Demystifying Objectivity with Operator Algebra Quantum Error Correction

ChunJun Cao; Gong Cheng; Marin Girard

arxiv: 2606.06588 · v2 · pith:COI4G6FJnew · submitted 2026-06-04 · 🪐 quant-ph

Demystifying Objectivity with Operator Algebra Quantum Error Correction

Marin Girard , Gong Cheng , ChunJun Cao This is my paper

Pith reviewed 2026-06-28 00:35 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum darwinismoperator algebra quantum error correctionobjectivitystabilizer codesdecoherencequantum error correctionclassicalityredundancy

0 comments

The pith

Objectivity emerges when quantum information becomes algebraically locally recoverable in error-correcting codes.

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

The paper connects quantum Darwinism, which seeks to explain the appearance of classical objectivity through decoherence, to the mathematical structure of operator algebra quantum error correction. It establishes that the emergence of objectivity corresponds directly to the algebraic local recoverability property of quantum codes. This identification supplies a single algebraic criterion that replaces several separate measures of objectivity used in prior work. A reader would care because the link turns an interpretive question about quantum-to-classical transition into a precise question about code properties that can be checked with existing computational tools.

Core claim

By connecting quantum Darwinism to operator algebra quantum error correction, we show that the emergence of objectivity can be identified with the algebraic local recoverability of quantum codes. Applying this algebraic framework to stabilizer codes yields a far more precise characterization of classicality and redundancy, unifies the traditional measures of objectivity, enables efficient classification via coding-theoretic tools, and supports large-scale Clifford simulations of decoherence dynamics.

What carries the argument

Algebraic local recoverability of quantum codes in the operator algebra quantum error correction framework, which detects when system information can be recovered from local subsystems without needing global access.

If this is right

Stabilizer codes receive a precise algebraic criterion for when they produce classicality and redundancy.
Separate existing measures of objectivity become special cases of a single recoverability condition.
Coding-theoretic algorithms can classify which codes exhibit objectivity.
Decoherence dynamics can be simulated at larger scales using Clifford-circuit methods.

Where Pith is reading between the lines

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

The same algebraic test might be applied to non-stabilizer codes to predict objectivity in a wider range of physical systems.
Existing quantum error correction software libraries could be repurposed to compute objectivity measures for many-body decoherence models.
The unification suggests that debates over which objectivity measure is fundamental may be settled by checking which one aligns most closely with algebraic recoverability.

Load-bearing premise

The operator algebra quantum error correction framework accurately captures the relevant aspects of objectivity in quantum Darwinism without introducing extraneous structure or missing key decoherence features.

What would settle it

An explicit quantum Darwinism model in which observers agree on a classical record yet the associated quantum code lacks algebraic local recoverability, or conversely a code with algebraic local recoverability but no emergent classical consensus among observers.

Figures

Figures reproduced from arXiv: 2606.06588 by ChunJun Cao, Gong Cheng, Marin Girard.

**Figure 2.** Figure 2: Algebraic objectivity regimes for a fragment family [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Schematic brickwork Clifford circuit with periodic [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

Quantum Darwinism extends the decoherence formalism to explain how objectivity emerges from quantum mechanics. However, existing approaches often capture only partial aspects of objectivity. By connecting quantum Darwinism to operator algebra quantum error correction, we show that the emergence of objectivity can be identified with the algebraic local recoverability of quantum codes. Applying this algebraic framework to stabilizer codes, we show that it yields a far more precise characterization of classicality and redundancy, unifies the traditional measures of objectivity, enables efficient classification via coding-theoretic tools, and supports large-scale Clifford simulations of decoherence dynamics.

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.

Desk Editor's Note private letter to a colleague

The paper identifies objectivity in quantum Darwinism with algebraic local recoverability in operator-algebra QEC and then uses stabilizer codes to classify redundancy and run Clifford simulations.

read the letter

The central move is equating the emergence of objectivity with algebraic local recoverability of quantum codes, then specializing to stabilizer codes for sharper classification and larger simulations. That framing is new on its face and gives a clean way to import coding tools into Darwinism questions.

What works is the unification claim: once you have the algebraic recoverability condition, several traditional objectivity measures fall out as different ways of reading the same algebra. The stabilizer-code application also looks practical for Clifford-circuit simulations of decoherence, which could let people check redundancy scaling on bigger systems than before.

The soft spot is whether the identification is derived or stipulated. The abstract states that objectivity “can be identified with” algebraic recoverability, but does not indicate a step that starts from a concrete decoherence Hamiltonian, extracts the pointer basis, and shows that the von Neumann algebra recoverability condition is necessary and sufficient for the usual Darwinism observables. If that step is missing or only sketched, the framework risks adding algebraic structure that is not forced by the dynamics. The stress-test note raises exactly this point, and without seeing the body it is hard to tell how much weight the claim carries.

This is for readers already working on quantum Darwinism or on operator-algebra approaches to open systems. Someone looking for a new simulation handle or a way to organize existing measures would find the stabilizer section useful. I would send it to peer review; the angle is distinct enough and the simulation payoff is concrete enough that referees should check the derivation and the concrete examples.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that the emergence of objectivity in quantum Darwinism can be identified with the algebraic local recoverability of quantum codes in the operator algebra quantum error correction framework. Specializing to stabilizer codes, the approach is said to yield a more precise characterization of classicality and redundancy, unify traditional objectivity measures, enable classification via coding-theoretic tools, and support large-scale Clifford simulations of decoherence dynamics.

Significance. If the central identification is valid and the algebraic framework faithfully reproduces the redundant encoding of classical information under realistic decoherence without extraneous structure, the work could offer a rigorous bridge between quantum Darwinism and quantum error correction, allowing the use of established coding tools for classification and efficient simulation. The explicit application to stabilizer codes for these purposes is a concrete strength that could enable falsifiable numerical tests of objectivity measures.

major comments (2)

[Abstract] Abstract: The identification of the emergence of objectivity with algebraic local recoverability is asserted without any derivation or argument showing that this algebraic condition is necessary or sufficient for the standard Darwinism observables (redundancy of classical information in the environment, pointer-basis selection) to arise from microscopic interaction Hamiltonians. This is load-bearing for the claim that the operator-algebra QEC framework captures the relevant aspects of quantum Darwinism.
[Abstract] The specialization to stabilizer codes (mentioned in the abstract): It is not demonstrated whether the choice of von Neumann algebra or the algebraic recoverability condition reproduces the interaction-specific decoherence features without adding extraneous structure not forced by the dynamics, which directly affects whether the framework avoids the weakest assumption noted in the stress-test.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address the major comments point by point below, indicating revisions where appropriate to strengthen the presentation of our arguments.

read point-by-point responses

Referee: [Abstract] Abstract: The identification of the emergence of objectivity with algebraic local recoverability is asserted without any derivation or argument showing that this algebraic condition is necessary or sufficient for the standard Darwinism observables (redundancy of classical information in the environment, pointer-basis selection) to arise from microscopic interaction Hamiltonians. This is load-bearing for the claim that the operator-algebra QEC framework captures the relevant aspects of quantum Darwinism.

Authors: We agree the abstract is concise and could more explicitly signal the supporting derivation. In the body of the manuscript the identification follows directly from the definition of objectivity as redundant encoding of a pointer observable, which algebraic local recoverability formalizes within the operator-algebra QEC framework; the construction ensures that the condition is both necessary and sufficient for the standard observables once the interaction Hamiltonian is mapped to the error model. To address the concern, we will revise the abstract and add a short paragraph in the introduction that sketches how the algebraic condition emerges from the microscopic dynamics, thereby making the load-bearing step explicit without changing the core claims. revision: yes
Referee: [Abstract] The specialization to stabilizer codes (mentioned in the abstract): It is not demonstrated whether the choice of von Neumann algebra or the algebraic recoverability condition reproduces the interaction-specific decoherence features without adding extraneous structure not forced by the dynamics, which directly affects whether the framework avoids the weakest assumption noted in the stress-test.

Authors: For stabilizer codes the von Neumann algebra is fixed by the logical operators that commute with the error operators induced by the given interaction Hamiltonian; the algebraic recoverability condition is then checked against that algebra. This choice is therefore dictated by the dynamics rather than imposed externally. We will add a clarifying paragraph in the revised manuscript (near the stabilizer-code application) that explicitly ties the algebra selection to the interaction Hamiltonian and references the stress-test to confirm that no extraneous structure is introduced, thereby addressing the concern directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity: identification presented as derived connection, not definitional reduction.

full rationale

The abstract frames the central claim as a shown identification arising from connecting quantum Darwinism to operator-algebra QEC, followed by application to stabilizer codes for classification and simulation. No quoted equations or steps in the provided text reduce the claimed emergence of objectivity to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The derivation chain is presented as external to the inputs, with the algebraic condition offered as a unifying characterization rather than a renaming or self-definition. This is the expected non-finding for a conceptual bridging paper whose load-bearing steps remain independent of the target observables.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5611 in / 948 out tokens · 26867 ms · 2026-06-28T00:35:28.792379+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 2 canonical work pages

[1]

For any operator ˜O∈M f and any state| ˜ψ⟩ ∈ C, there exists an operatorO f onH f such that ˜O| ˜ψ⟩= Of | ˜ψ⟩and ˜O†| ˜ψ⟩=O † f | ˜ψ⟩
[2]

In other words, accessibility of logical operators onf (statement 1) is equivalent to state-level recoverability fromf(statement 2)

We can decomposeH f =⊕ α(Hf α 1 ⊗ H f α 2 )such that for all basis states| gα, ij⟩there exists unitary transformationU f onfand orthonormal ancil- lary states|χ α,j⟩f α 2 ¯f ∈ H f α 2 ¯f such thatU f |gα, ij⟩= |α, i⟩f α 1 ⊗ |χα,j⟩f α 2 ¯f . In other words, accessibility of logical operators onf (statement 1) is equivalent to state-level recoverability fro...
[3]

\ i Z(M fi) =M fj ,∀j.(14) In other words, only classical information is found in each fragment, and the information is identical across all fragments inF

Strong algebraic objectivity (SAO) requires that all relevant fragmentsF={f i}have a common center that is maximal. \ i Z(M fi) =M fj ,∀j.(14) In other words, only classical information is found in each fragment, and the information is identical across all fragments inF. 5 no-cloning boundQDO LAO SAO = SQD Weak AO/No AO forbidden shared quantum info ℓ-red...
[4]

This means that, for some partitioning, each fragment only contains classical information of the system, but different fragments know about different parts of the decohered system

Localized algebraic objectivity (LAO), where all pro- liferated information is classical, i.e.M fi =Z(M fi) for alli, but the common center need not be maxi- mal across different fragments:M fi ∩M fj ⊂Z(M fi). This means that, for some partitioning, each fragment only contains classical information of the system, but different fragments know about differe...
[5]

The quantum information is ex- clusive to each fragment, but the classical information can remain redundant either globally or in a localized fashion like in LAO

Quantum doped objectivity (QDO), where we allow quantum information to be found in fragments such thatZ(M fi)⊂M fi. The quantum information is ex- clusive to each fragment, but the classical information can remain redundant either globally or in a localized fashion like in LAO. Systems showing QDO correspond to fragments that may contain some anticommutin...
[6]

Keeping only the rows that have no sup- port in the columns corresponding to qubits in ¯f and removing the zero columns, we obtain a re- ducedH ′ t = [f ′|R′]

We first shuffle the columns ofH t such that the columns (for both the Z and X sections) are ar- ranged as [ ¯f|f|R], then perform Gaussian elimina- tion overF 2 to obtain its reduced row echelon form (RREF). Keeping only the rows that have no sup- port in the columns corresponding to qubits in ¯f and removing the zero columns, we obtain a re- ducedH ′ t ...
[7]

Keeping only rows with support inR ′ and removing zero columns, we obtain a check ma- trixH ′′ t with the form [R ′′|f ′′] for both the Z and X sections

Next, we arrange the columns ofH ′ t with order [R′|f ′], then perform row elimination again to get its RREF. Keeping only rows with support inR ′ and removing zero columns, we obtain a check ma- trixH ′′ t with the form [R ′′|f ′′] for both the Z and X sections. This then eliminates any stabilizer gen- erators that don’t act non-trivially on the logical space
[8]

Finally, the rows of [f ′′] correspond to a logical ba- sis with support only inf. Performing the stan- dard row operation of [32], one can arrange it to the standard form with 2qrows that correspond to theqsymplectic pairs andcrows that correspond to elements in the center. By row counting, we can then determine precisely the qubits and bits recoverable ...
[9]

The most expensive step of the above algorithm is Gaussian elimination, which has complexity no more thanO((n+k) 3) for each time and fragment

using Choi states. The most expensive step of the above algorithm is Gaussian elimination, which has complexity no more thanO((n+k) 3) for each time and fragment. The over- all generation of such information dynamics is approxi- matelyO(T N f(n+k) 3) whereTis the total number of time steps andN f is the number of fragments we check per time step. ACKNOWLE...
[10]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Foundations of Physics35, 1857 (2005), arXiv:quant-ph/0408147

Pith/arXiv arXiv 2005
[11]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Physical Review A 73, 062310 (2006), arXiv:quant-ph/0505031

Pith/arXiv arXiv 2006
[12]

W. H. Zurek, Nature Physics5, 181 (2009)

2009
[13]

Horodecki, J

R. Horodecki, J. K. Korbicz, and P. Horodecki, Physical Review A91, 032122 (2015), arXiv:1312.6588 [quant-ph]

Pith/arXiv arXiv 2015
[14]

T. P. Le and A. Olaya-Castro, Physical Review Letters 122, 010403 (2019), arXiv:1803.08936 [quant-ph]

Pith/arXiv arXiv 2019
[15]

J. K. Korbicz, Quantum5, 571 (2021), arXiv:2007.04276 [quant-ph]

arXiv 2021
[16]

D. A. Chisholm, L. Innocenti, and G. M. Palma, Physical Review A110, 012218 (2024)

2024
[17]

E. Ryan, M. Paternostro, and S. Campbell, Physics Let- ters A416, 127675 (2021)

2021
[18]

W. H. Zurek, Entropy24, 1520 (2022), arXiv:2208.09019 [cond-mat, physics:quant-ph]

arXiv 2022
[19]

Balaneskovic and M

N. Balaneskovic and M. Mendler, The European Physical Journal D70, 177 (2016), arXiv:1606.09366 [quant-ph]

Pith/arXiv arXiv 2016
[20]

Fert´ e and X

B. Fert´ e and X. Cao, Physical Review Letters132, 110201 (2024), arXiv:2305.03694 [quant-ph]

arXiv 2024
[21]

Fert´ e, D

B. Fert´ e, D. Farci, and X. Cao, Decoherent histories with(out) objectivity in a (broken) apparatus (2025), arXiv:2508.16482 [quant-ph]

arXiv 2025
[22]

Harlow, Communications in Mathematical Physics 354, 865 (2017), arXiv:1607.03901 [gr-qc, physics:hep-th, physics:quant-ph]

D. Harlow, Communications in Mathematical Physics 354, 865 (2017), arXiv:1607.03901 [gr-qc, physics:hep-th, physics:quant-ph]

Pith/arXiv arXiv 2017
[23]

Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D

D. Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D. thesis, California Institute of Technology (1997), arXiv:quant-ph/9705052

Pith/arXiv arXiv 1997
[26]

[5, 13, 32, 34]

See Supplemental Material at [URL will be inserted by publisher] for detailed proofs, and supporting calcula- tions, which includes Refs. [5, 13, 32, 34]
[27]

A. M. Steane, Phys. Rev. A54, 4741 (1996), arXiv:quant- ph/9605021 [quant-ph]

arXiv 1996
[28]

Ioffe and M

L. Ioffe and M. M´ ezard, Phys. Rev. A75, 032345 (2007), arXiv:quant-ph/0606107 [quant-ph]

Pith/arXiv arXiv 2007
[29]

Fert´ e and X

B. Fert´ e and X. Cao, Physical Review A109, 032226 (2024), arXiv:2312.04284 [quant-ph]

arXiv 2024
[31]

Bacon and A

D. Bacon and A. Casaccino, arXiv e-prints , quant- ph/0610088 (2006), arXiv:quant-ph/0610088 [quant-ph]

arXiv 2006
[32]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A86, 10.1103/phys- reva.86.032324 (2012)

work page doi:10.1103/phys- 2012
[33]

Prange,Cyclic Error-Correcting Codes in Two Sym- bols, Tech

E. Prange,Cyclic Error-Correcting Codes in Two Sym- bols, Tech. Rep. TN-57-/03 (Air Force Cambridge Re- search Center, 1957)

1957
[34]

Girard, G

M. Girard, G. Cheng, and C. Cao, In preparation
[35]

We note that this is slightly different from the Zurekian definition where redundancy means the number of inde- pendent fragments of the environment that have access to almost all of the classical information. 8
[36]

Cao and B

C. Cao and B. Lackey, Quantum weight enumerators and tensor networks (2023), arXiv:2211.02756 [math-ph, physics:quant-ph]

arXiv 2023
[37]

Zwolak and W

M. Zwolak and W. H. Zurek, Phys. Rev. A95, 030101 (2017), arXiv:1703.10096 [quant-ph]

Pith/arXiv arXiv 2017
[38]

Aaronson and D

S. Aaronson and D. Gottesman, Physical Review A70, 10.1103/physreva.70.052328 (2004)

work page doi:10.1103/physreva.70.052328 2004
[39]

Cao and B

C. Cao and B. Lackey, Quantum lego power-up: De- signing transversal gates with tensor networks (2026), arXiv:2603.03542 [quant-ph]

arXiv 2026
[40]

Gidney, Quantum5, 497 (2021)

C. Gidney, Quantum5, 497 (2021)

2021
[41]

C. Cao, G. Cotardo, and B. Lackey, Quantum Anticodes (2025), arXiv:2512.13891 [quant-ph]

arXiv 2025
[42]

Steinberg, J

M. Steinberg, J. Fan, R. J. Harris, D. Elkouss, S. Feld, and A. Jahn, Quantum9, 1826 (2025)

2025
[43]

Bravyi, D

S. Bravyi, D. Fattal, and D. Gottesman, Journal of Mathematical Physics47, 062106 (2006), arXiv:quant- ph/0504208. SUPPLEMENT AL MA TERIAL Proof of Theorem 3 We represent logical Pauli operators modulo phases and stabilizers by binary vectors, with the usual symplec- tic form encoding commutation, adopting the definition and results from [32]. LetW f be t...

arXiv 2006

[1] [1]

For any operator ˜O∈M f and any state| ˜ψ⟩ ∈ C, there exists an operatorO f onH f such that ˜O| ˜ψ⟩= Of | ˜ψ⟩and ˜O†| ˜ψ⟩=O † f | ˜ψ⟩

[2] [2]

In other words, accessibility of logical operators onf (statement 1) is equivalent to state-level recoverability fromf(statement 2)

We can decomposeH f =⊕ α(Hf α 1 ⊗ H f α 2 )such that for all basis states| gα, ij⟩there exists unitary transformationU f onfand orthonormal ancil- lary states|χ α,j⟩f α 2 ¯f ∈ H f α 2 ¯f such thatU f |gα, ij⟩= |α, i⟩f α 1 ⊗ |χα,j⟩f α 2 ¯f . In other words, accessibility of logical operators onf (statement 1) is equivalent to state-level recoverability fro...

[3] [3]

\ i Z(M fi) =M fj ,∀j.(14) In other words, only classical information is found in each fragment, and the information is identical across all fragments inF

Strong algebraic objectivity (SAO) requires that all relevant fragmentsF={f i}have a common center that is maximal. \ i Z(M fi) =M fj ,∀j.(14) In other words, only classical information is found in each fragment, and the information is identical across all fragments inF. 5 no-cloning boundQDO LAO SAO = SQD Weak AO/No AO forbidden shared quantum info ℓ-red...

[4] [4]

This means that, for some partitioning, each fragment only contains classical information of the system, but different fragments know about different parts of the decohered system

Localized algebraic objectivity (LAO), where all pro- liferated information is classical, i.e.M fi =Z(M fi) for alli, but the common center need not be maxi- mal across different fragments:M fi ∩M fj ⊂Z(M fi). This means that, for some partitioning, each fragment only contains classical information of the system, but different fragments know about differe...

[5] [5]

The quantum information is ex- clusive to each fragment, but the classical information can remain redundant either globally or in a localized fashion like in LAO

Quantum doped objectivity (QDO), where we allow quantum information to be found in fragments such thatZ(M fi)⊂M fi. The quantum information is ex- clusive to each fragment, but the classical information can remain redundant either globally or in a localized fashion like in LAO. Systems showing QDO correspond to fragments that may contain some anticommutin...

[6] [6]

Keeping only the rows that have no sup- port in the columns corresponding to qubits in ¯f and removing the zero columns, we obtain a re- ducedH ′ t = [f ′|R′]

We first shuffle the columns ofH t such that the columns (for both the Z and X sections) are ar- ranged as [ ¯f|f|R], then perform Gaussian elimina- tion overF 2 to obtain its reduced row echelon form (RREF). Keeping only the rows that have no sup- port in the columns corresponding to qubits in ¯f and removing the zero columns, we obtain a re- ducedH ′ t ...

[7] [7]

Keeping only rows with support inR ′ and removing zero columns, we obtain a check ma- trixH ′′ t with the form [R ′′|f ′′] for both the Z and X sections

Next, we arrange the columns ofH ′ t with order [R′|f ′], then perform row elimination again to get its RREF. Keeping only rows with support inR ′ and removing zero columns, we obtain a check ma- trixH ′′ t with the form [R ′′|f ′′] for both the Z and X sections. This then eliminates any stabilizer gen- erators that don’t act non-trivially on the logical space

[8] [8]

Finally, the rows of [f ′′] correspond to a logical ba- sis with support only inf. Performing the stan- dard row operation of [32], one can arrange it to the standard form with 2qrows that correspond to theqsymplectic pairs andcrows that correspond to elements in the center. By row counting, we can then determine precisely the qubits and bits recoverable ...

[9] [9]

The most expensive step of the above algorithm is Gaussian elimination, which has complexity no more thanO((n+k) 3) for each time and fragment

using Choi states. The most expensive step of the above algorithm is Gaussian elimination, which has complexity no more thanO((n+k) 3) for each time and fragment. The over- all generation of such information dynamics is approxi- matelyO(T N f(n+k) 3) whereTis the total number of time steps andN f is the number of fragments we check per time step. ACKNOWLE...

[10] [10]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Foundations of Physics35, 1857 (2005), arXiv:quant-ph/0408147

Pith/arXiv arXiv 2005

[11] [11]

Blume-Kohout and W

R. Blume-Kohout and W. H. Zurek, Physical Review A 73, 062310 (2006), arXiv:quant-ph/0505031

Pith/arXiv arXiv 2006

[12] [12]

W. H. Zurek, Nature Physics5, 181 (2009)

2009

[13] [13]

Horodecki, J

R. Horodecki, J. K. Korbicz, and P. Horodecki, Physical Review A91, 032122 (2015), arXiv:1312.6588 [quant-ph]

Pith/arXiv arXiv 2015

[14] [14]

T. P. Le and A. Olaya-Castro, Physical Review Letters 122, 010403 (2019), arXiv:1803.08936 [quant-ph]

Pith/arXiv arXiv 2019

[15] [15]

J. K. Korbicz, Quantum5, 571 (2021), arXiv:2007.04276 [quant-ph]

arXiv 2021

[16] [16]

D. A. Chisholm, L. Innocenti, and G. M. Palma, Physical Review A110, 012218 (2024)

2024

[17] [17]

E. Ryan, M. Paternostro, and S. Campbell, Physics Let- ters A416, 127675 (2021)

2021

[18] [18]

W. H. Zurek, Entropy24, 1520 (2022), arXiv:2208.09019 [cond-mat, physics:quant-ph]

arXiv 2022

[19] [19]

Balaneskovic and M

N. Balaneskovic and M. Mendler, The European Physical Journal D70, 177 (2016), arXiv:1606.09366 [quant-ph]

Pith/arXiv arXiv 2016

[20] [20]

Fert´ e and X

B. Fert´ e and X. Cao, Physical Review Letters132, 110201 (2024), arXiv:2305.03694 [quant-ph]

arXiv 2024

[21] [21]

Fert´ e, D

B. Fert´ e, D. Farci, and X. Cao, Decoherent histories with(out) objectivity in a (broken) apparatus (2025), arXiv:2508.16482 [quant-ph]

arXiv 2025

[22] [22]

Harlow, Communications in Mathematical Physics 354, 865 (2017), arXiv:1607.03901 [gr-qc, physics:hep-th, physics:quant-ph]

D. Harlow, Communications in Mathematical Physics 354, 865 (2017), arXiv:1607.03901 [gr-qc, physics:hep-th, physics:quant-ph]

Pith/arXiv arXiv 2017

[23] [23]

Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D

D. Gottesman,Stabilizer Codes and Quantum Error Cor- rection, Ph.D. thesis, California Institute of Technology (1997), arXiv:quant-ph/9705052

Pith/arXiv arXiv 1997

[24] [26]

[5, 13, 32, 34]

See Supplemental Material at [URL will be inserted by publisher] for detailed proofs, and supporting calcula- tions, which includes Refs. [5, 13, 32, 34]

[25] [27]

A. M. Steane, Phys. Rev. A54, 4741 (1996), arXiv:quant- ph/9605021 [quant-ph]

arXiv 1996

[26] [28]

Ioffe and M

L. Ioffe and M. M´ ezard, Phys. Rev. A75, 032345 (2007), arXiv:quant-ph/0606107 [quant-ph]

Pith/arXiv arXiv 2007

[27] [29]

Fert´ e and X

B. Fert´ e and X. Cao, Physical Review A109, 032226 (2024), arXiv:2312.04284 [quant-ph]

arXiv 2024

[28] [31]

Bacon and A

D. Bacon and A. Casaccino, arXiv e-prints , quant- ph/0610088 (2006), arXiv:quant-ph/0610088 [quant-ph]

arXiv 2006

[29] [32]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A86, 10.1103/phys- reva.86.032324 (2012)

work page doi:10.1103/phys- 2012

[30] [33]

Prange,Cyclic Error-Correcting Codes in Two Sym- bols, Tech

E. Prange,Cyclic Error-Correcting Codes in Two Sym- bols, Tech. Rep. TN-57-/03 (Air Force Cambridge Re- search Center, 1957)

1957

[31] [34]

Girard, G

M. Girard, G. Cheng, and C. Cao, In preparation

[32] [35]

We note that this is slightly different from the Zurekian definition where redundancy means the number of inde- pendent fragments of the environment that have access to almost all of the classical information. 8

[33] [36]

Cao and B

C. Cao and B. Lackey, Quantum weight enumerators and tensor networks (2023), arXiv:2211.02756 [math-ph, physics:quant-ph]

arXiv 2023

[34] [37]

Zwolak and W

M. Zwolak and W. H. Zurek, Phys. Rev. A95, 030101 (2017), arXiv:1703.10096 [quant-ph]

Pith/arXiv arXiv 2017

[35] [38]

Aaronson and D

S. Aaronson and D. Gottesman, Physical Review A70, 10.1103/physreva.70.052328 (2004)

work page doi:10.1103/physreva.70.052328 2004

[36] [39]

Cao and B

C. Cao and B. Lackey, Quantum lego power-up: De- signing transversal gates with tensor networks (2026), arXiv:2603.03542 [quant-ph]

arXiv 2026

[37] [40]

Gidney, Quantum5, 497 (2021)

C. Gidney, Quantum5, 497 (2021)

2021

[38] [41]

C. Cao, G. Cotardo, and B. Lackey, Quantum Anticodes (2025), arXiv:2512.13891 [quant-ph]

arXiv 2025

[39] [42]

Steinberg, J

M. Steinberg, J. Fan, R. J. Harris, D. Elkouss, S. Feld, and A. Jahn, Quantum9, 1826 (2025)

2025

[40] [43]

Bravyi, D

S. Bravyi, D. Fattal, and D. Gottesman, Journal of Mathematical Physics47, 062106 (2006), arXiv:quant- ph/0504208. SUPPLEMENT AL MA TERIAL Proof of Theorem 3 We represent logical Pauli operators modulo phases and stabilizers by binary vectors, with the usual symplec- tic form encoding commutation, adopting the definition and results from [32]. LetW f be t...

arXiv 2006