---
title: Infinite-Order Filtered Stabilizer Renyi Entropy
url: https://www.emergentmind.com/topics/infinite-order-filtered-stabilizer-renyi-entropy
type: topic
---

# Infinite-Order Filtered Stabilizer Renyi Entropy

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 [2106.12587], [2507.02540].

## 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 $\{\Xi_P\}$ shifted by $-\ln d$. The standard properties stated in the literature are faithfulness, Clifford invariance, and additivity, with monotonicity under allowed operations established for $\alpha\ge 2$ in the formulation used for the purity-encoding algorithm [2507.02540].

The phrase “filtered stabilizer Rényi entropy” appears in several distinct senses. One construction restricts attention to a subset $F\subseteq\mathcal P_n$ of Pauli strings and renormalizes the surviving probabilities, for example by removing the identity operator. In that case,
$$
p_P^{(F)}(\lvert\psi\rangle)=\frac{\Xi_P(\lvert\psi\rangle)}{Z_F(\lvert\psi\rangle)},\qquad
Z_F(\lvert\psi\rangle)=\sum_{Q\in F}\Xi_Q(\lvert\psi\rangle),
$$
and one computes a Rényi entropy from the filtered distribution $\{p_P^{(F)}\}_{P\in F}$ [2106.12587].

A second construction defines filtered SRE by preprocessing the state with stabilizer-preserving filters and minimizing:
$$
M_{\alpha}^{\mathrm{filt}}(\lvert\psi\rangle)=\min_{\mathfrak F\in\mathcal C} M_{\alpha}(\mathfrak F(\lvert\psi\rangle)),
$$
where $\mathcal C$ 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 [2507.10656].

A third usage arises in mixed-state many-body settings, where long-range SRE is defined by subtracting local subsystem contributions,
$$
L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B),
$$
so that strictly local magic is filtered out and only the nonstabilizerness linking $A$ and $B$ remains. This is a filtered component of magic rather than a filtered pure-state SRE in the Pauli-subset sense [2405.04448].

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

For a fixed probability distribution, the Rényi-$\infty$ entropy is the min-entropy,
$$
S_\infty=-\ln\max_i p_i.
$$
Applied to the stabilizer distribution, this gives
$$
S_\infty^{\mathrm{stab}}(\lvert\psi\rangle)=-\ln\Big(\max_{P\in\mathcal P_n}\Xi_P(\lvert\psi\rangle)\Big).
$$
Because the identity Pauli is included in the standard definition, $\Xi_I=d^{-1}$ for every pure state, and since $\lvert\langle\psi|P|\psi\rangle\rvert\le 1$ for every Pauli $P$, one has $\Xi_P\le d^{-1}$ for all $P$. Hence
$$
S_\infty^{\mathrm{stab}}(\lvert\psi\rangle)=\ln d,\qquad
M_\infty(\lvert\psi\rangle)=0
$$
for every pure state. In the unfiltered formulation, the infinite-order magic measure therefore trivializes completely [2106.12587].

This trivialization is the main reason filtered variants become important at $\alpha=\infty$. If one chooses
$$
F=\mathcal P_n\setminus\{I\},
$$
then
$$
Z_F(\lvert\psi\rangle)=1-\Xi_I(\lvert\psi\rangle)=1-\frac1d=\frac{d-1}{d}.
$$
For a stabilizer state $\lvert\phi\rangle$, the non-identity elements of its stabilizer group contribute $\Xi_P(\lvert\phi\rangle)=1/d$, while all other non-identity Paulis contribute $0$. The filtered distribution is therefore uniform on $d-1$ outcomes, so
$$
S_\infty^{\mathrm{stab},F}(\lvert\phi\rangle)=\ln(d-1).
$$
A filtered infinite-order magic measure can then be shifted to vanish on stabilizer states:
$$
M_\infty^{(F)}(\lvert\psi\rangle)=S_\infty^{\mathrm{stab},F}(\lvert\psi\rangle)-\ln(d-1).
$$
This produces a nontrivial quantity that directly probes the largest non-identity Pauli overlap [2106.12587].

Resource-theoretic properties depend strongly on the filtering prescription. For Pauli-subset filtering, Clifford invariance survives if the subset $F$ is invariant under Clifford conjugation; $F=\mathcal P_n\setminus\{I\}$ has this property. The literature explicitly cautions, however, that additivity generally fails for filtered entropies because the normalization factor $Z_F$ need not factorize under tensor products. By contrast, in the preprocessing/minimization definition, monotonicity under stabilizer protocols is part of the construction for $\alpha\ge 2$, and the $\alpha\to\infty$ limit is interpreted through the survival of universal terms rather than through Pauli-subset renormalization [2507.10656].

From the purity-based formulation,
$$
M_\alpha(\ket{\psi})=(1-\alpha)^{-1}\ln A_\alpha(\ket{\psi}),\qquad
A_\alpha(\ket{\psi})=d^{-1}\sum_j\bra\psi P_j\ket\psi^{2\alpha},
$$
one also obtains the inferred pure-state infinite-order expression
$$
S_{\infty}(\ket{\psi})=\lim_{\alpha\to\infty}M_\alpha(\ket{\psi})
=-\ln\Big(\max_j \bra{\psi}P_j\ket{\psi}^2\Big),
$$
which is the standard Rényi-$\infty$ limit for the characteristic distribution. This inference is explicitly identified as such in the purity-encoding work [2507.02540].

## 3. Estimation through purity encoding

An operational route to finite-order stabilizer Rényi entropies is provided by a mixed-unitary channel acting on $\alpha$ copies of an unknown pure state:
$$
E\big(\psi^{\otimes\alpha}\big)=d^{-2}\sum_{j=0}^{d^2-1}\big(P_j\psi P_j\big)^{\otimes\alpha}.
$$
Its central property is
$$
\operatorname{tr}\Big[E\big(\psi^{\otimes\alpha}\big)^2\Big]=d^{-1}A_\alpha(\ket\psi),
$$
so the purity of the engineered state directly yields the moment $A_\alpha(\ket\psi)$ and therefore $M_\alpha(\ket\psi)$ after classical post-processing. A coherent implementation uses $2n$ ancilla qubits prepared by $H^{\otimes 2n}$ and a controlled Pauli-string unitary
$$
cU_{\mathcal P}=\sum_{j=0}^{d^2-1}\ket j\!\bra j_A\otimes P_j^{\otimes\alpha},
$$
after which tracing out the ancilla realizes the channel $E$. The same channel can also be implemented incoherently by uniformly sampling a Pauli string, applying it to all $\alpha$ copies, and then forgetting which string was applied [2507.02540].

The paper focuses on purity estimation by the swap test. Two copies of the output state yield the purity to additive error $\tau$ using $O(\tau^{-2})$ repetitions. Because each output copy consumes $\alpha$ copies of $\lvert\psi\rangle$, the total copy complexity is $O(\alpha\tau^{-2})$. To estimate $A_\alpha(\lvert\psi\rangle)$ to additive error $\epsilon$, one needs $\tau=\epsilon/d$, which gives $O(\alpha d^2\epsilon^{-2})$ copies, refined in the stated failure-probability form to $\lceil \alpha d^2\epsilon^{-2}\delta^{-1}\rceil$ copies. The swap-test realization requires two output states simultaneously, hence $2\alpha$ copies of $\lvert\psi\rangle$ in flight and an at least $2n(\alpha+1)+1$ qubit device including ancilla. Randomized measurements can estimate the same purity using a single output copy, reducing concurrency to $\alpha$ copies on an $n(\alpha+2)$-qubit device [2507.02540].

Relative to alternative algorithms, the purity-encoding protocol is reported to scale worse than quantum-state-tomography estimation of $\Gamma_\alpha^{\otimes n}$ for odd $\alpha$, but better for even $\alpha$. It matches the scaling of direct $2\alpha$-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 $A_\alpha(\lvert\psi\rangle)$ with $O(\alpha\epsilon^{-2})$ copies for odd $\alpha$ and $O(\alpha d\epsilon^{-2})$ for even $\alpha$ using only two-copy measurements [2507.02540].

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 $M_\alpha$ for increasing integers $\alpha>1$ and extrapolating $\alpha\to\infty$. The paper benchmarks the qubit family
$$
\ket{\psi_\theta}=\frac{1}{\sqrt2}\big(\ket0+e^{i\theta}\ket1\big),
$$
for which
$$
A_\alpha(\ket{\psi_\theta})=\frac12\big(1+\cos(\theta)^{2\alpha}+\sin(\theta)^{2\alpha}\big).
$$
In simulations of Algorithm 1 for $\alpha\in\{2,3,5,7\}$, additive error $\epsilon=0.05$, and failure probability $\delta=0.1$, the measured values match theoretical predictions within the plotted error bars, and “the accuracy of the algorithm appears to have no dependence on $\alpha$” over that tested range. No general finite-$\alpha$ error bound to $S_\infty$ is given [2507.02540].

The same work notes an inferred filtered adaptation: replacing the uniform Pauli mixture by a non-uniform distribution $\{p_j\}$ over a subset or weighted family of Pauli strings. The resulting purity becomes
$$
\operatorname{tr}\big[E_F(\rho^{\otimes\alpha})^2\big]
=\sum_{j,k}p_jp_k\,\operatorname{tr}\big[P_j\rho P_jP_k\rho P_k\big]^\alpha,
$$
which reduces to a weighted sum of $\langle\psi|P_\ell|\psi\rangle^{2\alpha}$ 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 [2507.02540].

## 4. Universal terms in critical systems: boundaries, defects, and filtered $\alpha\to\infty$

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 $L$ qubits, the global SRE is written as
$$
M_\alpha(\psi)=\frac{1}{1-\alpha}\ln\!\Big[\sum_{\vec m}\big(\operatorname{Tr}[\sigma^{\vec m}\psi]\big)^{2\alpha}\Big]-(\ln 2)L,
$$
equivalently as a Bell-basis participation entropy computed from a replica partition function $Z_{2\alpha}$. For factorising defects, or open boundaries, the replica free energy has the form
$$
-\ln(Z_{2\alpha}/Z^{2\alpha})=b_\alpha L+\gamma_\alpha\ln L+O(1),
$$
which implies
$$
M_\alpha(L)=M_\alpha^{\mathrm{bulk}}(L)+c_b(\alpha)\ln L+O(1),\qquad
c_b(\alpha)=\frac{\gamma_\alpha}{\alpha-1}.
$$
For topological defects $\mathfrak A$,
$$
-\ln(Z_{2\alpha}/Z^{2\alpha})=b_\alpha L-\ln g_\alpha^{\mathfrak A}+o(1),
$$
so that
$$
M_\alpha(L;\mathfrak A)=M_\alpha^{\mathrm{bulk}}(L)-C_{\mathrm{defect}}(\alpha;\mathfrak A)+o(1),\qquad
C_{\mathrm{defect}}(\alpha;\mathfrak A)=\frac{1}{\alpha-1}\ln g_\alpha^{\mathfrak A}.
$$
With multiple defects obeying fusion algebra $\mathfrak A\otimes\mathfrak B=\bigoplus_c \mathfrak C$, the universal constant is set by the dominant fusion channel in the ground-state limit [2507.10656].

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
$$
S_\infty^{\mathrm{filt}}(L;\text{defects,boundaries})
=
S_\infty^{\mathrm{bulk}}(L)+c_b(\infty)\ln L-C_{\mathrm{defect}}(\infty),
$$
with
$$
c_b(\infty)=\lim_{\alpha\to\infty}\frac{\gamma_\alpha}{\alpha-1},\qquad
C_{\mathrm{defect}}(\infty)=\lim_{\alpha\to\infty}\frac{\ln g_\alpha^{\mathfrak A}}{\alpha}.
$$
For multiple defects,
$$
C_{\mathrm{defect}}(\infty;a\otimes b)
=
\max_{c:N_{ab}^c\neq 0}\lim_{\alpha\to\infty}\frac{\ln g_\alpha^c}{\alpha},
$$
so the infinite-order limit selects the dominant channel, in direct analogy with min-entropy behavior [2507.10656].

The Ising universality class supplies explicit values. For all nine elementary factorising defects, the paper finds
$$
M_\alpha(L;\mathrm{open})=m_\alpha L-\frac14\ln L+O(1)
$$
for any $\alpha\ge 2$, so $c_b(\infty)=-1/4$. For closed chains, the reported $M_2$ defect constants are $c_2^{1}=\ln\sqrt2$, $c_2^\eta=0.755(1)$, and $c_2^{\mathfrak D}=0.020(3)$. For two duality defects, the fusion rule $\mathfrak D\otimes\mathfrak D=1\oplus\eta$ leads the ground state to select the identity channel, and the corresponding SRE constant becomes $c_2^{\mathfrak D\otimes\mathfrak D}=\ln\sqrt2$. The infinite-order filtered interpretation is that these universal constants survive stabilizer-preserving filtering, while the nonuniversal bulk term may change [2507.10656].

## 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 $N$ qubits evolved by a brick-wall circuit with two-qubit gates
$$
U_{e,o}=\exp(-iJ_{e,o}\,\sigma_z\otimes\sigma_z)\cdot\mathrm{SWAP},
$$
starting from a product of Bell pairs,
$$
\ket{\psi(0)}=\ket{\phi^+}^{\otimes N/2},\qquad
\ket{\phi^+}=\frac{1}{\sqrt2}\sum_{i=0}^1\ket{i\,i}.
$$
For pure states, the stabilizer moments are
$$
\zeta_n(\ket\psi)=\frac{1}{2^N}\sum_{P\in\mathcal P_N}\langle\psi|P|\psi\rangle^{2n},
\qquad
M_n(\ket\psi)=\frac{1}{1-n}\log\zeta_n(\ket\psi).
$$
For mixed states, the paper uses
$$
\tilde M_2(\rho)=-\log\!\Big(\frac{\zeta_2(\rho)}{\zeta_1(\rho)}\Big),
\qquad
\zeta_n(\rho)=\frac{1}{2^N}\sum_{P\in\mathcal P_N}\big(\operatorname{tr}(P\rho)\big)^{2n}.
$$
The long-range SRE,
$$
L(\rho_{AB})=\tilde M_2(\rho_{AB})-\tilde M_2(\rho_A)-\tilde M_2(\rho_B),
$$
is interpreted as the amount of magic that cannot be removed by short-depth local quantum circuits [2405.04448].

The paper derives exact formulas by ZX-calculus. After a half time step, the thermodynamic-limit SRE density is
$$
m_n\big(\ket{\psi(J,0,\tfrac12)}\big)
=
\frac{1}{2(1-n)}
\log\!\bigg(\frac{1+\cos^{2n}(2J)+\sin^{2n}(2J)}{2}\bigg).
$$
As $n\to\infty$, both trigonometric terms vanish for generic $J$, and the limit gives
$$
m_\infty=0.
$$
At the Clifford points $J=0$ and $J=\pi/4$, the limit is also $0$ [2405.04448].

For the specific partition
$$
B_0:\qquad N_A=N_B=2T,\quad d=2(T-1),\quad N=8T-4,
$$
the reduced states $\rho_A$ and $\rho_B$ are maximally mixed, so the filtering subtraction is trivial and
$$
L_{B_0}(J_o,J_e,T)=\tilde M_2(\rho_{AB}).
$$
The exact moments are
$$
\zeta_n(\rho_{AB})
=
\frac{1}{
2^{4T-2}\Big(1+2\big(f_n(J_o)f_n(J_e)\big)^T
+g_n(J_o,J_e)\big(f_n(J_o)f_n(J_e)\big)^{2T-4}\Big)},
$$
with
$$
f_n(J)=\cos^{2n}(2J)+\sin^{2n}(2J),
$$
and an explicit closed form for $g_n(J_o,J_e)$ given in the paper. At finite order $n=2$, the long-range SRE is nonzero and, for large $T$, saturates around $2\log 2$ for generic parameters, while vanishing at the Clifford points [2405.04448].

The infinite-order statement for this mixed-state construction is more delicate because the paper defines long-range SRE only at $n=2$. It does, however, provide $\zeta_n(\rho_{AB})$ for all integer $n$. A natural extension is
$$
\tilde M_n(\rho)=\frac{1}{1-n}\log\!\Big(\frac{\zeta_n(\rho)}{\zeta_1(\rho)}\Big),
\qquad
L_n(\rho_{AB})=\tilde M_n(\rho_{AB})-\tilde M_n(\rho_A)-\tilde M_n(\rho_B),
$$
but this extension is explicitly not adopted in the paper. Under that inferred extension, one finds for generic $J_o,J_e$ that $f_n(J_{e,o})\to 0$ and $g_n(J_o,J_e)\to 0$ as $n\to\infty$, so
$$
\lim_{n\to\infty}\zeta_n(\rho_{AB})=2^{-(4T-2)},
$$
which implies $\tilde M_\infty(\rho_{AB})=0$ and therefore $L_\infty=0$ in partition $B_0$. 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 [2405.04448].

## 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 $\alpha\to\infty$ 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 $\alpha\to\infty$. 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 [2106.12587], [2507.10656], [2405.04448].

Several points are firmly established. The pure-state infinite-order limit of the standard, identity-inclusive SRE gives $M_\infty\equiv 0$. 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 $-(1/4)\ln L$ boundary term and the defect constants determined by $g$-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 $\alpha\to\infty$ extension collapses to zero in partition $B_0$. The purity-encoding algorithm gives an explicit operational procedure for estimating $M_\alpha$ at finite integer $\alpha>1$ and therefore suggests, but does not rigorously provide, an estimation strategy for approaching the infinite-order regime [2507.02540].

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 $\alpha\to\infty$ limit. The same paper does not provide a detailed noise analysis or explicit finite-$\alpha$ bounds to $S_\infty$. The filtered Pauli-subset construction does not inherit additivity in general, and the long-range construction is rigorously defined only for $n=2$ 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 [2106.12587], [2507.02540], [2405.04448].

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 $\alpha\to\infty$ 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 $\alpha\to\infty$ 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.

Source: https://www.emergentmind.com/topics/infinite-order-filtered-stabilizer-renyi-entropy