In this work, we set out to study and define a fundamental separation between two classes of magic states in quantum computation. In order to reach this goal, we define the “left Clifford equivalence,” an equivalence relation that aims to properly distinguish the interplay between magic and entanglement in magic states. Utilizing such an equivalence relation, we define the notion of a “magic class” and show that there are two different magic classes under this notion, which we call T-magic and W-magic. The former displays an almost trivial relationship with entanglement, meaning they behave as completely independent resources. Meanwhile, the latter is not only exclusively a feature of entangled states, but we find that the non-stabilizer resources themselves of W-magic states are intrinsically related to entanglement. In order to further study the distinction between these classes, we discuss how the separation of these classes shows itself in the topic of universality. Then, we depict the whole space of magic states as defined by these classes, and we study the relationship between measurements and both magic classes.
In this work, we set out to study and define a fundamental separation between two classes of magic states in quantum computation. In order to reach this goal, we define the “left Clifford equivalence,” an equivalence relation that aims to properly distinguish the interplay between magic and entanglement in magic states. Utilizing such an equivalence relation, we define the notion of a “magic class” and show that there are two different magic classes under this notion, which we call T-magic and W-magic. The former displays an almost trivial relationship with entanglement, meaning they behave as completely independent resources. Meanwhile, the latter is not only exclusively a feature of entangled states, but we find that the non-stabilizer resources themselves of W-magic states are intrinsically related to entanglement. In order to further study the distinction between these classes, we discuss how the separation of these classes shows itself in the topic of universality. Then, we depict the whole space of magic states as defined by these classes, and we study the relationship between measurements and both magic classes.
We develop a Clifford-orbit framework for studying magic-protected entanglement, which we refer to as magical entanglement: the part of bipartite entanglement that remains after optimal stabilizer simplification. This construction leverages residual entanglement under Clifford reduction as a state-level organizing principle for magic state space. It defines canonical representatives, spectra, and ranks that characterize the Clifford-irreducible structure of a state. We identify two regimes of the magic-entanglement interplay. In the T-magic regime, local nonstabilizer resources can coexist with entanglement, but the protected component remains weak and state-dependent. In the W-magic regime, by contrast, entanglement is Clifford-irreducibly tied to nonstabilizerness, producing typical, strongly self-averaging behavior. Analytical examples and random-circuit numerics support a crossover from broad T-magic fluctuations to concentrated, Haar-like W-magic behavior. These results identify magical entanglement as an orbit-level diagnostic of how nonstabilizerness protects quantum correlations against Clifford reduction.
Abstract Magic, also known as nonstabilizerness, quantifies the distance of a quantum state to the set of stabilizer states, and it serves as a necessary resource for potential quantum advantage over classical computing. In this work, we study magic in a measurement-only quantum circuit with competing types of Clifford and non-Clifford measurements, where magic is injected through the non-Clifford measurements. This circuit can be mapped to a classical model that can be simulated efficiently, and the magic can be characterized using any magic measure that is additive to the tensor product of single-qubit states. Leveraging this observation, we study the magic transition in this circuit in both one- and two-dimensional lattices using large-scale numerical simulations. Our results demonstrate the presence of a magic transition between two different phases with extensive magic scaling, separated by a critical point in which the mutual magic exhibits scaling behavior analogous to entanglement. We further show that these two distinct phases can be distinguished by topological magic. In a different regime, with a vanishing rate of non-Clifford measurements, we find that the magic saturates in both phases. Our work sheds light on the behavior of magic and its linear combinations in quantum circuits, employing genuine magic measures.
Quantum magic resources, or nonstabilizerness, are a central ingredient for universal quantum computation. In noisy many-body systems, the interplay between these resources and errors leads to sharp magic phase transitions. However, the microscopic mechanism behind these critical phenomena is still an open question, especially since early empirical evidence showed conflicting results regarding their universality classes. In this work, we provide a comprehensive picture of magic phase transitions for the class of encoding-decoding quantum circuits to resolve these ambiguities. We analytically show that the behavior of magic resources is fundamentally dictated by the chosen measurement protocol. When we fix, or post-select, the class of measurement syndromes, the magic transition inherits the universal features of the error-resilience phase transition in the circuits. Interestingly, this clean transition survives even for fully random Haar encoders showing that it is a consequence of initial's state retrieval, and not an artifact of the Clifford encoders. On the other hand, if we consider realistic Born-rule sampling, the intrinsic statistical fluctuations of a given syndrome measurement act as a relevant perturbation. This brings in strong finite-size drifts and an apparent multifractality, which end up altering the critical behavior of the system. We reveal that magic phase transitions are actually direct manifestations of error-resilience thresholds, rather than independent critical phenomena, reconciling conflicting observations from the earlier literature. Ultimately, our framework clarifies how the quantum computational power can survive, or be irreversibly destroyed, due to the competition between scrambling, measurements, and errors.
Entanglement serves as a foundational pillar in quantum information theory, delineating the boundary between what is classical and what is quantum. The common assumption is that a higher degree of entanglement corresponds to a greater degree of “quantumness.” However, this folk belief is challenged by the fact that classically simulable operations, such as Clifford circuits, can create highly entangled states. The simulability of these states raises the question: What are the differences between “low-magic” entanglement and “high-magic” entanglement? To understand this interplay between entanglement and magic, we take an operational approach by studying tasks such as entanglement estimation, distillation, and dilution. We uncover a separation of Hilbert space into two distinct regimes: the entanglement-dominated (ED) phase, where entanglement surpasses magic, and the magic-dominated (MD) phase, where magic dominates entanglement. This separation induces a computational phase transition: entanglement-related tasks are efficiently solvable in the ED phase, but become intractable in the MD phase. Our results find applications in diverse areas such as quantum error correction, many-body physics, and the study of quantum chaos, providing a unifying framework for understanding the behavior of quantum systems. We also offer theoretical explanations for previous numerical observations, highlighting the broad implications of the ED-MD distinction across various subfields of physics.
In most stabilizer-based quantum computing schemes, so-called magic states are a necessary resource for implementing non-transversal quantum gates. With the resource theory of magic, it is possible to analyze and quantify the generation of the non-stabilizer states. The relative entropy is a measure used in various resource theories. For single qubits, we characterize magic states and their closest stabilizer states by applying analytical results known from the relative entropy of entanglement and show that the magic states and their closest stabilizer states are arranged symmetrically around the states at the centers of the faces of the stabilizer octahedron. For tensor products of single-qubit states, we prove analytically that the relative entropy of magic is nonadditive in almost all cases.
In distributed fault-tolerant quantum computing, entanglement and magic are essential resources for quantum communication and universal fault-tolerant computation, respectively. Although they are usually treated as distinct resource currencies, whether they admit a unified resource-theoretic description remains an open question. Here, we introduce the distributed resource theory of entanglement and magic (DREAM). In this framework, the free states are convex mixtures of product local stabilizer states, and the free operations are local stabilizer circuits assisted by classical communication (LSCC). We show that DREAM contains nontrivial resource states that are neither entanglement nor magic, so it is strictly richer than treating the two resources independently. Surprisingly, such resources can enable the teleportation of magic states between distant parties without consuming entanglement, revealing a counterintuitive form of resource teleportation mediated entirely by separable states. We further generalize this result to quantum networks and investigate general quantum-state teleportation under LSCC. We show that the shared resource under DREAM is closely related to the teleportation capability, quantified by the magic of the teleported state. Our work establish a systematic framework for studying distributed quantum resources and uncover intrinsic relations among distinct resources within a unified resource theory.