The PPT² conjecture through a qutrit: diagonal-unitary twirls when the coherence matrix has rank at most two

The PPT² conjecture asserts that composites of PPT (completely positive and completely copositive) maps are entanglement breaking. We test it on composites through a qutrit, Ψ ∘ Φ with Φ: M_d → M₃ and Ψ: M₃ → M_d, by twirling their Choi matrix W over the local diagonal unitaries D ⊗ D̄: the twirl is separable exactly when no D ⊗ D̄-invariant entanglement witness detects W. Our main result is that for every d the twirl is separable when the coherence matrix of W, its compression to the span of the vectors |ii⟩, has rank at most 2. In some cases the proof uses the theorem that composites of PPT maps through a qubit are entanglement breaking (the qubit theorem); given it, invariant witnesses cannot exhibit a counterexample at coherence rank at most 2. A peeling operation, which splits an entanglement-breaking part off one factor, extends the result to higher coherence rank whenever repeated peeling and removal of zero rows reach rank at most 2 or outer dimension at most 3. At d = 4 and coherence rank 3 the four matrices of the Gram form satisfy one linear relation; when it has a zero coefficient, the twirl is separable, and the proof uses a rigidity theorem for rank-four PPT entangled states of two qutrits.

Version 5 revises the exposition. The results and the numbering are unchanged, except that Tables 2–4 are now one table, Table 2; some statements and proofs change in notation or wording. The abstract and the introduction are rewritten. Section 4 opens with an outline of the proof of Theorem 1.1, Section 5 with a guide to the section, and Section 10, on the constant 1/5, is now a short account with pointers to the companion paper. Table 2 gives each computational fact (marked with a dagger) with its kind, where it is used and its folder in the software deposit; the deposit's files MAP.md and MAP.tsv list the programs and outputs with their SHA-256. The Lean counts in Section 9 are updated to the Lean development in version 5 of the deposit: 2850 theorems, 1995 of them checked by the kernel alone; the modules for the facts of this paper contain 2225 of these theorems, 1470 of them checked by the kernel alone. In the supplement, repeated material, including the proof of the pivot lemma (Lemma 3.1 of the paper) and the statement of each result of S7, now appears only once, and every computation is cited with its folder in the deposit. Points found by floating-point search are now called computed rather than exact. As before, no numerical result is used in a proof. The numerical remarks now give the values printed in the deposited outputs. A pairwise completely positive (PCP) decomposition of the twirl is called a PCP fit when it is computed in floating point; the supplement measures its error by the B residual and the A-excess. The remarks change as follows; what version 4 said is given in parentheses. S5, after Lemma T1: in the controls, an index with rank Tₐ = 1 has rank Mₐ = 2 (rank Mₐ equal to 2 or 3). S7, opening: the remark on about 4,000 PPT pairs is removed; no deposited output gives it. S7.1, after Corollary CC1: the remark gives one example, with Gaussian-integer data and the witness value −61555322235269/63423000000000 (several examples, one of them with this value). S7.4: the sub-case (b) of Proposition R3′ contains an exact PPT pair with integer data (exact rational pairs); at the two converged points of the Gauss–Newton searches, all Ãₓ and C̃ₓ are normal and commute, to within 1.5·10⁻¹¹ at one point and 1.9·10⁻⁹ at the other (at the one analysed point, with no tolerance stated). S7.5: the certificate of Theorem J holds in floating point at 911 computed pairs (923, a count that included 12 pairs twice): the separable decompositions of alternatives (i) and (ii) hold to 5.2·10⁻¹¹, and the certificate of Lemma PP in alternative (iii) to 1.1·10⁻¹³. PCP fits were computed at eight of the pairs, with B residual and A-excess up to 5.3·10⁻⁹ (fits to 10⁻¹⁰ at all pairs except boundary points, said to carry exact certificates of Lemma PP). The deposited certificates at the boundary points are in floating point, and the remark no longer singles these points out. For the 36 deposited controls the remark names the part of the proof that fails in each family (every step fails): the decompositions of alternative (i) have relative errors up to 24 in two families of 12, and the certificate of Lemma PP has errors up to 0.68 in the third. It also gives the PCP fits of their twirls, with B residual and A-excess up to 5.9·10⁻¹⁵ in the first two families and 6.0·10⁻⁶ in the third; it no longer says that the controls show the hypotheses to be needed for the proof but not for the conclusion. S7.6: at 178 (174) of the 221 computed points (about 220) the case distinction ends in the core configuration; there the decomposition of Lemma H3.8 has B residual up to 1.43·10⁻⁶ and A-excess up to 7.34·10⁻⁶, and its hypotheses hold to 1.91·10⁻⁵ (the certificate to about 10⁻⁵ at the unpolished points); at the 134 polished points the three values are at most 3.22·10⁻¹⁴, 3.80·10⁻¹⁴ and 3.76·10⁻¹³ (the certificate to about 10⁻¹⁴). S7.7: at the 678 pairs the remark names what the deposited programs check: the proof trees of Theorems A4, B2L and C4, with certificates to 1.2·10⁻⁶; Lemma SPLIT(a), to 2.9·10⁻⁷; and Lemma SP (all the statements of S7.7 were checked). S7.8: the PCP fits at 599 points have B residual up to 3.33·10⁻⁸ and A-excess up to 1.37·10⁻⁴ (residual 1.7·10⁻¹⁵); the value 1.7·10⁻¹⁵ is the squared misfit that the fit minimizes. The condition of Lemma SW holds up to floating-point error, with smallest margin −1.07·10⁻¹⁴, and with equality up to 10⁻¹² on some cycle at 319 of the 540 points with two lines and at 36 of the 56 with three (holds at every tested point with two lines). S7.9: at 81 points the identities of the proof hold to 3.1·10⁻¹³ (3·10⁻¹³), and the checks of the peeling steps hold to 5·10⁻⁵, or to 4.8·10⁻⁷ at 243 of the 245 steps (7·10⁻¹⁰). S7.10: the twirls have PCP fits at 122, 73, 71 and 208 points of the classes P1–P4 (23 for P1, 120 for P4 and further points for P2 and P3), with B residual up to 3.7·10⁻⁹ and A-excess up to 2.4·10⁻¹⁰ (residual at most 5·10⁻¹⁴, except at boundary points with exact certificates); the deposited run has no exact certificates, and its bounds hold at every point. The deposit holds the programs, data and outputs behind these remarks in folders under supp/, and its file PROVENANCE.tsv records for each output whether it was reproduced by a rerun (and with which differences), was not rerun, or was written for the deposit. The paper cites version 5 of the deposit (doi:10.5281/zenodo.23713731) and version 2 of the companion paper, now titled "The PPT² conjecture through a qutrit: barriers at 1/5 and a smaller constant when the first factor acts on M₃" (doi:10.5281/zenodo.23713736).

Mathematics Subject Classification (2020): 81P40; 15A69, 15B48, 47L07.