over the affine space via an isomorphism, in [7], Di Nasso introduced a new symmetric operation over \(\mathbb{Z}\), that we will discuss soon. In this subsection, we describe this operation briefly. First, we need to recall some basic facts from [7].

**Definition 2.4** (Elementary symmetric polynomial). For \(j = 1, 2, \ldots, n\), the elementary symmetric polynomial in \(n\) variables is the polynomial:

\[
e_j(X_1, X_2, \ldots, X_n) = \sum_{1 \leq i_1 \leq \cdots \leq i_j \leq n} X_{i_1} X_{i_2} \cdots X_{i_j} = \sum_{\emptyset \ne G \subseteq \{1, \ldots, n\}} \prod_{s \in G} X_s
\]

For all \(a_1, \ldots, a_n\), the product \(\prod_{j=1}^n (a_j + 1) = \sum_{j=1}^n e_j(a_1, \ldots, a_n) + 1\), and so

\[
c = \sum_{j=1}^n e_j(a_1, \ldots, a_n) \iff \prod_{j=1}^n (a_j + 1) = (c + 1)
\]

More generally, for \(l, k \ne 0\) it can be easily verified that

\[
\prod_{j=1}^n (l a_j + k) = l c + k \iff c = \sum_{j=1}^n l^{j-1} k^{n-j} e_j(a_1, \ldots, a_n) + \frac{k^n - k}{l}
\]

If \(k, l \in \mathbb{N}\), then \(c \in \mathbb{Z}\) if and only if \(l \mid k(k-1)\).
The function \(G_{l,k}(\cdot)\) in the next definition is precisely the same as the value of \(c\) in Definition 2.4, which gives a justification for our attention to this function.

**Definition 2.5** ((\(l, k\))-symmetric polynomial). For \(l, k \in \mathbb{Z}\) with \(l, k \ne 0\) the \((l, k)\)-symmetric polynomial in \(n\) variables is

\[
G_{l,k}(X_1, X_2, \ldots, X_n) = \sum_{j=1}^n l^{j-1} k^{n-j} e_j(X_1, X_2, \ldots, X_n) + \frac{k^n - k}{l}
\]

\[
= \sum_{\emptyset \ne G \subseteq \{1, \ldots, n\}} \left(l^{|G|-1} k^{n - |G|} \cdot \prod_{s \in G} X_s\right) + \frac{k^n - k}{l}
\]

Definition 2.6. For \( l, k \in \mathbb{Z} \) where \( l \neq 0 \) divides \( k(k-1) \), define

\[
a \odot_{l, k} b = c \iff (l a + k)(l b + k) = (l c + k).
\]

So,

\[
c = a \odot_{l, k} b = \frac{1}{l} \left[ (l a + k)(l b + k) - k \right] = l a b + k(a + b) + \frac{k^2 - k}{l}.
\]

Clearly, \( c \in \mathbb{Z} \) if and only if \( l \) divides \( k^2 - k = k(k-1) \).

It is easy to verify that \( (\mathbb{Z}, \odot_{l, k}) \) is an abelian group when \( l \mid k(k-1) \). In [7], Di Nasso analyzed the structure of this group to deduce new Ramsey theoretic patterns. In our work we analyze the algebraic structure of this group to deduce new patterns and new partition regular equations. Throughout our work, we will be concerned about the group \( (\mathbb{Z}, \odot_{1, t}) \), where \( t \in \mathbb{Z} \).

Let \( t \in \mathbb{Z} \), \( i : \mathbb{Z} \to \mathbb{Z} \) be the identity map defined by \( i(x) = x \) for all \( x \in \mathbb{Z} \). Let \( h_t : \mathbb{Z} \to \mathbb{Z} \) be the map defined by \( h_t(x) = x + t \) for all \( x \in \mathbb{Z} \). When \( t \) is fixed, we use \( h \) to denote \( h_t \). Let \( \tilde{i} \), and \( \tilde{h} \) be the map defined from \( \beta \mathbb{Z} \) to \( \beta \mathbb{Z} \), are the continuous extensions of \( i \), and \( h \) respectively. In this section, we study how much additive and multiplicative large sets behave like a symmetric large set. Basically we test the compatibility of the following diagram.

For every \( t \in \mathbb{Z} \), and \( a, b \in \mathbb{Z} \), define \( a \odot_t b = a \odot_{1, -t} b \). Define \( a \oplus_t b = a + b - t \). One can check that \( (\mathbb{Z}, \oplus_t, \odot_t) \) forms a ring, and the map \( h : (\mathbb{Z}, +, \cdot) \to (\mathbb{Z}, \oplus_t, \odot_t) \) defined by \( h(x) = x + t \) for all \( x \in \mathbb{Z} \) is a ring isomorphism. Surprisingly it shows that a translation of additive and multiplicative large sets are “large” (in some sense) in \( (\mathbb{Z}, \oplus_t, \odot_t) \). As for \( t = 0 \), \( x \odot_0 y = x \cdot y \), we can stress that: the operation \( \odot_t \) clearly generalizes the multiplicative operation. As \( \odot_t \) is also a symmetric operation, this leads to a natural question how many additive large sets are symmetrically large, and how many symmetrically large sets are additive large? This immediately generalizes a large studied theory on the interplay between “additive and multiplicative large sets”¹. For each \( t \in \mathbb{Z} \), the following theorem shows the interplay between the multiplicative and \( \odot_t \) syndetic, thick, and piecewise syndetic sets.

Proof. Let \( h : \mathbb{Z} \to \mathbb{Z} \) be defined by \( h(x) = x + t \). Then \( h \) is an isomorphism between \( (\mathbb{Z}, \cdot) \) and \( (\mathbb{Z}, \odot_t) \). Now the extension of \( h \) over \( \beta \mathbb{Z} \) is defined by \( \tilde{h} : \beta \mathbb{Z} \to \beta \mathbb{Z} \) be defined by \( \tilde{h}(p) = \tilde{h}(p-\lim x) = p-\lim h(x) = p-\lim x + t = p + t \). As \( h \) is a homomorphism, \( \tilde{h} \) is also a homomorphism from \( (\beta \mathbb{Z}, \cdot) \) and \( (\beta \mathbb{Z}, \odot_t) \). Now for every \( p \in \beta \mathbb{Z} \), \( \tilde{h}(p - t) = p \), and hence \( \tilde{h} \) is surjective. Again it is easy to verify that \( \tilde{h}(p) = \tilde{h}(q) \) implies \( p = q \), proving \( \tilde{h} \) is injective. Hence \( \tilde{h} \) is an isomorphism, and we have for every left ideal \( L \) in \( (\beta \mathbb{Z}, \cdot) \), \( \tilde{h}(L) = L + t \) is a left ideal of \( (\beta \mathbb{Z}, \odot_t) \).

---

¹ for history on this topic authors are referred to the article [3]

1. Now if A is multiplicatively thick, then there exist a left ideal L such that L ⊆ A. But then, h(L) = L + t ⊆ A + t, which implies A + t is thick in (ℤ, ⊙t).

2. Again if A is multiplicative syndetic, then for every left ideal L, we have L ∩ A ≠ ∅. But h̃ is an isomorphism. Hence for every left ideal M for (βℤ, ⊙t), h̃⁻¹(M) is a left ideal of (βℤ, ·). Hence, h̃⁻¹(M) ∩ A ≠ ∅, which implies M ∩ (A + t) ≠ ∅. This shows that A + t is syndetic in (ℤ, ⊙t).

The proof of 3 is similar, so we leave it to the reader.

The following theorem says how additive large sets are preserved in the group (ℤ, ⊕t).

Theorem 3.2. For every t ∈ ℤ, each piecewise syndetic set in (ℤ, ⊕t) is piecewise syndetic in (ℤ, +).

Proof. It will be sufficient to prove that K(βℤ, ⊕t) ⊆ K(βℤ, +). Let h : ℤ → ℤ defined by h(x) = x + t for all x ∈ ℤ. Then h is ring isomorphism between (ℤ, +, ·) into (ℤ, ⊕t, ⊙t). Now

K(βℤ, ⊕t) = h̃[K(βℤ, +)] = K(βℤ, +) + t ⊆ K(βℤ, +).

This completes the proof.

In [15], authors proved that K(βℤ, +) ∩ K(βℤ, ·) = ∅. The above Proposition immediately implies that for every t ∈ ℤ, K(βℤ, ⊕t) ∩ K(βℤ, ·) = ∅. It was proved in [4], that cl(K(βℤ, +)) ∩ K(βℤ, ·) ≠ ∅. The following theorem is the symmetric version of this result. In the following theorem ℤ* = ℤ \ {0}.

Theorem 3.3. For every t ∈ ℤ, E_t = cl(K(βℤ, +)) ∩ K((βℤ, ⊙t)) ≠ ∅. In fact, E_t is a left ideal of K(βℤ*, ⊙t).

Proof. For t = 0, the result is immediate from [4]. So we may assume t ≠ 0. We claim that cl(K(βℤ, +)) is a left ideal of (βℤ*, ⊙t). Then our proposition directly follows from our claim.

To prove our claim, let p ∈ βℤ*, and q ∈ cl(K(βℤ, +)). Let A ∈ p ⊙t q. Hence {n ∈ ℤ : n⁻¹A ∈ q} ∈ p. Choose n ∈ ℤ* such that n⁻¹A ∈ q, hence B = n⁻¹A is an additive piecewise syndetic set. But then A = n ⊙t B = {t(t + 1) − tn + nx − tx : x ∈ B} = {t(t + 1) − tn + (n − t)x : x ∈ B} is again an additive piecewise syndetic set. Hence p ⊙t q ∈ cl(K(βℤ, +)).

In the next section, we use the following theorem to find a new class of equations that are partition regular.

Theorem 3.4. For every t ∈ ℤ,
h̃[(cl(E(K(βℤ, +))) ∩ E(K(βℤ, ·))] = cl(E(K(βℤ, ⊕t))) ∩ E(K(βℤ, ⊙t)).

Proof. Let h : (ℤ, +, ·) → (ℤ, ⊕t, ⊙t) be the ring isomorphism defined by h(x) = x + t for all x ∈ ℤ. And let h̃ : βℤ → βℤ be the continuous extension of h. As h is a ring homomorphism, h̃ : (βℤ, +) → (βℤ, ⊕t) and h̃ : (βℤ, ·) → (βℤ, ⊙t) are semigroup homomorphisms. In fact, h̃(p) = p + t for all p ∈ βℤ shows that h̃ is also bijective. Hence, h̃ is a semigroup isomorphism from (βℤ, +) to (βℤ, ⊕t), and (βℤ, ·) to (βℤ, ⊙t). Now we claim that h̃(cl(E(K(βℤ, +)))) = cl(E(K(βℤ, ⊕t))).

Proof of the Claim: Let q ∈ h̃(cl(E(K(βℤ, +)))). Hence q = h̃(p) where p ∈ cl(E(K(βℤ, +))). Observe that h(ℤ) = ℤ is central in (ℤ, ⊕t). Hence from [3, Lemma 4.6.], [10, Theorem 2.6.], if B ∈ ℤ is central in (ℤ, +), then h(B) is central in (ℤ, ⊕t). Let us assume A ∈ q. It implies that there exists B ∈ p such that h(B) ⊆ A. But B ∈ p implies that B is central in (ℤ, +). Hence, A is central in (ℤ, ⊕t). So q ∈ cl(E(K(βℤ, ⊕t))). To prove the converse, we assume q ∈ cl(E(K(βℤ, ⊕t))). We define h₁ : (ℤ, ⊕t) → (ℤ, +) by h₁(x) = x − t for all x ∈ ℤ. Here h₁ is a semigroup isomorphism and its extension h̃₁ : (βℤ, ⊕t) → (βℤ, +) defined by h̃₁(p) = p − t is an isomorphism too. Clearly for every p ∈ βℤ, h̃₁[h̃(p)] = h̃[h̃₁(p)] = p. Again by similar reasoning, we have q ∈ h̃(cl(E(K(βℤ, +)))). This gives the reverse inclusion and proves our claim.

Now again h̃ : (βℤ, ·) → (βℤ, ⊙t) is an isomorphism. Hence, we have h̃[(K(βℤ, ·))] = K(βℤ, ⊙t), which implies h̃[E(K(βℤ, ·))] = E(K(βℤ, ⊙t)), as idempotents go to idempotents. Hence we have
h̃[(E(K(βℤ, +))) ∩ E(K(βℤ, ·))] = (E(K(βℤ, ⊕t)) ∩ E(K(βℤ, ⊙t)).

From [3, Theorem 6.1], we know that every multiplicative syndetic set is additively central. However for general (ℤ, ⊙t), we have the following weaker result.

Theorem 3.5. Let t ∈ Z, and A be a syndetic set in (Z, ⊙t). Then A is additively piecewise syndetic.

Proof. Let {t₁, t₂, …, tₙ} be a finite set such that Z = ⋃_{i=1}^n t_i^{-1}A. Then at least one of them is additively central. For some s ∈ {t₁, t₂, ..., tₙ}, let C = s^{-1}A is additively central. Then

A = s ⊙t C = t(t+1) - ts + {(s-t)·y : y ∈ C} = t(t+1) - ts + (s-t)·C.

As C is central, (s-t)·C is central, and so A, being a translation of a central set, is a piecewise syndetic set.
□

## 4 Combinatorial applications

In this section, we find several new monochromatic patterns. We prove for every t ∈ Z, certain ⊙t-large sets contain arithmetic progressions of arbitrary length, and its polynomial extensions which were known for t = 0. Then we find new equations which are partition regular.

### 4.1 New monochromatic patterns

In this section, we study how much symmetric structures are preserved in additive and multiplicative large sets. We show in most of the cases they contain symmetric patterns up to an additive translation.

Theorem 4.1. Let t ∈ Z, and A ∈ p for some p ∈ E(βZ, ·). Then there exists a sequence ⟨xₙ⟩ₙ such that σt(⟨xₙ⟩ₙ) − t ⊆ A.

Proof. From the previous discussion, it is clear that ĥ(p) ∈ E(βZ, ⊙t). Then if A ∈ p, h(A) ∈ ĥ(p), as h is an isomorphism. Hence there exists a sequence ⟨xₙ⟩ₙ such that σt(⟨xₙ⟩ₙ) ⊆ A + t. This completes the proof. □

As arithmetic progressions are invariant under multiplication, using van der Waerden’s theorem, one can easily deduce that every multiplicatively piecewise syndetic set contains A.P. of arbitrary length. The following theorem says that symmetric large sets also contain A.P. of arbitrary length.

Theorem 4.2. For every t ∈ Z, ⊙t-piecewise syndetic sets contain A.P. of arbitrary length.

Proof. Let A be a ⊙t-piecewise syndetic set. Hence there exists p ∈ K(βZ, ⊙t) such that A ∈ p. But then A − t is a multiplicative piecewise syndetic set. Hence for every l ∈ Z, there exists a, d ∈ Z such that {a, a+d, …, a+ld} ⊆ A − t. Hence A contains AP of arbitrary length. This completes the proof. □

Let AP = {p ∈ βZ : (∀A ∈ p) A contains A.P. of arbitrary length}. Clearly the above theorem says that ⋃_{t ∈ Z} cl(K(βZ, ⊙t)) ⊆ AP. But for the ultrafilters witnessing polynomial extensions of van der Waerden’s theorem, things are different. Let P be the set of all polynomials with rational coefficients, and with zero constant term. Let L = {p ∈ βZ : (∀A ∈ p) (∀F ∈ Pf(P)) (∃a, d ∈ Z) (∀p ∈ F) a + p(d) ∈ A}. From [5], we know that the following set is non-empty.

One can show that L is a left ideal of (βZ*, ·). The following Theorem shows that for every t ∈ Z, L is a left ideal in (βZ \ Z, ⊙t).

Theorem 4.3. For every t ∈ Z, L is a left ideal of (βZ \ Z, ⊙t).

Proof. Let p ∈ βZ \ Z, q ∈ L, and A ∈ p ⊙t q. Then {n : n^{-1}A ∈ q} ⊆ p. Now we choose any n ∈ Z \ {t} and let F ∈ Pf(P). Let F' = {1/(n-t)f : f ∈ F}. Then there exists a, d ∈ Z such that ∀P ∈ F',

a + (1/n) P(d) ∈ n^{-1}A.

i.e., n ⊙t (a + P(d)) = c + P(d) ∈ A for some c ∈ Z. As F ∈ Pf(P) is arbitrary, so we have p ⊙t q ∈ L. Hence, L is a left ideal. □

Before we proceed to our next result, we need the following technical definition.

Definition 4.4. For every t ∈ Z, define the followings:

**Corrected text:**

---

Till now we have discussed how additively large sets are preserved in $(\mathbb{Z}, \odot)$ for every $t \in \mathbb{Z}$. But we know if $l|k \cdot (k - 1)$, then $(\mathbb{Z}, \odot_{l,k})$ forms a group. As in the preceeding sections we were working with ring homomorphisms we restricted ourselves over $(\mathbb{Z}, \odot_t)$ only. In this section, we show that we can generalize some of the results discussed till now over the group $(\mathbb{Z}, \odot_{l,k})$ for some $l, k$, satisfying $l|(k-1)$. That means, we can show that some collections of ultrafilters "reach in an additive sense" are large in the group $(\mathbb{Z}, \odot_{l,k})$, whenever $l|(k-1)$.

Note that if $A$ is syndetic/ piecewise syndetic in $(\mathbb{Z}, +)$, then $n \cdot A$ is also syndetic/ piecewise syndetic for all $n \in \mathbb{Z}^*$. Again these largeness are preserved under additive translations. Hence for any $n \in \mathbb{Z}$, and if $l|(k-1)$ (where $l \neq 1$) we have $n \odot_{l,k} A = (kn + \frac{k^2 - k}{l}) + (ln+k) \cdot A$ is a syndetic/ piecewise syndetic set if $A$ is resp. Hence we have the following theorem.

Let $\mathcal{F}$ be a family of finite subsets of $\mathbb{Z}$. We call such a family, a Rado family if every additive or multiplicative piecewise syndetic subset of $\mathbb{Z}$ contains a member of $\mathbb{Z}$. For example, a collection of finite-length arithmetic progressions are Rado family. The following theorem mixes symmetric patterns with other monochromatic patterns.

**Theorem 4.5.** *Let $t \in \mathbb{Z}$ and $p \in cl(E(K(\beta \mathbb{Z}, \oplus_t))) \cap E(K(\beta \mathbb{Z}, \odot_t))$. Then for every $A \in p$, there exist a sequence $\{F_n\}_n \subset \mathcal{F}$, and a sequence $\langle x_n \rangle_n$ such that*

1. *$\langle F_n \rangle_n \subset A$,*
2. *$\sigma_t(\langle x_n \rangle_n) \cup \sigma_t(\langle F_n \cdot x_n \rangle_n) \cup \sigma_t(\langle F_n \rangle_n) \subset A$.*

**Proof.** As $A \in p$ and $p \odot_t p = p$, we have $A^{\star} = \{x \in A : x^{-1}A \in p\} \in p$. From [15, Lemma 4.14], we know if $x \in A^{\star}$, then $x^{-1}A^{\star} \in p$. Let $F_1 \in \mathcal{F}$ such that $F_1 \subset A^{\star}$. Then $B = A^{\star} \cap_{f \in F_1} f^{-1}A^{\star} \in p$. Now choose $x_1 \in B$. Choose $F_1 \cdot x_1 \subset A^{\star}$. Suppose for some $N \in \mathbb{N}$, we have a sequence $\{F_n\}_{n=1}^N \subset \mathcal{F}$, and $\langle x_n \rangle_{n=1}^N \subset A^{\star}$ such that $C = \sigma_t(\langle x_n \rangle_n) \cup \sigma_t(\langle F_n \cdot x_n \rangle_n) \cup \sigma_t(\langle F_n \rangle_n) \subset A^{\star}$. Let $D = A^{\star} \cap_{y \in C} y^{-1}A^{\star} \in p$. Then choose $F_{N+1} \in D^{\star}$, and then choose $x_{N+1} \in D^{\star} \cap F_{N+1}^{-1}D^{\star}$. This completes the induction. a sequence such that $\sigma_t(\langle x_n \rangle_n) \subset B$. This gives us the desired result. $\Box$

The following corollary is an immediate application of the above theorem.

**Corollary 4.6.** *Let $p \in (cl(E(K(\beta \mathbb{Z},+))) \cap E(K(\beta \mathbb{Z},\cdot))$, and $A \in p$. Then there exist $F \in \mathcal{F}$, and a sequence $\langle x_n \rangle_n$ such that $F \cup \sigma_t(\langle x_n \rangle_n) \cup \{f \odot_t y; f \in F \text{ and } y \in \sigma_t(\langle x_n \rangle_n)\} \subset A + t$*

**Proof.** Let $A \in p$, and so $\tilde{h}(A) \in \tilde{h}(p)$ where $\tilde{h}(p) \in cl(E(K(\beta \mathbb{Z}, \oplus_t))) \cap E(K(\beta \mathbb{Z}, \odot_t))$. Now our corollary follows from Theorem 4.5. $\Box$

In [4], authors proved for every finite partition of $\mathbb{Z}$, there exists two sequences $\langle x_n \rangle_n$, $\langle y_n \rangle_n$ such that $FS(\langle x_n \rangle_n) \cup FP(\langle x_n \rangle_n)$ is monochromatic. The following theorem is a symmetric variant, as well as an extension of this result.

**Theorem 4.7.** *For every finite partition of $\mathbb{Z}$, there exist $t \in \mathbb{Z}$, and three sequences $\langle x_n \rangle_n$, $\langle y_n \rangle_n$, $\langle z_n \rangle_n$, and $\langle w_n \rangle_n$, such that $FS(\langle x_n \rangle_n) \cup FP(\langle w_n \rangle_n) \cup (t + FS_{-t}(\langle y_n \rangle_n)) \cup (t + \sigma_{-t}(\langle z_n \rangle_n))$ is monochromatic, where $FS_{-t}(\langle y_n \rangle_n)$ denotes the finite sum over the semigroup $(\mathbb{Z}, \oplus_{-t})$. In fact such a collection of $t$ is infinite.*

**Proof.** Let $p \in cl(E(K(\beta \mathbb{Z},+))) \cap E(K(\beta \mathbb{Z},\cdot))$, and $A \in p$. Choose $q \in E(K(\beta \mathbb{Z}, +))$ be such that $A \in q$. Then $A^{\star} = \{t \in A : A - t \in q\} \in q$. Let $t \in A^{\star}$. Now $(\mathbb{Z},\oplus_{-t},\odot_{-t})$ is a commutative ring and $h(x) = x-t$ is a ring isomorphism between $(\mathbb{Z},+,\cdot)$ and $(\mathbb{Z},\oplus_{-t},\odot_{-t})$. Now from Theorem 3.4, $\tilde{h}(p) \in cl(E(K(\beta \mathbb{Z},\oplus_t)))\cap E(K(\beta \mathbb{Z},\odot_t))$, and $A-t \in \tilde{h}(p)$.

Now $A-t \in q$ implies there exists a sequence $\langle x_n \rangle_n$ such that $FS(\langle x_n \rangle_n) \subseteq A \cap A-t$. And $A-t \in \tilde{h}(p)$ implies there exists two sequences $\langle y_n \rangle_n$ and $\langle z_n \rangle_n$ such that $(t + FS_{-t}(\langle y_n \rangle_n)) \cup (t + \sigma_{-t}(\langle z_n \rangle_n)) \subseteq A-t$. Again from the choice of $p$, there exists a sequence $\langle w_n \rangle_n$ such that $FP(\langle w_n \rangle_n) \subseteq A$. $\Box$

---

**Definition snippet reconstructions:**

$\sigma_t(A_1, A_2, \ldots, A_n) = \{a_1 \odot_t a_2 \odot_t \ldots \odot_t a_n : a_i \in A_i \text{ for each } 1 \leq i \leq n\}$

$\sigma_t(\langle A_n \rangle_n) = \{\sigma_t(A_{i_1}, A_{i_2}, \ldots, A_{i_n}) : n \in \mathbb{N}, A_{i_j} \in \langle A_n \rangle_n \text{ for all } n\}$

Theorem 4.8. If l (≠ 1), k ∈ ℤ such that l | (k − 1), then for every n ∈ ℤ,

1. If A is piecewise syndetic/syndetic in (ℤ,+), then n ⊙_{l,k} A is piecewise syndetic/syndetic in (ℤ,+). Hence, we have the following
 (a) cl(K(βℤ,+)) is a left ideal in (βℤ, ⊙_{l,k}), and
 (b) if A is a syndetic set in (ℤ, ⊙_{l,k}), then A is syndetic in (ℤ,+).

2. Every ⊙_{l,k}-piecewise syndetic set contains A.P. of arbitrary length.

3. L is a left ideal in (βℤ, ⊙_{l,k}).

Proof.
1. Proof is similar to the proof of Theorems 3.3, 3.5.

2. Let p ∈ AP. Then it is easy to verify that for each n ∈ ℤ, n ⊙_{l,k} p ∈ AP. Hence taking closure we have βℤ ⊙_{l,k} AP ⊆ AP. Now choose p ∈ AP, q ∈ βℤ, and A ∈ p ⊙ q. Then again it is easy to verify that A contains A.P. of arbitrary length. Hence AP ⊙_{l,k} βℤ ⊆ AP. Hence AP is a two sided ideal of (βℤ, ⊙_{l,k}). Hence K(βℤ, ⊙_{l,k}) ⊆ AP. This completes the proof.

3. Proof is similar to the proof of Theorem 4.3.

## 4.3 New partition regular equations

In this section we give an affirmative answer to [11, Final remarks (4)]. To do so, we need the following theorem, which is an essential tool to formulate new partition regular equations from old equations.

Theorem 4.9. [8, Lemma 2.1] Let U ∈ βℕ, and let f(x₁, ..., xₙ), g(y₁, ..., yₘ) be two functions such that U ⊨ (f₁(x₁, ..., xₙ) = 0) ∧ (g(y₁, ..., yₘ) = 0). Then U ⊨ (f₁(x₁, ..., xₙ) = 0) ∧ (g(y₁, ..., yₘ) = 0) ∧ (x₁ = y₁).

From [2, 13], we know that every member of cl(E(K(βℤ,+))) ∩ E(K(βℤ,·)) witness the solution of the equation a + b = c · d. Later in [8], and [1], the authors extended this result, and proved partition regularity of several new classes of equations. In some upcoming theorems, we produce more new examples of partition regular equations. The following theorem is the first one.

Theorem 4.10. For every l, k ∈ ℤ with l | (k − 1), the equation (a + b)/2 = c ⊙_{l,k} d is P.R. In particular, for l = 1; k = −1, a + b = 2(cd + c + d) is P.R.

Proof. From Theorem 4.8, let p ∈ cl(K(βℤ,+)) ∩ E(K(βℤ, ⊙_{l,k})), and A ∈ p. Hence A is central in (ℤ, ⊙_{l,k}), and it is piecewise syndetic in (ℤ,+). As every piecewise syndetic set contains arithmetic progressions of arbitrary length, A contains the solution of the equation z = (a + b)/2. Again A is central in (ℤ, ⊙_{l,k}). Hence A contains solution of the equation z = c ⊙_{l,k} d. Now from the Theorem 4.9, A contains solution of the equation (a + b)/2 = c ⊙_{l,k} d. Now expanding this equation we have the desired solution.

The following theorem is a general version of the above theorem.

Theorem 4.11. Let l, k ∈ ℤ with l | (k−1), and G(z₁, x₁, x₂, ..., xₙ) = 0, and let H(z₂, y₁, y₂, ..., yₘ) = 0 be two equations such that every additively piecewise syndetic set contains a solution of G, and every ⊙_{l,k} central set contains solution of H. Then G(z, x₁, x₂, ..., xₙ) = H(z, y₁, y₂, ..., yₘ) is P.R.

Proof. From Theorem 3.3, pick p ∈ cl(K(βℤ,+)) ∩ E(K(βℤ, ⊙_{l,k})). Then each member of p is additively piecewise syndetic, and so p witness the P.R. of G(z₁, x₁, x₂, ..., xₙ) = 0. Again p ∈ E(K(βℤ, ⊙_{l,k})), hence p witness the P.R. of H(z₂, y₁, y₂, ..., yₘ) = 0. Hence from the Theorem 4.9, we have G(z, x₁, x₂, ..., xₙ) = H(z, y₁, y₂, ..., yₘ) is P.R.

The following corollary is the most general and covers many new classes of partition regular equations.

Corollary 4.12. Let l, k ∈ ℤ with l | (k − 1), and P be a polynomial with no constant term. Then the equation x + P(y − x) = z ⊙_{l,k} w, where x ≠ y is P.R.
In particular for l = 1; k = −1, x + P(y − x) = z + w + zw (x ≠ y) is P.R.

**Proof.** Every additively piecewise syndetic set contains patterns of the form \(\{a, a + d, a + P(d)\}\).
Then it witnesses the solution of the equation \(z = x + P(y - x)\), where \((x \neq y)\). Hence from Theorem 4.11 we have \(x + P(y - x) = z \oplus_{l,k} w\) is P.R.

**Example 4.13.** Letting \(n \in \mathbb{N}\), and \(P(z) = z^n\). Then \(x + (y - x)^n = z + w + zw\) is P.R.

**Corollary 4.14.** Let \(m, n \in \mathbb{N}\). Let \(P_1, P_2, \ldots, P_m\) be polynomials such that for every \(i \in \{1, 2, \ldots, m \}\), \(P_i(0) = 0\). Then the following system of equation is P.R.

\[
\begin{aligned}
&x_1 - P_1(y - x) = y_1 \oplus_{l,k} y_2 \oplus_{l,k} \cdots \oplus_{l,k} y_n \\
&x_2 - P_2(y - x) = y_1 \oplus_{l,k} y_2 \oplus_{l,k} \cdots \oplus_{l,k} y_n \\
&\vdots \\
&x_m - P_m(y - x) = y_1 \oplus_{l,k} y_2 \oplus_{l,k} \cdots \oplus_{l,k} y_n \\
\end{aligned}
\]

**Proof.** Every additively piecewise syndetic set contains witnesses the solution of the following system of equations.
\[
\begin{aligned}
&x_1 - P_1(y - x) = z_1 \\
&x_2 - P_2(y - x) = z_1 \\
&\vdots \\
&x_m - P_m(y - x) = z_1 \\
\end{aligned}
\]

Again every central set contains the solutions of the equation \(z_2 = y_1 \oplus_{l,k} y_2 \oplus_{l,k} \cdots \oplus_{l,k} y_n\). Hence from Theorem 4.11, our desired conclusion follows.

The authors thank Prof. Lorenzo Luperi Baglini, and Pintu Debnath for helpful comments on the previous draft. The second named author is supported by NBHM postdoctoral fellowship with reference no: 0204/27/(27)/2023/R & D-II/11927.

[9] R. L. Graham, B. L. Rothschild, and J. H. Spencer. Ramsey theory. John Wiley & Sons, Inc., New York, second edition, 1990.

[10] S. Goswami and S. Jana: A combinatorial viewpoint of preserving notions of largeness and an abstract Rado theorem, INTEGERS 21 (2021).

[11] N. Hindman: Finite sums from sequences within cells of partitions of N, J. Combin. Theory Ser. A, 17 (1974), 1–11.

[12] N. Hindman: Partitions and sums and products of integers. Trans. Amer. Math. Soc., 247:227–245, 1979.

[13] N. Hindman: Monochromatic sums equal to products in N, Integers 11A (2011), #A10.

[14] N. Hindman, I. Leader, and D. Strauss: Open problems in partition regularity. Combin. Probab. Comput., 12(5–6):571–583, 2003. Special issue on Ramsey theory.

[15] N. Hindman and D. Strauss: Algebra in the Stone-Čech Compactification: theory and applications, second edition, de Gruyter, Berlin, 2012.

[16] L. Luperi Baglini: Partition regularity of nonlinear polynomials: a nonstandard approach, Integers 14 (2014), #A30.

[17] J. Moreira: Monochromatic sums and products in N, Ann. of Math., 185 (2017), 1069–1090.

[18] I. Schur: Über die Kongruenz $$x^m + y^m \equiv z^m (\text{mod } p)$$, Jahresbericht der Deutschen Math.-Verein., 25 (1916), 114–117.

[19] B. van der Waerden: Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskunde, 19 (1927), 212–216.


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
En-Jui Chang¹,* and Ching-Yi Lai¹,†

¹Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan
(Dated: April 30, 2024)

Gaussian quantum information processing with continuous-variable (CV) quantum information carriers holds significant promise for applications in quantum communication and quantum internet. However, applying Gaussian state distillation and quantum error correction (QEC) faces limitations imposed by no-go results concerning local Gaussian unitary operations and classical communications. This paper introduces a Gaussian QEC protocol that relies solely on local Gaussian resources. A pivotal component of our approach is CV gate teleportation using entangled Gaussian states, which facilitates the implementation of the partial transpose operation on a quantum channel. Consequently, we can efficiently construct a two-mode noise-polarized channel from two noisy Gaussian channels. Furthermore, this QEC protocol naturally extends to a nonlocal Gaussian state distillation protocol.

The theoretical advantages of universal quantum computers and quantum communications over classical counterparts have been demonstrated in specific tasks, such as Shor’s factoring algorithm [1], Grover search [2], and quantum key distribution [3]. To fully exploit the potential of both quantum and classical technologies, it is necessary to develop a quantum internet [4, 5] that enables the transfer of classical or quantum data packets between classical and quantum devices [6, 7].

Recent quantum devices often suffer from imperfect state preparation, noisy quantum gates, and imprecise measurements, all of which degrade the performance of quantum schemes. Although recent studies have demonstrated that noisy, shallow circuits can still yield quantum advantages [8], circuits with large quantum depth are essential for achieving greater computational power [9]. Thus, incorporating quantum error correction (QEC) is essential to ensure efficient and reliable operation of a quantum network [10–14]. Additionally, QEC schemes for qubits based on quantum architectures of the near future are anticipated to require significant overheads [15]. Consequently, there is a need for low-resource, fault-tolerant quantum computing solutions.

One potential solution to address the issue of large overhead is using continuous-variable (CV) quantum information carriers, such as quantized harmonic oscillators, which are particularly suitable for quantum communication. CV QEC codes [16–25] serve as building blocks for quantum internet and transducers [26]. Particularly, Gaussian quantum information processing techniques have reached a high level of maturity [27].

(Gaussian no-go theorems.) However, limitations arise when applying Gaussian state distillation [28–30] and quantum error correction [31] with local Gaussian unitary operations and classical communications. The prevailing belief is that the Gaussian QEC struggles to effectively reduce noise in both the position and the momentum quadratures of a bosonic mode [27, 31]. A reduction in noise variance in one quadrature often results in a corresponding increase in the other, as specifically observed in single-mode squeezing, presenting a significant challenge in noise management strategies [32].

This paper presents a Gaussian QEC protocol comprising beam splitters, two-mode squeezers, and homodyne measurements. An essential aspect of our approach is using entangled Gaussian states to facilitate CV gate teleportation. This enables us to implement the partial transpose of a quantum channel, effectively conjugating a symplectic matrix by a nonsymplectic one. Consequently, we can transform two noisy Gaussian channels into a two-mode noise-polarized channel with one cleaner mode and one noisier mode, thus facilitating effective quantum communication on the cleaner mode.

Remarkably, our findings do not contravene the main theorem in [31] (if extended to the multi-mode case), and the entanglement degradation of the polarized channel is not decreased. Moreover, all the components of our protocol have been previously considered in [28–30], except for the ideal Gaussian EPR states that are locally prepared, which remain a Gaussian resource.

(Gaussian quantum information processing.) We briefly introduce Gaussian quantum information [33] and establish notation. To simplify, we set the Planck constant ℏ to be one. Let â and â† represent the annihilation and creation operators on a bosonic mode associated with a Hilbert space 𝓗, respectively, where â := ∑ₙ'₌₁^∞ √n' |n' − 1⟩⟨n'| in the number basis {|n⟩}ₙ₌₀^∞, and their commutator satisfies [â, â†] = 1. We will focus on the phase space with two quadratures defined by position and momentum operators: q̂ := (â + â†)/√2 and p̂ := (â − â†)/i√2, where i = √−1 and [q̂, p̂] = i.

For a quantum system comprising n bosonic modes in ⊗ⱼ₌₁ⁿ 𝓗ⱼ, the vector of canonical quadrature operators is defined as:

x̂ := (q̂₁, p̂₁, ..., q̂ₙ, p̂ₙ), (1)

* ejchang@nycu.edu.tw
† cylai@nycu.edu.tw

where q̂ᵢ and p̂ᵢ operate on the i-th mode. Additionally, let x̂ⱼ denote the j-th component of x̂.

An n-mode Gaussian state ρ can be fully characterized by its mean vector mⱼ = tr(ρx̂ⱼ) and the covariance matrix γ(ρ). For simplicity, we focus on Gaussian states with zero means mⱼ = 0 and thus the (j, k)-th entry of γ(ρ) is γ(ρ)ⱼ,ₖ = tr(ρx̂ⱼx̂ₖ). Adjusting the mean vector can be accomplished using a displacement operator D̂(α) := e^{αâ†-α*â}, where α ∈ ℂ.

An n-mode Gaussian quantum operation is a completely positive and trace-preserving operation that transforms one Gaussian state into another. Consequently, we care only about its effects on the first two moments of an input Gaussian state. A Gaussian operation will be denoted as 𝒢^{M,N}, where M and N are two 2n × 2n real matrices and N is symmetric and positive semidefinite. Its action on a covariance matrix γ is defined by

𝒢^{M,N} : γ → MγMᵀ + N. (2)

Thus, M characterizes a Gaussian operation, while N introduces additive Gaussian noise.

Let I = [1 0; 0 1], Z = [1 0; 0 -1], and 0 be a zero matrix of appropriate dimension. Herein, the noisy single-mode Gaussian channel under consideration is defined by

𝒩^{σ²} := 𝒢^{I,σ²I}, (3)

where σ² is the noise variance.

A noiseless Gaussian unitary operation is defined by 𝒢^M ≜ 𝒢^{M,0}, with M being symplectic. In this case, det{M} = 1 and MᵀΩₙM = Ωₙ, with Ω_{α+1} := Ω_α ⊕ Ω_1 and Ω_1 := [0 -1; 1 0] for α ∈ ℕ. Elementary noiseless Gaussian unitary operations include single-mode squeezers Ŝⱼ(r) := e^{r(âⱼ²-âⱼ†²)/2}, two-mode squeezers Ŝ_{j,k}(r) := e^{r(âⱼâₖ-âⱼ†âₖ†)} for r ∈ ℝ and (balanced) beam splitters B̂_{j,k} := e^{iπ(q̂ⱼp̂ₖ-p̂ⱼq̂ₖ)/4}, where the subscripts j and k denote the modes to which they are applied. Their corresponding symplectic matrices are Sⱼ(r) = [e^{-r} 0; 0 e^{r}],

S_{j,k}(r) = [cosh(r)I sinh(r)Z; sinh(r)Z cosh(r)I] and B_{j,k} = 1/√2 [I I; -I I],

respectively.

By applying a two-mode squeezer to the vacuum state, we obtain the following two-mode squeezed vacuum state

|ζ_r⟩_{1,2} = 1/cosh(r) ∑_{n'=0}^∞ tanh^{n'}(r) |n'⟩₁ |n'⟩₂, (4)

which is a two-mode Gaussian state with zero mean, also known as an Einstein-Podolsky-Rosen (EPR) state. In particular, an infinitely squeezed two-mode squeezed vacuum state will be denoted as |ζ_{EPR}⟩ := lim_{r→∞} |ζ_r⟩ and ρ_ζ := |ζ_{EPR}⟩⟨ζ_{EPR}|, which serves as an ideal resource for CV quantum teleportation [34, 35] as shown in Fig. 1.

(CV quantum teleportation.) The Bell measurement operation, combined with the EPR pair in CV teleportation, can be characterized by the Kraus operators K̂(q₁, p₂) : ℋ₁ → ℋ₃ for q₁, p₂ ∈ ℝ, given by

K̂(q₁, p₂) = ⟨ζ_{EPR}|_{1,2} D̂₃(-μ) |ζ_{EPR}⟩_{2,3}, (5)

where μ := √2(q₁+ip₂). Up to normalization, the output state of the gate teleportation scheme becomes

ρ_out ∝ K̂(q₁, p₂)ρ_in K̂†(q₁, p₂)
   = D̂₃(-μ) (∑_{m,n} ⟨m|₁ ρ_in |n⟩₁ |m⟩₃⟨n|₃) D̂₃(μ). (6)

The expression ∑_{m,n} ⟨m|₁ ρ_in |n⟩₁ |m⟩₃⟨n|₃ represents the quantum state teleported from the first mode to the third mode. Consequently, the resulting output state is the teleported state with a displacement dependent on the measurement outcome: ρ_out = D̂₃(-μ) ρ_in D̂₃(μ). Hence, we can counteract this extra displacement by implementing a recovery operation D̂₃(μ) as shown in Fig. 1. (See Appendix A for more details.)

This process can be extended to CV gate teleportation [36], where a target quantum operation is applied to the entangled resource state first, followed by a corresponding recovery operation dependent on the measurement outcomes. In particular, if a two-mode unitary operation Ŵ_{j,k} is to be applied to a single-mode state ρ_in and an ancillary state ρ_anc, Ŵ_{j,k} is first applied to one mode of the EPR state. Then, a Bell measurement is conducted on the other mode of the EPR state together with ρ_in. This effectively produces the following state before applying a recovery operation:

Ŵ_{3,4} D̂₃(-μ) ρ_in ⊗ ρ_anc D̂₃(μ) Ŵ_{3,4}†. (7)

The recovery operation is Ŵ_{3,4} D̂₃(μ) Ŵ_{3,4}† and the complete process is shown in Fig. 2. If a single-mode operator Ŵ_j is to be applied, we may ignore the ancillary mode in Fig. 2.

While this recovery operator may generally be complex, it simplifies to some displacement operators for linear gates Ŵ_{3,4}. (Refer to Appendix B.) A gate is said to

be linear if it is of the form $\hat{W}_{j,k} = e^{-i\hat{H}_{j,k} t}$, where $\hat{H}_{j,k}$ is a Hamiltonian composed of a linear combination of $\hat{I}$, $\hat{q}_j$, $\hat{p}_j$, $\hat{q}_k$, $\hat{p}_k$, $\hat{q}_j \hat{q}_k$, $\hat{q}_j \hat{p}_k$, $\hat{p}_j \hat{q}_k$, and $\hat{p}_j \hat{p}_k$ with real coefficients. In particular, we will consider $\hat{W}_{j, k}$ corresponding to a beam splitter, as shown in the following lemma. (For further elaboration, see Appendix C.)

Lemma 1 When a displacement operator $\hat{D}_j(\mu)$ evolves in the Heisenberg picture under a beam splitter $\hat{B}^\dagger_{j, k}$, it transforms into a composition of two displacements:

\[
\hat{B}_{j,k} \hat{D}_j(\mu) \hat{B}^\dagger_{j,k} = \hat{D}_j\left(\frac{\mu}{\sqrt{2}}\right) \otimes \hat{D}_k\left(-\frac{\mu}{\sqrt{2}}\right). \qquad (8)
\]

(Engineering Gaussian channels.) One can show that a bouncing property holds for an infinitely squeezed two-mode state, as shown in [36]. (See Appendix D.)

Lemma 2 (CV bouncing.) Let $\hat{W}$ be a single-mode operator. Then

\[
\hat{I} \otimes \hat{W} |\zeta_{EPR}\rangle = \hat{W}^T \otimes \hat{I} |\zeta_{EPR}\rangle. \qquad (9)
\]

Moreover, we generalize the above lemma to the case of partially bouncing operations. (See Appendix E.)

Lemma 3 (CV partial bouncing.) Let $|\psi\rangle$ be an arbitrary $n$-mode state, $n \in \mathbb{N}$. Then

\[
\left(\hat{I}_i \otimes \hat{W}_{j, k_1}\right) |\zeta_{EPR}\rangle_{i, j} \otimes |\psi\rangle_{k_1, \dots, k_n}
= \left(\hat{W}^T_{i, k_1} \otimes \hat{I}_j\right) |\zeta_{EPR}\rangle_{i, j} \otimes |\psi\rangle_{k_1, \dots, k_n}. \qquad (10)
\]

for any two-mode operator $\hat{W}$, where $T_i$ denotes the partial transpose with respect to the $i$-th mode.

Therefore, if we initially swap the two modes of the EPR state for gate teleportation, we can effectively implement the operation $\hat{W}^T_{j, k}$, using a corresponding recovery operation. This process is illustrated in Fig. 3.

Moreover, by combining Lemma 1 and Lemma 3, we can effectively implement the partial transpose of a balanced beam splitter through CV gate teleportation with recovery by displacement operators.

It is clear that $\hat{a}^T = \left(\sum_{n' = 1}^{\infty} \sqrt{n'} |n' - 1\rangle \langle n'|\right)^T = \hat{a}^\dagger$.
Thus, $\hat{q}^T = \hat{q}$ and $\hat{p}^T = -\hat{p}$. Consequently, if $\hat{W}_{j, k}$ is an operator that is a function of $\hat{x}$, then the component $\hat{p}_j$ of $\hat{W}_{j, k}$ is flipped in $\hat{W}^{T}_{j, k}$. We have the following observation.

Lemma 4 Given an $n$-mode Gaussian operation $\mathcal{G}^{M,N}$, its partial transpose with respect to the $i$-th mode is $\mathcal{G}^{\tilde{M}_i, \tilde{N}_i}$, where $\tilde{M}_i := Z_i M Z_i$ and $\tilde{N}_i := Z_i N Z_i$ and $Z_i = I^{\oplus i - 1} \oplus Z \oplus I^{\oplus n - i}$.

Proof.
Consider a Gaussian operator $\hat{W}(\hat{x})$ that is a function of $\hat{x}$ and let $\mathcal{G}^{M,N}$ be its corresponding symplectic-matrix representation. If $\hat{W}^{T_i}(\hat{x})$ is applied to $\rho$, the covariance matrix becomes $\gamma\left(\hat{W}^{T_i}(\hat{x}) \rho (\hat{W}^{T_i}(\hat{x}))^\dagger \hat{x}_j \hat{x}_k\right)$. In the Heisenberg picture, it can be expressed as

\[
\hat{x}_j \hat{x}_k \xrightarrow{\hat{W}^{T_i}(\hat{x})} (\hat{W}^{T_i}(\hat{x}))^\dagger \hat{x}_j \hat{x}_k \hat{W}^{T_i}(\hat{x}). \qquad (11)
\]

Let $f_i : \hat{q}_i, \hat{p}_i \mapsto \hat{q}_i, -\hat{p}_i$ be a function describing the action of partial transpose on the $i$-th mode in the Heisenberg picture for $i \in \mathbb{N}$. Then $\hat{W}^{T_i}(\hat{x}) = f_i(\hat{W}(\hat{x})) = \hat{W}(f_i(\hat{x}))$. Then (11) can written as

\[
\hat{x}_j \hat{x}_k \xrightarrow{\hat{W}^{T_i}(\hat{x})} (\hat{W}(f_i(\hat{x})))^\dagger \hat{x}_j \hat{x}_k \hat{W}(f_i(\hat{x}))
= f_i(\hat{W}(\hat{x})^\dagger f_i(\hat{x}_j \hat{x}_k) \hat{W}(\hat{x})). \qquad (12)
\]

Also, we have $f_i(\gamma(\rho)_{j k}) = \operatorname{tr}(\rho f_i(\hat{x}_j) f_i(\hat{x}_k))$, which implies $f_i(\gamma(\rho)) = Z_i \gamma(\rho) Z_i$. Thus (12) implies that

\[
\gamma(\rho) \xrightarrow{\hat{W}^{T_i}(\hat{x})} Z_i M Z_i \gamma(\rho) Z_i M Z_i + Z_i N Z_i. \qquad (13)
\]

In general, $\tilde{M}_i$ is not symplectic since $Z_i$ is not symplectic. For convenience, we define an operation $B_i$ at the level of covariance matrices:

\[
B_i : \mathcal{G}^{M,N} \mapsto \mathcal{G}^{\tilde{M}_i, \tilde{N}_i}. \qquad (14)
\]

In the following, we propose a noise-polarization protocol for a one-way CV quantum communication system.

Here are the corrected OCR texts for each crop in order:

---
1. By concatenating the GKP two-mode squeezing code with our noise-polarization scheme, we can further suppress the logical noise variance to
---
2. code is approximately given by
---
3.
\[
\sigma_L^2 \stackrel{\sigma \ll 1}{\longrightarrow} \frac{2\sigma^2}{\sqrt{\pi}} \ln \frac{\pi^{1.5}}{2\sigma^4}.
\]
(15)

---
4.
\[
\sigma_L^2 \stackrel{e^{-r}\sigma \ll 1}{\longrightarrow} e^{-2r}\frac{2\sigma^2}{\sqrt{\pi}} \sqrt{4r + \ln \frac{\pi^{1.5}}{2\sigma^4}}.
\]
(16)

---
5. This concatenation is useful when
---
6.
\[
\frac{4}{\ln \frac{\pi^{1.5}}{2\sigma^4}} + 1 < e^{4r}.
\]
(17)

---
7. For typical noise levels satisfying \(\sigma^2 < \frac{\pi^{3}}{4\sqrt{2e}} \approx 1.012\), it is clear that this concatenation always yields benefits for \(r > 0\).
---
8. (Application.) Noh et al. [20] demonstrated that on input the channel noise variance \(\sigma^2\), the optimal logical noise variance \(\sigma_L^2\) using the GKP two-mode squeezing
---
9. Moreover, using Theorem 5, we can construct a Gaussian entanglement distillation that circumvents the Gaussian QEC no-go result [31]. The idea involves substituting the noisy Gaussian channels in Fig. 4 (a) with noisy teleportation operations and subsequently implementing QEC on a shared EPR.
---
10.
Theorem 5 (Noise-polarization.) Consider independent and identically distributed noisy Gaussian channels \(\mathcal{N}_j^{\sigma^2} , j = 1, 2\). Then, the composite channel
---
11.
\[
\mathcal{G}_{1,2}^{I,\mathcal{N}^{polar}(r)} := \mathcal{B}_2(\mathcal{G}_{1,2}^B) \circ \mathcal{G}_{1,2}^{S(-r)} \circ \bigotimes_{j=1}^2 \mathcal{N}_j^{\sigma^2} \circ \mathcal{G}_{1,2}^{S(r)} \circ \mathcal{B}_2(\mathcal{G}_{1,2}^{B^T})
\]
---
12. is noise-polarized with \(\mathbf{N}^{polar}(r) = \sigma^2 \begin{bmatrix} e^{-2r}\mathbf{I} & 0 \\ 0 & e^{2r}\mathbf{I} \end{bmatrix}\)
---
13.
Proof. Recall that the action of a Gaussian channel is given in Eq. (2). Thus \(\mathcal{G}_{1,2}^{S(-r)} \circ (\mathcal{N}_1^{\sigma^2} \otimes \mathcal{N}_2^{\sigma^2}) = \mathcal{G}_{1,2}^{S(-r), \sigma^2 S(-r) S^T(-r)}\). Then we have
---
14.
\[
\begin{aligned}
\mathcal{G}_{1,2}^{I, \mathcal{N}^{polar}(r)}
&= \tilde{\mathcal{G}}_{1,2}^{B_2} \circ \mathcal{G}_{1,2}^{I, \sigma^2 S(-r) S^T(-r)} \circ \tilde{\mathcal{G}}_{1,2}^{B_2^T} \\
&= \mathcal{G}_{1,2}^{I, \sigma^2 B_2^2 S(-r) S^T(-r) B_2^T} \\
&= \mathcal{G}_{1,2}^{I, \sigma^2 Z_2^2 B_2 Z_2^2 S(-r) S^T(-r) Z_2 B_2^T Z_2}
\end{aligned}
\]

---
15. Therefore, \(\mathbf{N}^{polar}(r) = \sigma^2 \begin{bmatrix} e^{-2r}\mathbf{I} & 0 \\ 0 & e^{2r}\mathbf{I} \end{bmatrix}\) after some calculations (see Appendix F). ■
---
16. The implementation of this composite channel is illustrated in Fig. 4. Note that \(\mathbf{B}_{2,1} = \mathbf{B}_{1,2}^T\), and thus we have used the block \(\hat{B}_{(2,1)}\) in Fig. 4 (b).
---
17. By Theorem 5, this is equivalent to applying a Gaussian channel \(\mathbf{I}_{(1)} \otimes \mathcal{G}_{(2,3)}^{I, \mathcal{N}^{polar}(r)}\) to \(\sigma^{(1)}_{AB}\) and \(\rho_{anc}\). Consequently, the output state is
---
18.
\[
(\hat{I} \otimes \mathcal{N}^{e^{-2r}\sigma^2})(\rho_\zeta)_{1,2} \otimes (\mathcal{N}^{e^{2r}\sigma^2})(\rho_a)_{(3)}.
\]
(18)

---
19. Finally, Bob disposes of the third mode, and they end up sharing a Gaussian EPR pair with reduced noise.
---
20.
Corollary 6 (Entangled Gaussian state distillation.) Assume that Alice and Bob can locally prepare ideal EPR pairs \(\rho_\zeta\). Suppose that Alice and Bob share some noisy entangled states \(\sigma_{AB}^{(i)} = (\hat{I}_A \otimes \mathcal{N}_B^{\sigma^2})(\rho_\zeta)\). Then, a less noisy shared entangled Gaussian state \((\hat{I} \otimes \mathcal{N}^{e^{-2r}\sigma^2})(\rho_\zeta)_{1,2}\) can be distilled.
---
21.
Proof. We have a distillation protocol as follows. Firstly, Alice encodes \(\sigma_A^{(1)}\) along with some ancillary state \(\rho_{anc}\) using the circuit in Fig. 4 (b) with the help of one ideal local EPR. Subsequently, using two noisy EPR states \(\sigma_{AB}^{(2)}\) and \(\sigma_{AB}^{(3)}\), she teleports the two encoded modes to Bob. Upon receiving the measurement outcomes, Bob executes recovery operations on \(\sigma_B^{(2)}\) and \(\sigma_B^{(3)}\), accordingly. Then, he applies the decoding circuit in Fig. 4 (c) with the help of one ideal local EPR.
---
22.
(Finitely squeezed Gaussian EPR.) In the previous constructions, we assumed the availability of a two-mode infinitely squeezed vacuum state \(\rho_\zeta\). Now, we investigate the case where a finitely squeezed vacuum state \(|\zeta_r\rangle\) is available instead of an infinitely squeezed one. Specifically, we examine the fidelity between the states on the
---

left-hand and right-hand sides of Eq.(10). This fidelity approaches 1 as r tends to infinity. However, for finite r, higher-order terms can be truncated since |tanh(r)| < 1. Although summing up this series remains challenging, it can be inferred that the fidelity F is sufficiently high when 1 − tanh(r) = o(1). Further details are provided in Appendix G.

(Discussion.) Our Gaussian QEC method demonstrates the ability to engineer uncorrelated additive noisy Gaussian channels utilizing local Gaussian unitary operators, local quadrature measurements, and an arbitrary CV ancillary state. In particular, this process remains unaffected by the initial states, allowing the extension of our Gaussian QEC approach to non-Gaussian states [18, 22–25, 37].

While Gaussian QEC applied to specific types of correlated noisy Gaussian channels [16, 17], described by a single variance variable, has potential in specific physical platforms; addressing uncorrelated noisy Gaussian channels is still crucial. It is worth noting that the beam splitter alone, as considered in [16, 17], cannot convert two identical and independently distributed random variables into correlated ones within the context of [31]. However, as demonstrated in [20], the two-mode squeezers enable this transformation. To utilize the distinction between two-mode squeezers and beam splitters, we concatenated standard Gaussian channels with a channel that exhibits Gaussian behavior, thereby achieving noise polarization in the noisy Gaussian channels.

In summary, a fully Gaussian QEC combined with Gaussian quantum information processing techniques is expected to be more readily accessible and impactful in the near term.

CYL was supported by the National Science and Technology Council in Taiwan under Grants 111-2628-E-A49-024-MY2 and 113-2119-M-A49-008.

Proceedings of the 29th annual ACM symposium on theory of computing, STOC '97 (ACM, New York, NY, USA, 1997) pp. 176–

tonomous error-correction with low-order nonlinearity, Quantum Science and Technology 4, 035007 (2019).

Note that the two-mode squeezer can be decomposed into single-mode squeezers and beam splitters:

$$S_{j, k}(-r) = B_{j, k} \big(S(-r)_j \oplus S(r)_k \big) B^{T}_{j, k}. \tag{A1}$$

Then, the two-mode squeezed vacuum state can be written as

$$|\zeta_r\rangle_{1, 2} = \hat{B}_{1, 2} \circ (\hat{S}_1(-r) \otimes \hat{S}_2(r)) \circ \hat{B}^{\dagger}_{1, 2} |0\rangle_1 |0\rangle_2$$
$$= \hat{B}_{1, 2} \circ (\hat{S}_1(-r) \otimes \hat{S}_2(r)) |0\rangle_1 |0\rangle_2,$$

where the second equality is because applying a beam splitter on two vacuum modes has no effect. As $r$ approaches infinity, the two states $\lim_{r \to \infty} \hat{S}(r) |0\rangle$ and $\lim_{r \to \infty} \hat{S}(-r) |0\rangle$ emerge as eigenstates of quadrature operators $\hat{q}$ and $\hat{p}$, with eigenvalues 0, respectively. These states will be denoted as $|q = 0\rangle$ and $|p = 0\rangle$, respectively, and they should not be confused with the vacuum state $|0\rangle$ in the number basis. Consequently, the ideal EPR state $|\zeta_{EPR}\rangle$ can be written as [36]

$$|\zeta_{EPR}\rangle_{1, 2} = \hat{B}_{1, 2}\, |p = 0\rangle_1 |q = 0\rangle_2. \tag{A2}$$

The Bell measurement operation, combined with the EPR pair in CV teleportation, can be characterized by the Kraus operators $\hat{K}(q_1, p_2) : \mathcal{H}_1 \to \mathcal{H}_3$ for $q_1, p_2 \in \mathbb{R}$, given by

$$\hat{K}(q_1, p_2) = \langle q = q_1|_1 \langle p = p_2|_2\, \hat{B}_{1, 2} |\zeta_{EPR}\rangle_{2, 3}$$
$$= \langle q = 0|_1 \langle p = 0|_2\, \hat{D}^{\dagger}_1(q_1)\hat{D}^{\dagger}_2(ip_2)\hat{B}_{1, 2} |\zeta_{EPR}\rangle_{2, 3}$$
$$= \langle \zeta_{EPR}|_{1, 2}\, \hat{B}^{\dagger}_{1, 2} \hat{D}^{\dagger}_1(q_1)\hat{D}^{\dagger}_2(ip_2)\hat{B}_{1, 2} |\zeta_{EPR}\rangle_{2, 3}$$
$$= \langle \zeta_{EPR}|_{1, 2}\, \hat{D}^{\dagger}_1\left(\frac{\mu_+}{2}\right)\hat{D}^{\dagger}_2\left(\frac{\mu_-}{2}\right) |\zeta_{EPR}\rangle_{2, 3}, \tag{A3}$$

where $\mu_{\pm} := \sqrt{2}(q_1 \pm ip_2)$. In the Heisenberg picture, the last equality is to evolve the operator $\hat{D}^{\dagger}_1(q_1)\hat{D}^{\dagger}_2(ip_2)$ under

\(\hat{B}_{1,2}\), i.e. applying \(\mathbf{B}_{1,2}\) on the quadratures.

\[
\hat{K}(q_1, p_2) = \langle \zeta_{EPR} |_{1,2} \hat{D}_1^\dagger \left( \frac{\mu_+}{2} \right) \hat{D}_2^\dagger \left( \frac{\mu_-}{2} \right) | \zeta_{EPR} \rangle_{2,3},
\]
\[
= \langle \zeta_{EPR}|_{1,2} \hat{D}_2( -\mu_- ) | \zeta_{EPR} \rangle_{2,3},
\]
\[
= \langle \zeta_{EPR}|_{1,2} \hat{D}_3( -\mu_+ ) | \zeta_{EPR} \rangle_{2,3}.
\tag{A4}
\]

The second and the third equality utilize Lemma 2 with \(\langle \zeta_{EPR} |_{1,2}\) and \(| \zeta_{EPR} \rangle_{2,3}\), respectively, and the transpose of displacement operators is the complex conjugate of its inversion, i.e., \(\hat{D}^T(\alpha) = (\hat{D}^\dagger(\alpha))^* = \hat{D}(-\alpha^*)\).

Then, the output state is

\[
\rho_{out} = \frac{ \hat{K}(q_1, p_2) \rho_{in} \hat{K}^\dagger(q_1, p_2) }{ \mathrm{Pr}(q_1, p_2) },
\tag{A5}
\]

where \(\mathrm{Pr}(q_1, p_2) = \mathrm{tr}( \hat{K}(q_1, p_2) \rho_{in} \hat{K}^\dagger(q_1, p_2) )\).

Up to normalization, the output state of the gate teleportation scheme becomes

\[
\rho_{out} \propto \hat{K}(q_1, p_2) \rho_{in} \hat{K}^\dagger(q_1, p_2)
\]
\[
= \hat{D}_3(-\mu_+) \left( \sum_{m, n} \langle m|_1 \rho_{in} |n \rangle_1 |m\rangle_3 \langle n|_3 \right) \hat{D}_3(\mu_+).
\tag{A6}
\]

## Appendix B: Recovery operator

**Lemma 7** When a displacement operator \(\hat{D}_j(\mu)\) evolves in the Heisenberg picture under a linear gate \(\hat{W}_{j,k}^\dagger\), it transforms into a composition of two displacements.

Herein, we prove that when a gate \(\hat{W}_{j,k} = e^{-i \hat{H}_{j,k} t}\) is linear, then \(\hat{W}_{j,k} \hat{D}_j(\mu) \hat{W}_{j,k}^\dagger\) remains a product of displacement operators on the \(j\)-th and \(k\)-th modes. To be succinct, we define the operators \(\hat{O}_j^{(0)} := \hat{I}_j, \hat{O}_j^{(1)} := \hat{q}_j, \hat{O}_j^{(2)} := \hat{p}_j\) on the \(j\)-th mode. Consider a Hamiltonian \(\hat{H}_{j,k}\) of the form

\[
\hat{H}_{j,k} = \sum_{a,b=0}^2 c_{ab} \hat{O}_j^{(a)} \otimes \hat{O}_k^{(b)},
\tag{B1}
\]

where \(c_{ab} \in \mathbb{R}, \forall a, b \in \{0,1,2\}\), which is a linear combination of \(\{\hat{I}_j, \hat{q}_j, \hat{p}_j, \hat{q}_k, \hat{p}_k, \hat{q}_j\hat{q}_k, \hat{q}_j\hat{p}_k, \hat{p}_j\hat{q}_k, \hat{p}_j\hat{p}_k\}\).

Recall that the displacement operator \(\hat{D}_j(\alpha) = e^{i\sqrt{2} \left( \mathrm{Im}\{\alpha\} \hat{q}_j + \mathrm{Re}\{\alpha\} \hat{p}_j \right) }\). We denote \(\hat{D}_j(\alpha) = e^{-i\hat{G}_j t}\), where \(\hat{G}_j = -t^{-1} \sqrt{2} ( \mathrm{Im}\{\alpha\} \hat{q}_j + \mathrm{Re}\{\alpha\} \hat{p}_j )\).

Now consider the following identity for matrices

\[
e^X Y e^{-X} = Y + [X, Y] + \frac{1}{2!} [X, [X, Y]] + \frac{1}{3!} [X, [X, [X, Y]]] + \cdots.
\tag{B2}
\]

For our case, we take \(X = -i \hat{H}_{j,k} t\) and \(Y = \sum_{m=0}^{\infty} \frac{(-i \hat{G}_j t)^m}{m!}\). Then, the recovery operator can be expressed as

\[
e^X Y e^{-X} = e^X \sum_{m=0}^\infty \frac{(-i \hat{G}_j t)^m}{m!} e^{-X}
\tag{B3}
\]
\[
= \sum_{m=0}^\infty \frac{(-it)^m}{m!} e^X \hat{G}_j^m e^{-X}
\tag{B4}
\]
\[
= \sum_{m=0}^\infty \frac{(-it)^m}{m!} ( e^X \hat{G}_j e^{-X} )^m
\tag{B5}
\]
\[
= \sum_{m=0}^\infty \frac{ ( -i e^X \hat{G}_j e^{-X} t )^m }{ m! }
\tag{B6}
\]
\[
= \sum_{m=0}^\infty \frac{ ( -i \hat{H}^{\prime} t )^m }{ m! }.
\tag{B7}
\]

where $\hat{H}' = e^{\hat{X}}\hat{G}_j e^{-\hat{X}}$ is the effective Hamiltonian.

Next, we show that $\hat{H}'$ is a linear combination of $\{\hat{q}_j, \hat{p}_j, \hat{q}_k, \hat{p}_k\}$. We expand this Hamiltonian as

$$\hat{H}' = \hat{G}_j + (-it)[\hat{H}_{j,k}, \hat{G}_j] + \frac{(-it)^2}{2!}[\hat{H}_{j,k},[\hat{H}_{j,k},\hat{G}_j]] + \frac{(-it)^3}{3!}[\hat{H}_{j,k},[\hat{H}_{j,k},[\hat{H}_{j,k},\hat{G}_j]]] + \cdots. \tag{B8}$$

Since $\hat{G}_j$ is a linear combination of $\hat{q}_j$ and $\hat{p}_j$, we can use the commutation relations between these quadratures to deduce that the commutator $[\hat{H}_{j,k}, \hat{G}_j]$ is a linear combination of $\hat{q}_k, \hat{p}_k$ and also $\hat{I}_k$ with pure imaginary coefficients. This makes $(-it)[\hat{H}_{j,k}, \hat{G}_j]$ a real coefficient linear combination.

\[
\begin{aligned}
(-it)[\hat{H}_{j,k}, \hat{G}_j]
&= (-it) \left[\sum_{a,b=0}^2 c_{ab} \hat{O}_j^{(a)} \otimes \hat{O}_k^{(b)}, -t^{-1}\sqrt{2}(\text{Im}\{\alpha\}\hat{q}_j + \text{Re}\{\alpha\}\hat{p}_j)\right] \\
&= (i\sqrt{2}) \sum_{a,b=0}^2 c_{ab} \left[\hat{O}_j^{(a)}, (\text{Im}\{\alpha\}\hat{q}_j + \text{Re}\{\alpha\}\hat{p}_j)\right] \otimes \hat{O}_k^{(b)} \\
&= (i\sqrt{2}) \sum_{a,b=0}^2 c_{ab} \left(\text{Im}\{\alpha\}[\hat{O}_j^{(a)}, \hat{q}_j] + \text{Re}\{\alpha\}[\hat{O}_j^{(a)}, \hat{p}_j]\right) \otimes \hat{O}_k^{(b)} \\
&= (i\sqrt{2}) \sum_{b=0}^2 (c_{2b}\text{Im}\{\alpha\}[\hat{O}_j^{(2)}, \hat{q}_j] + c_{1b}\text{Re}\{\alpha\}[\hat{O}_j^{(1)}, \hat{p}_j]) \otimes \hat{O}_k^{(b)} \\
&= (i\sqrt{2}) \sum_{b=0}^2 (c_{2b}\text{Im}\{\alpha\} [\hat{p}_j,\hat{q}_j] + c_{1b}\text{Re}\{\alpha\} [\hat{q}_j,\hat{p}_j]) \otimes \hat{O}_k^{(b)} \\
&= \sqrt{2} \sum_{b=0}^2 (c_{2b}\text{Im}\{\alpha\} - c_{1b}\text{Re}\{\alpha\})\hat{I}_j \otimes \hat{O}_k^{(b)} \\
&= \sum_{b=0}^2 \text{Re}\{\alpha\beta_b\} \hat{I}_j \otimes \hat{O}_k^{(b)} \\
&= \text{Re}\{\alpha\beta_0\}\hat{I}_j \otimes \hat{O}_k^{(0)} + \hat{G}_k',
\end{aligned}
\tag{B9}
\]
where $\beta_b = -\sqrt{2}(c_{1b} + ic_{2b})$ and $\hat{G}_k' = \sum_{b=1}^2 \text{Re}\{\alpha\beta_b\}\hat{I}_j \otimes \hat{O}_k^{(b)}$. Note that the term $\hat{I}_j \otimes \hat{I}_k$ will not affect higher order commutators, $[\hat{H}_{j,k}, \hat{I}_j\otimes\hat{I}_k]=0$. Also, the gate $e^{-i\hat{I}_j\otimes\hat{I}_k \tau}$ for $\tau\in \mathbb{R}$ contributes only a global phase. Thus, we can ignore this term.

Since $\hat{G}_k'$ is expressed as a displacement operator on $k$-th mode, the higher order term $(-it)^2[\hat{H}_{j,k},[\hat{H}_{j,k},\hat{G}_j]]$ is a linear combination of $\{\hat{q}_j,\hat{p}_j\}$ again.

\[
\begin{aligned}
(-it)^2[\hat{H}_{j,k},[\hat{H}_{j,k},\hat{G}_j]]
&= (-it)[\hat{H}_{j,k},\hat{G}_k'] \tag{B10}\\
&= (-it)\left[\sum_{a,b=0}^2 c_{ab}\hat{O}_j^{(a)}\otimes \hat{O}_k^{(b)}, \sum_{d=1}^2 \text{Re}\{\alpha\beta_d\}\hat{I}_j \otimes \hat{O}_k^{(d)}\right] \tag{B11}\\
&= (-it) \sum_{a,b=0}^2 \sum_{d=1}^2 c_{ab} \text{Re}\{\alpha\beta_d\} \hat{O}_j^{(a)} \otimes [\hat{O}_k^{(b)}, \hat{O}_k^{(d)}] \tag{B12}\\
&= t \sum_{a=0}^2 (c_{a1} \text{Re}\{\alpha\beta_2\} - c_{a2} \text{Re}\{\alpha\beta_1\}) \hat{O}_j^{(a)}\otimes \hat{I}_k. \tag{B13}
\end{aligned}
\]

Iterating this argument, it is clear that $\hat{H}'$ is a linear combination of $\{\hat{q}_j, \hat{p}_j, \hat{q}_k, \hat{p}_k\}$. Hence, the Hamiltonian $\hat{H}'$ can be expressed as $\hat{H}' = \hat{H}_j' + \hat{H}_k'$, where $\hat{H}_j'$ and $\hat{H}_k'$ are linear combinations of $\{\hat{q}_j, \hat{p}_j\}$ and $\{\hat{q}_k, \hat{p}_k\}$, respectively. Thus, the Hamiltonian $\hat{H}'$ corresponds to displacement operators $\hat{D}_j(\alpha_j) \otimes \hat{D}_k(\alpha_k)$ for some $\alpha_j, \alpha_k \in \mathbb{C}$.

Let $\hat{C}:=(\hat{q}_j,\hat{p}_j,\hat{q}_k,\hat{p}_k)^T$ be a column vector of the quadratures. A real linear combination of $\{\hat{q}_j,\hat{p}_j,\hat{q}_k,\hat{p}_k\}$ can be written as $\mathbf{R}\hat{C}$ for some $4\times 1$ real row vector $\mathbf{R}$. For a linear operator $\hat{O}_{\text{lin}} = \mathbf{R}\hat{C}$ for some $\mathbf{R}$, the mapping $L: \hat{O}_{\text{lin}} \mapsto (-it)[\hat{H}_{j,k},\hat{O}_{\text{lin}}]$ is also a real linear transformation. Thus $(-it)[\hat{H}_{j,k},\hat{O}_{\text{lin}}] = \mathbf{R} \mathbf{S} \hat{C}$ for some real ...

matrix $\mathbf{S}$. Thus, the operator $L^{\circ n}$, which applies this linear transformation $n$ times, remains a linear transform, and $L^{\circ n} : \mathbf{R}\hat{C} \mapsto \mathbf{R} \mathbf{S}^n \hat{C}$.

Let $\hat{G}_j = \mathbf{R}\hat{C}$. Then, the Hamiltonian in Eq. (B8) can be determined as:

\[
\hat{H}' = \mathbf{R}\hat{C} + \mathbf{R}\mathbf{S}\hat{C} + \mathbf{R} \frac{\mathbf{S}^2}{2!}\hat{C} + \mathbf{R} \frac{\mathbf{S}^3}{3!}\hat{C} + \cdots \tag{B14}
\]

\[
= \mathbf{R}e^{\mathbf{S}}\hat{C}. \tag{B15}
\]

Since the Hamiltonian $\hat{H}' = \mathbf{R}e^{\mathbf{S}}\hat{C}$ is still a real linear combination of $\{\hat{q}_j, \hat{p}_j, \hat{q}_k, \hat{p}_k\}$, the recovery operator $e^{-i\hat{H}' t}$
can be expressed as

\[
e^{-i\hat{H}' t} = e^{-i(a\hat{q}_j + b\hat{p}_j)t} \otimes e^{-i(c\hat{q}_k + d\hat{p}_k)t} \tag{B16}
\]

\[
= \hat{D}_j\left(-\frac{b+ia}{\sqrt{2}}\right) \otimes \hat{D}_k\left(-\frac{d+ic}{\sqrt{2}}\right)  \tag{B17}
\]

for certain $a, b, c, d \in \mathbb{R}$.

## Appendix C: Recovery operator of a gate teleportation with beam splitter

In this context, we demonstrate that when a displacement operator $\hat{D}_j(\mu)$ evolves in the Heisenberg picture under a balanced beam splitter $\hat{B}^\dagger_{j,k}$, denoted as $\hat{B}_{j,k} \hat{D}_j(\mu)\hat{B}^\dagger_{j,k}$, the resulting product of displacement operators on the $j$-th and $k$-th modes can be directly inferred from the input.
Recall that $\gamma_{j,k} = \mathrm{tr} \left( \rho (\hat{x}^T \hat{x})_{j,k} \right)$ and a Gaussian channel $\mathcal{G}^{\mathbf{M},\mathbf{N}} : \gamma \mapsto \mathbf{M} \gamma \mathbf{M}^T + \mathbf{N}$. Thus a Gaussian channel $\mathcal{G}^{\mathbf{M},0}$
can be expressed as

\[
\mathcal{G}^{\mathbf{M},0} : \mathrm{tr} \big( \rho (\hat{x}^T \hat{x})_{j,k} \big) \mapsto \mathrm{tr} \big( \rho ( \mathbf{M} \hat{x}^T (\mathbf{M} \hat{x}^T)^T )_{j,k} \big),  \tag{C2}
\]

Or equivalently, $\mathcal{G}^{\mathbf{M},0}$ can be simply expressed as $\mathcal{G}^{\mathbf{M},0} : \hat{x}^T \rightarrow \mathbf{M} \hat{x}^T$.
For a beam splitter $\hat{B}_{j,k}^{\dagger}$, the corresponding Gaussian channel is $\mathcal{G}^{\mathbf{B}^{-1},0}_{j,k}$, i.e.

\[
(\hat{q}_j, \hat{p}_j, \hat{q}_k, \hat{p}_k)^T \mapsto \mathbf{B}_{j,k}^T(\hat{q}_j, \hat{p}_j, \hat{q}_k, \hat{p}_k)^T \tag{C3}
\]

\[
= \frac{1}{\sqrt{2}} (\hat{q}_j - \hat{q}_k, \hat{p}_j - \hat{p}_k, \hat{q}_j + \hat{q}_k, \hat{p}_j + \hat{p}_k)^T. \tag{C4}
\]

Therefore, the evolution of a displacement operator $\hat{D}_j(\mu)$ becomes

\[
e^{i\sqrt{2} (\mathrm{Im}\{\mu\} \hat{q}_j + \mathrm{Re}\{\mu\} \hat{p}_j)} \mapsto e^{i(\mathrm{Im}\{\mu\} (\hat{q}_j - \hat{q}_k) + \mathrm{Re}\{\mu\} (\hat{p}_j - \hat{p}_k))} \tag{C5}
\]

\[
\hat{D}_j(\mu) \mapsto \hat{D}_j\left( \frac{\mu}{\sqrt{2}} \right) \otimes \hat{D}_k\left( -\frac{\mu}{\sqrt{2}} \right). \tag{C6}
\]

## Appendix D: Proof of Lemma 2

Let $\hat{W} = \sum_{m', n' = 0}^\infty \alpha_{m' n'} |m' \rangle \langle n'|$ be an expansion of $\hat{W}$ in the number basis. By a direct calculation, one can show that

\[
\hat{I} \otimes \hat{W} \sum_{n'=0}^\infty |n'\rangle |n'\rangle
= \hat{W}^T \otimes \hat{I} \sum_{n'=0}^\infty |n'\rangle |n'\rangle.
\]

Thus, the assertion holds.

\[
\hat{W}_{j,k_1} = \sum_{n',m',n'',m''=0}^{\infty} \alpha_{n' m' n'' m''} |m'\rangle_j \langle n'| \otimes |m''\rangle_{k_1} \langle n''|
\]

be an expansion of \(\hat{W}\) in the number basis. Then

\[
\hat{W}_{i,k_1}^T = \sum_{n',m',n'',m''=0}^{\infty} \alpha_{n' m' n'' m''} |n'\rangle_i \langle m'| \otimes |m''\rangle_{k_1} \langle n''|.
\]

By a direct calculation, one can demonstrate that

\[
\hat{I}_i \otimes \hat{W}_{j,k_1} \sum_{n'=0}^{\infty} |n'\rangle_i |n'\rangle_j \otimes |\psi\rangle_{k_1,\ldots,k_n} = \hat{W}_{i,k_1}^T \otimes \hat{I}_j \sum_{n'=0}^{\infty} |n'\rangle_i |n'\rangle_j \otimes |\psi\rangle_{k_1,\ldots,k_n}.
\]

---

In this section, we explicitly derive \( N^{\mathrm{polar}}(r) \) in Theorem 5. We have shown in Eq.(15) that

\[
\mathcal{G}_{1,2}^{N^{\mathrm{polar}}(r)} = \mathcal{G}_{1,2}^{\sigma^2 Z_2 BZ_2 S(-r)S^T(-r)Z_2 B^T Z_2}.
\]

Therefore,
\[
N^{\mathrm{polar}}(r) = \sigma^2 Z_2 BZ_2 S(-r)S^T(-r)Z_2 B^T Z_2
\]
\[
= \frac{\sigma^2}{2}
\begin{bmatrix}
I & 0 \\ 0 & Z
\end{bmatrix}
\begin{bmatrix}
I & I \\ -I & I
\end{bmatrix}
\begin{bmatrix}
I & 0 \\ 0 & Z
\end{bmatrix}
\begin{bmatrix}
\cosh(-r)I & \sinh(-r)Z \\ \sinh(-r)Z & \cosh(-r)I
\end{bmatrix}^2
\begin{bmatrix}
I & 0 \\ 0 & Z
\end{bmatrix}
\begin{bmatrix}
I & -I \\ I & I
\end{bmatrix}
\begin{bmatrix}
I & 0 \\ 0 & Z
\end{bmatrix}
\]
\[
= \frac{\sigma^2}{2}
\begin{bmatrix}
I & Z \\ -Z & I
\end{bmatrix}
\begin{bmatrix}
\cosh(-r)I & \sinh(-r)Z \\ \sinh(-r)Z & \cosh(-r)I
\end{bmatrix}^2
\begin{bmatrix}
I & -Z \\ Z & I
\end{bmatrix}
\]
\[
= \frac{\sigma^2}{2}
\begin{bmatrix}
I & Z \\ -Z & I
\end{bmatrix}
\begin{bmatrix}
(\cosh^2(-r) + \sinh^2(-r))I & 2 \cosh(-r)\sinh(-r) Z \\
2 \cosh(-r)\sinh(-r) Z & (\cosh^2(-r) + \sinh^2(-r))I
\end{bmatrix}
\begin{bmatrix}
I & -Z \\ Z & I
\end{bmatrix}
\]
\[
= \frac{\sigma^2}{2}
\begin{bmatrix}
(\cosh(-r) + \sinh(-r))^2 I & (\cosh(-r) + \sinh(-r))^2 Z \\
-(\cosh(-r) - \sinh(-r))^2 Z & (\cosh(-r) - \sinh(-r))^2 I
\end{bmatrix}
\begin{bmatrix}
I & -Z \\ Z & I
\end{bmatrix}
\]
\[
= \frac{\sigma^2}{2}
\begin{bmatrix}
e^{-2r}I & e^{-2r}Z \\
-e^{2r}Z & e^{2r}I
\end{bmatrix}
\begin{bmatrix}
I & -Z \\ Z & I
\end{bmatrix}
\]
\[
= \sigma^2
\begin{bmatrix}
e^{-2r} I & 0 \\
0 & e^{2r} I
\end{bmatrix}.
\tag{F1}
\]

---

Let \( |\Psi_0\rangle := |\zeta_r\rangle_{i,j} \otimes |\psi\rangle_{k_1,\ldots,k_n} \), and \( \rho_0 := |\Psi_0\rangle \langle\Psi_0| \). Now, define

\[
|\Psi_L\rangle := (\hat{I}_i \otimes \hat{W}_{j,k_1}) |\Psi_0\rangle,
\]
\[
|\Psi_R\rangle := (\hat{W}_{i,k_1}^T \otimes \hat{I}_j) |\Psi_0\rangle, \tag{G1}
\]

and let \( \rho_L = |\Psi_L\rangle \langle\Psi_L| \), \( \rho_R = |\Psi_R\rangle \langle\Psi_R| \). Hence, the fidelity between \( \rho_L \) and \( \rho_R \) is

\[
= \left| \operatorname{tr}(\rho_0 \hat{V}) \right|^2. \tag{G2}
\]

1.
\[
|\psi\rangle_{k_1,...,k_n} = \sum_{n_{k_1},...,n_{k_n}=0}^{\infty} \beta_{n_{k_1},...,n_{k_n}} |n_{k_1},\ldots,n_{k_n}\rangle
\]

2.
be an expansion of \(\hat{W}\) in the number basis. Then \(\hat{V}\) can be explicitly expanded in the number basis

3.
Suppose that

4.
where \(\beta_{n_{k_1},\ldots,n_{k_n}} \in \mathbb{C}\). Therefore,

5.
\[
\hat{W}_{j,k_1} = \sum_{n',m',n'',m''=0}^{\infty} \alpha_{n' m' n'' m''} |m'\rangle_j \langle n'| \otimes |m''\rangle_{k_1} \langle n''|
\]

6.
\[
\hat{V} = \sum_{n',m',n'',m'',n''',m''',n''''=0}^{\infty} \alpha_{n' m' n'' m_{k_1}} \alpha_{n''' m''' n'''' n''} |n'\rangle_i\langle m'| \otimes |m'''\rangle_j\langle n''| \otimes |m_{k_1}\rangle_{k_1} \langle n''''|
\]

7.
\[
\text{tr}\left(\rho_0 \hat{V}\right) = \sum_{n',m',n'',m'',n_{k_1},m_{k_1},\ldots,n_{k_n},m_{k_n}=0}^{\infty} \frac{\tanh^{m'+n'}(r)}{\cosh^2(r)} \alpha_{n' m' n'' n_{k_1}} \alpha_{m' n' m_{k_1} n''} \beta_{m_{k_1},...,n_{k_n}} \beta^*_{n_{k_1},...,n_{k_n}}. \tag{G3}
\]

8.
where \(\hat{V} = \left(\hat{W}^{T}_{i,k_1} \otimes \hat{I}_j\right)^\dagger \left(\hat{I}_i \otimes \hat{W}_{j,k_1}\right)\). Let

9.
\[
\frac{\tanh^{m'+n'}(r)}{\cosh^2(r)} \alpha_{n' m' n'' n_{k_1}} \alpha_{m' n' m_{k_1} n''} \beta_{m_{k_1},...,n_{k_n}} \beta^*_{n_{k_1},...,n_{k_n}}. \tag{G3}
\]