Quantum Homogenization as a Quantum Steady State Protocol on NISQ Hardware

Aditya Iyer; Alexander Yosifov; Daniel Ebler; Vlatko Vedral

REVIEW 4 major objections 5 minor 68 references

Quantum Homogenization as a Quantum Steady State Protocol on NISQ Hardware

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that (SWAP)^alpha quantum homogenization, implementable with four CNOT and six single-qubit gates, encodes logical states into a reservoir code subspace that is approximately correctable, making the protocol a…

desk verdict The circuit is fine, but the correctability proof assumes the very condition it needs to prove; Eq. (14) is also malformed. read the letter →

arxiv 2412.14544 v1 pith:XBBDEM2B submitted 2024-12-19 quant-ph

Alexander Yosifov , Aditya Iyer , Daniel Ebler , Vlatko Vedral This is my paper · ORCID

classification quant-ph MSC 81P6881P7081P45 PACS 03.67.-a03.67.Pp

keywords quantumhomogenizationpartialSWAP(SWAP)^alphagateHeisenbergexchangeinteractionNISQhardwareerrorcorrectionapproximaterecoverychannelsteady-statestabilization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes using quantum homogenization—a sequential, reservoir-based protocol in which an input qubit repeatedly collides with reservoir qubits via a partial-SWAP interaction—as a platform for stabilizing and protecting quantum information, not just for transforming states. To make the protocol hardware-friendly, it rewrites the partial SWAP as the $(\mathrm{SWAP})^\alpha$ operator generated by Heisenberg exchange interactions and gives a shallow circuit of four CNOT and six single-qubit gates. It then argues that this iterated interaction is a noisy encoding channel whose complementary channel is approximately constant, so by the generalized Knill-Laflamme conditions an approximate recovery channel exists. If the argument is right, the protocol would be a measurement-free, dissipation-driven steady-state quantum memory that can run on current noisy processors.

What carries the argument

The central object is the $(\mathrm{SWAP})^\alpha$ gate, a two-qubit Heisenberg-exchange operator that leaves all Bell states unchanged except $|\Psi^-\rangle$, which acquires a phase $e^{i\pi\alpha}$; it is equal, up to a global phase, to the partial-SWAP homogenizer interaction. The load-bearing mechanism is the complementary channel $\tilde{\zeta}(\rho_S)=\mathrm{tr}_R(M\rho_S M^\dagger)$, which quantifies how much input information leaks into the environment during encoding. The paper uses the information-disturbance tradeoff and a subsystem decoupling theorem to argue that if this complementary channel is approximately constant, then an approximate recovery channel exists, with the generalized Knill-Laflamme conditions as the criterion. Convergence of the homogenizer to a fixed steady state supplies the constancy; the four-CNOT, six-single-qubit circuit supplies the practical implementation.

What would settle it

Calculate the complementary channel $\tilde{\zeta}(\rho_S)=\mathrm{tr}_R(M\rho_S M^\dagger)$ for the $(\mathrm{SWAP})^\alpha$ encoding with finite reservoir size $N$ and weak coupling $\eta$, and evaluate $\lVert \tilde{\zeta} - P \rVert_\diamond$; if this diamond distance is not $\ll 1$ in the regime the paper targets, the claimed approximate recovery channel is not guaranteed to exist.

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Extended reading notes

Core claim

On its own terms, the central discovery is that the iterative input-reservoir collisions of the $(\mathrm{SWAP})^\alpha$ homogenizer encode logical states into a subsystem of the reservoir Hilbert space through a completely positive, trace-preserving map that is approximately correctable. More precisely, the paper claims there exists a recovery channel $R$ with $\lVert R \circ \zeta - \mathcal{F} \rVert_\diamond \le \delta$ for small $\delta$, where $\zeta$ is the encoding channel, $\mathcal{F}$ is the target channel, and the diamond norm is a channel-distance measure that accounts for entanglement with an ancilla. The reasoning is that the homogenizer drives every input state to a fixed reservoir steady state, making the complementary channel $\tilde{\zeta}(\rho_S) \approx \varphi\,\mathrm{tr}(\rho_S)$ approximately constant; subsystem decoupling then guarantees a recovery channel. The paper also establishes that the partial SWAP $U_{SR}$ equals $e^{i\eta}(\mathrm{SWAP})^\alpha$ up to a global phase with $\alpha=1/n$ and $-2\eta=\pi/n$, so the exchange-interaction gate is dynamically equivalent. It focuses on qubit logical states and notes that higher-dimensional logical states can quickly saturate the channel capacity.

Load-bearing premise

The argument hinges on the unproved assertion that, once the homogenizer converges, the channel describing how much input information leaks to the environment is effectively constant; if that constancy fails, the generalized Knill-Laflamme conditions do not guarantee any recovery channel.

Editorial extensions

If this is right

The homogenizer can act as a passive quantum memory: input states converge to a reservoir steady state while retaining recoverable information, without measurement feedback.
The four-CNOT, six-single-qubit circuit means the proposed stabilization can be tested directly on current superconducting NISQ processors.
The dynamical equivalence of $(\mathrm{SWAP})^\alpha$ to the partial SWAP transfers known homogenization results, including contractive convergence, to exchange-interaction hardware controlled by a single coupling parameter.
Because tuning $\alpha$ by a global external field replaces individual-qubit control, the protocol reduces hardware overhead compared with feedback-based stabilization schemes.
Constructing the promised recovery channel would give an explicit decoding map from reservoir to system, making the steady-state protection usable for state retrieval.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the central claim holds, the same pattern—convergent reservoir dynamics plus a forgetful complementary channel—could serve as a general recipe for turning collision models into approximate error-correcting codes.
The paper leaves the code subspace and recovery map abstract; identifying which reservoir subsystem carries the logical information and computing $R$ explicitly is the natural next step.
Because the argument is presented for one-dimensional logical states, extending the proof to qudit inputs would require checking whether the forgetfulness condition survives higher-dimensional channel-capacity constraints.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper has a correct algebraic core and a plausible circuit, but the advertised result—correctable encoding into a reservoir subspace—is not proved. The central argument in Section IV mistakes which channel becomes forgetful.

What is good: Eq. (9) shows the partial SWAP is equivalent (up to a global phase) to e^{iη}(SWAP)^{1/n}. That is a known result from Fan et al., but the presentation is clear. The circuit with four CNOTs and six single-qubit gates looks correct and is a reasonable NISQ proposal. The idea of applying Beny-Oreshkov conditions to the homogenizer is interesting and worth exploring.

The soft spot is Section IV. They define ζ as the encoding channel from system to reservoir, then claim ζ has fixed single-point output and therefore its complement bζ is approximately constant. Their own Eq. (5) shows the reservoir output after one interaction depends on the input state, so ζ is not constant. The channel that homogenization drives to a constant is the complementary channel, i.e., the system output after tracing the reservoir. So they are assuming exactly the condition the Beny-Oreshkov theorem requires, not deriving it. They also never construct the code subspace, the noise model, the Stinespring isometry, or the recovery map. Equation (14) compares a channel norm to a state, which is malformed.

This is a load-bearing flaw, not a minor gap. Without a proof that bζ is forgetful in diamond norm, the correctability claim collapses. A salvage is plausible: use the contractivity bounds already known for the homogenizer to prove the complementary channel is approximately constant. But as written, the paper overclaims.

I would not cite the correctability result. The paper might be useful for a reading group as an example of why correctability arguments need the right channel. If I were an editor, I would send it to a serious referee because the topic is relevant and the circuit is concrete, but the referee should be told the main theorem is currently unsupported.

Best, [Name]

Referee Report

4 major / 5 minor

Summary. The paper proposes a quantum homogenization protocol based on the (SWAP)^alpha two-qubit gate, claims dynamical equivalence with the standard partial-SWAP homogenizer, gives a 4-CNOT/6-single-qubit circuit implementation, and argues via the Beny-Oreshkov generalization of the Knill-Laflamme conditions that the protocol is a CPTP map under which information encoded in a reservoir code subspace is approximately correctable against noise.

Significance. If the central correctability claim were proven, the paper would offer an interesting NISQ-friendly route to dissipation-driven quantum information protection. The (SWAP)^alpha formulation and the explicit circuit decomposition are useful and appear correct. However, the advertised main result--that the homogenizer yields a correctable code under CPTP dynamics--is not actually derived: the key condition on the complementary channel is asserted rather than proven, and no code subspace, recovery map, or quantitative error bound is constructed. The paper therefore does not currently deliver its central claim.

major comments (4)

[Section IV, after Eq. (15)] The assertion that 'given that zeta is a channel with fixed single-point output, its complementary bzeta is also an approximately constant output channel' is not generally true and is not derived for the homogenizer. A constant channel can have a non-constant complement (e.g., an isometry that maps every input to the same reservoir state while the environment retains full information about the input). For the homogenizer itself, the single-collision map in Eq. (4) is not fixed single-point output, and no proof is given that the complementary channel becomes approximately constant in diamond norm for finite N. Since this step is the load-bearing condition for the Beny-Oreshkov theorem, the correctability claim collapses without it.
[Section IV, Eq. (14)] The inequality || bzeta - P || <= || R composed with zeta - xi(0) || is not well-formed as written: the right-hand side compares a channel (R composed with zeta) to a state (xi(0)), and no identification of xi(0) with a constant channel is stated. Moreover, the cited information-disturbance theorem does not yield this precise inequality without additional assumptions on the norms and the recovery channel. This makes the logical chain from Eq. (14) to Eq. (15) and the final conclusion unclear.
[Section IV, Definition 1 and following paragraph] The code subspace, the Stinespring isometry M, the noise operators N_i, and the recovery channel R are never explicitly defined. The parameters of the protocol (number of collisions N and coupling eta) are never connected to the correctability parameters delta and k. Consequently, the paper does not demonstrate that the homogenizer satisfies the generalized KL conditions; it merely asserts that the required complementary channel is approximately constant. A quantitative statement, such as an explicit bound delta(N,eta), is missing.
[Section IV, paragraph after Eq. (15)] The claim that the complementary channel bzeta is 'simply the restriction of zeta to act on S(H_tildeE)' conflates the complementary channel with a restriction of the original channel. These are different objects in general: bzeta maps the input to the environment Hilbert space, whereas zeta maps it to the reservoir Hilbert space. This conflation obscures what is being assumed and what needs to be proven.

minor comments (5)

[Section III, Eq. (10)] The expression for RZ(-2eta) appears to be incorrect under the convention RZ(theta)=diag(e^{-i theta}, e^{i theta}) used in the same equation: RZ(-2eta) should be diag(e^{2i eta}, e^{-2i eta}), not diag(e^{2i eta}, e^{2i eta}).
[Section III, text after Eq. (10)] The statement 'for alpha = pi (mod 2 pi) the operator U^alpha can act as SWAP' is not compatible with alpha = 1/n for integer n, since alpha in (0,1] in the protocol; the intended value is presumably alpha = 1.
[Section II, Eq. (2)] The approximation sign in Eq. (2) is used without specifying the distance measure or the convergence rate; a precise statement in terms of, e.g., diamond norm or fidelity, with the dependence on N and eta, would help support later arguments.
[Section IV, notation] The notation for the complementary recovery channel, rendered as '\R composed with F' and 'bR', is not defined consistently; this makes the already technical text harder to follow.
[References] Reference [21] is cited as an arXiv preprint; a published version of the quantum homogenizer protocol (Ziman et al., Phys. Rev. A 65, 042105 (2002), which is also [28]) could be cited here for clarity.

Circularity Check

1 steps flagged · score 7.0 of 10

Section IV's correctability proof asserts the Beny-Oreshkov condition (bζ ≈ φtr) instead of deriving it, so the central information-protection claim reduces to its own premise.

self definitional [Section IV ('Robustness of Quantum Homogenization'), around Eqs. (14)-(15) and the paragraph following Eq. (15)]
"If ∀ρS ∈ S(HS) there exists a channel with approximately constant output to the environment, such as bζ(ρS) ≈ φtr(ρS) ... then R must exist. ... From (13–15), and given that ζ is a channel with fixed single-point output, its complementary bζ is also an approximately constant output channel bζ(ρS) ≈ φtr(ρS) ... Thus, it straightforwardly follows from the subsystem decoupling theorem [60] ... that ... an approximate recovery channel R must exist."

The first quoted sentence is the paper's own statement that the existence of a recovery channel R is guaranteed by the condition bζ(ρS) ≈ φtr(ρS). The proof then introduces Eq. (15) as a supposition that bζ is approximately k-forgetful, i.e. ∥bζ − P∥⋄ ≤ δ, restates that supposition in the next sentence as 'bζ is also an approximately constant output channel bζ(ρS) ≈ φtr(ρS),' and concludes that R must exist. Thus the conclusion (recoverability) is assumed as the premise; the condition equivalent to the existence of R is asserted, not derived from the partial SWAP/(SWAP)^α map. The attempted justification 'ζ is a channel with fixed single-point output' is not proved for the finite-N encoding channel — Eq.

full rationale

The paper's independent contributions — the equivalence USR = e^{iη}U^α (Eq. 9), the CNOT/single-qubit decomposition (Fig. 1), and the use of homogenization as a steady-state protocol — are not circular: they are explicit algebraic identities and an explicit circuit construction checkable against standard two-qubit gate decompositions. However, the central advertised result, that the protocol protects quantum information in a reservoir code subspace under CPTP dynamics, is not derived. Section IV invokes the Beny-Oreshkov criterion and then asserts the very condition that the criterion requires. Eq. (15) is introduced as a supposition ('suppose ζ is such that its complementary bζ is approximately k-forgetful'), and the next sentence restates this as 'bζ is also an approximately constant output channel bζ(ρS) ≈ φtr(ρS)' before concluding that a recovery channel R exists. Since the paper had already stated that bζ(ρS) ≈ φtr(ρS) is the condition under which 'R must exist,' the conclusion is the premise. The additional justification 'ζ is a channel with fixed single-point output' is not a derived property of the partial SWAP encoding: for N=1, Eq. (4) gives ρ_S^(1) = cos²η ρ_S + sin²η ξ + i cosη sinη [ξ, ρ_S], which depends on ρ_S, and the asymptotic convergence in Eq. (2) concerns the joint state approaching ξ⊗(N+1), which would make the reservoir output constant and therefore information-destroying, not a correctable encoding. No code subspace, isometry M, noise model Ni, or recovery map R is constructed, so the recoverability claim cannot be checked independently. The correctability conclusion therefore reduces by construction to the assumed approximate constancy of the complementary channel. Score 7: the central claim is circular and assumption-laden, while the circuit and equivalence results remain self-contained.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the tunable coupling parameter eta, on prior contractivity results, and critically on the unproved assumption that zeta has fixed single-point output and bzeta is approximately constant. No explicit code subspace, noise operators, or recovery channel are provided, so the correctability claim is not independently grounded.

free parameters (1)

coupling parameter eta (equivalently alpha = 1/n) = not fitted; tunable via external field
The swap strength is chosen by hand through eta with -2 eta = pi/n. It controls the convergence rate and the circuit rotations, but it is not fitted to data.

assumptions (5)

standard math The (SWAP)^alpha operator has the Bell-basis form of Eq. (8).
Taken from Fan et al. [31] and used to establish the equivalence to partial SWAP in Eq. (9).
standard math The homogenization map is contractive and drives all input states toward the reservoir state xi.
Invoked from prior work [21,37,38] via Eqs. (2)-(6); the paper does not reprove it.
ad hoc to paper The encoding channel zeta has a fixed single-point output and its complementary channel bzeta is approximately constant.
This is the condition that guarantees recovery. The paper asserts it from homogenization dynamics without proof, making the correctability conclusion depend on a target-like assumption.
standard math The Beny-Oreshkov conditions and the subsystem decoupling theorem are applicable to this setup.
Invoked from [32,60], but the paper does not verify all hypotheses, in particular the approximate forgetfulness of bzeta.
domain assumption Noise elements N_i can be absorbed into the encoding channel zeta.
A Stinespring dilation assumption in Section IV; no specific noise model is given, so the claim is not quantitative.

invented entities (1)

reservoir code subspace of dimension at most k
purpose: The purported subspace in which logical states are encoded and from which a recovery channel restores them.
No explicit projector, basis, logical states, or recovery map is constructed. The subspace is only referenced as existing because bzeta is assumed to map to a small subsystem of the environment.

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Cite this review

Pith. "Pith review of Quantum Homogenization as a Quantum Steady State Protocol on NISQ Hardware." pith.science (2026). https://pith.science/paper/XBBDEM2B

@misc{pith2026241214544,
  author       = {Pith},
  title        = {Pith review of: Quantum Homogenization as a Quantum Steady State Protocol on NISQ Hardware},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBBDEM2B}},
  note         = {Machine review of arXiv:2412.14544}
}

abstract

Quantum homogenization is a reservoir-based quantum state approximation protocol, which has been successfully implemented in state transformation on quantum hardware. In this work we move beyond that and propose the homogenization as a novel platform for quantum state stabilization and information protection. Using the Heisenberg exchange interactions formalism, we extend the standard quantum homogenization protocol to the dynamically-equivalent ($\mathtt{SWAP}$)$^\alpha$ formulation. We then demonstrate its applicability on available noisy intermediate-scale quantum (NISQ) processors by presenting a shallow quantum circuit implementation consisting of a sequence of $\mathtt{CNOT}$ and single-qubit gates. In light of this, we employ the Beny-Oreshkov generalization of the Knill-Laflamme (KL) conditions for near-optimal recovery channels to show that our proposed ($\mathtt{SWAP}$)$^\alpha$ quantum homogenization protocol yields a completely positive, trace preserving (CPTP) map under which the code subspace is correctable. Therefore, the protocol protects quantum information contained in a subsystem of the reservoir Hilbert space under CPTP dynamics.

Figures

Figures reproduced from arXiv: 2412.14544 by the authors.

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Works this paper leans on

68 extracted references · 67 canonical work pages

[1]

Muralidharan et al., Sci

S. Muralidharan et al., Sci. Rep. 6, 20463 (2016)

work page 2016
[2]

Kimble, Nature 453, 1023–1030 (2008)

H. Kimble, Nature 453, 1023–1030 (2008)

work page 2008
[3]

Ekert, Phys

A. Ekert, Phys. Rev. Lett. 67, 661 (1991)

work page 1991
[4]

Giovannetti et al., Science 306, 5700 (2004)

V. Giovannetti et al., Science 306, 5700 (2004)

work page 2004
[5]

Sayrin et al., Nature 477, 73–77 (2011)

C. Sayrin et al., Nature 477, 73–77 (2011)

work page 2011
[6]

Vijay et al., Nature 490, 77–80 (2012)

R. Vijay et al., Nature 490, 77–80 (2012)

work page 2012
[7]

Wiseman, Phys

H. Wiseman, Phys. Rev. A 49, 2133 (1994)

work page 1994
[8]

Doherty et al., Phys

A. Doherty et al., Phys. Rev. A 62, 012105 (2000)

work page 2000

Show all 68 references

[9]

Carvalho et al., Phys

A. Carvalho et al., Phys. Rev. A 78, 012334 (2008)

2008
[10]

Pereira et al., Sci

F. Pereira et al., Sci. Rep. 13, 10540 (2023)

2023
[11]

Kastner, Found

R. Kastner, Found. Phys. 47, 697–707 (2017)

2017
[12]

Ladd et al., Nature 490, 77–80 (2012)

T. Ladd et al., Nature 490, 77–80 (2012)

2012
[13]

Reiter et al., Nat

F. Reiter et al., Nat. Commun. 8, 1822 (2017)

2017
[14]

Verstraete, et al., Nat

F. Verstraete, et al., Nat. Phys. 5, 633–636 (2009)

2009
[15]

Fujii et al., Phys

K. Fujii et al., Phys. Rev. X 4, 041039 (2014)

2014
[16]

Krauter et al., Phys

H. Krauter et al., Phys. Rev. Lett. 107, 080503 (2011)

2011
[17]

Shankar et al., Nature 504, 419–422 (2013)

S. Shankar et al., Nature 504, 419–422 (2013)

2013
[18]

Loss and D

D. Loss and D. DiVincenzo, Phys. Rev. A 57, 120 (1998)

1998
[19]

Bacon et al., Phys

D. Bacon et al., Phys. Rev. Lett. 85, 1758 (2000)

2000
[20]

Kempe et al., Phys

J. Kempe et al., Phys. Rev. A 63, 042307 (2001)

2001
[21]

Ziman et al., arXiv:quant-ph/0110164

M. Ziman et al., arXiv:quant-ph/0110164

arXiv
[22]

Ciccarello et al., Phys

F. Ciccarello et al., Phys. Rev. A 87, 040103(R) (2013)

2013
[23]

Cattaneo et al., Phys

M. Cattaneo et al., Phys. Rev. Lett. 126, 130403 (2021)

2021
[24]

De Chiara and M

G. De Chiara and M. Antezza, Phys. Rev. Res. 2, 033315 (2020)

2020
[25]

Lewis-Swan et al., Nat

R. Lewis-Swan et al., Nat. Revs. Phys. 1, 627–634 (2019)

2019
[26]

Cazalilla and M

M. Cazalilla and M. Rigol, New J. Phys. 12, 055006 (2010)

2010
[27]

Scarani et al., Phys

V. Scarani et al., Phys. Rev. Lett. 88, 097905 (2002)

2002
[28]

Ziman et al., Phys

M. Ziman et al., Phys. Rev. A 65, 042105 (2002)

2002
[29]

Violaris et al., Phys

M. Violaris et al., Phys. Rev. A 103, 022414 (2021)

2021
[30]

Marletto et al., Phys

C. Marletto et al., Phys. Rev. Lett. 128, 080401 (2022)

2022
[31]

Fan et al., Phys

H. Fan et al., Phys. Rev. A 72, 052323 (2005)

2005
[32]

B´ eny and O

C. B´ eny and O. Oreshkov, Phys. Rev. Lett. 104, 120501 (2010)

2010
[33]

Kim et al., J

I. Kim et al., J. High Energy Phys. 06, 031 (2020)

2020
[34]

Schrauwen et al., Proc

B. Schrauwen et al., Proc. Eur. Symp. ANN 471 (2007)

2007
[35]

Markovi´ c and J

D. Markovi´ c and J. Grollier, Appl. Phys. Lett. 117, 150501 (2020)

2020
[36]

Ruderman, arXiv:2106.00962

M. Ruderman, arXiv:2106.00962

arXiv
[37]

Nagaj et al., Phys

D. Nagaj et al., Phys. Rev. A 66, 062307 (2002)

2002
[38]

Qi-Ping Su et al., Phys. Rev. A 104, 032412 (2021)

2021
[39]

Beever et al., Phys

A. Beever et al., Phys. Rev. A 110, 012464 (2024)

2024
[40]

Burgarth and V

D. Burgarth and V. Giovannetti, Phys. Rev. A 76, 062307 (2007)

2007
[41]

Raginsky, Phys

M. Raginsky, Phys. Rev. A 65, 032306 (2002)

2002
[42]

Leghtas et al., Phys

Z. Leghtas et al., Phys. Rev. Lett. 111, 120501 (2013)

2013
[43]

Terhal and D

B. Terhal and D. DiVincenzo, Phys. Rev. A 61, 022301 (2000)

2000
[44]

Brodin and M

G. Brodin and M. Marklund, New J. Phys. 9, 277 (2007)

2007
[45]

Gilles et al., Phys

L. Gilles et al., Phys. Rev. A 49, 2785 (1994)

1994
[46]

Mirrahimi et al., New J

M. Mirrahimi et al., New J. Phys. 16, 045014 (2014)

2014
[47]

Wolinsky and H

M. Wolinsky and H. Carmichael, Phys. Rev. Lett. 60, 1836 (1988)

1988
[48]

Turkpence et al., Phys

D. Turkpence et al., Phys. Lett. A 13, 384 (2019)

2019
[49]

Mohseni and D

M. Mohseni and D. Lidar, Phys. Rev. Lett. 94, 040507 (2005)

2005
[50]

Levy, Phys

J. Levy, Phys. Rev. Lett. 89, 147902 (2002)

2002
[51]

Porras and J

D. Porras and J. Cirac, Phys. Rev. Lett. 92, 207901 (2004)

2004
[52]

Cirac and P

J. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995)

1995
[53]

Kribs et al., Phys

D. Kribs et al., Phys. Rev. Lett. 94, 180501 (2005)

2005
[54]

Stinespring, Proc

W. Stinespring, Proc. Amer. Math. Soc. 6, (1955)

1955
[55]

O’Connor et al., Phys

T. O’Connor et al., Phys. Rev. A 90, 032305 (2014)

2014
[56]

Gilchrist et al., Phys

A. Gilchrist et al., Phys. Rev. A 71, 062310 (2005)

2005
[57]

Note the generalized KL conditions are broadly applicable for subsystem and algebraic codes and the recovery operation may be taken with respect to an arbitrary target channel [58]

as ∥R ◦ζ − F ∥⋄ ≤ δ (12) for δ ≪ 1. Note the generalized KL conditions are broadly applicable for subsystem and algebraic codes and the recovery operation may be taken with respect to an arbitrary target channel [58]. To simplify things, we can convert Definition 1 into a cond...
[58]

Benenti and G

G. Benenti and G. Strini, J. Phys. B: At. Mol. Opt. Phys. 43 215508 (2010)

2010
[59]

B´ enyet al., Phys

C. B´ enyet al., Phys. Rev. Lett. 98, 100502 (2007)

2007
[60]

Kretschmann et al., IEEE Trans

D. Kretschmann et al., IEEE Trans. Info. Th. 54, (2008)

2008
[61]

Hayden and G

P. Hayden and G. Penington, Comm. Math. Phys. 374, 369 (2020)

2020
[62]

Cerf et al., Phys

N. Cerf et al., Phys. Rev. Lett. 88, 127902 (2002)

2002
[63]

Andersen, et al., Nat

U. Andersen, et al., Nat. Phys. 11, 713–719 (2015)

2015
[64]

Plenio and S

M. Plenio and S. Huelga, Phys. Rev. A 93, 032123 (2016)

2016
[65]

Martinez et al., Nature 534, 516–519 (2016)

E. Martinez et al., Nature 534, 516–519 (2016)

2016
[66]

Yun-Feng Huang et al., Phys. Rev. A 64, 012315 (2001)

2001
[67]

Giovannetti and D

V. Giovannetti and D. Burgarth, Phys. Rev. Lett. 96, 030501 (2006)

2006
[68]

Coecke et al., Info

B. Coecke et al., Info. and Comp. 250, 59–86 (2016)

2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Muralidharan et al., Sci

S. Muralidharan et al., Sci. Rep. 6, 20463 (2016)

work page 2016

[2] [2]

Kimble, Nature 453, 1023–1030 (2008)

H. Kimble, Nature 453, 1023–1030 (2008)

work page 2008

[3] [3]

Ekert, Phys

A. Ekert, Phys. Rev. Lett. 67, 661 (1991)

work page 1991

[4] [4]

Giovannetti et al., Science 306, 5700 (2004)

V. Giovannetti et al., Science 306, 5700 (2004)

work page 2004

[5] [5]

Sayrin et al., Nature 477, 73–77 (2011)

C. Sayrin et al., Nature 477, 73–77 (2011)

work page 2011

[6] [6]

Vijay et al., Nature 490, 77–80 (2012)

R. Vijay et al., Nature 490, 77–80 (2012)

work page 2012

[7] [7]

Wiseman, Phys

H. Wiseman, Phys. Rev. A 49, 2133 (1994)

work page 1994

[8] [8]

Doherty et al., Phys

A. Doherty et al., Phys. Rev. A 62, 012105 (2000)

work page 2000