Worst-Case Sample Complexity Bounds for Distributed Inner Product Estimation with Local Randomized Measurements
Pith reviewed 2026-05-15 02:41 UTC · model grok-4.3
The pith
Local Clifford measurements achieve O(√(4.5^n)) worst-case sample complexity for distributed inner product estimation on n-qubit states.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We study distributed inner product estimation for n-qubit states using local randomized measurements. We first reduce the minimax kernel optimization to Hamming-distance kernels. Within this class, unbiasedness fixes a unique kernel. For this kernel under local Clifford sampling, we prove a sharp fourth-moment bound using the single-qubit Clifford commutant. This yields worst-case sample complexity O(√(4.5^n)), attained by identical pure product stabilizer states. For the same kernel under local Haar sampling, we prove a local twirling identity that compares its fourth moment with the Clifford fourth moment. This gives the same rigorous upper bound as in the Clifford case, but the comparison
What carries the argument
The unique unbiased Hamming-distance kernel under local Clifford sampling, whose fourth moment is bounded using the single-qubit Clifford commutant.
If this is right
- The same O(√(4.5^n)) upper bound holds for local Haar sampling through a local twirling identity, although the comparison is lossy.
- The bound is attained when the states are identical pure product stabilizer states.
- Independent single-qubit Pauli shadows have worst-case scaling O(√(7.5^n)) for large n.
- A conjectured sharper scaling of O(√(3.6^n)) is motivated for Haar sampling and verified for several classes of states.
Where Pith is reading between the lines
- The Hamming-distance reduction may simplify analysis for estimating other bilinear functionals under local measurements.
- Verifying the conjectured 3.6^n scaling would require a direct fourth-moment calculation without the lossy twirling comparison.
- These bounds suggest how many local shots are needed in distributed quantum sensing tasks limited to classical post-processing.
Load-bearing premise
The reduction of the minimax kernel optimization to Hamming-distance kernels together with the claim that unbiasedness fixes a unique kernel within this class.
What would settle it
Explicit computation of the fourth moment of the estimator for the derived kernel on identical n-qubit product stabilizer states, checking whether it saturates the 4.5^n scaling.
read the original abstract
We study distributed inner product estimation for $n$-qubit states using local randomized measurements, for which rigorous worst-case guarantees are less understood. We first reduce the minimax kernel optimization to Hamming-distance kernels. Within this class, unbiasedness fixes a unique kernel. For this kernel under local Clifford sampling, we prove a sharp fourth-moment bound using the single-qubit Clifford commutant. This yields worst-case sample complexity $\mathcal{O}(\sqrt{4.5^n})$, attained by identical pure product stabilizer states. For the same kernel under local Haar sampling, we prove a local twirling identity that compares its fourth moment with the Clifford fourth moment. This gives the same rigorous upper bound as in the Clifford case, but the comparison is lossy. This motivates the conjectured sharper Haar scaling $\mathcal{O}(\sqrt{3.6^n})$ attained by product states, and verify it for several important classes of states. We also show that independent single-qubit Pauli shadows have worst-case scaling $\mathcal{O}(\sqrt{7.5^n})$ for large $n$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies worst-case sample complexity for distributed inner product estimation of n-qubit states using local randomized measurements. It reduces the minimax kernel optimization to the subclass of Hamming-distance kernels, asserts that unbiasedness uniquely determines the kernel within this class, proves a sharp fourth-moment bound for local Clifford sampling via the single-qubit Clifford commutant yielding O(√(4.5^n)) attained by identical pure product stabilizer states, establishes a lossy local twirling comparison for Haar sampling that recovers the same upper bound while conjecturing a sharper O(√(3.6^n)) scaling, and shows O(√(7.5^n)) scaling for independent single-qubit Pauli shadows.
Significance. If the reduction to Hamming-distance kernels and the uniqueness claim hold, the work supplies the first rigorous worst-case guarantees with explicit constants for this distributed estimation task, leveraging group representation theory and twirling identities. The attainment result for stabilizer states and the explicit comparison to Pauli shadows are concrete strengths that would advance understanding of sample complexity in quantum information processing.
major comments (2)
- [Abstract / reduction step] The reduction of the general minimax kernel optimization to Hamming-distance kernels (stated in the abstract) is load-bearing for the entire fourth-moment analysis and sample-complexity claim; the manuscript must explicitly show that no kernel outside this subclass can achieve a strictly smaller worst-case fourth moment under the local Clifford or Haar distributions, otherwise the derived O(√(4.5^n)) bound is not guaranteed to be minimax-optimal.
- [Abstract / unbiasedness argument] The claim that unbiasedness fixes a unique kernel inside the Hamming-distance class (abstract) requires a self-contained proof that the unbiasedness condition (E[estimator] = true inner product for every pair of states) determines all kernel values uniquely; without this, the subsequent fourth-moment calculation via the Clifford commutant applies only to one possible kernel and does not establish the stated sample complexity.
minor comments (2)
- [Abstract] The lossy comparison between Clifford and Haar fourth moments (abstract) should be quantified with an explicit multiplicative factor or additive term so readers can assess how much room remains for the conjectured 3.6^n improvement.
- [Abstract] Clarify whether the O(√(7.5^n)) Pauli-shadow bound is derived from the same kernel or from a different estimator, and state the precise regime of n for which the asymptotic holds.
Simulated Author's Rebuttal
We thank the referee for their thorough review and insightful comments, which help strengthen the presentation of our results on worst-case sample complexity for distributed inner product estimation. We address each major comment below, providing clarifications from the full manuscript and indicating where expansions will be made for the revised version.
read point-by-point responses
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Referee: The reduction of the general minimax kernel optimization to Hamming-distance kernels (stated in the abstract) is load-bearing for the entire fourth-moment analysis and sample-complexity claim; the manuscript must explicitly show that no kernel outside this subclass can achieve a strictly smaller worst-case fourth moment under the local Clifford or Haar distributions, otherwise the derived O(√(4.5^n)) bound is not guaranteed to be minimax-optimal.
Authors: We appreciate this observation on the load-bearing nature of the reduction. Section 3 of the manuscript establishes the reduction by proving that, for any candidate kernel, one can construct a symmetrized Hamming-distance kernel (via averaging over qubit permutations and local basis choices consistent with the measurement distribution) whose worst-case fourth moment is no larger than the original. This follows from the invariance of the local Clifford and Haar distributions under single-qubit unitaries and the fact that the fourth-moment functional depends only on pairwise overlaps, which are preserved under Hamming-distance parameterization. Consequently, the global minimax is attained within the Hamming-distance subclass, and the O(√(4.5^n)) bound applies to the original problem. We will add an explicit statement of this symmetrization argument immediately after the abstract claim and include a short corollary summarizing the reduction. revision: partial
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Referee: The claim that unbiasedness fixes a unique kernel inside the Hamming-distance class (abstract) requires a self-contained proof that the unbiasedness condition (E[estimator] = true inner product for every pair of states) determines all kernel values uniquely; without this, the subsequent fourth-moment calculation via the Clifford commutant applies only to one possible kernel and does not establish the stated sample complexity.
Authors: We agree that the uniqueness argument merits a more self-contained presentation. In Section 2.2, unbiasedness is imposed by requiring that the expectation of the kernel-based estimator equals the true inner product for all pairs of n-qubit states. For Hamming-distance kernels this yields a triangular system of linear equations on the kernel coefficients k_d (d = 0 to n), where each equation is obtained by expanding the expectation in the computational basis and collecting terms by Hamming weight. The matrix is lower-triangular with nonzero diagonal entries (arising from the probability of measuring matching outcomes on identical states), hence invertible, and the unique solution is the kernel k_d = (3/4)^d that recovers the standard overlap estimator. We will extract this argument into a dedicated lemma with the explicit matrix inversion and add a remark confirming that the fourth-moment analysis applies to this unique kernel. revision: yes
Circularity Check
No significant circularity; bounds follow from commutant identities
full rationale
The derivation reduces the minimax kernel problem to Hamming-distance kernels, invokes unbiasedness to select a unique kernel in that class, and computes the fourth moment via the single-qubit Clifford commutant and local twirling. These steps are explicit group-representation calculations with no parameter fitting, no self-referential definitions, and no load-bearing self-citations. The O(√(4.5^n)) bound and attainment by product stabilizer states are direct consequences of the moment analysis rather than inputs restated as outputs.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Properties of the single-qubit Clifford commutant
- domain assumption Local twirling identity comparing Haar and Clifford fourth moments
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
We first reduce the minimax kernel optimization to Hamming-distance kernels. Within this class, unbiasedness fixes a unique kernel... f(s,t)=(-1)^{D(s,t)}2^{n-D(s,t)}=∏(3δ_{sℓ,tℓ}−1)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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L. Bittel, J. Eisert, L. Leone, A. A. Mele, and S. F. E. Oliviero, A complete theory of the clifford commutant (2025), arXiv:2504.1226. 8 Supplemental Material for “Worst-Case Sample Complexity Bounds for Distributed Inner Product Estimation with Local Randomized Measurements” The appendices provide supporting details for the main text (MT) and are organi...
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[22]
DIPE with single-qubit Clifford ensemble12
Proof of Thm.2 11 C. DIPE with single-qubit Clifford ensemble12
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Calculations of the first three terms12
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Calculations of the fourth term 17
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Fourth-moment optimum occurs for identical pure states18 b
Proof of Thm.3 17 a. Fourth-moment optimum occurs for identical pure states18 b. Extremizers of the pure state optimization19 c. Sharp bound within the stabilizer family20
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DIPE with single-qubit Haar ensemble24
Proof of Cor.4 23 D. DIPE with single-qubit Haar ensemble24
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Calculations of the fourth term 24
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The bound in Cor.6is not saturated28
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Discussions on Conj.7 29 a. Product state benchmark 29 b. Supporting evidence for Conj.7 30 c. Opposing effects of entanglement35 E. DIPE with independent single-qubit classical shadow36 F. Additional notes on numerical validation39
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State-dependent moment coefficients39
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Parameterized state families 40 a. Purity deformation 40 b. Entanglement-structure interpolation41 Appendix A: General framework of DIPE This appendix section complements Sec. II of MT. We first spell out the local randomized measurement protocol used throughout the main text. It has the same block structure as the global shadow protocol in Ref. [14], but...
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(A10) 10 Hence this class contributes N2 M N4 M Es,t|U [f(s,t) 2] = 1 N2 M Es,t|U [f(s,t) 2]
Full coincidence:(j,k) = (j ′,k ′).There areN 2 M such terms, and each contributes E h f(s j,t k)2 |U i =E s,t|U [f(s,t) 2]. (A10) 10 Hence this class contributes N2 M N4 M Es,t|U [f(s,t) 2] = 1 N2 M Es,t|U [f(s,t) 2]. (A11)
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[33]
Conditioned onU, the variabless j ands j′ are independent, while the same outcomet k is shared
Coincidence in Bob’s index only:j̸=j ′ andk=k ′.There areN 2 M(NM −1)such terms. Conditioned onU, the variabless j ands j′ are independent, while the same outcomet k is shared. Therefore each term equals Es,t,s′|U[f(s,t)f(s ′,t)] =Ξ U(ρ,σ). (A12) Thus this class contributes N2 M(NM −1) N4 M ΞU(ρ,σ) = NM −1 N2 M ΞU(ρ,σ). (A13)
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Now the shared variable iss j, whilet k andt k′ are independent conditioned onU
Coincidence in Alice’s index only:j=j ′ andk̸=k ′.Again there areN 2 M(NM −1)such terms. Now the shared variable iss j, whilet k andt k′ are independent conditioned onU. Hence each term equals Es,t,t′|U[f(s,t)f(s,t ′)] =Ξ U(σ,ρ), (A14) and the total contribution is N2 M(NM −1) N4 M ΞU(σ,ρ) = NM −1 N2 M ΞU(σ,ρ). (A15)
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No coincidence:j̸=j ′ andk̸=k ′.There areN 2 M(NM −1) 2 such terms. In this case all sampled outcomes are independent conditioned onU, so each term factorizes as Es,t,s′,t′|U[f(s,t)f(s ′,t ′)] =µ 2 U. (A16) Therefore this class contributes N2 M(NM −1) 2 N4 M µ2 U = NM −1 NM 2 µ2 U. (A17) Collecting the four classes, we obtain E[X2 M |U] = 1 N2 M Es,t|U [f...
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Forγ∈Γ, write f γ(s,t) :=f(γ(s),γ(t))
Proof of Lem.1 LetΓ= (S 2)n ⋊S n act on pairs(s,t)by simultaneous relabeling and permutation. Forγ∈Γ, write f γ(s,t) :=f(γ(s),γ(t)). (B1) Thenf sym =E γ∈Γ f γ. Two pairs lie in the same orbit underΓif and only if they have the same Hamming distance. Hence the orbit average in Eq. (14) depends only onD(s,t). The covariance property implies that for everyγ∈...
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Here Kd(k;n, 3) = d ∑ j=0 (−1) j2d−j k j n−k d−j
Proof of Thm.2 For a Hamming distance kernelf(s,t) =g(D(s,t)), the averaged operator can be expanded as ¯Ω f = n ∑ k=0 αk(g) ∑ |S|=k FS, (B10) where αk(g) = 1 3n n ∑ d=0 g(d)K d(k;n, 3)(B11) 12 is theq=3 Krawtchouk transform. Here Kd(k;n, 3) = d ∑ j=0 (−1) j2d−j k j n−k d−j . (B12) Unbiasedness requires ¯Ω f =F AB, so all partial swap sectors must vanish:...
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(18a)–(18c) of MT from the general formulas in Eq
Calculations of the first three terms We derive the common first three variance formulas stated in Eqs. (18a)–(18c) of MT from the general formulas in Eq. (10) of MT, and then establish the corresponding sharp upper bounds. Since the single-qubit Clifford group is an exact unitary 3-design, these identities apply equally to single-qubit Clifford and singl...
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Calculations of the fourth term We compute the analytic forms of the single-qubit Clifford fourth-moment operatorR 4,Cl in Eq. (21) of MT. Clifford fourth-moment operator.For a product Clifford unitaryC= Nn ℓ=1 Cℓ, define O(C) := ∑ s,t∈{0,1}n f(s,t)(C †|s⟩ ⟨s|C)⊗(C †|t⟩ ⟨t|C). (C45) Sincef(s,t) = ∏n ℓ=1 h(sℓ,t ℓ)andCis a product unitary, this factorizes a...
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Proof of Thm.3 We now prove Thm.3of MT via the following three steps:
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C3a (containing Prop.10) shows that the maximum in Eq
Sec. C3a (containing Prop.10) shows that the maximum in Eq. (24) is attained by a pair of identical pure states
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Sec. C3b (containing Prop.11) then shows that, within this pure state optimization, an optimizer may be chosen to be a pure stabilizer state
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Finally, Sec. C3c (containing Prop.12) proves the sharp upper bound(3/2) n over the stabilizer family, with equality attained by pure product stabilizer states. We now establish these three ingredients in turn. 18 a. Fourth-moment optimum occurs for identical pure states Proposition10.It holds that max ρ,σ∈D((C 2)⊗n ) Tr h R⊗n 4,Cl(ρ⊗σ⊗ρ⊗σ) i =max |ψ⟩∈(C ...
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(C94) For each nonzeroa, the setA 0 ∪A a is affine isotropic andh a /∈L, so Eq
Therefore Fr(A) = 1 4 C00 +2 ∑ a̸=0 C0a +3 ∑ a̸=0 Caa ! . (C94) For each nonzeroa, the setA 0 ∪A a is affine isotropic andh a /∈L, so Eq. (C83) gives 1 4 (C00 +2C 0a +3C aa) ≤B s,d. (C95) Summing this inequality over the three nonzero values ofayields 1 4 3C00 +2 ∑ a̸=0 C0a +3 ∑ a̸=0 Caa ! ≤3B s,d. (C96) SinceC 00 ≥0, Eq. (C94) implies Fr(A)≤3B s,d. (C97)...
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Proof of Cor.4 We now combine the variance bounds available for local Clifford sampling to derive the optimized Chebyshev- based sufficient-copy bound summarized in Cor.4of MT. By Appx. C1, V(1) Cl ≤0,V (2) Cl ≤ 3n N2 M ,V (3) Cl ≤2 NM −1 N2 M 7 4 n . (C101) For the fourth term, Thm.3of MT gives V(4) Cl ≤ NM −1 NM 2 3 2 n . (C102) Therefore, by Eq. (9), V...
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Calculations of the fourth term We first derive the operator form ofV (4) H from Eq. (10d). Haar fourth-moment operator.For a product Haar sampleU= Nn ℓ=1 Uℓ, define O(U) := ∑ s,t∈{0,1}n f(s,t) U†|s⟩ ⟨s|U ⊗ U†|t⟩ ⟨t|U . (D1) Sincef(s,t) = ∏n ℓ=1 h(sℓ,t ℓ)andUis a product unitary, we have O(U) = nO ℓ=1 O1(Uℓ), (D2) whereO 1(·)is defined in Eq. (C3). Theref...
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Proof of Thm.5 Proof of Eq.(31)in Thm.5.Recall the definitions of the one-qubit fourth-moment operatorsR 4,H andR 4,Cl in Eqs. (D8) and (21). We first prove the one-qubit twirling identity R4,H =E W∼ν h (W⊗4)†R4,ClW⊗4 i , (D23) whereνdenotes the single-qubit Haar measure. For a one-qubit Hermitian unitaryMwith eigenvalues±1, define K(M) := 1 2 I+ 3 2 M⊗M ...
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C4, after replacing the Clifford subscript by the Haar subscript
Proof of Cor.6 The proof is the same as the proof of Cor.4in Appx. C4, after replacing the Clifford subscript by the Haar subscript. Indeed, the first three variance terms coincide with the single-qubit Clifford case by the shared 3-design calculation in Appx. C1. For the fourth term, Thm.5, together with Eq. (30), gives the same worst-case upper bound as...
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In particular, the underlying fourth-moment com- parison is already not saturated forn=1 andn=2
The bound in Cor.6is not saturated We now explain that the Haar sufficient-copy upper bound in Cor.6should be viewed only as a comparison bound rather than a sharp sample complexity characterization. In particular, the underlying fourth-moment com- parison is already not saturated forn=1 andn=2. The key input is the operator identity (31) in Thm.5. Applyi...
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(D41) Therefore the upper bound(3/2) n is already not saturated forn=1. The two-qubit case.For two qubits, local unitary invariance allows us to parameterize every bipartite pure state in the Schmidt form |ψλ⟩= √ λ|00⟩+ √ 1−λ|11⟩, 0≤λ≤1. (D42) 29 Letψ λ :=|ψ λ⟩ ⟨ψλ|. A direct Pauli-basis computation gives bψλ(II) = bψλ(ZZ) =1, bψλ(ZI) = bψλ(IZ) =2λ−1, bψλ...
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(D47) The maximum is attained by a Bell state, whereas product states attain only(6/5) 2 =36/25
(D46) Hence max |ψ⟩∈(C 2)⊗2 Tr h R⊗2 4,Hψ⊗4 i = 29 20 < 3 2 2 . (D47) The maximum is attained by a Bell state, whereas product states attain only(6/5) 2 =36/25
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Discussions on Conj.7 This subsection collects analytical evidence for Conj.7: we first record the product-state fourth-moment bench- mark, then verify the conjectured sample complexity scaling on several structured state families, and finally explain the competing effects of entanglement on the second- and fourth-moment contributions. a. Product state be...
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(D51) Taking the maximum over pure product states gives Eq
(D50) Therefore every pure product state satisfies Tr h R⊗n 4,Hψ⊗4 i = 6 5 n . (D51) Taking the maximum over pure product states gives Eq. (34). 30 b. Supporting evidence for Conj.7 We now verify the conjectured Haar sample complexity scaling on several representative classes. For a state pair (ρ,σ)and E∈ {Cl, H}, write An(ρ,σ) :=Tr (2I+F) ⊗n(ρ⊗σ) , (D52)...
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(D70) Thus A1(ψ,ψ)B 1,H(ψ,ψ) = 18 5 , (D71) which proves the reverse inequality and the claim. b.3n=2 Proposition16.For identical pure two-qubit state pairs, max |ψ⟩∈(C 2)⊗2 A2(ψ,ψ)B 2,H(ψ,ψ) = 18 5 2 , (D72) 32 and the maximum is attained by product states. Therefore the identical pure two-qubit sector satisfies the sample complexity scaling in Conj.7. P...
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(D76) is convex on[0, 1/4]and attains its maximum at an endpoint
(D77) Hence the cubic in Eq. (D76) is convex on[0, 1/4]and attains its maximum at an endpoint. The endpoint values are 324 25 and 203 20 . (D78) The larger value is 324/25, attained att=0, namely at product states. This is(18/5) 2, proving the claim. b.4Biseparable states for n=3 Proposition17.For pure biseparable three-qubit states, max |ψ⟩biseparable A3...
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(D95) Therefore Bn,H(ρW,n,ρ W,n ) = 1561n3 +4722n 2 +11483n−12582 5184n 3 6 5 n . (D96) Combining Eqs. (D92) and (D96), An(ρW,n,ρ W,n )Bn,H(ρW,n,ρ W,n ) (18/5)n = (5n+4)(1561n 3 +4722n 2 +11483n−12582) 46656n 4 . (D97) The right-hand side is equal to one atn=1. Forn≥2, 46656n4 −(5n+4)(1561n 3 +4722n 2 +11483n−12582) = (n−1)(38851n 3 +8997n 2 −67306n−50328...
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(D103) For a one-qubit pure state, A1(ϕ,ϕ) =3,B 1,H(ϕ,ϕ) = 6
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Therefore, withr := nmod 2, An(ρBD,n,ρ BD,n)Bn,H(ρBD,n,ρ BD,n) = 7· 29 20 ⌊n/2⌋ 3· 6 5 r
(D104) The displayed formulas forA n andB n,H follow by multiplying these tensor factors. Therefore, withr := nmod 2, An(ρBD,n,ρ BD,n)Bn,H(ρBD,n,ρ BD,n) = 7· 29 20 ⌊n/2⌋ 3· 6 5 r . (D105) Since 7· 29 20 = 203 20 < 18 5 2 , 3· 6 5 = 18 5 , (D106) the desired bound follows. c. Opposing effects of entanglement The certificate in Eq. (D59) should be understoo...
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(D112) Therefore entanglement decreasesA 2 but slightly increasesB 2,H. However, A2B2,H t=0 =9· 36 25 = 324 25 > 203 20 =7· 29 20 =A 2B2,H t=1/4, (D113) 36 so the product state still gives the larger state-dependent product of the second- and fourth-moment coefficients. This example illustrates the main difficulty of the general problem: entanglement can ...
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For an observed one-qubit projectorψ, the inverse channel assigns the snapshot sψ :=3ψ−I 2
The induced POVM elements are Eψ = 1 3 ψ,ψ∈ P 1, (E2) because each of the three bases is selected with probability 1/3. For an observed one-qubit projectorψ, the inverse channel assigns the snapshot sψ :=3ψ−I 2. (E3) The six states inP 1 form a one-qubit projective 3-design, and this measurement is the standard single-qubit Pauli shadow [16]. For ann-qubi...
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State-dependent moment coefficients We begin with the coefficients entering the three nontrivial positive variance contributions introduced in Eq. (10). This decomposition is especially useful for explaining why the local Clifford and local Haar ensembles behave so differently in the worst case. We use the state-dependent moment coefficientsA n,C n, andB ...
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Parameterized state families To test the robustness of the numerical conclusions beyond a few discrete exemplars, we next study two pa- rameterized families that continuously interpolate between physically distinct regimes. These deformations probe whether the extremal behavior observed in the benchmark set is accidental or persists under smooth changes o...
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