Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.02942

Resource Theory of Aquaternionicity: Every Pure State Is a Replicable Resource for Exact Simulation of Arbitrary Quantum Operations

Hayato Arai, Yasuaki Nakayama

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $d\ge4$, based on a quaternionic structure represented by an antiunitary $Θ$ with $Θ^2=-I$. 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 $\dim\mathcal M_g\le 2d-4,$ 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.

Comments
37pages, 1figure

Related papers

Inverse Problem of Alchemical Resource Theory: Replication and Universal Simulation Single Out Imaginarity and Parity Asymmetry

Yasuaki Nakayama, Hayato Arai

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.

quant-pharXiv:2609.12988•Sep 14, 2026•0 citations

Uniqueness of imaginarity-assisted exact transformation from real orthogonal operations to arbitrary unitary operations

Yasuaki Nakayama, Yuki Takeuchi, Seiseki Akibue

The computational universality with an elementary gate set $\{H,CCZ\}$ can be transformed to the strict universality by using a maximally imaginary state $|+i \rangle$ 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 $|+i \rangle$ 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 $|+i \rangle$ 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).

quant-pharXiv:2603.11812•Scientific Reports•Mar 13, 2026•0 citations

The power of the resource theory of genuine quantum coherence

Julio I. de Vicente, Alexander Streltsov

Any quantum resource theory is based on free states and free operations, i.e., states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. As an example for such a problem, a quantum operation can be incoherent in one particular experimental realization, but it still can create large amount of coherence in another. This problem is resolved by the framework of genuine quantum coherence. Here, we investigate this concept further and study other more general sets of operations which cannot create coherence in any experimental realization. We analyze in detail the mathematical structure of this class of maps and use this to study possible state transformations under these operations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e., there is no single state from which all other states can be created deterministically with the free operations. On the other hand, we show that these theories allow for a richer structure in what comes to stochastic manipulation of states. In particular, we determine the optimal probability of conversion among pure states with genuinely incoherent operations. Still, the aforementioned limitations suggest that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.

arXiv:1604.08031•Apr 27, 2016•8 citations

A General Theory of Multi-Resource Theories Involving Finite-Group Asymmetry

Yosuke Mitsuhashi, Hiroyasu Tajima

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.

quant-pharXiv:2609.40255•Oct 1, 2026•0 citations

Quantification of the Resource Theory of Imaginarity Revisited

Yue Han, Naihong Hu

This work studies the decay of three imaginarity measures, namely the $l_1$-norm-based imaginarity, the robustness of imaginarity, and the relative-entropy-based imaginarity, under typical noise channels. For arbitrary single-qubit pure states subject to real quantum channels, we prove an exact attenuation theorem: the $l_1$-norm-based imaginarity and the robustness exhibit identical attenuation governed by a common channel-dependent factor, whereas the relative-entropy-based imaginarity additionally depends on the input-state orientation relative to the reference basis. We extend the investigation to two-qubit entangled and dual-rail states, establishing a quantitative link between photon-loss probability and residual imaginarity for photonic dual-rail encodings. For separable two-qubit states, we define a real-operation resource preorder and show that $Ω_{++}=|+\rangle\langle+|\otimes|+\rangle\langle+|$ is operationally maximal. We further distinguish operational de-imaginary power from reference-state imaginarity loss. For several real two-qubit channels we derive exact operational de-imaginary powers, while for other channels we provide analytic reference-state losses serving as benchmarks and rigorous lower bounds.

quant-pharXiv:2603.13980•Journal of Physics A Mathematical and Theoretical•Mar 17, 2026•0 citations

Quantifying the imaginarity of quantum mechanics

Alexander Hickey, Gilad Gour

Abstract The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-generating (RNG) operations, and stochastically RNG operations. Moreover, the resource theory of imaginarity does not have a self-adjoint resource destroying map. After introducing and characterizing the free operations, we provide several measures of imaginarity, and give necessary and sufficient conditions for pure state transformations via physically consistent free operations in the single shot regime.

arXiv:1801.05123•Journal of Physics A Mathematical and Theoretical•Apr 17, 2018•130 citations
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≥4d\ge4d≥4, based on a quaternionic structure represented by an antiunitary ΘΘΘ with Θ2=−IΘ^2=-IΘ2=−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 ddd-dimensional resource theory and pure alchemical state ggg, the admissible pure-mixing set satisfies dim⁡Mg≤2d−4,\dim\mathcal M_g\le 2d-4,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.
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.
The computational universality with an elementary gate set {H,CCZ}\{H,CCZ\}{H,CCZ} can be transformed to the strict universality by using a maximally imaginary state ∣+i⟩|+i \rangle∣+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⟩|+i \rangle∣+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⟩|+i \rangle∣+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).
Any quantum resource theory is based on free states and free operations, i.e., states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. As an example for such a problem, a quantum operation can be incoherent in one particular experimental realization, but it still can create large amount of coherence in another. This problem is resolved by the framework of genuine quantum coherence. Here, we investigate this concept further and study other more general sets of operations which cannot create coherence in any experimental realization. We analyze in detail the mathematical structure of this class of maps and use this to study possible state transformations under these operations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e., there is no single state from which all other states can be created deterministically with the free operations. On the other hand, we show that these theories allow for a richer structure in what comes to stochastic manipulation of states. In particular, we determine the optimal probability of conversion among pure states with genuinely incoherent operations. Still, the aforementioned limitations suggest that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.
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.
This work studies the decay of three imaginarity measures, namely the l1l_1l1​-norm-based imaginarity, the robustness of imaginarity, and the relative-entropy-based imaginarity, under typical noise channels. For arbitrary single-qubit pure states subject to real quantum channels, we prove an exact attenuation theorem: the l1l_1l1​-norm-based imaginarity and the robustness exhibit identical attenuation governed by a common channel-dependent factor, whereas the relative-entropy-based imaginarity additionally depends on the input-state orientation relative to the reference basis. We extend the investigation to two-qubit entangled and dual-rail states, establishing a quantitative link between photon-loss probability and residual imaginarity for photonic dual-rail encodings. For separable two-qubit states, we define a real-operation resource preorder and show that Ω++=∣+⟩⟨+∣⊗∣+⟩⟨+∣Ω_{++}=|+\rangle\langle+|\otimes|+\rangle\langle+|Ω++​=∣+⟩⟨+∣⊗∣+⟩⟨+∣ is operationally maximal. We further distinguish operational de-imaginary power from reference-state imaginarity loss. For several real two-qubit channels we derive exact operational de-imaginary powers, while for other channels we provide analytic reference-state losses serving as benchmarks and rigorous lower bounds.
Abstract The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-generating (RNG) operations, and stochastically RNG operations. Moreover, the resource theory of imaginarity does not have a self-adjoint resource destroying map. After introducing and characterizing the free operations, we provide several measures of imaginarity, and give necessary and sufficient conditions for pure state transformations via physically consistent free operations in the single shot regime.