Infinite-Order Filtered Stabilizer Renyi Entropy

• The topic defines infinite-order filtered stabilizer Renyi entropy as a measure that isolates the largest nonidentity Pauli expectation to quantify magic.
• Different filtering prescriptions—including omitting the identity, state preprocessing, or subtracting local contributions—yield nontrivial yet variant entropy constructions.
• Operational strategies like purity encoding and swap tests enable estimation, while the infinite-order limit highlights dominant contributions and raises open research challenges.

Infinite-order filtered stabilizer Rényi entropy is the $\alpha\to\infty$ limit of a stabilizer Rényi-entropy construction after a filtering prescription has been imposed on the Pauli-overlap data or on the state itself. In the stabilizer Rényi framework, an $n$-qubit pure state $\lvert\psi\rangle$ is assigned a probability distribution built from squared Pauli expectation values, and the corresponding Rényi entropies quantify nonstabilizerness, or magic. The infinite-order limit isolates the largest weight in that distribution. Across the recent literature, however, “filtered” is not a single standardized operation: it may mean excluding the identity Pauli, minimizing over stabilizer-preserving preprocessing, or subtracting local short-range contributions in a mixed-state setting. As a result, infinite-order filtered stabilizer Rényi entropy is best understood as a family of closely related, but not identical, constructions rather than a unique invariant (Leone et al., 2021, Stratton, 3 Jul 2025).

1. Stabilizer Rényi entropy and the competing meanings of filtering

For an $n$-qubit Hilbert space of dimension $d=2^n$, with Pauli strings $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$, the characteristic distribution of a pure state $\lvert\psi\rangle$ is

$$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$

which satisfies $\sum_{P\in\mathcal P_n}\Xi_P=1$. The stabilizer Rényi entropy used as a magic measure is

$$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$

or equivalently the Rényi entropy of the distribution $n$0 shifted by $n$1. The standard properties stated in the literature are faithfulness, Clifford invariance, and additivity, with monotonicity under allowed operations established for $n$2 in the formulation used for the purity-encoding algorithm (Stratton, 3 Jul 2025).

The phrase “filtered stabilizer Rényi entropy” appears in several distinct senses. One construction restricts attention to a subset $n$3 of Pauli strings and renormalizes the surviving probabilities, for example by removing the identity operator. In that case,

$n$4

and one computes a Rényi entropy from the filtered distribution $n$5 (Leone et al., 2021).

A second construction defines filtered SRE by preprocessing the state with stabilizer-preserving filters and minimizing:

$n$6

where $n$7 may include local Clifford unitaries, Pauli measurements with classical feedforward, and stabilizer-preserving CPTP maps. In this usage, filtering acts on the state rather than on the Pauli-indexed probability distribution (Hoshino et al., 14 Jul 2025).

A third usage arises in mixed-state many-body settings, where long-range SRE is defined by subtracting local subsystem contributions,

$n$8

so that strictly local magic is filtered out and only the nonstabilizerness linking $n$9 and $\lvert\psi\rangle$0 remains. This is a filtered component of magic rather than a filtered pure-state SRE in the Pauli-subset sense (López et al., 2024).

2. Infinite-order limit and the nontrivial role of filtering

For a fixed probability distribution, the Rényi-$\lvert\psi\rangle$1 entropy is the min-entropy,

$\lvert\psi\rangle$2

Applied to the stabilizer distribution, this gives

$\lvert\psi\rangle$3

Because the identity Pauli is included in the standard definition, $\lvert\psi\rangle$4 for every pure state, and since $\lvert\psi\rangle$5 for every Pauli $\lvert\psi\rangle$6, one has $\lvert\psi\rangle$7 for all $\lvert\psi\rangle$8. Hence

$\lvert\psi\rangle$9

for every pure state. In the unfiltered formulation, the infinite-order magic measure therefore trivializes completely (Leone et al., 2021).

This trivialization is the main reason filtered variants become important at $n$0. If one chooses

$n$1

then

$n$2

For a stabilizer state $n$3, the non-identity elements of its stabilizer group contribute $n$4, while all other non-identity Paulis contribute $n$5. The filtered distribution is therefore uniform on $n$6 outcomes, so

$n$7

A filtered infinite-order magic measure can then be shifted to vanish on stabilizer states:

$n$8

This produces a nontrivial quantity that directly probes the largest non-identity Pauli overlap (Leone et al., 2021).

Resource-theoretic properties depend strongly on the filtering prescription. For Pauli-subset filtering, Clifford invariance survives if the subset $n$9 is invariant under Clifford conjugation; $d=2^n$0 has this property. The literature explicitly cautions, however, that additivity generally fails for filtered entropies because the normalization factor $d=2^n$1 need not factorize under tensor products. By contrast, in the preprocessing/minimization definition, monotonicity under stabilizer protocols is part of the construction for $d=2^n$2, and the $d=2^n$3 limit is interpreted through the survival of universal terms rather than through Pauli-subset renormalization (Hoshino et al., 14 Jul 2025).

From the purity-based formulation,

$d=2^n$4

one also obtains the inferred pure-state infinite-order expression

$d=2^n$5

which is the standard Rényi-$d=2^n$6 limit for the characteristic distribution. This inference is explicitly identified as such in the purity-encoding work (Stratton, 3 Jul 2025).

3. Estimation through purity encoding

An operational route to finite-order stabilizer Rényi entropies is provided by a mixed-unitary channel acting on $d=2^n$7 copies of an unknown pure state:

$d=2^n$8

Its central property is

$d=2^n$9

so the purity of the engineered state directly yields the moment $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$0 and therefore $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$1 after classical post-processing. A coherent implementation uses $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$2 ancilla qubits prepared by $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$3 and a controlled Pauli-string unitary

$\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$4

after which tracing out the ancilla realizes the channel $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$5. The same channel can also be implemented incoherently by uniformly sampling a Pauli string, applying it to all $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$6 copies, and then forgetting which string was applied (Stratton, 3 Jul 2025).

The paper focuses on purity estimation by the swap test. Two copies of the output state yield the purity to additive error $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$7 using $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$8 repetitions. Because each output copy consumes $\mathcal P_n=\{P_j\}_{j=0}^{d^2-1}$9 copies of $\lvert\psi\rangle$0, the total copy complexity is $\lvert\psi\rangle$1. To estimate $\lvert\psi\rangle$2 to additive error $\lvert\psi\rangle$3, one needs $\lvert\psi\rangle$4, which gives $\lvert\psi\rangle$5 copies, refined in the stated failure-probability form to $\lvert\psi\rangle$6 copies. The swap-test realization requires two output states simultaneously, hence $\lvert\psi\rangle$7 copies of $\lvert\psi\rangle$8 in flight and an at least $\lvert\psi\rangle$9 qubit device including ancilla. Randomized measurements can estimate the same purity using a single output copy, reducing concurrency to $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$0 copies on an $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$1-qubit device (Stratton, 3 Jul 2025).

Relative to alternative algorithms, the purity-encoding protocol is reported to scale worse than quantum-state-tomography estimation of $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$2 for odd $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$3, but better for even $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$4. It matches the scaling of direct $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$5-copy estimation of all Pauli terms, and it is outperformed in both copy complexity and concurrency by the algorithm of Phys. Rev. Lett. 132, 240602, which estimates $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$6 with $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$7 copies for odd $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$8 and $$\Xi_P(\lvert\psi\rangle)=d^{-1}\langle\psi|P|\psi\rangle^2,$$9 for even $\sum_{P\in\mathcal P_n}\Xi_P=1$0 using only two-copy measurements (Stratton, 3 Jul 2025).

For the infinite-order problem, the purity-encoding paper does not define a filtered SRE. It instead suggests, by implication, a practical strategy of estimating $\sum_{P\in\mathcal P_n}\Xi_P=1$1 for increasing integers $\sum_{P\in\mathcal P_n}\Xi_P=1$2 and extrapolating $\sum_{P\in\mathcal P_n}\Xi_P=1$3. The paper benchmarks the qubit family

$\sum_{P\in\mathcal P_n}\Xi_P=1$4

for which

$\sum_{P\in\mathcal P_n}\Xi_P=1$5

In simulations of Algorithm 1 for $\sum_{P\in\mathcal P_n}\Xi_P=1$6, additive error $\sum_{P\in\mathcal P_n}\Xi_P=1$7, and failure probability $\sum_{P\in\mathcal P_n}\Xi_P=1$8, the measured values match theoretical predictions within the plotted error bars, and “the accuracy of the algorithm appears to have no dependence on $\sum_{P\in\mathcal P_n}\Xi_P=1$9” over that tested range. No general finite-$$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$0 error bound to $$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$1 is given (Stratton, 3 Jul 2025).

The same work notes an inferred filtered adaptation: replacing the uniform Pauli mixture by a non-uniform distribution $$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$2 over a subset or weighted family of Pauli strings. The resulting purity becomes

$$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$3

which reduces to a weighted sum of $$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$4 only under suitable symmetry conditions. The paper gives no formal guarantees, normalization, or monotonicity theory for such filtered purity encodings, and treats this as an open question (Stratton, 3 Jul 2025).

4. Universal terms in critical systems: boundaries, defects, and filtered $$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$5

In one-dimensional critical spin chains, the stabilizer Rényi entropy admits a boundary-conformal-field-theory and replica formulation in which open boundaries and topological defects contribute universal corrections. For a chain of $$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$6 qubits, the global SRE is written as

$$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$7

equivalently as a Bell-basis participation entropy computed from a replica partition function $$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$8. For factorising defects, or open boundaries, the replica free energy has the form

$$M_{\alpha}(\ket{\psi})=(1-\alpha)^{-1}\ln\bigg(\frac{1}{d}\sum_{j=0}^{d^2-1}\bra{\psi}P_j\ket{\psi}^{2\alpha}\bigg),$$9

which implies

$n$00

For topological defects $n$01,

$n$02

so that

$n$03

With multiple defects obeying fusion algebra $n$04, the universal constant is set by the dominant fusion channel in the ground-state limit (Hoshino et al., 14 Jul 2025).

In this setting, filtered SRE is defined by minimizing over stabilizer-preserving filters. The crucial statement is that the universal BCFT terms are unchanged by such filtering. Finite-depth local Clifford circuits cannot change the replicated central charge or the boundary-condition changing dimensions, so the boundary logarithm is unaffected; likewise, moving or fusing topological defects by Clifford circuits does not alter the universal defect constants. For the infinite-order filtered case, the universal form is stated as

$n$05

with

$n$06

For multiple defects,

$n$07

so the infinite-order limit selects the dominant channel, in direct analogy with min-entropy behavior (Hoshino et al., 14 Jul 2025).

The Ising universality class supplies explicit values. For all nine elementary factorising defects, the paper finds

$n$08

for any $n$09, so $n$10. For closed chains, the reported $n$11 defect constants are $n$12, $n$13, and $n$14. For two duality defects, the fusion rule $n$15 leads the ground state to select the identity channel, and the corresponding SRE constant becomes $n$16. The infinite-order filtered interpretation is that these universal constants survive stabilizer-preserving filtering, while the nonuniversal bulk term may change (Hoshino et al., 14 Jul 2025).

5. Long-range filtered magic in the dual-unitary XXZ circuit

A different filtered construction appears in the exactly solvable dual-unitary XXZ Floquet circuit. The system is a chain of $n$17 qubits evolved by a brick-wall circuit with two-qubit gates

$n$18

starting from a product of Bell pairs,

$n$19

For pure states, the stabilizer moments are

$n$20

For mixed states, the paper uses

$n$21

The long-range SRE,

$n$22

is interpreted as the amount of magic that cannot be removed by short-depth local quantum circuits (López et al., 2024).

The paper derives exact formulas by ZX-calculus. After a half time step, the thermodynamic-limit SRE density is

$n$23

As $n$24, both trigonometric terms vanish for generic $n$25, and the limit gives

$n$26

At the Clifford points $n$27 and $n$28, the limit is also $n$29 (López et al., 2024).

For the specific partition

$n$30

the reduced states $n$31 and $n$32 are maximally mixed, so the filtering subtraction is trivial and

$n$33

The exact moments are

$n$34

with

$n$35

and an explicit closed form for $n$36 given in the paper. At finite order $n$37, the long-range SRE is nonzero and, for large $n$38, saturates around $n$39 for generic parameters, while vanishing at the Clifford points (López et al., 2024).

The infinite-order statement for this mixed-state construction is more delicate because the paper defines long-range SRE only at $n$40. It does, however, provide $n$41 for all integer $n$42. A natural extension is

$n$43

but this extension is explicitly not adopted in the paper. Under that inferred extension, one finds for generic $n$44 that $n$45 and $n$46 as $n$47, so

$n$48

which implies $n$49 and therefore $n$50 in partition $n$51. The paper presents this as an extrapolative conclusion rather than an explicit theorem. The significance is that finite-order long-range filtered magic can be nonzero and even saturating, while the infinite-order extension suppresses it entirely in this solvable setting (López et al., 2024).

6. Conceptual status, misconceptions, and open problems

A recurrent misconception is that infinite-order filtered stabilizer Rényi entropy denotes a single canonical quantity. The literature does not support that reading. One line of work shows that the unfiltered $n$52 stabilizer Rényi entropy is trivial on all pure states because the identity Pauli fixes the largest probability weight. Another introduces a nontrivial infinite-order quantity by filtering out the identity or another Clifford-invariant subset. A third studies filtering as minimization over stabilizer-preserving operations and emphasizes the preservation of universal BCFT data under $n$53. A fourth isolates a filtered long-range component in mixed states by subtracting local contributions, and in an exactly solvable partition its inferred infinite-order extension vanishes. This suggests that “infinite-order filtered SRE” is a family resemblance term rather than a unique invariant (Leone et al., 2021, Hoshino et al., 14 Jul 2025, López et al., 2024).

Several points are firmly established. The pure-state infinite-order limit of the standard, identity-inclusive SRE gives $n$54. Excluding the identity yields a nontrivial filtered min-entropy that measures the inverse of the largest non-identity Pauli overlap. In critical Ising chains, the infinite-order filtered SRE preserves the universal $n$55 boundary term and the defect constants determined by $n$56-factors and dominant fusion channels. In the dual-unitary XXZ circuit, finite-order long-range filtered magic is exactly solvable and nonzero, yet the natural $n$57 extension collapses to zero in partition $n$58. The purity-encoding algorithm gives an explicit operational procedure for estimating $n$59 at finite integer $n$60 and therefore suggests, but does not rigorously provide, an estimation strategy for approaching the infinite-order regime (Stratton, 3 Jul 2025).

The open problems are correspondingly structural rather than merely technical. The purity-encoding work explicitly leaves open how to choose a non-uniform Pauli filter so that purity remains a simple linear functional of a filtered stabilizer moment, what resource-theoretic meaning and monotonicity such a filtered SRE would have, and how to control the bias introduced by filtering in the $n$61 limit. The same paper does not provide a detailed noise analysis or explicit finite-$n$62 bounds to $n$63. The filtered Pauli-subset construction does not inherit additivity in general, and the long-range construction is rigorously defined only for $n$64 in the exactly solvable circuit setting. These limitations are not incidental: they indicate that the infinite-order limit is exceptionally sensitive to the exact choice of filtration, because it depends only on the single dominant contribution that survives the filter (Leone et al., 2021, Stratton, 3 Jul 2025, López et al., 2024).

Taken together, the current literature supports a precise but plural picture. Infinite-order filtered stabilizer Rényi entropy can mean a nonidentity-filtered min-entropy of the Pauli-overlap distribution, a stabilizer-protocol-minimized $n$65 entropy retaining universal boundary and defect signatures, or an inferred infinite-order limit of a long-range filtered mixed-state construction. What unifies these variants is that $n$66 always amplifies the dominant surviving contribution after filtering. What differentiates them is the object being filtered: Pauli strings, states under free operations, or local subsystem contributions.