Clifford Entropies in Quantum Information

• Clifford entropies are a family of entropy-like measures adapted to the action of Clifford operations, capturing non-stabilizerness and circuit dynamics.
• They include stabilizer Rényi entropies, rank-based metrics, and generalized stabilizer purities that leverage the invariant Pauli/Weyl structures in monitored circuits.
• These measures provide practical insights for diagnosing quantum circuit behavior, phase transitions, and magic quantification in both pure and mixed states.

Searching arXiv for papers on Clifford entropies, stabilizer entropies, and monitored Clifford dynamics. In current quantum-information literature, “Clifford entropies” does not denote a single universally fixed scalar. The phrase is used for several entropy-like quantities that are adapted to Clifford structure: rank-based entropies in monitored Clifford circuits, stabilizer Rényi entropies and generalized stabilizer purities for pure states, filtered stabilizer Rényi entropies defined from Pauli spectra, and an operator-level Clifford entropy for unitaries. What unifies these constructions is that Clifford operations preserve a distinguished algebraic structure—Pauli/Weyl operators, stabilizer groups, commutant sectors, or Weyl–Heisenberg phase-space distributions—so the relevant entropy is not an entropy of the eigenvalue spectrum alone, but an entropy of how a state or channel sits relative to the Clifford/stabilizer sector (Magni et al., 7 Jul 2026, Haug et al., 2023, Bittel et al., 16 Apr 2025, Magni et al., 2 Jun 2025, Maity et al., 11 May 2026, Cuffaro et al., 28 Dec 2025).

1. Taxonomy and algebraic setting

The literature uses several closely related notions, each tied to a specific Clifford-adapted object rather than to ordinary bipartite entanglement alone.

Notion	Defining quantity	Context
Rank-based Clifford entropy	$r=\log_q \operatorname{rank}\rho$	Monitored Clifford mixed states
Stabilizer entropy / SRE	$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$	Pure-state nonstabilizerness
Generalized stabilizer entropy	$\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$	Intrinsic Clifford-commutant sectors
Filtered stabilizer Rényi entropy	$M_\infty=-\ln(r_1^2)$	Pauli-spectrum magic and Clifford ergotropy
Clifford entropy of a unitary	$H_\alpha(U)$ from $\mathfrak D_{ab}(U)$	Non-Cliffordness of channels

For monitored Clifford circuits on $L$ qudits of prime dimension $q$, the Pauli strings are labeled by vectors $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$ with symplectic product

$$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$

A stabilizer group $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$0 with $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$1 independent commuting generators defines a codespace of dimension $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$2, and the associated mixed stabilizer state is a flat projector of rank $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$3. The decisive simplification is that for such a state all Rényi entropies coincide exactly: $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$4 Thus the integer $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$5 is itself an entropy variable (Magni et al., 7 Jul 2026).

For pure states, the standard stabilizer-entropy construction begins from Pauli or Weyl expectation values. In the qubit formulation,

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$6

while a closely related qudit formulation writes

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$7

These quantities vanish exactly on pure stabilizer states and are invariant under Clifford unitaries because Cliffords permute Pauli or Weyl labels (Haug et al., 2023, Haug et al., 2024, Bittel et al., 16 Apr 2025, Erew et al., 22 Dec 2025).

2. Rank-flat entropies in monitored Clifford circuits

The most literal “Clifford entropy” in monitored many-body dynamics is the rank exponent

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$8

because a mixed stabilizer state has completely flat spectrum: there are $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$9 equal eigenvalues $\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$0, so

$\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$1

Purification then becomes a problem of rank decay rather than of replica analytic continuation (Magni et al., 7 Jul 2026).

In the analytically solvable global model, monitored dynamics alternates a uniformly random global Clifford unitary with a projective measurement of a Pauli string. If the current state has rank parameter $\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$2, only measured strings in $\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$3 reduce the rank. The exact one-step rank-drop probability is

$\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$4

At fixed $\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$5 and large $\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$6,

$\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$7

On the rescaled time

$\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$8

with $\mathcal M_\Omega(\Psi)=-\log \zeta_\Omega(\Psi)$9 in the global model, the discrete process converges to a continuous-time pure-death chain with generator

$M_\infty=-\ln(r_1^2)$0

The process starts from $M_\infty=-\ln(r_1^2)$1, which becomes a process descending from infinity in the scaling limit (Magni et al., 7 Jul 2026).

Because entropy equals rank trajectory by trajectory, all averaged Rényi entropies collapse onto one universal function: $M_\infty=-\ln(r_1^2)$2 The paper derives exact moment hierarchies and asymptotics. At small scaled time,

$M_\infty=-\ln(r_1^2)$3

where $M_\infty=-\ln(r_1^2)$4 is unit-periodic with zero mean. At late times,

$M_\infty=-\ln(r_1^2)$5

Two effects are identified as genuinely Clifford-specific fingerprints of rank quantization: an $M_\infty=-\ln(r_1^2)$6 short-time entropy variance and a log-periodic modulation in $M_\infty=-\ln(r_1^2)$7. The variance saturates instead of vanishing, and the modulation arises from the discrete scale invariance

$M_\infty=-\ln(r_1^2)$8

Exact stabilizer simulations at $M_\infty=-\ln(r_1^2)$9 confirm the theory, with no fitted parameters in the global model and a single nonuniversal $H_\alpha(U)$0 in local brick-wall circuits (Magni et al., 7 Jul 2026).

3. Stabilizer entropies for pure states

For pure-state magic, the standard entropy family is the stabilizer Rényi entropies. They are Rényi entropies not of the eigenvalue spectrum of $H_\alpha(U)$1, but of the Pauli-spectrum distribution

$H_\alpha(U)$2

for qubits, or its Weyl-Heisenberg analogue in qudits. The associated moments

$H_\alpha(U)$3

yield

$H_\alpha(U)$4

The Tsallis form

$H_\alpha(U)$5

is also used. These quantities are faithful on pure states, Clifford-invariant, and for $H_\alpha(U)$6 additive under tensor products (Haug et al., 2023, Haug et al., 2024).

A major development is efficient estimation. The Bell-measurement protocol of “Efficient quantum algorithms for stabilizer entropies” measures stabilizer entropies for integer Rényi index $H_\alpha(U)$7 with $H_\alpha(U)$8 copies and $H_\alpha(U)$9 classical computational time. For odd $\mathfrak D_{ab}(U)$0, the protocol works directly from Bell measurements on pairs of copies. For even $\mathfrak D_{ab}(U)$1, efficient estimation is recovered by using $\mathfrak D_{ab}(U)$2 to sample the Pauli spectrum. The same work gives efficiently measurable bounds on stabilizer fidelity, stabilizer extent, and robustness of magic, and experimentally measures $\mathfrak D_{ab}(U)$3 and $\mathfrak D_{ab}(U)$4 on the IonQ quantum computer for doped Clifford circuits (Haug et al., 2023).

The dynamical meaning of these entropies depends strongly on the Rényi index. In random Clifford circuits doped with $\mathfrak D_{ab}(U)$5-gates and in random Hamiltonian evolution, stabilizer Rényi entropies saturate their maximum value at a critical density or time. For random Clifford circuits doped with $\mathfrak D_{ab}(U)$6-gates, the critical $\mathfrak D_{ab}(U)$7-gate density scales independently of $\mathfrak D_{ab}(U)$8. For random Hamiltonian evolution, the critical time is a constant for $\mathfrak D_{ab}(U)$9 and scales linearly with qubit number for $L$0 in the paper’s formulation. The same work argues that $L$1 probes Clifford simulation complexity, while $L$2 probes distance to the closest stabilizer state and approximate state-certification cost by Pauli measurements (Haug et al., 2024).

4. Clifford commutant and generalized stabilizer purities

The modern structural theory of Clifford entropies is organized by the Clifford commutant,

$L$3

“A complete theory of the Clifford commutant” gives an explicit orthogonal basis for this commutant, computes its dimension for arbitrary $L$4 and $L$5, introduces an alternative basis formed by isotropic sums of Pauli operators, and develops a graphical calculus. In that framework, stabilizer entropy appears as the logarithm of the expectation value of a primitive Pauli monomial: $L$6

$L$7

The same paper generalizes this to

$L$8

the generalized stabilizer purity associated with a commutant element $L$9. It proves that any measurable magic measure admitting an unbiased $q$0-copy estimator is a Pauli polynomial of degree $q$1, and that the optimal six-copy stabilizer property test is governed by $q$2. This yields the operational interpretation

$q$3

so $q$4 directly controls the optimal six-copy stabilizer-testing advantage (Bittel et al., 16 Apr 2025).

For odd-prime qudits, “Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits” elevates this commutant viewpoint into a family of generalized stabilizer entropies

$q$5

where $q$6 lies in the intrinsic part of the Clifford commutant, absent from Haar moments. The same work emphasizes that for odd prime $q$7, the Clifford and Haar commutants differ already at $q$8, so three replicas are enough to detect non-Cliffordness in qudit systems. In the large-$q$9 doped-circuit regime, the averaged purity obeys the universal scaling

$$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$0

hence

$$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$1

This turns generalized stabilizer entropies into explicit diagnostics of how doped Clifford circuits leave the stabilizer regime (Magni et al., 2 Jun 2025).

5. Dynamical saturation, transport, and phase transitions

Several recent works show that Clifford-adapted entropies exhibit sharp dynamical structure rather than smooth generic growth. In long-range random Clifford circuits with a $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$2 conservation law, bipartite entanglement is directly controlled by the hydrodynamic transport exponent: $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$3 with

$$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$4

for one-dimensional long-range gates distributed as $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$5. Because stabilizer states have flat entanglement spectra, all Rényi entropies coincide in these circuits, so the single stabilizer entanglement entropy $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$6 represents the entire Rényi family. The paper interprets the resulting scaling as evidence for higher-$$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$7 Rényi entropies in generic systems with conservation laws, while also stressing a Clifford-specific microscopic mechanism: under $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$8-symmetric Clifford evolution, operators become dominated by slowly spreading $$\mathbf v=(\mathbf a|\mathbf b)\in \mathbb F_q^{2L}$$9-components, and the corresponding light cones are understood in terms of classical Lévy flights (Richter et al., 2022).

In doped Clifford circuits, universal saturation of stabilizer entropies sharpens into dynamical phase transitions. “Probing quantum complexity via universal saturation of stabilizer entropies” shows that the derivative of the SRE crosses at the same point independent of the number of qubits and can be rescaled onto a single curve. The critical point depends non-trivially on the Rényi index $$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$0, and the work uses a two-peak approximation to the Pauli spectrum under random evolution to explain why $$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$1 and $$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$2 probe different aspects of complexity (Haug et al., 2024).

The many-qudit doped-Clifford theory makes the transition explicit. With doping ratio

$$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$3

the magic transition occurs at

$$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$4

while the chaos transition extracted from six-point OTOCs obeys

$$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$5

This yields an intermediate regime in which states are already maximally magical in generalized stabilizer entropy, yet operators are not fully chaotic in the unitary-design sense. The same work further shows that some low-order observables such as inverse participation ratios and entanglement purities become Haar-like already with $$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$6, whereas generalized stabilizer entropies and OTOCs require finite-density doping. For qutrit brickwork circuits, the local numerical data collapse onto the same global formula, suggesting that locality plays only a limited role in state-level magic spreading (Magni et al., 2 Jun 2025).

6. Unitary, thermodynamic, and topological extensions

The operator-level generalization is explicit in “Clifford entropy.” Fixing Weyl–Heisenberg displacement operators $$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$7, the channel characteristic matrix is

$$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$8

and the $$\langle\mathbf v,\mathbf v'\rangle=\sum_{j=1}^{L}\big(a_j b_j'-b_j a_j'\big)\bmod q.$$9-Clifford entropy is

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$00

For Clifford unitaries, $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$01 is a permutation matrix, so $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$02. The paper proves faithfulness, invariance under pre- and post-composition by Clifford unitaries, and the tensor-product law

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$03

It then rewrites $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$04 in terms of the stabilizer entropy of the Choi state and derives, for $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$05, an exact Haar-average formula that tends to $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$06. Using concentration of measure, it further shows that for a generic unitary $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$07, the ratio $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$08 lower-bounds the depth of a $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$09-doped Clifford implementation with probability approaching unity in large dimension (Cuffaro et al., 28 Dec 2025).

A thermodynamic variant appears in “Clifford Ergotropy.” There the central entropy-like quantity is the infinite-order filtered stabilizer Rényi entropy

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$10

where $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$11 is the largest nonidentity Pauli coefficient in the filtered Pauli spectrum. This yields the universal bound

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$12

for Clifford ergotropy. The interpretation is that larger magic, as quantified by $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$13, lowers the universal upper bound on work extractable under Clifford control alone. For Haar-random pure states, $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$14 is extensive, which the paper uses to derive a second-law-type statement: Clifford ergotropy is exponentially small or subextensive while the ergotropy gap remains extensive (Maity et al., 11 May 2026).

A topological realization is developed in $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$15 Chern-Simons theory. There Pauli and Clifford operators are represented by path integrals over 3-manifolds with Wilson loop insertions, and entanglement entropies are computed by topological gluing. For $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$16 states, the $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$17-party entropy is

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$18

and Clifford evolution is identified with modular transformations and Dehn twists. This work extends the Clifford-compatible topological toolkit from stabilizer states to non-stabilizer families such as $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$19 and Dicke states (Munizzi et al., 16 Oct 2025).

7. Extremality, non-stabilizer families, and scope

The extremal geometry of Clifford-adapted entropies is clarified by “Extremizing Measures of Magic on Pure States by Clifford-stabilizer States.” For a finite subgroup $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$20, the paper defines $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$21-stabilizer states as generators of one-dimensional $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$22-stabilizer codes and proves that if a pure state is fixed by a finite subgroup of the Clifford group, then it is an extremal point of a broad class of Clifford-covariant functionals. The theorem applies to symmetric functionals, max-type functionals, and Rényi-type sums

$$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$23

under variations orthogonal to the stabilized subspace. Specializing to Pauli and Clifford groups, the paper concludes that Clifford-stabilizer states extremize mana, stabilizer Rényi entropies, and stabilizer fidelity. It then classifies such states for qubits, qutrits, ququints, and two-qubit systems, identifying, among other examples, the two-qubit $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$24 family that maximizes $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$25 (Erew et al., 22 Dec 2025).

A complementary perspective comes from Dicke states. “Entropy Cones and Entanglement Evolution for Dicke States” studies a non-stabilizer family whose subsystem entropies are exactly computable: $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$26 The same work identifies Pauli and two-qubit Clifford stabilizers of Dicke states, constructs reachability graphs, and shows that restricted two-qubit Clifford orbits contain only finitely many entropy vectors: $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$27 for $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$28 and $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$29, and $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$30 for $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$31. This demonstrates that exact entropy control by Clifford orbit data extends beyond pure stabilizer states, though in a more limited and orbit-dependent way (Munizzi et al., 2023).

Taken together, these works suggest a precise but plural usage. In monitored Clifford circuits, Clifford entropy can literally be the rank exponent $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$32. In magic-resource theory, it most often means a stabilizer Rényi entropy, a generalized stabilizer purity, or a filtered Pauli-spectrum entropy. In operator theory, it has become an explicit unitary measure $$M_\alpha(\psi)=\frac{1}{1-\alpha}\log \Delta_{2\alpha}(\psi)$$33. Across all of these settings, the decisive structural feature is the same: Clifford operations reduce the relevant entropy problem to a Pauli, Weyl, stabilizer, or commutant object that is discrete, algebraically constrained, and often exactly tractable.