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arxiv: 2606.29443 · v1 · pith:O5NGNAJUnew · submitted 2026-06-28 · 🪐 quant-ph

Stabilizer entropy is trustworthy for mixed states

Gianluca Esposito , Michele Viscardi , Alioscia Hamma This is my paper

Pith reviewed 2026-06-30 07:35 UTC · model grok-4.3

classification 🪐 quant-ph
keywords stabilizer entropynon-stabilizernessmixed statesClifford channelsmonotonicityquantum resource theorymany-body systemspartial measurements
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The pith

Linear stabilizer entropy acts as a non-stabilizerness monotone for mixed states with overwhelming probability under non-adaptive Clifford channels on flat states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a linear Stabilizer Entropy to quantify non-stabilizerness in mixed quantum states. Strict monotones require superexponential time to compute, but this linear version satisfies the required monotonicity property with probability that approaches certainty exponentially fast in system size N, provided the states are flat mixed stabilizer states and the operations are non-adaptive Clifford channels. Analytical and numerical evidence for Haar-random states, Clifford orbits, and random matrix product states confirms the exponential decay of violation probability. The measure is also shown to be valid under partial measurements in many-body systems, where the resource amount never increases for individual outcomes or on average over outcomes. This makes the linear Stabilizer Entropy a practical and justified resource measure in the regimes where exact alternatives are intractable.

Core claim

We propose a linear Stabilizer Entropy that acts as a proper non-stabilizerness monotone with overwhelming probability when restricted to non-adaptive Clifford channels acting on flat mixed stabilizer states. Analytical and numerical results for Haar-random states, Clifford orbits, and random matrix product states show that monotonicity violation probabilities decay as exp−ηN. We also prove the validity of Stabilizer Entropy in specific many-body systems undergoing partial measurements, where the amount of resource never increases for each measurement outcome as well as when averaged over outcome probabilities.

What carries the argument

The linear Stabilizer Entropy, an efficiently computable function proposed as a non-stabilizerness monotone that holds with high probability in the restricted setting.

If this is right

  • Monotonicity holds with overwhelming probability for Haar-random states.
  • The same exponential decay of violations occurs for states in Clifford orbits.
  • Random matrix product states exhibit the identical decay behavior.
  • In many-body systems, partial measurements leave the stabilizer entropy non-increasing both per outcome and on average.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical studies of non-stabilizerness in large mixed-state systems become feasible using this measure where exact monotones cannot be evaluated.
  • The distinction between worst-case and typical-case monotonicity may apply to other resource measures in quantum information.
  • The result suggests checking whether similar high-probability monotonicity holds when the restriction to flat states is relaxed in controlled ways.

Load-bearing premise

The central claim depends on restricting attention to non-adaptive Clifford channels and flat mixed stabilizer states.

What would settle it

An explicit family of flat mixed stabilizer states and non-adaptive Clifford channels for which the probability of monotonicity violation fails to decay exponentially with system size N.

Figures

Figures reproduced from arXiv: 2606.29443 by Alioscia Hamma, Gianluca Esposito, Michele Viscardi.

Figure 2
Figure 2. Figure 2: FIG. 2. Behavior of the analytical violation probability bound [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Behavior of the analytical violation probability bound [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Chebyshev upper bound on the violation probability for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Chebyshev upper bound on the violation probability for [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Phase diagrams of the outcome-averaged post [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 1
Figure 1. Figure 1: Average magic gap of M˜ lin (left) and its standard deviation (right) under complete dephasing in the computational basis for MPS in OBC, plotted as a function of the bond dimension χ ∈ [2, 14] for system sizes N ∈ [2, 16]. For each pair (N, χ), 3.5 × 104 random MPS are sampled. F. SE gaps for MPS in OBC Having established the behavior of SE gaps for MPS in PBC, we now turn to the Open Boundary Condition (… view at source ↗
Figure 2
Figure 2. Figure 2: Average magic gap of M˜ lin (left) and its standard deviation (right) under the partial trace of the first qubit for MPS in OBC, plotted as a function of the bond dimension χ ∈ [2, 14] for system sizes N ∈ [2, 16]. For each pair (N, χ), 3.5 × 104 random MPS are sampled. 2 3 4 5 6 7 8 9 10 11 12 13 14 Â 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ­ ¹M(h alf) ® 2 3 4 5 6 7 8 9 10 11 12 13 14 Â 10¡2 10¡1 s t d ¹M(h alf) N=2 … view at source ↗
Figure 3
Figure 3. Figure 3: Average magic gap of M˜ lin (left) and its standard deviation (right) under the partial trace of the first half of the system for MPS in OBC, plotted as a function of the bond dimension χ ∈ [2, 14] for system sizes N ∈ [2, 16]. For each pair (N, χ), 3.5 × 104 random MPS are sampled. channels. This confirms their reliability as magic proxies in MPS ensembles that are most relevant to quantum many-body syste… view at source ↗
Figure 4
Figure 4. Figure 4: Average magic gaps (top row) and their standard deviation (bottom row) versus the [PITH_FULL_IMAGE:figures/full_fig_p049_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average 2-SE gap M¯ 2 (left) and its standard deviation (right) under complete dephasing in the computational basis for MPS in OBC, with the same parameters as [PITH_FULL_IMAGE:figures/full_fig_p049_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average 2-SE gap M¯ 2 (left) and its standard deviation (right) under partial trace of the first qubit for MPS in OBC, with the same parameters as [PITH_FULL_IMAGE:figures/full_fig_p050_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average 2-SE gap M¯ 2 (left) and its standard deviation (right) under partial trace of the first half of the system for MPS in OBC, with the same parameters as [PITH_FULL_IMAGE:figures/full_fig_p050_7.png] view at source ↗
read the original abstract

Quantifying non-stabilizerness in mixed states is provably intractable, as any strict monotone requires superexponential time. We propose a linear Stabilizer Entropy that acts as a proper non-stabilizerness monotone with overwhelming probability when restricted to non-adaptive Clifford channels acting on flat mixed stabilizer states. Analytical and numerical results for Haar-random states, Clifford orbits, and random matrix product states show that monotonicity violation probabilities decay as $\exp-\eta N$. We also prove the validity of Stabilizer Entropy in specific many-body systems undergoing partial measurements, where the amount of resource never increases for each measurement outcome as well as when averaged over outcome probabilities. Given the hardness of strict alternatives, Stabilizer Entropy emerges as a practical and theoretically justified resource measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a linear Stabilizer Entropy as a practical non-stabilizerness monotone for mixed states. It claims this measure acts as a proper monotone with overwhelming probability when restricted to non-adaptive Clifford channels on flat mixed stabilizer states, supported by analytical and numerical results showing exponential decay (exp(−ηN)) of monotonicity violation probabilities for Haar-random states, Clifford orbits, and random matrix product states. It also proves validity for partial measurements in specific many-body systems, where the resource does not increase for each outcome or on average.

Significance. If the restricted claims hold with the stated exponential bounds, the work supplies a computationally feasible resource quantifier for non-stabilizerness in mixed states, where strict monotones are known to be intractable. The combination of analytical derivations for specific ensembles, numerical verification across multiple state classes, and proofs for measurement scenarios provides concrete evidence that the measure can be trustworthy under the stated operational restrictions, addressing a practical gap in quantum resource theories.

major comments (2)
  1. [Numerical results and analytical derivations sections] The central claim of trustworthiness 'with overwhelming probability' is explicitly restricted to non-adaptive Clifford channels on flat mixed stabilizer states, yet the exponential decay is shown only for the three listed ensembles (Haar-random, Clifford orbits, random MPS). No argument is supplied that these ensembles are representative or that the bound extends to arbitrary flat mixed stabilizer states under the allowed channels.
  2. [Partial measurements section] The separate proof that Stabilizer Entropy is non-increasing under partial measurements (both per outcome and averaged) is stated to hold only for 'specific many-body systems.' This limits the scope of the measurement result and leaves open whether the property extends to general flat mixed stabilizer states.
minor comments (1)
  1. [Title] The title states that Stabilizer Entropy 'is trustworthy for mixed states' without qualification, while the abstract and body heavily restrict the regime; aligning the title with the actual scope would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, proposing clarifications and partial revisions to better delineate the scope of our results.

read point-by-point responses

  1. Referee: [Numerical results and analytical derivations sections] The central claim of trustworthiness 'with overwhelming probability' is explicitly restricted to non-adaptive Clifford channels on flat mixed stabilizer states, yet the exponential decay is shown only for the three listed ensembles (Haar-random, Clifford orbits, random MPS). No argument is supplied that these ensembles are representative or that the bound extends to arbitrary flat mixed stabilizer states under the allowed channels.

    Authors: The three ensembles were deliberately chosen as standard, representative classes in quantum information: Haar-random states model generic high-dimensional behavior, Clifford orbits capture structure under the relevant symmetry group, and random MPS represent physically relevant low-entanglement many-body states. The consistent exponential decay across these diverse classes supports the practical trustworthiness claim. We acknowledge that no general proof for every conceivable flat mixed stabilizer state is supplied. We will add a dedicated paragraph in the discussion section explaining the choice of ensembles, their representativeness for typical cases, and explicitly noting that extension to fully arbitrary states remains open. revision: partial

  2. Referee: [Partial measurements section] The separate proof that Stabilizer Entropy is non-increasing under partial measurements (both per outcome and averaged) is stated to hold only for 'specific many-body systems.' This limits the scope of the measurement result and leaves open whether the property extends to general flat mixed stabilizer states.

    Authors: The analytical proof is restricted to specific many-body systems because these admit explicit calculations based on the stabilizer structure and measurement operators. This provides a concrete demonstration of non-increase (per outcome and on average) in relevant physical settings. We will revise the manuscript to emphasize the restricted scope of this result, clarify that it does not claim generality over all flat mixed stabilizer states, and note the generalization as an interesting direction for future work. revision: partial

Circularity Check

0 steps flagged

No circularity; proposal verified on restricted ensembles via independent analytics and numerics

full rationale

The paper defines linear Stabilizer Entropy and shows it acts as a monotone with probability 1-exp(-ηN) only under the explicit restriction to non-adaptive Clifford channels on flat mixed stabilizer states. Support comes from direct calculations on Haar-random states, Clifford orbits, random MPS, and a separate proof for partial measurements in specific many-body systems. These are external verifications, not self-definitional fits or predictions that reduce to the definition by construction. No load-bearing self-citations or imported uniqueness theorems appear in the abstract or described chain. The claim is self-contained within its stated narrow regime.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies insufficient detail to enumerate free parameters, axioms, or invented entities; none are explicitly identified.

pith-pipeline@v0.9.1-grok · 5650 in / 1112 out tokens · 45754 ms · 2026-06-30T07:35:18.732165+00:00 · methodology

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Reference graph

Works this paper leans on

60 extracted references · 30 canonical work pages · 6 internal anchors

  1. [1]

    (62) we get ∆2 ˜Mlin(ψ) = 96(d−1) (d+ 3) 2(d+ 5)(d+ 6)(d+ 7) =O 1 d4 .(65)

    Variance of ˜Mlin(ψ) ∆2 ˜Mlin(ψ)reads ∆2 ˜Mlin(ψ) =E U[(1−SP(ψ U))2]−[E U ˜Mlin(ψU)]2 = 1 +E U[(SP(ψU))2]−2E U(SP(ψU))−[E U ˜Mlin(ψU)]2 = 1 +d 2 Tr (Q⊗Q)E U(ψ⊗8 U ) −2dTr QEU(ψ⊗4 U ) − 1− 4 d+ 3 2 1 +d 2 Tr (Q⊗Q)E U(ψ⊗8 U ) −2dTr QEU(ψ⊗4 U ) −1− 16 (d+ 3) 2 + 8 d+ 3 =d 2 Tr (Q⊗Q)E U(ψ⊗8 U ) − 8 d+ 3 − 16 (d+ 3) 2 + 8 d+ 3 (62) Let us focus on the last ter...

  2. [2]

    Variance of ˜Mlin(ψA) The variance of ˜Mlin(ψA)reads ∆2( ˜Mlin(ψ)) =E U[(Purψ A −SP(ψ A))2]−E U[Purψ A −SP(ψ A)]2 =E U[Pur2 ψA] +E U[SP(ψA)2]−2E U[SP(ψA) PurψA]− (dA −1)(d A + 1)dB (dAdB + 1)(dAdB + 3) 2 . (66) a Average square purity of reduce density matrix The average value of the square of the reduced density matrix purity reads EU Pur2(ψA U ) =E U Tr...

  3. [3]

    Covariance between ˜Mlin(ψ)and ˜Mlin(ψA) The covariance reads cov( ˜Mlin(ψ), ˜Mlin(ψA)) =E U[(1−SP(ψ))(Purψ A −SP(ψ A))]−E U[1−SP(ψ)]E U[Purψ A −SP(ψ A)] = (((((((((( EU[Purψ A −SP(ψ A)]−E U[SP(ψ) Purψ A] +E U[SP(ψ) SP(ψA)] − (((((((((( EU[Purψ A −SP(ψ A)] +E U[SP(ψ)(Purψ A −SP(ψ A))] =E U[SP(ψ) SP(ψA)] +E U[SP(ψ)]EU[Purψ A] −E U[SP(ψ) Purψ A]−E U[SP(ψ)]E...

  4. [4]

    Violation probability of SE under partial trace in Haar ensemble It is possible to obtain violation probability bounds either Chebyshev and/or Lévy typicality arguments, and we now show both concentration results for the SE gap for partial trace in the Haar ensemble. As seen before, the variance of the SE gap is exponentially vanishing asN→ ∞: ∆2 ¯M A =O ...

  5. [5]

    (92) Again, we split each piece in a separate calculation

    Variance of ˜Mlin(DBψ) The variance of the SE of the dephased state reads ∆2 ˜Mlin(DBψ) =E U h ˜Mlin(DBψU)2 i −E U h ˜Mlin(DBψU) i2 =E U Tr (T⊗T)(D Bψ⊗4 U ) +d 2EU Tr (Q⊗Q)(D BψU)⊗8 −2dE U Tr (T⊗Q)(D BψU)⊗6 − d−1 (d+ 1)(d+ 3) 2 . (92) Again, we split each piece in a separate calculation. a Average ofTr (T⊗T)(D Bψ⊗4 U ) We show computation of this object: ...

  6. [6]

    Final expression of the variance of SE gap for dephasing in the Haar ensemble Putting the pieces together, we finally get the following expression for∆¯M DB: ∆ ¯M DB = 4(d−1) d4 + 33d3 + 59d2 −15d+ 18 (d+ 1) 2(d+ 2)(d+ 3) 2(d+ 5)(d+ 6)(d+ 7) =O 1 d3 (106) F. Violation probability of SE under dephasing in the Haar ensemble Again, to establish monotonicity ...

  7. [7]

    Average ofPur 2(ψA C) We show computation of the following object: EC Pur2(ψA C) =E C Tr (T⊗2 A ⊗1 ⊗4 B )ψ⊗4 C =αTr (T⊗2 A ⊗1 ⊗4 B )QΠsym +βTr (TA ⊗1 ⊗4 B )Πsym = α 24 X π∈S4 Tr (TA ⊗T A)QATA π Tr QBTB π + β 24 X π∈S4 Tr T⊗2 A TA π Tr TB π (121) and direct evaluation of these traces yields EC Pur2(ψA C) = (d−1) d2 + 3dd2 A +d 4 A +d 2 A + (d2 A −1)(d 2 A ...

  8. [8]

    Average ofPur(ψ A C) SP(ψA C) We now show computation of the following object: EC[Pur(ψA C) SP(ψA C)] =d AEC Tr{[(QA ⊗T A)⊗1 ⊗6 B ]ψ⊗6 C }.(123) To compute this object, we show the Clifford average of 6 copies of a pure state, shown in [21]: ECψ⊗6 C = X Ω∈P6 pΩ(ψ) ΠsymΩΠsym Tr[ΩΠsym] (124) withP 6 ={Ω (6) 0 ,Ω (6) 1 ,Ω (6) 2 ,Ω (6) 3 }being the set of alg...

  9. [9]

    Average ofSP 2(ψA C) We now show computations of the following object: EC[SP2(ψA C)] =d 2 AEC Tr (Q⊗2 A ⊗1 ⊗8 B )ψ⊗8 C . (130) To compute this object, we show the Clifford average of 8 copies of a pure state, shown in [21]: ECψ⊗8 C = X Ω∈P8 pΩ(ψ) ΠsymΩΠsym Tr[ΩΠsym] (131) withP 8 ={Ω 0,Ω 1,Ω 2,Ω 3,Ω 4,Ω 5,Ω 6,Ω 7,Ω 8,Ω 9,Ω 10,Ω 11,Ω 12}being set of algebr...

    2052

  10. [10]

    Violation probability of SE under partial trace in the Clifford ensemble Since the Clifford orbit of a quantum state is a discrete set, we are not able to use Lévy lemma to establish typicality, hence we will Chebyshev inequality to bound the probability of violations. This is the reason why we computed the variance of the partial trace SE gap on the Clif...

  11. [11]

    (150) exploiting the usual properties of factorization, direct evaluation of these traces yield the result: EC[Pur2(DBψC)] = 2(3− ˜Mlin) (d+ 1)(d+ 2) (151)

    Average ofPur 2(DBψC) We show computation of the following object: EC[Pur2(DBψC)] =E C Tr T⊗T(D BψC)⊗4 =E C Tr (D⊗4 B T⊗T)ψ ⊗4 C =αTr (D⊗4 B T⊗T)QΠ sym +βTr (D⊗4 B T⊗T)Π sym = α 24 X π∈S4 Tr (D⊗4 B T⊗T)QT π + β 24 X π∈S4 Tr (D⊗4 B T⊗T)T π . (150) exploiting the usual properties of factorization, direct evaluation of these traces yield the result: EC[Pur2(...

  12. [12]

    Average ofPur(D BψC) SP(DBψC) In this section, we show complete computation of the following object: EC Pur(DBψC) SP(DBψC) =dE C Tr T⊗Q(D BψC)⊗6 =E C Tr (D⊗6 B T⊗Q)ψ ⊗6 C (152) Substituting the expression of the six-copy Clifford average of a state shown in Eq. (124) [21] we get the following: EC Pur(DBψC) SP(DBψC) =d X Ω∈P6 pΩ(ψ) Tr(ΩΠsym) X π,σ∈S 6 Tr (...

  13. [13]

    (155) 30 Substituting the expression of the 8-copy Clifford average of a pure state shown in Eq

    Average ofSP 2(DBψC) In this section we compute the following object: EC SP2(DBψC) =d 2EC Tr Q⊗Q(D BψC)⊗8 =d 2EC Tr (D⊗8 B Q⊗Q)ψ ⊗8 C . (155) 30 Substituting the expression of the 8-copy Clifford average of a pure state shown in Eq. (131) [21], we get EC SP2(DBψC) =d 2 X Ω∈P8 pΩ(ψ) Tr[ΩΠsym] Tr (D⊗8 B Q⊗Q)Π symΩΠsym = d2 8!2 X Ω∈P8 pΩ(ψ) Tr[ΩΠsym] X π,σ∈S...

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    Final expression of∆ 2 ¯M DB Summing all the elements together yields the final expression of∆2 ¯M DB, which reads ∆2 ¯M DB = (6 + 2d−4d 2)(1− ˜Mlin(ψ))− ˜Mlin(ψ)2 + (−4d−4) SP 3(ψ) + (d+ 1) SP4(ψ) + (d+ 1) Tr Ω8ψ⊗8 (d+ 1) 2(d+ 2) ≤ (−4d2 −2d+ 2(d+ 1) + 2)(1− ˜Mlin(ψ))− ˜Mlin(ψ)2 (d+ 1) 2(d+ 2) =O ˜Mlin(ψ) d ! (158) withSP 3(ψ)andSP 4(ψ)being the stabiliz...

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    NY s=1 Ais[s] # TrA

    Violation probability of SE under dephasing in the Clifford orbit We have seen that ∆2 C ¯M DB =O ˜Mlin(ψ) d ! (159) Hence, using Chebyshev’s inequality, we obtain an upper bound to the probability of violations establishing ˜Mlin as aη−magic proxy for dephasing withη=O(1)in the Clifford orbit. VI. SE as a proxy in random MPS: techniques A. Random MPS Giv...

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    (183): the operatorial part of this expression is factorized on each qubit, paired with the fact thatQ=Q (1) 1 ⊗ · · · ⊗Q (1) N , with Q(1) i = 1 4 P P∈{1,X,Y,Z} P ⊗4

    Average stabilizer purity of random MPS Objective of this section is to show the result of the following computation: Eψ∼χ SP(ψ) := 2N Eψ∼χ Tr Qψ⊗4 (184) 35 We will exploit an important fact about the expression Eq. (183): the operatorial part of this expression is factorized on each qubit, paired with the fact thatQ=Q (1) 1 ⊗ · · · ⊗Q (1) N , with Q(1) i...

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    Tr h T(2) πNB i Tr h T(χ) πNB T(χ) σNB −1 i W (2χ) πNB ,σNB | {z } ˜W (2) σNB −1 ,σNB × X πNB +1,...,πN W (2χ) πNB +1,σNB +1

    Average purity of traced out Random MPS The partial trace of the firstNB of the average2−copy average RMPS reads Eψ∼χψ⊗2 A = TrB Eψ∼χψ⊗2 = 1 N2 X σ1,...,σN X π1,...,πNB Tr h T(2) π1 i Tr h T(χ) π1 T(χ) σN i W (2χ) π1,σ1 | {z } ˜W (2) σN ,σ1 . . . . . .Tr h T(2) πNB i Tr h T(χ) πNB T(χ) σNB −1 i W (2χ) πNB ,σNB | {z } ˜W (2) σNB −1 ,σNB × X πNB +1,...,πN W...

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    W(2χ) πN ,σN Tr h T(χ) πNB +1T(χ) σNB i

    Average stabilizer purity of traced out RMPS By the same procedure utilized in the partial trace of the2−copy average RMPS, the4−copy partial trace of the average RMPS reads TrB Eψ∼χ(ψ⊗4) = 1 N4 X πNB +1,...,πN ,σNB ,...,σN ( ˜W (4))NB σN ,σNB W (2χ) πNB +1,σNB +1 . . . W(2χ) πN ,σN Tr h T(χ) πNB +1T(χ) σNB i . . .Tr h T(χ) πN T(χ) σ1 i (T(2) πNB +1 ⊗ · ·...

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    Average purity of dephased RMPS In this section we show how to compute the following object: Eψ∼χ[Pur(DBψ)] =E ψ∼χ Tr T(DBψ)⊗2 (192) Let us first recall the definition of the dephasing channel overNqubits: DBψ= X i1,...iN (Πi1 ⊗. . .Π iN )ψ(Πi1 ⊗. . .Π iN ) = (DB)(1) ⊗. . .⊗(D B)(N) ψ(193) with (DB(·))(k) = 1X ik=0 Πik(·)Πik (194) as we see, the dephasing...

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    ˜W (Q)(DB) σN−1 ,σN = 1 N4 Tr h ( ˜W (Q)(DB))N i (200) E

    Average stabilizer purity of dephased RMPS In this section we will show the derivation of the following object: Eψ∼χ[SP(DBψ)] = 2N Eψ∼χ Tr Q(DBψ)⊗4 (199) 38 Again, using the factorization of the dephasing channel and of theQoperator, we get Eψ∼χ[SP(DBψ)] = 2N N4 NY s=1 X πs σs∈S4 W (2χ) πs,σs Tr h T(χ) πs T(χ) σs−1 i Tr h Q(D⊗4 B T(2) πs ) i = 2N N4 NY s=...

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