Magic (non-stabilizerness) is a key resource for achieving universal fault-tolerant quantum computation beyond classical computation. While previous studies have primarily focused on magic in single systems, its interactions and distribution in multipartite settings remain largely unexplored. In this work, we introduce mutual mana as a measure of magic correlations defined in close analogy with quantum mutual information. Our definition builds upon mana, which is the established quantifier of magic based on discrete Wigner function negativity. We characterize magic correlations generated by discrete beamsplitters, whose Gaussian counterparts are fundamental components in quantum optics and quantum technologies. We show that coupling a magic state with a stabilizer vacuum state via a discrete beamsplitter will induce a full conversion of local magic into mutual mana, thereby establishing a mechanism for redistributing magic resources as magic correlations. We reveal the fundamental properties of mutual mana and derive its explicit expressions for several prototypical qutrit states subject to a discrete beamsplitter. We make a comparative study of mutual mana with several established quantifiers of correlations generated by the qutrit beamsplitter, including quantum mutual information, mutual $L^1$-norm magic, and mutual stabilizer 2-Rényi entropy.
Magic (non-stabilizerness) is a key resource for achieving universal fault-tolerant quantum computation beyond classical computation. While previous studies have primarily focused on magic in single systems, its interactions and distribution in multipartite settings remain largely unexplored. In this work, we introduce mutual mana as a measure of magic correlations defined in close analogy with quantum mutual information. Our definition builds upon mana, which is the established quantifier of magic based on discrete Wigner function negativity. We characterize magic correlations generated by discrete beamsplitters, whose Gaussian counterparts are fundamental components in quantum optics and quantum technologies. We show that coupling a magic state with a stabilizer vacuum state via a discrete beamsplitter will induce a full conversion of local magic into mutual mana, thereby establishing a mechanism for redistributing magic resources as magic correlations. We reveal the fundamental properties of mutual mana and derive its explicit expressions for several prototypical qutrit states subject to a discrete beamsplitter. We make a comparative study of mutual mana with several established quantifiers of correlations generated by the qutrit beamsplitter, including quantum mutual information, mutual L1-norm magic, and mutual stabilizer 2-Rényi entropy.
Non-stabilizerness – commonly known as magic – measures the extent to which a quantum state deviates from stabilizer states and is a fundamental resource for achieving universal quantum computation. In this work, we investigate the behavior of non-stabilizerness around criticality in quantum spin chains. To quantify non-stabilizerness, we employ a monotone called mana, based on the negativity of the discrete Wigner function. This measure captures non-stabilizerness for both pure and mixed states. We introduce Rényi generalizations of mana, which are also measures of non-stabilizerness for pure states, and utilize it to compute mana in large quantum systems. We consider the three-state Potts model and its non-integrable extension and we provide strong evidence that the mutual mana exhibits universal logarithmic scaling with distance in conformal field theory, as is the case for entanglement.
Abstract Beam‐splitters are basic and instrumental tools not only in the study of foundational issues in quantum mechanics, but also in various applications of quantum information processing. In the continuous‐variable scenario involving quantum optical states and Gaussian quantum information, interference phenomena are widely investigated and characterized via beam‐splitters. In this work, within the framework of stabilizer quantum computation in finite quantum systems, entanglement and magic correlations generated by discrete beam‐splitters are explored. Two types of magic correlations are distinguished: mutual magic (analogous to quantum mutual information) and non‐local magic. Explicit formulas for these correlations are derived in the qubit scenario where the beam‐splitter is implemented by the controlled‐NOT gate, with the ‐type magic states playing a particularly prominent role as input states. Extending to prime‐dimensional qudit systems, it is shown that fiducial states of mutually unbiased bases also play a remarkable role. Basic properties of these magic correlations are revealed and illustrated through some typical examples.
Non-stabilizerness, also known as ``magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer Rényi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number N of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits (d=2) and the mana for qutrits (d=3) for pure states given as state vectors. Our methods compute SRE and mana at cost O(Nd2N), providing an exponential improvement over the naive O(d3N) scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG ( https://github.com/bsc-quantic/HadaMAG.jl/ ), which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.
The advent of quantum technologies brought forward much attention to the theoretical characterization of the computational resources they provide. A method to quantify quantum resources is to use a class of functions called magic monotones and stabilizer entropies, which are, however, notoriously hard and impractical to evaluate for large system sizes. In recent studies, a fundamental connection between information scrambling, the magic monotone mana and 2-Renyi stabilizer entropy was established. This connection simplified magic monotone calculation, but this class of methods still suffers from exponential scaling with respect to the number of qubits. In this work, we establish a way to sample an out-of-time-order correlator that approximates magic monotones and 2-Renyi stabilizer entropy. We numerically show the relation of these sampled correlators to different non-stabilizerness measures for both qubit and qutrit systems and provide an analytical relation to 2-Renyi stabilizer entropy. Furthermore, we put forward and simulate a protocol to measure the monotonic behaviour of magic for the time evolution of local Hamiltonians.
Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.
To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum 'magic' or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.