Quantum resource theories usually begin with a prescribed free structure and ask what tasks become possible when a resource is supplied. We study the inverse problem of alchemical resource theories, which we define as resource theories admitting a resource that can both replicate itself exactly and universally simulate quantum instruments. Under natural consistency assumptions and branchwise complete freeness, we show that for multi-qubit systems only two nontrivial theories survive, up to a common local unitary change of basis: parity asymmetry and imaginarity. We further show that exact universality exhibits an all-or-nothing trade-off: any nonmaximal resource state can exactly simulate only free unitaries. These results demonstrate that prescribed operational capabilities can strongly constrain the underlying resource structure and reveal a fundamental trade-off between computational capability and the precision required for physical implementation.
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42 pages, 2 figures. Revised and retitled. Updated the terminology and presentation
Quantum resource theories usually begin with a prescribed free structure and ask what tasks become possible when a resource is supplied. We study the inverse problem of alchemical resource theories, which we define as resource theories admitting a resource that can both replicate itself exactly and universally simulate quantum instruments. Under natural consistency assumptions and branchwise complete freeness, we show that for multi-qubit systems only two nontrivial theories survive, up to a common local unitary change of basis: parity asymmetry and imaginarity. We further show that exact universality exhibits an all-or-nothing trade-off: any nonmaximal resource state can exactly simulate only free unitaries. These results demonstrate that prescribed operational capabilities can strongly constrain the underlying resource structure and reveal a fundamental trade-off between computational capability and the precision required for physical implementation.
An alchemical resource theory admits a nonfree state that can be exactly replicated by free operations and whose finitely many copies enable exact simulation of arbitrary quantum instruments. The only known nontrivial examples, imaginarity and parity asymmetry, exhaust the qubit case (arXiv:2609.12988) and exhibit an all-or-nothing property: any nonmaximal resource exactly simulates only free unitaries. We introduce aquaternionicity for even-dimensional qudits d≥4, based on a quaternionic structure represented by an antiunitary Θ with Θ2=−I. It is extremal in two senses: every pure state is alchemical, and alchemicality is maximally abundant under mixing. Within this framework, for any nontrivial d-dimensional resource theory and pure alchemical state g, the admissible pure-mixing set satisfies dimMg≤2d−4, which aquaternionicity saturates for every pure state. Conversely, these two extremal properties characterize the quaternionic structure. Because every pure state is alchemical, the initial resource can be chosen to match the known noise structure. Moreover, when the noisy resource states remain within a common isotropic subspace, the same free processor works exactly for all of them while returning the resource unchanged. This enables recalibration-free exact simulation, including universal quantum computation, and is realized for standard dephasing and dissipative noise, where aquaternionicity substantially outperforms imaginarity and parity asymmetry.
The computational universality with an elementary gate set {H,CCZ} can be transformed to the strict universality by using a maximally imaginary state ∣+i⟩ and some non-imaginary ancillary qubits. From the viewpoint of operational resource theory, it would be intriguing to elucidate a resource for the universality transformation. In this paper, we consider the exact universality transformation in which arbitrary real orthogonal matrices can be applied and a supplied resource state is used to simulate unitary operations exactly and deterministically. Within this operational model, we explore a necessary and sufficient condition for resource states to realize the universality transformation under free real operations. We show that ∣+i⟩ is a unique resource state up to the free operations. Moreover, we obtain a stronger conclusion. If a given resource state cannot be used for the universality transformation, then realizable quantum gates are restricted to unitary matrices proportional to real orthogonal matrices. Therefore, we can tell that ∣+i⟩ is unique (up to the free operations) not only as a state whose resource measure of imaginarity is maximal, but also as a state which empowers real operations with the ability to apply at least one non-real quantum gate (regardless of the magnitudes of its imaginary parts).
Resource theories provide a universal framework for quantifying properties of quantum states, such as entanglement, magic, athermality, and asymmetry, as resources under various constraints encountered in quantum science. Since realistic settings typically impose several constraints at once, it is important to study multi-resource theories that combine them. Among these, combinations with asymmetry are especially natural, as symmetry is fundamental and ubiquitous in physics. Here we develop a general theory that treats combinations of finite-group asymmetry with a broad range of resource theories in a unified manner. Our key observation is that the effect of combining the two constraints is governed by the accessibility of symmetry information under the underlying resource theory: whether the information needed for the conversion can be extracted from the input and used to control free operations. If this can be done with asymptotically vanishing error, we show that the conversion rate of the combined theory equals the smaller of the rates of the two constituent theories, so combining them incurs no additional cost. This condition holds for LOCC on arbitrary mixed multipartite states, magic-state conversion of multiqubit systems under Clifford symmetries, and Gibbs-preserving operations. Thermal operations, being time-translation covariant, can access only limited symmetry information, giving an additional compatibility condition between the two symmetries. We also analyze symmetry imposed on elementary operations rather than only on the resulting channel. We establish analogous results for LOCC and thermal operations under suitable assumptions, and for magic when the symmetry is represented by Pauli operators. Finally, we show that finite-group symmetry does not reduce the asymptotic ergotropy per copy and hence introduces no new completely passive states.
Recently, the resource theory of imaginarity was proposed, viewing the imaginary part of a quantum state as a basic resource leading to quantum advantages. As with other quantum resources such as correlations, coherence, and asymmetry, a natural question arises: Can imaginarity resource be broadcast by a real operation which is free in the resource theory of imaginarity? In this work, we introduce two concepts of imaginarity broadcasting. The first pertains to nonreal states, while the second is related to the imaginarity resource. For nonreal states, we demonstrate that a specific type of nonreal states can be broadcast. However, when it comes to broadcasting the imaginarity resource, we get a surprising result that any imaginarity resource can be perfectly broadcast via a real operation. By comparing different (no-)broadcasting theorems, we distinguish the resource theory of imaginarity from the resource theories of correlations, coherence, and asymmetry, and reveal certain distinctive features of the resource theory of imaginarity.
We derive a trade-off relation between accuracy of implementing a desired unitary evolution using a restricted set of free unitaries and the size of the assisting system, in terms of the resource generating and losing capacity of the target unitary. In particular, this relation implies that, for any theory equipped with a resource measure satisfying lenient conditions, any resource changing unitary cannot be perfectly implemented by a free unitary applied to a system and an environment if the environment has finite dimensions. Our results are applicable to a wide class of resources including energy, asymmetry, coherence, entanglement, and magic, imposing ultimate limitations inherent in such important physical settings, as well as providing insights into operational restrictions in general resource theories.
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.