Pairwise detection, chiral conversion: Resource theory of quantum multistates

arXiv:2610.04930v1 Announce Type: new Abstract: Basis-independent coherence of a quantum multistate is fixed by whether its constituent states commute, and for qubits it is detected by pairwise state overlaps alone. What governs its deterministic transformation has been open. We give the exact resource-conversion theory of quantum multistates under common commutativity-preserving operations (CPOs). For qubits, pairwise Bargmann invariants determine the singular values of the unique candidate channel, while a single third-order Bargmann phase determines its orientation. Complete positivity then yields a necessary and sufficient criterion for exact conversion. The criterion identifies a chirality-sensitive shell $\mathcal C_{\rm chir}$, within which targets with identical pairwise geometry can differ in convertibility solely by the sign of their cubic invariant. This yields an order separation, $k_{\rm detect}=2<3=k_{\rm convert}$, for qubit multistates of every size $n\geqslant3$. The chirality-sensitive shell occupies exactly one third of the singular-value simplex, while mirror reflection obstructs two thirds of the admissible orientation-preserving transformations by volume. For $d\geqslant3$, by contrast, the structure is rigid: every resourceful conversion is realized by either an isotropic depolarizing or a transpose-depolarizing channel. Convertibility is then decided by centered Bargmann invariants of degree at most $d^2$.

Abstract:Basis-independent coherence of a quantum multistate is fixed by whether its constituent states commute, and for qubits it is detected by pairwise state overlaps alone. What governs its deterministic transformation has been open. We give the exact resource-conversion theory of quantum multistates under common commutativity-preserving operations (CPOs). For qubits, pairwise Bargmann invariants determine the singular values of the unique candidate channel, while a single third-order Bargmann phase determines its orientation. Complete positivity then yields a necessary and sufficient criterion for exact conversion. The criterion identifies a chirality-sensitive shell $\mathcal C_{\rm chir}$, within which targets with identical pairwise geometry can differ in convertibility solely by the sign of their cubic invariant. This yields an order separation, $k_{\rm detect}=2<3=k_{\rm convert}$, for qubit multistates of every size $n\geqslant3$. The chirality-sensitive shell occupies exactly one third of the singular-value simplex, while mirror reflection obstructs two thirds of the admissible orientation-preserving transformations by volume. For $d\geqslant3$, by contrast, the structure is rigid: every resourceful conversion is realized by either an isotropic depolarizing or a transpose-depolarizing channel. Convertibility is then decided by centered Bargmann invariants of degree at most $d^2$.

arXiv:2610.04930 [quant-ph]

(or arXiv:2610.04930v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.04930 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Pairwise detection, chiral conversion: Resource theory of quantum multistates

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