A negative Clifford third-moment expectation

Abstract

Zhu, Mao and Yi define the third-moment expectation kappa of a normalized qudit state against a stochastic Lagrangian subspace. The pointwise bound in their Conjecture 2 fails in prime dimension eleven: a one-qudit state has expectation -1196/64000. This refutes that clause and hence the conjecture as printed; the aggregate clauses are not separately refuted.

Definition 1.1 (Stochastic Lagrangian subspaces).

$$\forall d:\mathbb{N},\forall T:\mathrm{Submodule}\left(\mathrm{ZMod}\left(d\right),((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\times ((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\right),(\mathrm{IsStochasticLagrangian}\left(d,T\right))\iff ((\forall p:((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\times ((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right)),(p\in T)\Rightarrow (\sum _{k:\mathrm{Fin}\left(3\right)}(\mathrm{fst}\left(p\right)\left(k\right)\cdot \mathrm{fst}\left(p\right)\left(k\right))-\sum _{k:\mathrm{Fin}\left(3\right)}(\mathrm{snd}\left(p\right)\left(k\right)\cdot \mathrm{snd}\left(p\right)\left(k\right))=0))\wedge ((\mathrm{finrank}\left(\mathrm{ZMod}\left(d\right),T\right)=3)\wedge (((k:\mathrm{Fin}\left(3\right)\mapsto 1),(k:\mathrm{Fin}\left(3\right)\mapsto 1))\in T)))$$

Formalization. D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.IsStochasticLagrangian (✓ std3).

Citation. H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Commentary.

For three copies, the quadratic form is x dot x minus y dot y. A stochastic Lagrangian subspace is isotropic for this form, has dimension three, and contains the pair of all-ones vectors.

Definition 1.2 (The tensor-power operator).

$$\forall d:\mathbb{N},\forall n:\mathbb{N},\forall T:\mathrm{Submodule}\left(\mathrm{ZMod}\left(d\right),((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\times ((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\right),\forall X:(\mathrm{Fin}\left(3\right))\to (\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right),\forall Y:(\mathrm{Fin}\left(3\right))\to (\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right),R\left(d,n,T,X,Y\right)=\prod _{j:\mathrm{Fin}\left(n\right)}(\mathrm{ite}\left(((k:\mathrm{Fin}\left(3\right)\mapsto X\left(k\right)\left(j\right)),(k:\mathrm{Fin}\left(3\right)\mapsto Y\left(k\right)\left(j\right)))\in T,1,0\right))$$

Formalization. D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.R (✓ std3).

Citation. H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Commentary.

The source convention is r(T) = sum over (x;y) in T of |x><y|. Regrouping r(T) tensor n into three n-qudit copies gives the matrix R: each column j contributes the indicator that its pair of three-component vectors belongs to T.

Definition 1.3 (Three copies of a pure-state density matrix).

$$\forall d:\mathbb{N},\forall n:\mathbb{N},\forall Psi:((\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right))\to \mathbb{C},\forall X:(\mathrm{Fin}\left(3\right))\to (\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right),\forall Y:(\mathrm{Fin}\left(3\right))\to (\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right),\mathrm{stateCube}\left(Psi,X,Y\right)=\prod _{k:\mathrm{Fin}\left(3\right)}(Psi\left(X\left(k\right)\right)\cdot \mathrm{star}\left(Psi\left(Y\left(k\right)\right)\right))$$

Formalization. D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.stateCube (✓ std3).

Citation. H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Commentary.

The matrix of (|Psi><Psi|) tensor three has the displayed product of amplitudes and conjugate amplitudes.

Definition 1.4 (The trace expectation).

$$\forall d:\mathbb{N},\forall n:\mathbb{N},(d\ne 0)\Rightarrow (\forall Psi:((\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right))\to \mathbb{C},\forall T:\mathrm{Submodule}\left(\mathrm{ZMod}\left(d\right),((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\times ((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\right),\mathrm{kappa}\left(d,n,Psi,T\right)=\mathrm{trace}\left(R\left(d,n,T\right)\cdot \mathrm{stateCube}\left(Psi\right)\right))$$

Formalization. D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.kappa (✓ std3).

Citation. H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Commentary.

The expectation is the trace of R times stateCube, in the source bra-ket convention. The nonzero dimension hypothesis makes the computational basis finite.

Definition 1.5 (The pointwise bound of Conjecture 2).

$$(claim)\iff (\forall d:\mathbb{N},(\mathrm{Prime}\left(d\right))\Rightarrow ((d\ne 2)\Rightarrow (\forall n:\mathbb{N},\forall Psi:((\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right))\to \mathbb{C},(\sum _{x:(\mathrm{Fin}\left(n\right))\to \mathrm{ZMod}\left(d\right)}({\Vert Psi\left(x\right)\Vert }^2)=1)\Rightarrow (\forall T:\mathrm{Submodule}\left(\mathrm{ZMod}\left(d\right),((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\times ((\mathrm{Fin}\left(3\right))\to \mathrm{ZMod}\left(d\right))\right),(\mathrm{IsStochasticLagrangian}\left(d,T\right))\Rightarrow ((0\le \mathrm{kappa}\left(d,n,Psi,T\right))\wedge (\mathrm{kappa}\left(d,n,Psi,T\right)\le 1))))))$$

Formalization. D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.claim (✓ std3).

Citation. H. Zhu, C. Mao, C. Yi (2024). Third moments of qudit Clifford orbits and 3-designs based on magic orbits. DOI: 10.48550/arXiv.2410.13575. URL: https://arxiv.org/abs/2410.13575v1.

Commentary.

The assertion ranges over every odd prime dimension, every number of qudits, every normalized complex amplitude vector and every stochastic Lagrangian subspace. The complex order requires a zero imaginary part and bounds the real part between zero and one.

Theorem 1.6 (A normalized state violates the lower bound).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.result (✓ std3). ∎

Resolves. Problems/zhu-2024-clifford-third-moment-kappa-negativity (refuted) by D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.result.

Source. Repository-derived.

Commentary.

Over ZMod 11, let O have diagonal entries 7 and off-diagonal entries 8, and let T be the range of y mapped to (Oy,y). The identities transpose(O) O = I and O 1 = 1 make T stochastic Lagrangian; injectivity gives dimension three. Let v = (-2,0,-2i,1-2i,-1-i,1+2i,-1-2i,-2,-2i,2-i,1-i), whose squared norm is 40, and set Psi(x) = v(x(0))/sqrt(40). The trace reduces to 40^(-3) times the sum, over all y in (ZMod 11)^3, of the product of v(y(k)) conjugate(v((Oy)(k))). This Gaussian-integer sum is exactly -1196, with zero imaginary part. Thus kappa = -1196/64000 < 0.

References

• Truth anchor: D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.IsStochasticLagrangian
• Truth anchor: D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.R
• Truth anchor: D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.claim
• Truth anchor: D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.kappa
• Truth anchor: D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.result
• Truth anchor: D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.stateCube