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arXiv:2610.04930

Pairwise detection, chiral conversion: Resource theory of quantum multistates

Yuan-Hong Tao, Yu-Xin Wu, Bing Xie, Lin Zhang

quant-phSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
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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 $d^2$.

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11 pages, 4 figures

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Traditionally, the characterization of quantum resources has focused on individual quantum states. Recent literature, however, has increasingly explored the characterization of resources in states—ordered collections of states indexed by a varying parameter. In this work, we provide a unitary-invariant framework to pinpoint imaginarity and coherence in sets of qubit states: We prove that Bloch vectors must be coplanar to be imaginarity free and collinear to be incoherent, yielding exact rank-based tests of coherence and imaginarity and closed-form bounds for existing robustness quantifiers, all based on two-state overlaps only. We also show that the set of imaginarity-free multistates is not convex, and that third-order invariants completely characterize multistate imaginarity of single-qubits but not of higher-dimensional systems. As our main technical result, we show that Bargmann invariant of single-qubit states is determined (up to conjugation) by two-state overlaps. Beyond qubits, we give purity and system-agnostic coherence witnesses from equality constraints on higher-order invariants and connect our results to practical protocols—characterization of partial distinguishability, spin-chirality detection, and subchannel discrimination.
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Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.