Relative Entropy of Magic in Quantum Computation

• Relative Entropy of Magic is defined as the quantum-relative entropy distance to the set of stabilizer states, quantifying nonstabilizerness essential for universal quantum computation.
• It is analyzed through geometric methods on the Bloch sphere, optimization over the stabilizer polytope, and Rényi divergence generalizations, highlighting both additivity and subadditivity properties.
• The measure serves as a benchmark across many-body and fermionic systems, linking resource theory to practical applications like magic-state distillation and non-Clifford gate synthesis.

Searching arXiv for recent and foundational papers on relative entropy of magic and closely related measures. Relative entropy of magic (REM) is the quantum-relative-entropy distance from a state to the convex set of stabilizer states. In the stabilizer resource theory, it is a canonical information-theoretic monotone for quantifying nonstabilizerness, or “magic,” as the resource beyond the stabilizer–Clifford framework that is necessary for universal quantum computation. The modern literature treats REM along several complementary lines: as an optimization-based resource monotone on the stabilizer polytope, as a geometric problem that is analytically tractable for single qubits, as a quantity with a subtle additivity landscape on tensor products, and as a reference point for proxy measures such as stabilizer Rényi entropies and phase-space Rényi divergences (Leone et al., 2021, Rubboli et al., 2023, Deckers et al., 21 May 2026).

1. Definition within the stabilizer resource theory

For an $n$-qubit density operator $\rho$, the relative entropy of magic is defined by

$$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$

Here $\mathrm{STAB}$ is the convex hull of pure stabilizer states, equivalently the stabilizer polytope. In the qubit setting, pure stabilizer states are the Clifford orbit of $|0\rangle^{\otimes n}$, and the free mixed states are their convex hull (Leone et al., 2021).

REM has the standard resource-theoretic properties expected of a relative-entropy distance to a convex free set. It is faithful, in the sense that $M_{\mathrm{rel}}(\rho)=0$ if and only if $\rho\in\mathrm{STAB}$. It is monotone under stabilizer-preserving operations, including Clifford unitaries, addition of stabilizer ancilla, and Pauli measurements with classical randomness and feedforward. It is convex by joint convexity of quantum relative entropy, and subadditive because the free set is closed under tensor products: $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$ It also obeys the simple upper bound $M_{\mathrm{rel}}(\rho)\le \log d$, with $d=2^n$, by choosing $\rho$0 (Leone et al., 2021).

For pure states, operator concavity of the logarithm yields a useful lower bound in terms of stabilizer fidelity. If

$\rho$1

then

$\rho$2

This bound is often easier to evaluate than the defining minimization. By contrast, direct computation of REM is generally difficult: the optimization is over a convex set with exponentially many extreme points, and no efficient general algorithm is known (Leone et al., 2021).

Operationally, REM is used as a converse-type monotone in magic-state distillation and non-Clifford gate synthesis. A plausible implication is that its conceptual role parallels that of the relative entropy of entanglement in entanglement theory, but the stabilizer geometry makes explicit computation much more dependent on symmetry and special structure.

2. Single-qubit geometry and closest stabilizer states

For a single qubit, REM admits a detailed geometric description in the Bloch sphere. Any state can be written as

$\rho$3

and the stabilizer polytope is the octahedron

$\rho$4

Thus single-qubit magic states are exactly the Bloch vectors outside this octahedron and inside the Bloch ball (Deckers et al., 21 May 2026).

A central analytic tool is the supporting-hyperplane method adapted from the relative entropy of entanglement. If $\rho$5 is a full-rank boundary stabilizer state and $\rho$6 is a supporting hyperplane at $\rho$7, then every resource state for which $\rho$8 is closest in relative entropy has the form

$\rho$9

where $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$0 is the Hadamard product in the eigenbasis of $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$1, and $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$2 is the divided-difference matrix of the logarithm. This converts the minimization over $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$3 into a converse construction from boundary points inward (Deckers et al., 21 May 2026).

On faces of the octahedron, the geometry becomes especially rigid. For the face with vertices $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$4, $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$5, and $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$6, the unique supporting hyperplane induces magic-state trajectories that tilt symmetrically toward the face center. More generally, magic states and their closest stabilizer states are arranged symmetrically around the centers of the faces of the stabilizer octahedron. The face centers are

$$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$7

with an even number of minus signs (Deckers et al., 21 May 2026).

Commuting optimal pairs occupy distinguished locations. The condition $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$8 is equivalent to $$M_{\mathrm{rel}}(\rho)=\min_{\sigma\in\mathrm{STAB}} S(\rho\Vert\sigma), \qquad S(\rho\Vert\sigma)=\mathrm{Tr}\!\left[\rho(\log\rho-\log\sigma)\right].$$9 in the supporting-hyperplane construction, and for qubits this occurs exactly at the centers of faces and edges. The face center $\mathrm{STAB}$0 is closest to the $\mathrm{STAB}$1-state

$\mathrm{STAB}$2

while the edge center $\mathrm{STAB}$3 is closest to the $\mathrm{STAB}$4-like state

$\mathrm{STAB}$5

For commuting pure–mixed pairs with common Bloch direction $\mathrm{STAB}$6, if the closest stabilizer has Bloch length $\mathrm{STAB}$7, then

$\mathrm{STAB}$8

This yields the explicit single-qubit values

$\mathrm{STAB}$9

All eight $|0\rangle^{\otimes n}$0-like states attain the maximal single-qubit value reported in the study (Deckers et al., 21 May 2026).

3. Additivity, subadditivity, and generic nonadditivity

REM is always subadditive, but exact additivity is a much sharper property. This distinction is now central to the subject. Subadditivity,

$|0\rangle^{\otimes n}$1

follows directly from the definition and closure of $|0\rangle^{\otimes n}$2 under tensor products. The question is when equality holds, and recent work shows that equality is exceptional rather than generic (Leone et al., 2021, Deckers et al., 21 May 2026).

A broad family of additive cases was identified for single-qubit products using $|0\rangle^{\otimes n}$3-$|0\rangle^{\otimes n}$4 Rényi relative entropies of magic. If $|0\rangle^{\otimes n}$5 lies in the data-processing region $|0\rangle^{\otimes n}$6, and for all but at most one tensor factor there exists an optimal stabilizer $|0\rangle^{\otimes n}$7 such that $|0\rangle^{\otimes n}$8, then

$|0\rangle^{\otimes n}$9

For $M_{\mathrm{rel}}(\rho)=0$0, this includes the ordinary REM. The same work proved a stronger statement on the boundary line $M_{\mathrm{rel}}(\rho)=0$1: for arbitrary products of single-qubit states,

$M_{\mathrm{rel}}(\rho)=0$2

This boundary line includes $M_{\mathrm{rel}}(\rho)=0$3, $M_{\mathrm{rel}}(\rho)=0$4, and $M_{\mathrm{rel}}(\rho)=0$5, and in particular implies multiplicativity of the stabilizer fidelity for tensor products of arbitrary single-qubit states (Rubboli et al., 2023).

The single-qubit states whose optimizers commute with the input are exactly the depolarized axis families $M_{\mathrm{rel}}(\rho)=0$6, $M_{\mathrm{rel}}(\rho)=0$7, and $M_{\mathrm{rel}}(\rho)=0$8. Consequently, REM is additive whenever all single-qubit factors but at most one belong to a symmetry axis of the stabilizer octahedron. This is the positive side of the additivity landscape (Rubboli et al., 2023).

The negative side is substantially stronger. For tensor products of single-qubit magic states, REM is nonadditive in almost all cases. More precisely, let $M_{\mathrm{rel}}(\rho)=0$9 be single-qubit magic states with closest stabilizers $\rho\in\mathrm{STAB}$0, and suppose that at least two pairs $\rho\in\mathrm{STAB}$1 do not commute and that at least one of those $\rho\in\mathrm{STAB}$2 lies in the relative interior of a face of the stabilizer octahedron. Then

$\rho\in\mathrm{STAB}$3

The additive cases, in this formulation, form a lower-dimensional subset associated with commuting-axis configurations. In particular,

$\rho\in\mathrm{STAB}$4

because each single-copy optimizer is the face center and commutes with the $\rho\in\mathrm{STAB}$5-state (Deckers et al., 21 May 2026).

Concrete examples illustrate the scale of the effect. A specific pure single-qubit state with Bloch vector

$\rho\in\mathrm{STAB}$6

satisfies

$\rho\in\mathrm{STAB}$7

while the two-qubit state

$\rho\in\mathrm{STAB}$8

has

$\rho\in\mathrm{STAB}$9

This suggests that maximizing REM over multi-qubit systems is not equivalent to simply tensoring single-qubit maximizers (Rubboli et al., 2023, Deckers et al., 21 May 2026).

4. Rényi generalizations and phase-space relative-entropic analogues

A major extension of REM replaces Umegaki relative entropy by the $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$0-$$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$1 Rényi relative entropy

$$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$2

and defines the corresponding magic monotone by minimizing over stabilizer states: $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$3 Special cases include the Petz Rényi family ($$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$4), the sandwiched Rényi family ($$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$5), the min-relative entropy in the limit $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$6, the Umegaki REM at $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$7, and the max-relative entropy in the limit $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$8. These monotones organize a continuous family of additivity and one-shot conversion statements. Closed-form expressions are available for depolarized axis states and for several depolarized two- and three-qubit states, including Toffoli, Hoggar, and CS states, precisely because their optimal stabilizers commute with the input (Rubboli et al., 2023).

A distinct but closely related development occurs in odd-prime qudits, where the paper on Wigner negativity constructs relative-entropic magic monotones directly in phase space. For a state $$M_{\mathrm{rel}}(\rho\otimes\tau)\le M_{\mathrm{rel}}(\rho)+M_{\mathrm{rel}}(\tau).$$9 with discrete Wigner quasi-distribution $M_{\mathrm{rel}}(\rho)\le \log d$0 and a full-rank stabilizer reference $M_{\mathrm{rel}}(\rho)\le \log d$1 with strictly positive $M_{\mathrm{rel}}(\rho)\le \log d$2, the Rényi divergences

$M_{\mathrm{rel}}(\rho)\le \log d$3

are well-defined for

$M_{\mathrm{rel}}(\rho)\le \log d$4

These divergences are nonnegative, additive on tensor powers, and obey data processing under free operations that keep $M_{\mathrm{rel}}(\rho)\le \log d$5 in the interior of the stabilizer set. They are therefore valid “relative-entropy-like” magic monotones in the Wigner representation, even though they are not the operator REM itself (Koukoulekidis et al., 2021).

This phase-space framework also provides a sharp bridge to mana and to distillation bounds. The paper establishes

$M_{\mathrm{rel}}(\rho)\le \log d$6

so mana appears as the residue of the divergence of Wigner Rényi entropy in the Shannon limit. For qutrit Strange-state distillation under unital protocols, the resulting majorization and Rényi-divergence bounds are tighter than the corresponding mana and max-thauma bounds. A common misconception is to identify these quantities with the operator REM; the literature is explicit that they are instead REM-like monotones in phase space, designed to capture magic under Clifford processing in the Wigner representation (Koukoulekidis et al., 2021).

5. Many-body settings, stabilizer Rényi entropies, and REM as a benchmark

Much of the many-body literature does not compute REM directly. Instead, it uses stabilizer Rényi entropies (SREs) as computable and experimentally accessible proxies for nonstabilizerness. For pure $M_{\mathrm{rel}}(\rho)\le \log d$7-qubit states $M_{\mathrm{rel}}(\rho)\le \log d$8, the basic construction starts from

$M_{\mathrm{rel}}(\rho)\le \log d$9

and defines

$d=2^n$0

This family is faithful, Clifford invariant, and additive for pure states, and it avoids the global minimization over $d=2^n$1 that makes REM hard to evaluate. The same work places REM and SRE in a common hierarchy through stabilizer fidelity and robustness bounds, but does not provide a direct closed-form inequality linking $d=2^n$2 to $d=2^n$3 (Leone et al., 2021).

The transverse-field Ising-model study is explicit on this point: it defines REM for context,

$d=2^n$4

but does not compute it. Instead it analyzes the second-order stabilizer Rényi entropy

$d=2^n$5

For the TFIM ground state, $d=2^n$6 scales extensively as

$d=2^n$7

for $d=2^n$8, with $d=2^n$9, and an internal SRE bound gives

$\rho$00

In the gapped phase, the local density of nonstabilizerness is localized: already $\rho$01 gives relative error $\rho$02. At criticality, by contrast,

$\rho$03

These results are about SRE, not REM, but they are often used to motivate how REM bounds might behave in local many-body systems (Oliviero et al., 2022).

A similar distinction appears in generalized Rokhsar–Kivelson wavefunctions. There the SRE $\rho$04 can be mapped to a classical free-energy difference,

$\rho$05

with $\rho$06 the partition function of a coupled four-replica classical model. Across the phase diagram, $\rho$07 is relatively featureless at the underlying critical point, though singular in derivatives according to the order of the transition, while its maximum occurs generically at a cusp at $\rho$08. The same work derives rigorous bounds on the min-relative entropy of magic $\rho$09 using overlaps with specific stabilizer states, but does not compute REM itself (Tarabunga et al., 2023).

A recurrent misconception is therefore that many-body papers on “magic” are studying REM directly. In the cited TFIM and generalized Rokhsar–Kivelson analyses, the primary objects are SRE and $\rho$10, with REM serving as a conceptual benchmark rather than the computed quantity (Oliviero et al., 2022, Tarabunga et al., 2023).

6. Fermionic non-Gaussian magic as a relative-entropy analogue

An important generalization replaces stabilizer states by fermionic Gaussian states and studies “magic” as fermionic non-Gaussianity. In that setting the free set is

$\rho$11

and the relative entropy of magic is defined by

$\rho$12

Under parity superselection, first moments vanish, and the minimizer is the Gaussian state $\rho$13 with the same covariance matrix as $\rho$14. The REM then takes the closed form

$\rho$15

and for pure $\rho$16,

$\rho$17

This is a direct relative-entropy distance to a free set, but the free set is Gaussian rather than stabilizer (Coffman et al., 10 Jan 2025).

The same work proves that three Gaussification procedures coincide: the Gaussian state with the same covariance matrix, the fixed point of iterated fermionic convolution, and the closest Gaussian in relative entropy. The convolution satisfies

$\rho$18

and the corresponding central-limit theorem gives

$\rho$19

The resulting fermionic REM is monotone under Gaussian- or matchgate-preserving channels and additive on product states: $\rho$20 This additivity stands in marked contrast to the generic nonadditivity now known for stabilizer REM on products of single-qubit magic states (Coffman et al., 10 Jan 2025).

The fermionic theory also connects REM to violations of Wick’s theorem and of the matchgate identity. A plausible implication is that relative-entropy constructions for magic are not confined to the stabilizer setting; they extend naturally to other notions of classical simulability, with the free set chosen as the appropriate Gaussian or stabilizer manifold.