A Software Approach to the PPT2 Conjecture

Novšak, Noah (author), Zalar, Aljaž (mentor), Klep, Igor (comentor)

The PPT2 conjecture asserts that the composition of any two PPT maps is entanglement breaking. It is proven for maps on matrices of size up to $3 times 3$ and for several structured families, but the general case remains open; the smallest open case, $4 times 4$, is the one this thesis attacks computationally. We build a reproducible Julia pipeline that (i) mass-produces provably indecomposable entanglement witnesses via the Klep--McCullough--Šivic--Zalar construction of positive but not completely positive maps, rationalizing each certificate after the semidefinite program is solved so that every stored witness is exact -- 10,000 witnesses in under an hour, orders of magnitude faster than comparable implementations; (ii) generates bound entangled PPT candidates by generic random sampling, partial-transpose-invariant sampling, and witness-guided extraction; and (iii) tests the conjecture both by screening tens of thousands of composed channels with witness and DPS criteria, and by a see-saw SDP that searches the manifold of composed PPT maps directly. No counterexample is found. The witness library proves to be a collection of single-state detectors: each witness detects essentially only the state extracted from it. The central finding is a sharp contrast: every one of the 10,000 witnesses attains a negative optimum somewhere on the PPT cone, yet not one fires on the composition manifold -- precisely the signature expected if the conjecture holds in dimension four. This indicates that a counterexample, if one exists at all, must inhabit a measure-zero subset of the composition manifold that a random search cannot reach. As a by-product, the construction yields explicit biquadratic $4 times 4$-forms that are positive but not sums of squares, which is a notoriously difficult problem in real algebraic geometry.

• Language: English
• Keywords: PPT2 conjecture, quantum entanglement, positive maps, semidefinite programming, bound entanglement
• Work type: Master's thesis/paper
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2026
• PID: 20.500.12556/RUL-184600
• Publication date in RUL: 10.07.2026

Secondary language

• Language: Slovenian
• Title: Programski pristop k domnevi PPT2

Abstract

Domneva PPT2 trdi, da kompozitum poljubnih dveh PPT-preslikav, uniči prepletenost. Dokazana je za preslikave na matrikah velikosti do $3 times 3$ in za več strukturiranih družin, v splošnem pa ostaja odprta; najmanjši odprti primer, $4 times 4$, v tem delu napademo računsko. Razvijemo ponovljiv cevovod v jeziku Julia, ki (i) s konstrukcijo Klepa, McCullougha, Šivica in Zalarja množično izdeluje dokazano nerazcepne priče prepletenosti iz pozitivnih, a ne popolnoma pozitivnih preslikav; vsak certifikat po rešitvi semidefinitnega programa racionaliziramo, tako da je vsaka shranjena priča eksaktna -- 10.000 prič zgradimo v manj kot uri, za rede velikosti hitreje od primerljivih implementacij; (ii) generira mejno prepletene PPT-kandidate z generičnim naključnim vzorčenjem, z vzorčenjem, invariantnim na delno transpozicijo, ter z ekstrakcijo iz prič; in (iii) domnevo preizkusi s presejanjem več deset tisoč kompozitumov ter z izmenično optimizacijo, ki mnogoterost kompozitumov PPT-preslikav preiskuje neposredno. Protiprimera ne najdemo. Knjižnica prič se izkaže za zbirko detektorjev enega samega stanja: vsaka priča zazna v bistvu le stanje, ki je bilo iz nje ekstrahirano. Osrednja ugotovitev je ostro nasprotje: vsaka od 10.000 prič doseže negativni optimum nekje na stožcu PPT, na mnogoterosti kompozitumov pa se ne sproži nobena -- natanko kar pričakujemo, če domneva v dimenziji štiri drži. To nakazuje, da morebitni protiprimer, če sploh obstaja, leži na podmnožici kompozitumov z mero nič, ki je naključno iskanje ne doseže. Spotoma konstrukcija ustvari eksplicitne bikvadratne $4 times 4$-forme, ki so pozitivne, a niso vsote kvadratov, kar je znano težek problem v realni algebraični geometriji.

• Keywords: domneva PPT2, kvantna prepletenost, pozitivne preslikave, semidefinitno programiranje, mejna prepletenost