Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the Rényi entropy of order 2, $M_2$ , and entanglement by the concurrence $Δ$ . We find that the Pareto frontier of maximal magic $M_2^{(max)}(Δ)$ is composed of three separate segments, while the boundary of minimal magic $M_2^{(min)}(Δ)$ is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.Magic and entanglement are two measures that are widely used to characterize quantum resources. We study the interplay between magic and entanglement in two-qubit systems, focusing on the two extremes: maximal magic and minimal magic for a given level of entanglement. We quantify magic by the Rényi entropy of order 2, $M_2$, and entanglement by the concurrence $Δ$. We find that the Pareto frontier of maximal magic $M_2^{(max)}(Δ)$ is composed of three separate segments, while the boundary of minimal magic $M_2^{(min)}(Δ)$ is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.

quant-pharXiv:2603.24902Mar 27, 20260 citations

Magic-protected entanglement and Clifford-irreducible structure in magic state space

Alejandro Borda Kuhlmann, Julian Rincon

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.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.

quant-pharXiv:2607.18400Jul 22, 20260 citations

The Essence of Entanglement

Časlav Brukner, Marek Żukowski, Anton Zeilinger

arXiv:quant-ph/0106119Fundamental theories of physicsJan 1, 202178 citations

An entanglement-based separation of magic states

Alejandro Borda Kuhlmann

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.

May 30, 20250 citations

Untangling Entanglement

ScienceFeb 5, 20040 citations

M_2

, and entanglement by the concurrence

Δ

. We find that the Pareto frontier of maximal magic

M_2^{(max)}(Δ)

is composed of three separate segments, while the boundary of minimal magic

M_2^{(min)}(Δ)

is a single continuous line. We derive simple analytical formulas for all these four cases, and explicitly parametrize all distinct quantum states of maximal or minimal magic at a given level of entanglement.

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.