Negativity is not the partial transpose distance to the PPT states

Abstract

Ganardi, Miller, Paterek and Zukowski (arXiv:2111.11887, Quantum 6, 654) conjecture that the partial transpose distance d_T(rho, sigma) = ||rho^{T_B} - sigma^{T_B}||_1 / 2 from every state rho to the PPT states has infimum equal to the negativity N(rho) = (||rho^{T_B}||_1 - 1)/2. The equality fails: a rank-five two-qutrit state has negativity 1/34, while its distance to every PPT state is at least 2/51.

Definition 1.1 (The conjectured equality).

$$(claim)\iff (\forall d:\mathbb{N},\forall rho:\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),(\mathrm{IsDensity}\left(rho\right))\Rightarrow (\mathrm{sInf}\left(\{t\mid \exists sigma:\mathrm{Matrix}\left((\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),(\mathrm{Fin}\left(d\right)\times \mathrm{Fin}\left(d\right)),\mathbb{C}\right),(\mathrm{IsDensity}\left(sigma\right))\wedge ((\mathrm{PosSemidef}\left(\mathrm{partialTransposeB}\left(sigma\right)\right))\wedge (t=\frac{\mathrm{traceNorm}\left(\mathrm{partialTransposeB}\left(rho\right)-\mathrm{partialTransposeB}\left(sigma\right)\right)}{2}))\}\right)=\frac{\mathrm{traceNorm}\left(\mathrm{partialTransposeB}\left(rho\right)\right)-1}{2}))$$

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

Citation. Ray Ganardi; Marek Miller; Tomasz Paterek; Marek Żukowski (2022). Hierarchy of correlation quantifiers comparable to negativity. DOI: 10.22331/q-2022-02-16-654. URL: https://arxiv.org/abs/2111.11887v2.

Commentary.

For a state rho on C^d (x) C^d, rho^{T_B} is the partial transposition on the second factor (the existing partialTransposeB) and ||A||_1 = Re Tr sqrt(A^dagger A) is the existing trace norm. The partial transpose distance is d_T(rho, sigma) = ||rho^{T_B} - sigma^{T_B}||_1 / 2, the PPT states are the density matrices sigma with sigma^{T_B} positive semidefinite, and the negativity is N(rho) = (||rho^{T_B}||_1 - 1)/2. Conjecture 1 of the paper states that the infimum of d_T(rho, sigma) over the PPT states equals N(rho) for every density matrix rho; the displayed statement is its case of equal local dimensions.

Definition 1.2 (The counterexample state).

$$witness=\frac{1}{34}\cdot ((|00⟩+|11⟩+4\cdot |22⟩)(⟨00|+⟨11|+4\cdot ⟨22|)+4\cdot (|02⟩⟨02|+|20⟩⟨20|+|12⟩⟨12|+|21⟩⟨21|))$$

Formalization. D5/S3/Quantum/Entanglement/NegativityPartialTransposeDistanceRefutation.witness (✓ std3).

Source. Repository-derived.

Acknowledgement. Ray Ganardi; Marek Miller; Tomasz Paterek; Marek Żukowski (2022). Hierarchy of correlation quantifiers comparable to negativity. DOI: 10.22331/q-2022-02-16-654. URL: https://arxiv.org/abs/2111.11887v2.

Commentary.

The state is R/34 on two qutrits, with R = |v><v| + 4 (|02><02| + |20><20| + |12><12| + |21><21|) and v = |00> + |11> + 4 |22>. It is positive semidefinite with trace one and rank five.

Theorem 1.3 (The conjectured equality fails).

$$\neg claim$$

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

Resolves. Problems/ganardi-2022-negativity-ppt-distance (refuted) by D5/S3/Quantum/Entanglement/NegativityPartialTransposeDistanceRefutation.result.

Source. Repository-derived.

Acknowledgement. Ray Ganardi; Marek Miller; Tomasz Paterek; Marek Żukowski (2022). Hierarchy of correlation quantifiers comparable to negativity. DOI: 10.22331/q-2022-02-16-654. URL: https://arxiv.org/abs/2111.11887v2.

Commentary.

Let c = |01> - |10>. The partial transpose of the state is P - |c><c|/68, where P is a sum of positive rank-one and diagonal terms of trace 35/34, so the triangle inequality bounds its trace norm by 36/34 and the negativity by 1/34. The matrices U_1 (the signs -1 on 00, 01, 10, 11, 22 and +1 on 02, 20, 12, 21), U_2 (equal to -1 except for the swaps of 02 with 20 and of 12 with 21) and U_3 (equal to U_2 except for +1 on 22) are unitary, so Re Tr(U_k X) is at most ||X||_1 for every X, and so is Re Tr(F X) for F = U_1/3 + U_2/6 + U_3/2. Against the partial transpose of the state, F has trace 7/17. With b = 2|00> + 2|11> - |22>, every matrix sigma satisfies Tr(F sigma^{T_B}) = Tr(sigma)/3 - <b|sigma|b>/3 - 4<c|sigma^{T_B}|c>/3, so for every PPT state the real part is at most 1/3. Hence the partial transpose distance to every PPT state is at least (7/17 - 1/3)/2 = 2/51, the set of PPT states contains |00><00|, and the infimum is at least 2/51 > 1/34.

References

• Truth anchor: D5/S3/Quantum/Entanglement/NegativityPartialTransposeDistanceRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/NegativityPartialTransposeDistanceRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/NegativityPartialTransposeDistanceRefutation.witness
• Dependency: D5/S3/Quantum/Entanglement/StructuredNegativityCoincidenceRefutation
• Dependency: D5/S3/Quantum/Foundation/FiniteTraceDistance