A local measure-and-reset refutes the LOCC monotonicity of I2

Abstract

A local measure-and-reset raises the mutually unbiased correlation I2 from 1 to 3/2, refuting its monotonicity under LOCC in the fixed and the optimized reading.

Definition 1.1 (A pair of local mutually unbiased bases).

$$\forall d:\mathbb{N},\forall s:\mathrm{Setting}\left(d\right),(A\left(s\right)\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right)\wedge A^{\prime }(s)\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right)\wedge B\left(s\right)\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right)\wedge B^{\prime }(s)\in \mathrm{unitaryGroup}\left(\mathrm{Fin}\left(d\right),\mathbb{C}\right))\wedge (\forall ij:\mathrm{Fin}\left(d\right),\mathrm{normSq}\left((A{\left(s\right)}^H\times A^{\prime }(s))(i,j)\right)=1/d)\wedge (\forall ij:\mathrm{Fin}\left(d\right),\mathrm{normSq}\left((B{\left(s\right)}^H\times B^{\prime }(s))(i,j)\right)=1/d)$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.Setting (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

Alice’s two bases are the columns of the unitary matrices A and A’, Bob’s those of B and B’; the bases of each party are mutually unbiased, |⟨a_i|a’_j⟩|² = 1/d, as in §II of arXiv:2509.24045v1 (“{a’, b’} ∈ B_2 which is mutually unbiased to B_1”).

Definition 1.2 (The product basis vector).

$$\forall d:\mathbb{N},\forall XY:\mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall i:\mathrm{Fin}\left(d\right),\forall ab:\mathrm{Fin}\left(d\right),\mathrm{prodVec}\left(X,Y,i,(a,b)\right)=X(a,i)\times Y(b,i)$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.prodVec (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

The i-th column of X tensored with the i-th column of Y, so that ⟨i_a ⊗ i_b|ρ|i_a ⊗ i_b⟩ is the joint probability P_{a,b}(i,i) of Eq. (pAB).

Definition 1.3 (The correlation I2 for a setting).

$$\forall d:\mathbb{N},\forall s:\mathrm{Setting}\left(d\right),\forall \rho :\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),I2\left(s,\rho \right)=\sum _i\mathrm{re}\left(\mathrm{star}\left(\mathrm{prodVec}\left(A\left(s\right),B\left(s\right),i\right)\right)\cdot (\rho \cdot \mathrm{prodVec}\left(A\left(s\right),B\left(s\right),i\right))\right)+\sum _i\mathrm{re}\left(\mathrm{star}\left(\mathrm{prodVec}\left(A^{\prime }(s),B^{\prime }(s),i\right)\right)\cdot (\rho \cdot \mathrm{prodVec}\left(A^{\prime }(s),B^{\prime }(s),i\right))\right)$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.I2 (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

I2 = C_{a,b} + C_{a’,b’} with C_{a,b} = sum over i of P_{a,b}(i,i) (Eq. (cAB)); the expression is linear in ρ, so p_k I2(ρ_k) equals I2 of the unnormalized branch.

Definition 1.4 (The optimized correlation).

$$\forall d:\mathbb{N},\forall \rho :\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),I2max\left(\rho \right)={\sup }_{s:\mathrm{Setting}\left(d\right)}I2\left(s,\rho \right)$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.I2max (✓ std3).

Source. Repository-derived.

Commentary.

The supremum of I2 over all local settings: the paper evaluates I2 of a pure state in its Schmidt basis, and this is the state-independent version of that choice.

Definition 1.5 (The unnormalized branch of a local instrument).

$$\forall d:\mathbb{N},\forall E:\mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall \rho :\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),\mathrm{branch}\left(E,\rho \right)=\mathrm{kronecker}\left(E,1\right)\times \rho \times (\mathrm{kronecker}\left(E,1\right){)}^H$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.branch (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

Eq. (10) uses ρ_k = (E_k ⊗ I) ρ (E_k ⊗ I)† / p_k; branch E ρ is the numerator, so p_k ρ_k = branch E_k ρ, including p_k = 0.

Definition 1.6 (Monotonicity for every fixed setting).

$$claimFixed\iff (\forall d:\mathbb{N},2\le d\Rightarrow \forall s:\mathrm{Setting}\left(d\right),\forall \rho :\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),\forall E1E2:\mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{PosSemidef}\left(\rho \right)\Rightarrow \mathrm{trace}\left(\rho \right)=1\Rightarrow E{1}^H\times E1+E{2}^H\times E2=1\Rightarrow I2\left(s,\mathrm{branch}\left(E1,\rho \right)\right)+I2\left(s,\mathrm{branch}\left(E2,\rho \right)\right)\le I2\left(s,\rho \right))$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.claimFixed (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

The Conjecture of §II, “The quantity I_2 is monotonically non-increasing under LOCC operations”, for the binary local instruments of Eq. (10), with I2 taken for any fixed pair of local mutually unbiased bases in any dimension d ≥ 2.

Definition 1.7 (Monotonicity of the optimized correlation).

$$claimOptimized\iff (\forall d:\mathbb{N},2\le d\Rightarrow \forall \rho :\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),\forall E1E2:\mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{PosSemidef}\left(\rho \right)\Rightarrow \mathrm{trace}\left(\rho \right)=1\Rightarrow E{1}^H\times E1+E{2}^H\times E2=1\Rightarrow I2max\left(\mathrm{branch}\left(E1,\rho \right)\right)+I2max\left(\mathrm{branch}\left(E2,\rho \right)\right)\le I2max\left(\rho \right))$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.claimOptimized (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

The same inequality for the optimized correlation I2max in every dimension d ≥ 2.

Definition 1.8 (The conjecture in either reading).

$$claim\iff (claimFixed\vee claimOptimized)$$

Formalization. D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.claim (✓ std3).

Citation. S. Nandi (2025). Genuine multipartite entanglement detection with mutually unbiased bases (MUBs). DOI: 10.48550/arXiv.2509.24045. URL: https://arxiv.org/abs/2509.24045v1.

Commentary.

The conjecture is read as the disjunction of the fixed and the optimized meaning of the quantity I2.

Theorem 1.9 (The conjecture fails in both readings).

$$\neg claim$$

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

Resolves. Problems/nandi-2025-mub-correlation-locc-refutation (refuted) by D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.result.

Source. Repository-derived.

Commentary.

Take d = 2, ρ = (1/2) I ⊗ |0⟩⟨0|, and Alice’s instrument E₁ = |0⟩⟨0|, E₂ = |0⟩⟨1|, which measures and resets outcome 1 to |0⟩. Both branches equal (1/2)|00⟩⟨00|. For every setting I2 of ρ is 1, because each basis term is the sum over i of (1/2)|⟨b_i|0⟩|², which is 1/2 by unitarity of Bob’s basis; hence I2max of ρ is 1. In the computational/Hadamard setting each branch has I2 = (1/2)(1 + 1/2) = 3/4, so the branches sum to 3/2 > 1, and the same lower bound holds for I2max.

References

• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.I2
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.I2max
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.Setting
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.branch
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.claimFixed
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.claimOptimized
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.prodVec
• Truth anchor: D5/S3/Quantum/Entanglement/NandiMutualPredictabilityLOCCRefutation.result
• Dependency: D5/S3/Quantum/Information/BinaryStabilizerLocalInequivalence