Matrix Elements of Local Operators in the Excitation Ansatz Overview of the ProofOverview of the Proof

QUANTUM ERROR-DETECTION AT LOW ENERGIES

2.6 AQEDC at Low Energies: The Excitation Ansatz

2.6.4 Matrix Elements of Local Operators in the Excitation Ansatz Overview of the ProofOverview of the Proof

The norm of the vector (2.6.3) can be bounded as

kΨ𝑗 , 𝑝(𝛼, 𝛽) k²=

= tr

𝐸^𝑗−1𝐸

𝐵(𝑝)𝐵(𝑝)𝐸^𝐿−𝑗(|𝛽ih𝛼| ⊗ |𝛽ih𝛼|)

≤ k𝐸

𝐵(𝑝)𝐵(𝑝)k𝐹 · k𝐸^𝑗−1k𝐹 · k𝐸^𝐿−𝑗k𝐹

≤ k𝐸

𝐵(𝑝)𝐵(𝑝)k𝐹 .

In the first inequality, we have used (2.5), together with the fact that k |𝛽ih𝛼| ⊗ |𝛽ih𝛼| k𝐹 =1.

In the second inequality, we have used Lemma 2.4.2, along with the fact 𝜌(𝐸) =1.

The claim (2.6.3) follows from this.

With a completely analogous proof, we also have k𝐸_𝐹(𝑗 , 𝑝) k𝐹 ≤ 𝐷²k𝐹k√︃

k𝐸

𝐵(𝑝)𝐵(𝑝)k𝐹, and k𝐸_𝐹k𝐹 ≤ 𝐷²k𝐹k, which are claims (2.6.3) and (2.6.3).

2.6.4 Matrix Elements of Local Operators in the Excitation Ansatz

for different momenta 𝑝 ≠ 𝑝⁰. For this purpose, we need to identify the leading order term in the expressionh𝜙_𝑝|𝐹|𝜙_𝑝i. Higher order terms are again small by the properties of the transfer operator.

To establish these bounds, first observe that an unnormalized excitation ansatz state

|Φ𝑝(𝐵;𝐴)iis a superposition of the “position space” states{|Φ𝑗 , 𝑝i}^𝑛_𝑗=

1, where each state|Φ𝑗 , 𝑝iis given by a simple tensor network with an “insertion” of an operator at site 𝑗⁰. Correspondingly, we first study matrix elements of the formhΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i. Bounds on these matrix elements are given in Lemma 2.6.5. The idea of the proof of this statement is simple: in the tensor network diagram for the matrix element, subdiagrams associated with powers 𝐸^Δ with sufficiently largeΔmay be replaced by the diagram associated with the map |𝑟iihhℓ|, with an error scaling term scaling as𝑂(𝜆^Δ/2

2 ). This is due to the Jordan decomposition of the transfer operator. Thanks to the gauge condition (2.6.1), the resulting diagrams then simplify, allowing us to identify the leading order term.

To realize this approach, a key step is to identify suitable subdiagrams corresponding to powers𝐸^Δ in the diagram associated withhΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i. These are associated with connected regions of sizeΔwhere the operator𝐹 acts trivially, and there is no insertion of 𝐵(𝑝) (respectively𝐵(𝑝⁰)), meaning that 𝑗 and 𝑗⁰do not belong to the region. Lemma 2.6.5 provides a careful case-by-case analysis depending on, at the coarsest level of detail, whether or not 𝑗 and 𝑗⁰belong to aΔ-neighborhood of the support of𝐹.

Some subleties that arise are the following: to obtain estimates on the leading-order terms for the diagonal matrix elements (see (2) above) as well as related expressions, a bound on the magnitude of the matrix elementhΦ𝑗 , 𝑝⁰|𝐹|Φ𝑗 , 𝑝ionly is not sufficient.

The lowest-order approximating expression to hΦ𝑗 , 𝑝⁰|𝐹|Φ𝑗 , 𝑝iobtained by making the above substitutions of the transfer operators a priori seems to depend on the exact site location 𝑗. This is awkward because the termhΦ𝑗 , 𝑝⁰|𝐹|Φ𝑗 , 𝑝iappears as a summand (with sum taken over 𝑗) when computing matrix elements of excitation ansatz states. We argue that in fact, the leading order term of hΦ𝑗 , 𝑝⁰|𝐹|Φ𝑗 , 𝑝i is identical for all values of 𝑗 not belonging to the support of 𝐹. This statement is formalized in Lemma 2.6.6 and allows us to subsequently estimate sums of interest without worry about the explicit dependence on 𝑗.

Finally, we require a strengthening of the estimates obtained in Lemma 2.6.5 because we are ultimately interested in excitation ansatz states: these are superpositions of

the states |Φ𝑗 , 𝑝i, with phases of the form 𝑒^{𝑖 𝑝 𝑗}. Estimating only the magnitude of matrix elements of the form hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i is not sufficient to establish our results. Instead, we need to treat the phases “coherently,” which leads to certain cancellations. The corresponding statement is given in Lemma 2.6.7.

The Proof

We will envision the sites {1, . . . , 𝑛}as points on a ring, i.e., using periodic boundary conditions, and measure the distance between sites 𝑗 , 𝑗⁰by

dist(𝑗 , 𝑗⁰) :=min

𝑘∈Z

|𝑗− 𝑗⁰+𝑘 ·𝑛|. ForΔ∈ {0, . . . , 𝑛} and a subsetF ⊂ {1, . . . , 𝑛}, let

B^Δ(F )= {𝑗 ∈ {1, . . . , 𝑛} | ∃ 𝑗⁰ ∈ F such that^dist(𝑗 , 𝑗⁰) ≤Δ}

be theΔ-thickeningofF.

We say that 𝑗⁰∈ {1, . . . , 𝑛}is aleft neighbor of(or isleft-adjacent to) 𝑗 ∈ {1, . . . , 𝑛} if 𝑗⁰ = 𝑗 −1 for 𝑗 > 1, or 𝑗⁰ = 𝑛for 𝑗 = 1. A connected region R ⊂ {1, . . . , 𝑛} is said to lie on the left of (or be left-adjacent to) 𝑗 ∈ {1, . . . , 𝑛} if it is of the formR = {𝑗

1, . . . , 𝑗_𝑟}, with 𝑗_𝛼+

1left-adjacent to 𝑗_𝛼 for𝛼 ∈ {0, . . . , 𝑟 −1} with the convention that 𝑗

0 = 𝑗_𝑟. Analogous definitions hold for right-adjacency.

For an operator 𝐹 acting on (C^p)^⊗𝑛, let ^supp(𝐹) ⊂ {1, . . . , 𝑛} denote its support, i.e., the sites of the system that the operator acts on non-trivially. We say that𝐹 is 𝑑-local if |supp(𝐹) | = 𝑑. Let us assume that ^supp(𝐹) decomposes into 𝜅 disjoint connected components

supp(𝐹) =

𝜅−1

Ø

𝛼=0

F𝛼 . (2.56)

We may, without loss of generality, assume that this gives a partition of{1, . . . , 𝑛} into disjoint connected sets

{1, . . . , 𝑛}=A₀∪ F₀∪ A₁∪ F₁∪ · · · ∪ A𝜅−1∪ F𝜅−1

whereA𝛼is left-adjacent toF𝛼for𝛼∈ {0, . . . , 𝜅−1},A𝛼+1is right-adjacent toF𝛼

for𝛼 ∈ {0, . . . , 𝜅−2}, andA₀ is right-adjacent toF𝜅−1. We may then decompose the operator𝐹 as

𝐹 =∑︁

𝑖 𝜅−1

Ì

𝛼=0

(𝐼_A

𝛼 ⊗𝐹_𝑖,𝛼),

𝐹 =

𝐹(𝜏(𝑗

1))=𝐹(𝜏(𝑗

4)) =𝐹(0) = ,

𝐹(𝜏(𝑗

2)) =𝐹(1) = ,

𝐹(𝜏(𝑗

3)) =𝐹(𝜅−1) = .

Figure 2.6: Example for𝐹and sites𝑗

1, 𝑗

2, 𝑗

3, 𝑗

4 ∈ {1, . . . , 𝑛}with𝜄(𝑗

1) =𝜄(𝑗

4) =7, 𝜄(𝑗

2) =19, and𝜄(𝑗

3) =35.

where we write𝐹 as a sum of decomposable tensor operators (indexed by𝑖), with each𝐹_𝑖,𝛼 being an operator acting on the componentF𝛼.

Let us define a function𝜏 :{1, . . . , 𝑛}\supp(𝐹) → {0, . . . , 𝜅−1}which associates to every site 𝑗 ∉ supp(F ) the unique index 𝜏(𝑗) for the component A_𝜏(𝑗) of the complement of^supp(𝐹)such that 𝑗 ∈ A𝜏(𝑗).

It is also convenient to introduce the following operators {𝐹(𝜏)}^𝜅−1

𝜏=0. The operator 𝐹(𝜏)is obtained by removing the identity factor on the sitesA𝜏 of𝐹, and cyclically permuting the remaining components in such a way that F𝜏 ends up on the sites {1, . . . ,|F𝜏|}. More precisely, we define𝐹(𝜏) ∈ B ( (C^p)^{⊗(𝑛−|A𝜏|)})by

𝐹(𝜏) =∑︁

𝑖

𝐹_𝑖,𝜏 ⊗

𝜏+𝜅−1

Ì

𝛼=𝜏+1

𝐼

⊗|A𝛼(mod𝜅)|

C^p ⊗ 𝐹_{𝑖,𝛼 (}

mod𝜅)

!

, (2.57)

for𝜏 ∈ {0, . . . , 𝜅−1}. We note that 𝑗 ↦→ 𝐹(𝜏(𝑗)) associates a permuted operator to each site 𝑗 not belonging to the support of 𝐹. Let us also define𝜄(𝑗) to be the index of the site which gets cyclically shifted to the first site when defining 𝐹_𝜏(𝑗). An example is shown diagrammatically in Figure 2.6.

For two excitation ansatz states |Φ𝑝iand |Φ𝑝⁰i, and an operator 𝐹 on (C^p)^⊗𝑛, we

may write the corresponding matrix element as hΦ𝑝⁰|𝐹|Φ𝑝i =

𝑛

∑︁

𝑗 , 𝑗⁰=1

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i, (2.58)

where|Φ𝑗 , 𝑝iare the “position space” states introduced in Equation (2.6.1). We are interested in bounding the magnitude of this quantity.

We begin by bounding the individual terms in the sum (2.6.4).

Lemma 2.6.5. Let 𝑗 , 𝑗⁰ ∈ {1, . . . , 𝑛} and let 𝑝, 𝑝⁰be arbitrary non-zero momenta.

Consider the states|Φ𝑗 , 𝑝iand|Φ𝑗⁰, 𝑝⁰idefined by(2.6.1). LetΔ = Δ(𝑛)and𝑑 =𝑑(𝑛) be monotonically increasing functions of𝑛. Suppose further that we have

10Δ𝑑 < 𝑛 .

Assume 𝐹 is a 𝑑-local operator of unit norm on (C^p)^⊗𝑛 whose support has 𝜅 connected components as in(2.6.4). Then we have the following:

(i) There is some fixed𝑞 ∈ [𝑛]such that for all 𝑗 , 𝑗⁰∈ B^Δ(supp(𝐹)), we have hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hhℓ|𝐸

𝐼^⊗Δ⊗𝐹𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii +𝑂(𝜆^Δ

2), where 𝑗ˆ= 𝑗 −𝜄(𝑞) +Δ+1(mod𝑛)and 𝑗ˆ⁰= 𝑗⁰−𝜄(𝑞) +Δ+1 (mod𝑛).

Furthermore,

hΦ𝑗⁰, 𝑝⁰|Φ𝑗 , 𝑝i = Δ𝑗 , 𝑗⁰𝑐_{𝑝 𝑝}0+𝑂(𝜆^Δ

2). (2.59)

(ii) If 𝑗 , 𝑗⁰∉B^Δ(supp(𝐹)), then (a) |hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|=𝑂(𝜆^Δ/2

2 )if 𝑗 ≠ 𝑗⁰.

(b) hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=hhℓ|𝐸_{𝐹(𝜏(𝑗))}|𝑟ii ·𝑐_{𝑝 𝑝}0+𝑂(𝜆^Δ/2

2 ).

Here the operator𝐹(𝜏(𝑗))is defined by Equation(2.6.4). (iii) If 𝑗 ∈ B^Δ(supp(𝐹)) and 𝑗⁰∉B^Δ(supp(𝐹)), then

(a) |hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|=𝑂(𝜆^Δ/2

2 )if 𝑗⁰∉B^2Δ(supp(𝐹)).

(b) There exists some fixed𝑞 ∈ [𝑛] such that, for all 𝑗 ∈ B^Δ(supp(𝐹)) and 𝑗⁰ ∈ B^2Δ(supp(𝐹))\B^Δ(supp(𝐹)), we have

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hhℓ|𝐸_{𝐹(𝜏(𝑞))}(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰,2Δ) |𝑟ii +𝑂(𝜆^2Δ

2 ), where 𝑗ˆ= 𝑗−𝜄(𝑞) +2Δ+1(mod𝑛) and 𝑗ˆ⁰= 𝑗⁰−𝜄(𝑞) +2Δ+1 (mod𝑛).

(iv) If 𝑗⁰∈ B^Δ(supp(𝐹))and 𝑗 ∉B^Δ(supp(𝐹)), then (a) |hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|=𝑂(𝜆^Δ/2

2 )if 𝑗 ∉B^2Δ(supp(𝐹)).

(b) There exists some fixed𝑞 ∈ [𝑛] such that, for all 𝑗⁰∈ B^Δ(supp(𝐹))) and 𝑗 ∈ B^2Δ(supp(𝐹))\B^Δ(supp(𝐹)), we have

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hhℓ|𝐸_{𝐹(𝜏(𝑞))}(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰,2Δ) |𝑟ii +𝑂(𝜆^2Δ

2 ), where 𝑗ˆ= 𝑗−𝜄(𝑞) +2Δ+1(mod𝑛) and 𝑗ˆ⁰= 𝑗⁰−𝜄(𝑞) +2Δ+1 (mod𝑛). Proof. For the proof of (i), suppose that 𝑗 , 𝑗⁰ ∈ B^Δ(supp(𝐹)). Pick any site 𝑞 ∉B^2Δ(supp(𝐹)). We note that such a site always exists since

|B^2Δ(supp(𝐹)) | ≤5Δ|supp(𝐹) | =5Δ𝑑 <10Δ𝑑 < 𝑛 by assumption. Let us define the shifted indices

ˆ

𝑗 = 𝑗 −𝜄(𝑞) +Δ+1(mod𝑛), and 𝑗ˆ⁰= 𝑗⁰−𝜄(𝑞) +Δ+1 (mod𝑛). Then we may write

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=tr(𝐸_𝐹(𝑗 , 𝑝, 𝑗⁰, 𝑝⁰))

=tr

𝐸^𝑠𝐸

𝐼^⊗Δ⊗𝐹_𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰)

(2.60) where 𝑠 ≥ 2Δ. This is because by the choice of 𝑞, there are at least 2Δsites not belonging to ^supp(𝐹) both on the left and the right of 𝑞. Each of these 4Δsites contributes a factor𝐸 =𝐸_𝐼 (i.e., a single transfer operator) to the expression within the trace. The term 𝐸

𝐼^⊗Δ⊗𝐹𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) incorporates Δ of the associated transfer operators𝐸 =𝐸_𝐼 on the left- and right of𝑞, respectively, such that at least 2Δfactors of𝐸 remain. By the cyclicity of the trace, these can be consolidated into a single term𝐸^𝑠 with 𝑠 ≥ 2Δ. The operator 𝐼^⊗Δ ⊗ 𝐹_𝜏(𝑞) ⊗ 𝐼^⊗Δ (i.e., the additional 𝐼^⊗Δfactors) in the term𝐸

𝐼^⊗Δ⊗𝐹_𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰)is used to ensure that 𝑗and 𝑗⁰are correctly “retained” when going from the first to the second line in (2.6.4). Inserting the Jordan decomposition𝐸 =|𝑟iihhℓ| ⊕𝐸˜, we obtain

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=hhℓ|𝐸

𝐼^⊗Δ⊗𝐹_𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii +tr

˜ 𝐸^𝑠𝐸

𝐼^⊗Δ⊗𝐹_𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) . (2.61)

By Lemma 2.4.2(ii) and Lemma 2.6.3, we have the bound

tr

˜ 𝐸^𝑠𝐸

𝐼^⊗Δ⊗𝐹_𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰)

≤ k𝐸˜^𝑠k𝐹 · k𝐸

𝐼^⊗Δ⊗𝐹_𝜏(𝑞)⊗𝐼^⊗Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) k𝐹

≤ 𝜆^𝑠/2

2 ·𝐷²k𝐹k ·√︃

k𝐸

𝐵(𝑝⁰)𝐵(𝑝⁰)k𝐹k𝐸

𝐵(𝑝⁰)𝐵(𝑝)k𝐹

=𝑂(𝜆^Δ

2), where we have used the fact that𝜆

𝑠/2 2 ≤ 𝜆^Δ

2 in the last line. We have also absorbed the dependence on the constants 𝐷, k𝐹k, and

√︃k𝐸

𝐵(𝑝⁰)𝐵(𝑝⁰)k𝐹k𝐸

𝐵(𝑝⁰)𝐵(𝑝)k𝐹 into the big-O notation. Inserting this into (2.6.4) gives the first claim of (i).

Now consider the inner product hΦ𝑗⁰, 𝑝⁰|Φ𝑗 , 𝑝i = tr(𝐸(𝑗 , 𝑝, 𝑗⁰, 𝑝⁰)), which corre- sponds to the case where 𝐹 is the identity. By the cyclicity of the trace, this can be written as hΦ𝑗⁰, 𝑝⁰|Φ𝑗 , 𝑝i = tr(𝐸^𝑠𝐸(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰)) for some 𝑠 ≥ 2Δ and suitably defined ˆ𝑗 ,𝑗ˆ⁰. Repeating the same argument as above and using the fact that

hhℓ|𝐸(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii = Δ_{𝑗 ,ˆ𝑗ˆ}0𝑐_{𝑝 𝑝}0 = Δ𝑗 , 𝑗⁰𝑐_{𝑝 𝑝}0

by definition of 𝐸(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰), Equation (2.4.1) (i.e., the fact that |ℓii and |𝑟ii are left, respectively right, eigenvectors of𝐸), and the gauge identities (2.6.1) of𝐸_𝐵(𝑝) and𝐸

𝐵(𝑝), we obtain the claim (i).

Now consider claim (ii). Suppose that 𝑗 , 𝑗⁰ ∉ B^Δ(supp(𝐹)). We consider the following two cases:

(iia) If 𝑗 ≠ 𝑗⁰, then there is a connected region of at least Δ sites not belonging to^supp(𝐹)to either the left of 𝑗⁰and not containing 𝑗, or the left of 𝑗 and not containing 𝑗⁰. Without loss of generality, we assume the former is the case.

By the cyclicity of the trace, we may also assume without loss of generality that 𝑗⁰= Δ+1, 𝑗 > 𝑗⁰, and that𝐹is supported on the sites{2Δ+2, . . . , 𝑛}. Let

ˆ

𝐹denote the restriction of𝐹to the sites{Δ+2, . . . , 𝑛}, and let ˆ𝑗 := 𝑗− (Δ+1). Then we may write

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=tr

𝐸^Δ𝐸

𝐵(𝑝⁰)𝐸

ˆ

𝐹(𝑗 , 𝑝ˆ ) . Substituting the Jordan decomposition𝐸^Δ =|𝑟iihhℓ| ⊕𝐸˜^Δ, we have

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hhℓ|𝐸

𝐵(𝑝⁰)𝐸

ˆ

𝐹(𝑗 , 𝑝ˆ ) |𝑟ii +tr

˜ 𝐸^Δ𝐸

𝐵(𝑝⁰)𝐸

ˆ 𝐹(𝑗 , 𝑝ˆ )

.

Since we assume that𝑝 ≠ 0, the gauge condition (2.6.1) states thathhℓ|𝐸

𝐵(𝑝) =0, hence the first term vanishes and it follows that

|hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|= tr

˜ 𝐸^Δ𝐸

𝐵(𝑝⁰)𝐸

ˆ

𝐹(𝑗 , 𝑝ˆ )

≤ k𝐸˜^Δk𝐹 · k𝐸

𝐵(𝑝⁰)k𝐹 · k𝐸

ˆ

𝐹(𝑗 , 𝑝ˆ ) k𝐹

≤𝜆^Δ/2

2 k𝐸

𝐵(𝑝⁰)k𝐹 ·𝐷²k𝐹ˆk√︃

k𝐸

𝐵(𝑝)𝐵(𝑝)k𝐹

=𝑂(𝜆^Δ/2

2 ) ,

as claimed in (iia). In the last line, we have again absorbed the constants into the big-𝑂-expression. This proves part (iia) of Claim (ii).

(iib) If 𝑗 = 𝑗⁰, then there are at leastΔsites to the left and right of 𝑗 which do not belong to^supp(𝐹). Therefore we may write

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=tr

𝐸^𝑠𝐸

𝐵(𝑝⁰)𝐵(𝑝)𝐸^𝑡𝐸_{𝐹(𝜏(𝑗))}

,

where𝑠 and𝑡 are integers greater thanΔ, representing the sites surrounding 𝑗 which are not in the support of𝐹.

Applying the Jordan decomposition 𝐸^Δ = |𝑟iihhℓ| ⊕ 𝐸˜^Δ twice (for 𝐸^𝑠 and 𝐸^𝑡) then gives four terms

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hhℓ|𝐸

𝐵(𝑝⁰)𝐵(𝑝)|𝑟iihhℓ|𝐸_{𝐹(𝜏(𝑗))}|𝑟ii +tr

|𝑟iihhℓ|𝐸

𝐵(𝑝⁰)𝐵(𝑝)𝐸˜^𝑠𝐸_{𝐹(𝜏(𝑗))}

+tr

˜ 𝐸^𝑡𝐸

𝐵(𝑝⁰)𝐵(𝑝)|𝑟iihhℓ|𝐸_{𝐹(𝜏(𝑗))}

+tr

˜ 𝐸^𝑡𝐸

𝐵(𝑝⁰)𝐵(𝑝)𝐸˜^𝑠𝐸_{𝐹(𝜏(𝑗))}

.

Since 𝑠 and 𝑡 are both larger than Δ, by the same arguments from before, it is clear that the last three terms can each be bounded by𝑂(𝜆^Δ/2

2 ). The claim follows sincehhℓ|𝐸

𝐵(𝑝⁰)𝐵(𝑝)|𝑟ii=𝑐_{𝑝 𝑝}0.

Next, we give the proof of claim (iii). Let us consider the situation where 𝑗 ∈ B^Δ(supp(𝐹)) and 𝑗⁰ ∉B^Δ(supp(𝐹)). The proof of the other setting is analogous.

We consider two cases:

(iiia) Suppose 𝑗⁰∉B^2Δ(supp(𝐹)). Let us define the shifted index ˆ𝑗 = 𝑗−𝜄(𝑗⁰) +Δ+ 1(mod𝑛). Then we may write

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i =tr

𝐸^𝑠𝐸

𝐵(𝑝⁰)𝐸^𝑡𝐸

𝐼^⊗Δ⊗𝐹(𝜏(𝑗⁰))⊗𝐼^⊗Δ(𝑗 , 𝑝ˆ ) ,

where𝑠and𝑡are integers larger thanΔ, representing the number of sites adjacent to 𝑗⁰ on the left and right which are not in B^Δ(supp(𝐹)). We use the Jordan decomposition𝐸 =|𝑟iihhℓ| ⊕𝐸˜ on𝐸^𝑠 to get

tr

𝐸^𝑠𝐸

𝐵(𝑝⁰)𝐸^𝑡𝐸

𝐼^⊗Δ⊗𝐹(𝜏(𝑗⁰))⊗𝐼^⊗Δ(𝑗 , 𝑝ˆ )

=hhℓ|𝐸

𝐵(𝑝⁰)𝐸^𝑡𝐸

𝐼^⊗Δ⊗𝐹(𝜏(𝑗⁰))⊗𝐼^⊗Δ(𝑗 , 𝑝ˆ ) |𝑟ii +tr

˜ 𝐸^𝑠𝐸

𝐵(𝑝⁰)𝐸^𝑡𝐸

𝐼^⊗Δ⊗𝐹(𝜏(𝑗⁰))⊗𝐼^⊗Δ(𝑗 , 𝑝ˆ )

=tr

˜ 𝐸^𝑠𝐸

𝐵(𝑝⁰)𝐸^𝑡𝐸

𝐼^⊗Δ⊗𝐹(𝜏(𝑗⁰))⊗𝐼^⊗Δ(𝑗 , 𝑝ˆ ) ,

where the first term vanishes due to the gauge condition (2.6.1). From Lemma 2.4.2(ii) we have k𝐸^𝑡k𝐹 ≤ 1, and repeating the same arguments as before, we get the bound

|hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|= tr

˜ 𝐸^𝑠𝐸

𝐵(𝑝⁰)𝐸^𝑡𝐸

𝐼^⊗Δ⊗𝐹(𝜏(𝑗⁰))⊗𝐼^⊗Δ(𝑗 , 𝑝ˆ )

≤ k𝐸˜^𝑠k𝐹 · k𝐸

𝐵(𝑝⁰)k𝐹 · k𝐸^𝑡k𝐹· 𝐷²k𝐹k · k𝐸

𝐵(𝑝)𝐵(𝑝)k𝐹

≤𝜆

𝑠/2 2 k𝐸

𝐵(𝑝⁰)k𝐹 · 𝐷²k𝐹k√︃

k𝐸

𝐵(𝑝)𝐵(𝑝)k𝐹 . Since𝑠 ≥ Δ, we conclude that

|hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|=𝑂

𝜆^Δ/2

2

.

(iiib) Suppose now that 𝑗⁰ ∈ B^2Δ(supp(𝐹)). Then by repeating the argument for case (i), withΔreplaced by 2Δ, we obtain

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hhℓ|𝐸

𝐼^⊗2Δ⊗𝐹(𝜏(𝑞))⊗𝐼^⊗2Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii +𝑂(𝜆^2Δ

2 ),

where we now have𝑞 ∉B^4Δ(F ). Again, the existence of such a𝑞is guaranteed by the condition 10Δ𝑑 < 𝑛.

We note that (iv) follows immediately from (iii) by interchanging the roles of (𝑗 , 𝑝)and (𝑗⁰, 𝑝⁰). Note that we can write

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=hΦ𝑗 , 𝑝|𝐹^†|Φ𝑗⁰, 𝑝⁰i.

The last expression within the parentheses is precisely what we had calculated in (iii), so this implies the following:

(iva) If 𝑗 ∉B^2Δ(supp(𝐹)), then hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i

=

hΦ𝑗 , 𝑝|𝐹^†|Φ𝑗⁰, 𝑝⁰i

=𝑂(𝜆^Δ/2

2 ),

where we note that the exact same bound holds for𝐹and𝐹^†sincek𝐹k = k𝐹^†k. (ivb) If 𝑗 ∈ B^2Δ(supp(𝐹)), then

hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i=hΦ𝑗 , 𝑝|𝐹^†|Φ𝑗⁰, 𝑝⁰i

=hhℓ|𝐸

𝐼^⊗2Δ⊗𝐹^†(𝜏(𝑞))⊗𝐼^⊗2Δ(𝑗ˆ⁰, 𝑝⁰, 𝑗 , 𝑝ˆ ) |𝑟ii +𝑂(𝜆^2Δ

2 )

=hhℓ|𝐸

𝐼^⊗2Δ⊗𝐹^†(𝜏(𝑞))⊗𝐼^⊗2Δ(𝑗ˆ⁰, 𝑝⁰, 𝑗 , 𝑝ˆ ) |𝑟ii +𝑂(𝜆^2Δ

2 )

=hhℓ|𝐸

𝐼^⊗2Δ⊗𝐹(𝜏(𝑞))⊗𝐼^⊗2Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii +𝑂(𝜆^2Δ

2 ). This proves the claim.⁸

Note that in the statement (iib), the dependence on𝑗in the expressionhhℓ|𝐸_{𝐹(𝜏(𝑗))}|𝑟ii can be eliminated as follows:

Lemma 2.6.6. Suppose 𝑗

1, 𝑗

2∉B^Δ(supp(𝐹)). Then

|hhℓ|𝐸_{𝐹(𝜏(𝑗}

1))|𝑟ii − hhℓ|𝐸_{𝐹(𝜏(𝑗}

2))|𝑟ii| =𝑂(𝜆^Δ

2). (2.62)

In particular, for any fixed 𝑗

0∉B^Δ(supp(𝐹)) we have hΦ𝑗 , 𝑝⁰|𝐹|Φ𝑗 , 𝑝i =hhℓ|𝐸_{𝐹(𝜏(𝑗}

0))|𝑟ii ·𝑐_{𝑝 𝑝}0 +𝑂(𝜆^Δ/2

2 ), (2.63)

for all 𝑗 ∉B^Δ(supp(𝐹)).

8To clarify how the termhhℓ|𝐸_𝐼⊗Δ⊗𝐹^†(𝜏(𝑞)) ⊗𝐼^⊗Δ(𝑗ˆ⁰, 𝑝⁰,𝑗 , 𝑝ˆ ) |𝑟iiis complex conjugated, first write

hhℓ|𝐸_𝐼⊗Δ⊗𝐹^†(𝜏(𝑞)) ⊗𝐼^⊗Δ(𝑗ˆ⁰, 𝑝⁰,𝑗 , 𝑝ˆ ) |𝑟ii=hΦ^𝐿

ˆ

𝑗⁰, 𝑝⁰|𝐼⊗𝐼^⊗2Δ⊗𝐹^†

𝜏(𝑞)⊗𝐼^⊗2Δ⊗𝐼|Φ^𝐿

ˆ 𝑗 , 𝑝i, where|Φ^𝐿

ˆ

𝑗 , 𝑝iare the states defined by (2.6.3), for some appropriate length𝐿. Then we can proceed to conjugate the matrix element, giving us

hΦ^𝐿

ˆ

𝑗⁰, 𝑝⁰|𝐼⊗𝐼^⊗2Δ⊗𝐹^†

𝜏(𝑞)⊗𝐼^⊗2Δ⊗𝐼|Φ^𝐿

ˆ

𝑗 , 𝑝i=hΦ^𝐿

ˆ

𝑗 , 𝑝|𝐼⊗𝐼^2Δ⊗𝐹_𝜏(𝑞)⊗𝐼^2Δ⊗𝐼|Φ^𝐿

ˆ 𝑗⁰, 𝑝⁰i

=hhℓ|𝐸_𝐼⊗2Δ⊗𝐹(𝜏(𝑞)) ⊗𝐼^⊗2Δ(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii.

Proof. The claim (2.6.6) follows immediately from (2.6.6) and claim (iib) of Lemma 2.6.5 since |𝑐_{𝑝 𝑝}0|=𝑂(1).

If𝜏(𝑗

1) =𝜏(𝑗

2), there is nothing to prove. Suppose𝜏(𝑗

1) ≠𝜏(𝑗

2). Without loss of generality, assume that𝜏(𝑗

1) =0 and𝜏(𝑗

2) =𝜉. Then we may write 𝐹(𝜏(𝑗

1))=∑︁

𝑖

𝐹_𝑖,

0⊗𝐼^⊗𝑎1 ⊗𝐹_𝑖,

1⊗𝐼^⊗𝑎2· · · ⊗𝐼^{⊗𝑎𝜅−1} ⊗𝐹_𝑖,𝜅−

1, and 𝐹(𝜏(𝑗

2))=∑︁

𝑖

𝐹_𝑖,𝜉 ⊗𝐼^{⊗𝑎𝜉+1} ⊗ 𝐹_𝑖,𝜉+

1⊗ 𝐼^{⊗𝑎𝜉+2}· · · ⊗ 𝐼^⊗𝑎𝜅 ⊗ 𝐹_𝑖,𝜅−

1⊗ 𝐼^⊗𝑎0

⊗𝐹_𝑖,

0⊗𝐼^⊗𝑎1 ⊗𝐹_𝑖,

1⊗𝐼^⊗𝑎2 ⊗ · · · ⊗𝐹_𝑖,𝜉−

1, where𝑎_𝛼 =|A𝛼|for𝛼 ∈ {0, . . . , 𝜅}. Defining the operators

ˆ

𝐹_𝑖 =𝐹_𝑖,𝜉 ⊗ 𝐼^{⊗𝑎𝜉+1} ⊗𝐹_𝑖,𝜉+

1⊗ 𝐼^{⊗𝑎𝜉+2}· · · ⊗ 𝐼^{⊗𝑎𝜅−1} ⊗𝐹_𝑖,𝜅−

1, ˆ

𝐺_𝑖 =𝐹_𝑖,

0⊗ 𝐼^⊗𝑎1 ⊗ 𝐹_𝑖,

1⊗ 𝐼^⊗𝑎2 ⊗ · · · ⊗ 𝐹_𝑖,𝜉−

1, we have

𝐹(𝜏(𝑗

1)) =∑︁

𝑖

ˆ

𝐺_𝑖 ⊗𝐼^⊗𝑎𝜉 ⊗𝐹ˆ_𝑖, and 𝐹(𝜏(𝑗

2))=∑︁

𝑖

ˆ

𝐹_𝑖⊗ 𝐼^⊗𝑎0 ⊗𝐺ˆ_𝑖.

(We give an example for the operator 𝐹, 𝐹(𝜏(𝑗

1)), and 𝐹(𝜏(𝑗

2)) in Figure 2.7.) Therefore we can write

hhℓ|𝐸_{𝐹(𝜏(𝑗}

1))|𝑟ii=∑︁

𝑖

hhℓ|𝐸

ˆ 𝐺_𝑖𝐸^𝑎𝜉𝐸

ˆ 𝐹𝑖|𝑟ii, hhℓ|𝐸_{𝐹(𝜏(𝑗}

2))|𝑟ii=∑︁

𝑖

hhℓ|𝐸

ˆ 𝐹𝑖

𝐸^𝑎0𝐸

ˆ 𝐺_𝑖|𝑟ii. Inserting the Jordan decomposition𝐸 = |𝑟iihhℓ| ⊕𝐸˜ gives

hhℓ|𝐸_{𝐹(𝜏(𝑗}

1))|𝑟ii=∑︁

𝑖

hhℓ|𝐸

ˆ

𝐺_𝑖|𝑟iihhℓ|𝐸

ˆ

𝐹𝑖|𝑟ii + hhℓ|𝐸

ˆ 𝐺_𝑖𝐸˜^𝑎𝜉𝐸

ˆ 𝐹𝑖|𝑟ii

, hhℓ|𝐸_{𝐹(𝜏(𝑗}

2))|𝑟ii=∑︁

𝑖

hhℓ|𝐸

ˆ

𝐹𝑖|𝑟iihhℓ|𝐸

ˆ

𝐺_𝑖|𝑟ii + hhℓ|𝐸

ˆ 𝐹𝑖𝐸˜^𝑎0𝐸

ˆ 𝐺_𝑖|𝑟ii

. Taking the difference, the first terms of the sums cancel, and we are left with hhℓ|𝐸_{𝐹(𝜏(𝑗}

1))|𝑟ii − hhℓ|𝐸_{𝐹(𝜏(𝑗}

2))|𝑟ii =

∑︁

𝑖

hhℓ|𝐸

ˆ 𝐺_𝑖𝐸˜^𝑎𝜉𝐸

ˆ

𝐹𝑖|𝑟ii −∑︁

𝑖

hhℓ|𝐸

ˆ 𝐹𝑖𝐸˜^𝑎0𝐸

ˆ 𝐺_𝑖|𝑟ii

≤

∑︁

𝑖

hhℓ|𝐸

ˆ 𝐺𝑖𝐸˜^𝑎𝜉𝐸

ˆ 𝐹𝑖|𝑟ii

+

∑︁

𝑖

hhℓ|𝐸

ˆ 𝐹𝑖𝐸˜^𝑎0𝐸

ˆ 𝐺𝑖|𝑟ii

. (2.64)

𝐹 = ,

𝐹(𝜏(𝑗

1)) = ,

𝐹(𝜏(𝑗

2)) = .

Figure 2.7: Example for the operator 𝐹 and the corresponding 𝐹(𝜏(𝑗

1)) and 𝐹(𝜏(𝑗

2)).

We can bound the first term

Í

𝑖hhℓ|𝐸

ˆ 𝐺_𝑖𝐸˜^𝑎𝜉𝐸

ˆ 𝐹𝑖|𝑟ii

as follows. First, we write

∑︁

𝑖

hhℓ|𝐸

ˆ 𝐺𝑖

˜ 𝐸^𝑎𝜉𝐸

ˆ 𝐹𝑖

|𝑟ii

=tr 𝐸˜^𝑎𝜉

∑︁

𝑖

𝐸𝐹ˆ𝑖

|𝑟iihhℓ|𝐸

ˆ 𝐺𝑖

!

≤ k𝐸˜^𝑎𝜉k𝐹

∑︁

𝑖

𝐸𝐹ˆ_𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺𝑖

𝐹

≤𝜆^Δ

2

∑︁

𝑖

𝐸𝐹ˆ𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺_𝑖

𝐹

, where the last inequality comes from the fact that 𝑗

2 ∉B^Δ(supp(𝐹)) and 𝑗

2 ∈ A𝜉

implies that 𝑎_𝜉 ≥ 2Δ, so Lemma 2.4.2(ii) gives k𝐸˜^𝑎𝜉k𝐹 ≤ 𝜆^Δ

2. Proceeding as we did in the proof of Lemma 2.6.3, we can write the latter Frobenius norm as

∑︁

𝑖

𝐸𝐹ˆ_𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺𝑖

2

𝐹

=

𝐷

∑︁

𝛼1,𝛼

2, 𝛽

1, 𝛽

2=1

h𝛼

1|h𝛼

2| ∑︁

𝑖

𝐸𝐹ˆ_𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺𝑖

!

|𝛽

1i|𝛽

2i

2

.

The individual terms in the sum can be depicted diagrammatically as h𝛼

1|h𝛼

2| ∑︁

𝑖

𝐸𝐹ˆ𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺_𝑖

!

|𝛽

1i|𝛽

2i = .

Defining the vectors

|Ψ(𝛼, 𝛽)i = ,

we can then write h𝛼

1|h𝛼

2| ∑︁

𝑖

𝐸𝐹ˆ𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺_𝑖

!

|𝛽

1i|𝛽

2i =hΨ(𝛼

1, 𝛽

1) | ∑︁

𝑖

ˆ

𝐹_𝑖 ⊗ 𝐼_𝐷 ⊗ 𝐼_𝐷 ⊗𝐺ˆ_𝑖

!

|Ψ(𝛼

2, 𝛽

2)i.

Applying the Cauchy-Schwarz inequality, we get

h𝛼

1|h𝛼

2| ∑︁

𝑖

𝐸𝐹ˆ𝑖

|𝑟iihhℓ|𝐸

ˆ 𝐺𝑖

!

|𝛽

1i|𝛽

2i

2

≤ kΨ(𝛼

1, 𝛽

1) k²· kΨ(𝛼

2, 𝛽

2) k²·

∑︁

𝑖

ˆ

𝐹_𝑖 ⊗𝐼_𝐷 ⊗𝐼_𝐷 ⊗𝐺ˆ_𝑖

2

.

The norm of the vector|Ψ(𝛼, 𝛽)iis given by

kΨ(𝛼, 𝛽) k²= = = h𝛼|𝑟|𝛼ih𝛽|ℓ|𝛽i,

where in the second equality, we have used the fixed-point equations (2.4.1). There- fore we have

∑︁

𝑖

𝐸𝐹ˆ_𝑖|𝑟iihhℓ|𝐸

ˆ 𝐺𝑖

2

𝐹

≤

∑︁

𝑖

ˆ

𝐹_𝑖 ⊗𝐼_𝐷 ⊗𝐼_𝐷 ⊗𝐺ˆ_𝑖

2 𝐷

∑︁

𝛼1,𝛼

2, 𝛽

1, 𝛽

2=1

h𝛼

1|𝑟|𝛼

1ih𝛼

2|𝑟|𝛼

2ih𝛽

1|ℓ|𝛽

1ih𝛽

2|ℓ|𝛽

2i

=

∑︁

𝑖

ˆ

𝐹_𝑖 ⊗𝐼_𝐷 ⊗𝐼_𝐷 ⊗𝐺ˆ_𝑖

2

· |tr(𝑟)tr(ℓ) |²= 𝐷²

∑︁

𝑖

ˆ

𝐹_𝑖⊗ 𝐼_𝐷 ⊗ 𝐼_𝐷 ⊗𝐺ˆ_𝑖

2

,

where the last equality follows from the fact that we gauge-fix the left and right fixed-points such that𝑟 =𝐼

C^𝐷 and tr(ℓ) =1. Finally, we note that since the operator

norm is multiplicative over tensor products, i.e., k𝐴⊗𝐵k =k𝐴k · k𝐵k, we have

∑︁

𝑖

ˆ

𝐹_𝑖⊗ 𝐼_𝐷 ⊗ 𝐼_𝐷 ⊗𝐺ˆ_𝑖

=

∑︁

𝑖

ˆ 𝐹_𝑖 ⊗𝐺ˆ_𝑖

=k𝐹k. Therefore, we have

∑︁

𝑖

hhℓ|𝐸

ˆ 𝐺𝑖

˜ 𝐸^𝑎𝜉𝐸

ˆ 𝐹𝑖|𝑟ii

≤ 𝐷k𝐹k𝜆^Δ

2. The term involving𝑎

0in (2.6.4) can be bounded identically, and so hhℓ|𝐸

𝐹(𝜏(𝑗

1))|𝑟ii − hhℓ|𝐸

𝐹(𝜏(𝑗

2))|𝑟ii

≤2𝐷k𝐹k𝜆^Δ

2 , which proves (2.6.6).

We also need a different version of statement (i), as well as statements (iiib) and (ivb) derived from it.

Lemma 2.6.7. ForΩ ⊂ [𝑛]², let us define 𝜎_{𝑝 𝑝}0(Ω) = ∑︁

(𝑗 , 𝑗⁰)∈Ω

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i.

Let us write F :=supp(𝐹) andA^𝑐 = [𝑛]\A for the complement of a subsetA ⊂ [𝑛]. Then:

𝜎_{𝑝 𝑝}0(B^Δ(F ) × B^Δ(F ))

≤ |B^Δ(F ) | · k𝐹k√

𝑐_𝑝𝑐_𝑝0+𝑂 √

𝑛 𝜆^Δ/2

2

, (2.65)

𝜎_{𝑝 𝑝}0(B^Δ(F ) × B^Δ(F )^𝑐)

≤ |B^2Δ(F ) | · k𝐹k√

𝑐_𝑝𝑐_𝑝0 +𝑂

𝑛²𝜆^Δ/2

2

, (2.66)

𝜎_{𝑝 𝑝}0(B^Δ(F )^𝑐× B^Δ(F ))

≤ |B^2Δ(F ) | · k𝐹k√

𝑐_𝑝𝑐_𝑝0 +𝑂

𝑛²𝜆^Δ/2

2

. (2.67) Finally, we have the following: There exists some fixed 𝑗

0 ∈ [𝑛]such that for 𝑝= 𝑝⁰, we have

𝜎_{𝑝 𝑝}(B^Δ(F )^𝑐× B^Δ(F )^𝑐) =|B^Δ(F )^𝑐| · hhℓ|𝐸_𝐹

𝜏(𝑗

0)|𝑟ii𝑐_𝑝+𝑂

𝑛²𝜆^Δ/2

2

.(2.68) For 𝑝 ≠ 𝑝⁰, we have

𝜎_{𝑝 𝑝}0(B^Δ(F )^𝑐× B^Δ(F )^𝑐)

≤ |B^Δ(F ) | · k𝐹k√

𝑐_𝑝𝑐_𝑝0+𝑂

𝑛²𝜆^Δ/2

2

. (2.69) We observe that the first expression on the right-hand side of the above bound scales linearly with the support size of F instead of the support size of F^𝑐, as may be naively expected. For (2.6.7), this is due to a cancellation of phases, see (2.6.4) below.

Proof. For the proof of (2.6.7), let us first define the vectors

|Ψ(𝑝)i = ∑︁

𝑗∈B^Δ(F )

𝑒^{𝑖 𝑝 𝑗}|Φ𝑗 , 𝑝i. Then we can write

|𝜎_{𝑝 𝑝}0(B^Δ(F ) × B^Δ(F )) | = |hΨ(𝑝⁰) |𝐹|Ψ(𝑝)i|

≤ k𝐹k · kΨ(𝑝) k · kΨ(𝑝⁰) k , (2.70) where the last inequality follows by Cauchy-Schwarz along with the definition of the operator normk𝐹k. The vector norm is given by

kΨ(𝑝) k² = ∑︁

𝑗 , 𝑗⁰∈B^Δ(F )

𝑒^{𝑖 𝑝(𝑗−𝑗}

0)hΦ𝑗⁰, 𝑝|Φ𝑗 , 𝑝i,

and together with Equation (i), we get

kΨ(𝑝) k²= |B^Δ(F ) | ·𝑐_𝑝+𝑂(𝜆^Δ/2

2 ) . Taking the square root and inserting into Equation (2.6.4), we get

|𝜎_{𝑝 𝑝}0(B^Δ(F ) × B^Δ(F )) | =k𝐹k

√︃

|B^Δ(F ) | ·𝑐_𝑝+𝑂(𝜆^Δ/2

2 ) √︃

|B^Δ(F ) | ·𝑐_𝑝0 +𝑂(𝜆^Δ/2

2 )

=|B^Δ(F ) | · k𝐹k√

𝑐_𝑝𝑐_𝑝0 +𝑂

√︁|B^Δ(F ) | ·𝜆^Δ/2

2

. Using the bound

B^Δ(F )

≤ 5𝑑Δ< 𝑛gives (2.6.7).

Next, let us look at (2.6.7). We have 𝜎_{𝑝 𝑝}0(B^Δ(F ) × B^Δ(F )^𝑐) = ∑︁

𝑗∈B^Δ(F )

∑︁

𝑗⁰∈B^Δ(F )^𝑐

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i= Σ₁+Σ₂,

where we define

Σ₁ := ∑︁

𝑗∈B^Δ(F )

∑︁

𝑗⁰∈B^2Δ(F )\B^Δ(F )

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i,

and

Σ₂:= ∑︁

𝑗∈B^Δ(F )

∑︁

𝑗⁰∈B^2Δ(F )^𝑐

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i.

The norm of the second sum can be bounded using Lemma 2.6.5(iiia), giving us

|Σ₂| ≤ ∑︁

𝑗∈B^Δ(F )

∑︁

𝑗⁰∈B^2Δ(F )^𝑐

|hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i|

≤ |B^Δ(F ) | · |B^2Δ(F )^𝑐| ·𝑂(𝜆^Δ/2

2 )

=𝑂(𝑛²𝜆^Δ/2

2 ) , (2.71)

where we again use the trivial bound

B^Δ(F ) ,

B^2Δ(F )^𝑐

≤ 𝑛in the last line. Using Lemma 2.6.5 (iiib), we can express the first sum, with some fixed𝑞 ∈ [𝑛], as

Σ₁= ∑︁

𝑗∈B^Δ(F )

∑︁

𝑗⁰∈B^2Δ(F )\B^Δ(F )

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hhℓ|𝐸_{𝐹(𝜏(𝑞))}(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii + |B^Δ(F ) | · |B^2Δ(F )\B^Δ(F ) | ·𝑂(𝜆^Δ/2

2 )

= ∑︁

𝑗∈B^Δ(F )

∑︁

𝑗⁰∈B^2Δ(F )\B^Δ(F )

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hhℓ|𝐸_{𝐹(𝜏(𝑞))}(𝑗 , 𝑝,ˆ 𝑗ˆ⁰, 𝑝⁰) |𝑟ii +𝑂

𝑛²𝜆^Δ/2

2

, where the indices ˆ𝑗 and ˆ𝑗⁰are defined as in Lemma 2.6.5. To bound the remaining sum, let us introduce the states

|Ψ₁(𝑝)i := ∑︁

𝑗∈B^Δ(F )

𝑒^{𝑖 𝑝 𝑗}|Φ^𝐿

ˆ

𝑗 , 𝑝i, and

|Ψ₂(𝑝⁰)i := ∑︁

𝑗⁰∈B^2Δ(F )\B^Δ(F )

𝑒^{𝑖 𝑝}

0𝑗⁰

|Φ^𝐿

ˆ 𝑗⁰, 𝑝⁰i,

where we set𝐿 =|supp(𝐹(𝜏(𝑞))) |. Here,|Φ^𝐿_{𝑗 , 𝑝}iare as defined in (2.6.3). Then we can write

Σ₁=hΨ₂(𝑝⁰) |𝐹(𝜏(𝑞)) |Ψ₁(𝑝)i +𝑂

𝑛²𝜆^Δ/2

2

.

By the Cauchy-Schwarz inequality and the orthogonality relations (2.6.4), we have

|hΨ₂(𝑝⁰) |𝐹(𝜏(𝑞)) |Ψ₁(𝑝)i| ≤ k𝐹k · kΨ₁(𝑝) k · kΨ₂(𝑝⁰) k

= k𝐹k

√︃

𝑐_𝑝𝑐_𝑝0|B^Δ(F ) | · |B^2Δ(F )\B^Δ(F ) | , where we bound the states|Ψ₁,2(𝑝)iin exactly the same way as we did in the proof of (2.6.7). Using the fact that|B^Δ(F ) |,|B^2Δ(F )\B^Δ(F ) | ≤ |B^2Δ(F ) |, we conclude that

|Σ₁| ≤ |B^2Δ(F ) | · k𝐹k√

𝑐_𝑝𝑐_𝑝0+𝑂

𝑛²𝜆^Δ/2

2

.

Combining this with (2.6.4) gives the claim (2.6.7). The proof of (2.6.7) is analo- gous, using Lemma 2.6.5(iv).

Finally, consider (2.6.7) and (2.6.7). We have 𝜎_{𝑝 𝑝}0(B^Δ(F )^𝑐× B^Δ(F )^𝑐) = ∑︁

𝑗∈B^Δ(F )^𝑐

𝑒^{𝑖 𝑗(𝑝−𝑝}

0)hΦ𝑗 , 𝑝⁰|𝐹|Φ𝑗 , 𝑝i

| {z }

=:Θ₁

+ ∑︁

𝑗 , 𝑗⁰∈B^Δ(F )^𝑐 𝑗≠𝑗⁰

𝑒^{𝑖(𝑝 𝑗−𝑝}

0𝑗⁰)hΦ𝑗⁰, 𝑝⁰|𝐹|Φ𝑗 , 𝑝i

| {z }

=:Θ₂

.

Using Lemma 2.6.5(iia), we have

|Θ₂| ≤ k𝐹k ·𝑂(𝑛²𝜆^Δ/2

2 ). (2.72)

On the other hand, by Lemma 2.6.5(iib), or more precisely its refinement in the form of Equation (2.6.6) from Lemma 2.6.6, we have

Θ₁=©

­

«

∑︁

𝑗∈B^Δ(F )^𝑐

𝑒^{𝑖 𝑗(𝑝−𝑝}

0)ª

®

¬

hhℓ|𝐸_{𝐹(𝜏(𝑗}

0))|𝑟ii𝑐_{𝑝 𝑝}0 +𝑂(𝑛𝜆^Δ/2

2 ) for some fixed 𝑗

0 ∈ B^Δ(F )^𝑐. For 𝑝⁰ = 𝑝, the sum above is given trivially by Í

𝑗∈B^Δ(F )^𝑐1=

B^Δ(F )^𝑐

. For𝑝 ≠ 𝑝⁰, we haveÍ

𝑗∈[𝑛]𝑒^{𝑖 𝑗(𝑝−𝑝}

0) =0, and hence

∑︁

𝑗∈B^Δ(F )^𝑐

𝑒^{𝑖 𝑗(𝑝−𝑝}

0)

=

∑︁

𝑗∈B^Δ(F )

𝑒^{𝑖 𝑗(𝑝−𝑝}

0)

≤ |B^Δ(F ) |. (2.73)

Therefore, for 𝑝 =𝑝⁰, we have Θ₁=

B^Δ(F )^𝑐

hhℓ|𝐸_{𝐹(𝜏(𝑗}

0))|𝑟ii𝑐_𝑝+𝑂(𝑛𝜆^Δ/2

2 ) , and for𝑝 ≠ 𝑝⁰, we have

|Θ₁| ≤

B^Δ(F )

hhℓ|𝐸_{𝐹(𝜏(𝑗}

0))|𝑟ii𝑐_{𝑝 𝑝}0+𝑂(𝑛𝜆^Δ/2

2 )

≤

B^Δ(F )

· k𝐹k𝑐_{𝑝 𝑝}0+𝑂(𝑛𝜆^Δ/2

2 ). Note that we also have𝑐_{𝑝 𝑝}0 ≤ √

𝑐_𝑝𝑐_𝑝0by the Cauchy-Schwarz inequality. Combining these results with (2.6.4) proves claims (2.6.7) and (2.6.7).