Asymptotic Limits of Entanglement Distribution

Aby Philip; Alexander Streltsov; Piotr Masajada

arxiv: 2605.23443 · v1 · pith:AXC3DDHRnew · submitted 2026-05-22 · 🪐 quant-ph

Asymptotic Limits of Entanglement Distribution

Piotr Masajada , Aby Philip , Alexander Streltsov This is my paper

Pith reviewed 2026-05-25 04:46 UTC · model grok-4.3

classification 🪐 quant-ph

keywords entanglement distributionquantum repeaterscorrectable subspaceLOCCquantum channelsasymptotic limitsparallel channels

0 comments

The pith

Long-distance entanglement preservation is possible if and only if the quantum channel has a correctable subspace.

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

The paper establishes a dichotomy for entanglement distribution using repeater stations and LOCC: asymptotic preservation over arbitrary distances holds exactly when the channel admits a correctable subspace. Without such a subspace the transmitted state converges exponentially to the set of separable states. Maintaining nonzero entanglement then requires the number of parallel channel uses per link to grow at least logarithmically with the number of intermediate stations.

Core claim

The central claim is a strict if-and-only-if statement: asymptotic preservation of entanglement across arbitrarily long networks is possible precisely when the underlying quantum channel admits a correctable subspace. For channels lacking this subspace the transmitted state converges exponentially fast to the separable set under LOCC, rendering standard filtering insufficient; counteracting the decay demands that parallel channel uses per link scale at least logarithmically with the number of repeater stations.

What carries the argument

The correctable subspace of a quantum channel, which permits perfect recovery of a nontrivial set of input states despite the noise.

If this is right

Channels possessing a correctable subspace allow entanglement to be preserved indefinitely with fixed resources per link.
Channels without a correctable subspace cause exponential loss, so standard LOCC post-processing cannot restore entanglement at large distances.
Sustaining entanglement without a correctable subspace requires the number of parallel uses per link to grow at least as log of the number of stations.
Quantum error-correcting codes capable of saturating the logarithmic scaling become necessary for scalable repeater architectures.

Where Pith is reading between the lines

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

The same dichotomy supplies a channel-classification test that could be applied to common noise models to decide whether they support long-range distribution without extra resources.
The logarithmic scaling bound implies that any architecture relying solely on post-selection or filtering will fail for non-correctable channels once the network exceeds a fixed size.
Integrating error correction directly into the channel model rather than at the repeater level may be required to meet the derived resource lower bound.

Load-bearing premise

The dichotomy and scaling bounds assume only local operations and classical communication at repeater stations together with memoryless channels.

What would settle it

An explicit calculation or experiment that exhibits slower than exponential decay of entanglement for a concrete channel proven to lack any correctable subspace.

Figures

Figures reproduced from arXiv: 2605.23443 by Aby Philip, Alexander Streltsov, Piotr Masajada.

**Figure 1.** Figure 1: Entanglement distribution through a noisy channel with intermediate repeater stations. An initial bipartite state ρ AB is shared between parties A and B. Subsystem A is transmitted through a sequence of n noisy channel uses Λ, which may be understood as propagation through n successive transmission segments with intermediate stations located between Alice and Bob. After each channel use, the parties, poss… view at source ↗

read the original abstract

Reliable distribution of quantum entanglement over long distances is a central challenge in quantum information science, fundamentally limited by decoherence in noisy communication channels. In this work, we investigate the asymptotic limits of entanglement distribution across quantum networks utilizing intermediate repeater stations and local operations and classical communication (LOCC). We establish a strict dichotomy: the asymptotic preservation of entanglement over arbitrarily long distances is possible if and only if the underlying quantum channel admits a correctable subspace. For channels lacking such a subspace, we prove that the transmitted state converges exponentially fast to the set of separable states, rendering standard LOCC filtering insufficient. To counteract this exponential degradation, we analyze networks employing parallel channel uses per link. We derive a fundamental lower bound on the physical resource requirements, proving that for channels without a correctable subspace, the number of parallel channels per link must scale at least logarithmically with the number of intermediate stations to sustain a non-zero amount of entanglement. This theoretical limit serves as a stringent benchmark for quantum repeater architectures and underscores the necessity of advanced quantum error-correcting codes, such as qLDPC codes, which show promise in saturating this optimal resource scaling.

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 proves an if-and-only-if condition for asymptotic entanglement preservation under LOCC repeaters plus a log scaling lower bound on parallel channels when no correctable subspace exists.

read the letter

The central result is a clean dichotomy: with LOCC at memoryless repeater stations, you can preserve entanglement over arbitrary distance if and only if the channel has a correctable subspace; otherwise the output converges exponentially to separable states. They also show that, without the subspace, the number of parallel uses per link must grow at least logarithmically with the number of stations to keep nonzero entanglement. Both the dichotomy and the scaling bound are new in this precise form.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a strict dichotomy for asymptotic entanglement distribution in quantum repeater networks restricted to LOCC at stations and i.i.d. memoryless channels: preservation over arbitrary distances is possible if and only if the channel admits a correctable subspace; otherwise the output state converges exponentially to the set of separable states. For the latter case it further derives a lower bound showing that the number of parallel channel uses per link must grow at least logarithmically with the number of repeater stations to maintain nonzero entanglement, positioning qLDPC codes as potentially optimal.

Significance. If the central claims hold, the work supplies a clean, parameter-free theoretical benchmark that quantifies the fundamental resource overhead imposed by the absence of a correctable subspace under LOCC. The explicit logarithmic scaling lower bound and the if-and-only-if characterization constitute falsifiable predictions that can directly inform repeater architecture design.

minor comments (2)

The abstract and introduction should explicitly restate that all claims are derived under the LOCC + memoryless restriction, to preempt misinterpretation of the dichotomy as applying to arbitrary quantum operations.
Notation for the correctable subspace and the exponential convergence rate should be introduced with a single forward reference to the relevant theorem or proposition in the main text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the central if-and-only-if dichotomy and the logarithmic scaling lower bound. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No circularity; mathematical proof of channel dichotomy under explicit LOCC and memoryless assumptions

full rationale

The paper states theorems establishing an if-and-only-if condition between asymptotic entanglement preservation and the existence of a correctable subspace, plus exponential convergence and logarithmic scaling bounds. These are presented as derived results from the definitions of correctable subspaces, LOCC operations, and separability, with no fitted parameters, no self-citation load-bearing the central claim, and no reduction of outputs to inputs by construction. The reader's provided circularity score of 2.0 aligns with a minor or zero finding for a self-contained proof. The skeptic note concerns scope of assumptions rather than circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard axioms of quantum mechanics and quantum information theory; no free parameters or new entities are introduced.

axioms (1)

standard math Properties of quantum channels, LOCC operations, and the definition of correctable subspaces in quantum information theory.
The paper builds upon established concepts in quantum information without introducing new axioms.

pith-pipeline@v0.9.0 · 5724 in / 1285 out tokens · 74560 ms · 2026-05-25T04:46:03.665461+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish a strict dichotomy: the asymptotic preservation of entanglement over arbitrarily long distances is possible if and only if the underlying quantum channel admits a correctable subspace. For channels lacking such a subspace, we prove that the transmitted state converges exponentially fast to the set of separable states
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4... min_{sigma in S} ||Lambda^n[rho] - sigma||_1 <= 4 kappa^{n/2} (d_A-1) (sqrt(d_A-1))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

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R. Pal, S. Bandyopadhyay, and S. Ghosh, Entanglement sharing through noisy qubit channels: One-shot optimal singlet frac- tion, Physical Review A90, 052304 (2014)

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Streltsov, R

A. Streltsov, R. Augusiak, M. Demianowicz, and M. Lewen- stein, Progress towards a unified approach to entanglement dis- tribution, Physical Review A92, 012335 (2015)

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C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Physical Review Letters76, 722 (1996)

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Deutsch, A

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy amplification and the security of quantum cryptography over noisy channels, Physical Review Letters77, 2818 (1996)

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Dür and H

W. Dür and H. J. Briegel, Entanglement purification and quan- tum error correction, Reports on Progress in Physics70, 1381 (2007)

work page 2007
[16]

Rozp˛ edek, T

F. Rozp˛ edek, T. Schiet, L. P. Thinh, D. Elkouss, A. C. Doherty, and S. Wehner, Optimizing practical entanglement distillation, Physical Review A97, 062333 (2018)

work page 2018
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Chitambar, D

E. Chitambar, D. Leung, L. Man ˇcinska, M. Ozols, and A. Win- ter, Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask), Communications in Mathematical Physics328, 303 (2014)

work page 2014
[18]

Azuma, S

K. Azuma, S. E. Economou, D. Elkouss, P. Hilaire, L. Jiang, H.-K. Lo, and I. Tzitrin, Quantum repeaters: From quantum networks to the quantum internet, Review of Modern Physics 95, 045006 (2023)

work page 2023
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W. J. Munro, A. M. Stephens, S. J. Devitt, K. A. Harrison, and K. Nemoto, Quantum communication without the necessity of quantum memories, Nature Photonics6, 777 (2012)

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[20]

S. A. Hill and W. K. Wootters, Entanglement of a pair of quan- tum bits, Physical Review Letters78, 5022 (1997)

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[21]

W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Physical Review Letters80, 2245 (1998)

work page 1998
[22]

Choi, Completely positive linear maps on complex ma- trices, Linear Algebra and its Applications10, 285 (1975)

M.-D. Choi, Completely positive linear maps on complex ma- trices, Linear Algebra and its Applications10, 285 (1975)

work page 1975
[23]

W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entan- gled in two inequivalent ways, Physical Review A62, 062314 (2000)

work page 2000
[24]

Konrad, F

T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, and A. Buchleitner, Evolution equation for quantum entangle- ment, Nature Physics4, 99 (2008)

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Blume-Kohout, H

R. Blume-Kohout, H. K. Ng, D. Poulin, and L. Viola, Charac- terizing the structure of preserved information in quantum pro- cesses, Physical Review Letters100, 030501 (2008)

work page 2008
[26]

Vidal, Entanglement monotones, Journal of Modern Optics 47, 355 (2000)

G. Vidal, Entanglement monotones, Journal of Modern Optics 47, 355 (2000)

work page 2000
[27]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cambridge University Press, 2010)

work page 2010
[28]

Guo, Strict entanglement monotonicity under local oper- ations and classical communication, Physical Review A99, 022338 (2019)

Y . Guo, Strict entanglement monotonicity under local oper- ations and classical communication, Physical Review A99, 022338 (2019)

work page 2019
[29]

Muralidharan, J

S. Muralidharan, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Ultrafast and fault-tolerant quantum communication across long distances, Physical Review Letters112, 250501 (2014)

work page 2014
[30]

Muralidharan, L

S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Optimal architectures for long distance quantum 6 communication, Scientific Reports6, 20463 (2016)

work page 2016
[31]

A. G. Fowler, D. S. Wang, C. D. Hill, T. D. Ladd, R. Van Meter, and L. C. L. Hollenberg, Surface code quantum communica- tion, Physical Review Letters104, 180503 (2010)

work page 2010
[32]

Dennis, A

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002)

work page 2002
[33]

Fault-Tolerant Quantum Computation with Constant Overhead

D. Gottesman, Fault-tolerant quantum computation with con- stant overhead (2014), arXiv:1310.2984 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[34]

Panteleev and G

P. Panteleev and G. Kalachev, Quantum ldpc codes with almost linear minimum distance, IEEE Transactions on Information Theory68, 213 (2022)

work page 2022
[35]

Leverrier and G

A. Leverrier and G. Zémor, Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes, inProceedings of the 2023 Annual ACM-SIAM Sympo- sium on Discrete Algorithms (SODA), pp. 1216–1244

work page 2023
[36]

Khatri and M

S. Khatri and M. M. Wilde, Principles of Quantum Communi- cation Theory: A Modern Approach (2024), arXiv:2011.04672 [quant-ph]

work page arXiv 2024
[37]

Gour, Family of concurrence monotones and its applications, Physical Review A71, 012318 (2005)

G. Gour, Family of concurrence monotones and its applications, Physical Review A71, 012318 (2005)

work page 2005
[38]

R. A. Horn and C. R. Johnson,Topics in Matrix Analysis(Cam- bridge University Press, 1991)

work page 1991
[39]

transition probability

A. Uhlmann, The “transition probability” in the state space of a *-algebra, Reports on Mathematical Physics9, 273 (1976)

work page 1976
[40]

Philip, S

A. Philip, S. Rethinasamy, V . Russo, and M. M. Wilde, Schrödinger as a Quantum Programmer: Estimating Entangle- ment via Steering, Quantum8, 1366 (2024)

work page 2024
[41]

C. A. Fuchs and J. van de Graaf, Cryptographic distinguishabil- ity measures for quantum-mechanical states, IEEE Transactions on Information Theory45, 1216 (1999). SUPPLEMENTAL MA TERIAL Concurrence For a two-qubit stateρthe concurrence is defined as [20] C(ρ)=max{ξ 1 −ξ 2 −ξ 3 −ξ 4,0},(36) whereξ i are the square roots of the eigenvalues ofρ˜ρ, arranged ...

work page 1999

[1] [1]

Horodecki, P

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Reviews of Modern Physics81, 865 (2009)

work page 2009

[2] [2]

C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, inProceedings of IEEE In- ternational Conference on Computers, Systems and Signal Pro- cessing(Bangalore, India, 1984) pp. 175–179

work page 1984

[3] [3]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Physical Review Letters70, 1895 (1993)

work page 1993

[4] [4]

H. J. Kimble, The quantum internet, Nature453, 1023 (2008)

work page 2008

[5] [5]

R. V . Meter and S. J. Devitt, The Path to Scalable Distributed Quantum Computing, Computer49, 31 (2016)

work page 2016

[6] [6]

R. Pal, S. Bandyopadhyay, and S. Ghosh, Entanglement sharing through noisy qubit channels: One-shot optimal singlet frac- tion, Physical Review A90, 052304 (2014)

work page 2014

[7] [7]

Streltsov, R

A. Streltsov, R. Augusiak, M. Demianowicz, and M. Lewen- stein, Progress towards a unified approach to entanglement dis- tribution, Physical Review A92, 012335 (2015)

work page 2015

[8] [8]

Krisnanda, Distribution of quantum entanglement: Principles and applications (2020), arXiv:2003.08657 [quant-ph]

T. Krisnanda, Distribution of quantum entanglement: Principles and applications (2020), arXiv:2003.08657 [quant-ph]

work page arXiv 2020

[9] [9]

Zuppardo, T

M. Zuppardo, T. Krisnanda, T. Paterek, S. Bandyopadhyay, A. Banerjee, P. Deb, S. Halder, K. Modi, and M. Paternostro, Excessive distribution of quantum entanglement, Physical Re- view A93, 012305 (2016)

work page 2016

[10] [10]

Lami and V

L. Lami and V . Giovannetti, Entanglement-breaking indices, Journal of Mathematical Physics56, 092201 (2015)

work page 2015

[11] [11]

Rahaman, S

M. Rahaman, S. Jaques, and V . I. Paulsen, Eventually entan- glement breaking maps, Journal of Mathematical Physics59, 062201 (2018)

work page 2018

[12] [12]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woot- ters, Mixed-state entanglement and quantum error correction, Physical Review A54, 3824 (1996)

work page 1996

[13] [13]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noisy entanglement and faithful teleportation via noisy channels, Physical Review Letters76, 722 (1996)

work page 1996

[14] [14]

Deutsch, A

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy amplification and the security of quantum cryptography over noisy channels, Physical Review Letters77, 2818 (1996)

work page 1996

[15] [15]

Dür and H

W. Dür and H. J. Briegel, Entanglement purification and quan- tum error correction, Reports on Progress in Physics70, 1381 (2007)

work page 2007

[16] [16]

Rozp˛ edek, T

F. Rozp˛ edek, T. Schiet, L. P. Thinh, D. Elkouss, A. C. Doherty, and S. Wehner, Optimizing practical entanglement distillation, Physical Review A97, 062333 (2018)

work page 2018

[17] [17]

Chitambar, D

E. Chitambar, D. Leung, L. Man ˇcinska, M. Ozols, and A. Win- ter, Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask), Communications in Mathematical Physics328, 303 (2014)

work page 2014

[18] [18]

Azuma, S

K. Azuma, S. E. Economou, D. Elkouss, P. Hilaire, L. Jiang, H.-K. Lo, and I. Tzitrin, Quantum repeaters: From quantum networks to the quantum internet, Review of Modern Physics 95, 045006 (2023)

work page 2023

[19] [19]

W. J. Munro, A. M. Stephens, S. J. Devitt, K. A. Harrison, and K. Nemoto, Quantum communication without the necessity of quantum memories, Nature Photonics6, 777 (2012)

work page 2012

[20] [20]

S. A. Hill and W. K. Wootters, Entanglement of a pair of quan- tum bits, Physical Review Letters78, 5022 (1997)

work page 1997

[21] [21]

W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Physical Review Letters80, 2245 (1998)

work page 1998

[22] [22]

Choi, Completely positive linear maps on complex ma- trices, Linear Algebra and its Applications10, 285 (1975)

M.-D. Choi, Completely positive linear maps on complex ma- trices, Linear Algebra and its Applications10, 285 (1975)

work page 1975

[23] [23]

W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entan- gled in two inequivalent ways, Physical Review A62, 062314 (2000)

work page 2000

[24] [24]

Konrad, F

T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, and A. Buchleitner, Evolution equation for quantum entangle- ment, Nature Physics4, 99 (2008)

work page 2008

[25] [25]

Blume-Kohout, H

R. Blume-Kohout, H. K. Ng, D. Poulin, and L. Viola, Charac- terizing the structure of preserved information in quantum pro- cesses, Physical Review Letters100, 030501 (2008)

work page 2008

[26] [26]

Vidal, Entanglement monotones, Journal of Modern Optics 47, 355 (2000)

G. Vidal, Entanglement monotones, Journal of Modern Optics 47, 355 (2000)

work page 2000

[27] [27]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cambridge University Press, 2010)

work page 2010

[28] [28]

Guo, Strict entanglement monotonicity under local oper- ations and classical communication, Physical Review A99, 022338 (2019)

Y . Guo, Strict entanglement monotonicity under local oper- ations and classical communication, Physical Review A99, 022338 (2019)

work page 2019

[29] [29]

Muralidharan, J

S. Muralidharan, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Ultrafast and fault-tolerant quantum communication across long distances, Physical Review Letters112, 250501 (2014)

work page 2014

[30] [30]

Muralidharan, L

S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Optimal architectures for long distance quantum 6 communication, Scientific Reports6, 20463 (2016)

work page 2016

[31] [31]

A. G. Fowler, D. S. Wang, C. D. Hill, T. D. Ladd, R. Van Meter, and L. C. L. Hollenberg, Surface code quantum communica- tion, Physical Review Letters104, 180503 (2010)

work page 2010

[32] [32]

Dennis, A

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002)

work page 2002

[33] [33]

Fault-Tolerant Quantum Computation with Constant Overhead

D. Gottesman, Fault-tolerant quantum computation with con- stant overhead (2014), arXiv:1310.2984 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2014

[34] [34]

Panteleev and G

P. Panteleev and G. Kalachev, Quantum ldpc codes with almost linear minimum distance, IEEE Transactions on Information Theory68, 213 (2022)

work page 2022

[35] [35]

Leverrier and G

A. Leverrier and G. Zémor, Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes, inProceedings of the 2023 Annual ACM-SIAM Sympo- sium on Discrete Algorithms (SODA), pp. 1216–1244

work page 2023

[36] [36]

Khatri and M

S. Khatri and M. M. Wilde, Principles of Quantum Communi- cation Theory: A Modern Approach (2024), arXiv:2011.04672 [quant-ph]

work page arXiv 2024

[37] [37]

Gour, Family of concurrence monotones and its applications, Physical Review A71, 012318 (2005)

G. Gour, Family of concurrence monotones and its applications, Physical Review A71, 012318 (2005)

work page 2005

[38] [38]

R. A. Horn and C. R. Johnson,Topics in Matrix Analysis(Cam- bridge University Press, 1991)

work page 1991

[39] [39]

transition probability

A. Uhlmann, The “transition probability” in the state space of a *-algebra, Reports on Mathematical Physics9, 273 (1976)

work page 1976

[40] [40]

Philip, S

A. Philip, S. Rethinasamy, V . Russo, and M. M. Wilde, Schrödinger as a Quantum Programmer: Estimating Entangle- ment via Steering, Quantum8, 1366 (2024)

work page 2024

[41] [41]

C. A. Fuchs and J. van de Graaf, Cryptographic distinguishabil- ity measures for quantum-mechanical states, IEEE Transactions on Information Theory45, 1216 (1999). SUPPLEMENTAL MA TERIAL Concurrence For a two-qubit stateρthe concurrence is defined as [20] C(ρ)=max{ξ 1 −ξ 2 −ξ 3 −ξ 4,0},(36) whereξ i are the square roots of the eigenvalues ofρ˜ρ, arranged ...

work page 1999