Fidelity Determines the Minimum Error of Quantum Recovery

Dario Picozzi

REVIEW

Fidelity Determines the Minimum Error of Quantum Recovery

Dario Picozzi

Quantum Physics · Submitted 5 Nov 2026 · 20 references

Pith's verdict: Cleanly resolves the open equality of optimized entanglement fidelity and diamond distance without shared randomness; the proof is elegant and the paper is thorough.

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arXiv ID2611.07344 (version 1)

Pith IDEZGGZWOM, Pith's permanent record of this paper and its review

CitationDario Picozzi. "Fidelity Determines the Minimum Error of Quantum Recovery." arXiv:2611.07344 (2026). https://pith.science/paper/EZGGZWOM

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Editor's note

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This paper resolves the open question from Holdsworth, Singh, and Wilde (2023) by proving that, for finite-dimensional noise channels and fixed encodings, the minimum normalized diamond distance and the minimum worst-input entanglement infidelity are equal over all recovery operations. No shared randomness is needed, and the construction is a deterministic final unitary. This is a genuinely new result with significance for quantum error correction and channel discrimination.

The key technical tool is the Unitary Recovery Principle, which shows that one can always append a final unitary to any decoder so that the diamond distance is bounded above by the infidelity. The proof uses polar decomposition and operator inequalities, and it is clean and dimension-independent. The paper also provides an SDP for finding the optimal final unitary when the noise is not too large, characterizes the families of optimal corrections, and gives robustness bounds showing that the polar correction is especially stable. The examples—especially the three-qubit calibration and the stabilizer-code benchmark—are concrete and validate the theory.

Soft spots are few. The equality holds only for decoder families that are closed under appending final logical unitaries. The authors state this condition explicitly, and the unrestricted CPTP family satisfies it, but some restricted decoder families (e.g., implementing only a fixed circuit family) would not directly enjoy the equality. This is an acknowledged boundary condition, not a flaw. The unpublished reference [17] for the stabilizer code is a minor concern; the example is not load-bearing and the code can be verified independently. The code and data are available on request, which is good practice.

This paper deserves a serious referee. I would bring it to a reading group and would cite it in my own work on quantum recovery. Recommendation: accept after minor comments—chiefly, ask the authors to cite a published source for the stabilizer code or note that the example does not depend on that specific reference.

Referee report

Significance

If correct, the paper establishes a fundamental equivalence: the often-used infidelity and the operationally meaningful diamond distance have exactly the same optimum over CPTP recovery. This tightens the standard Fuchs-van de Graaf inequality from a square-root gap to equality after optimization, and it provides a clean single-number characterization of the best possible recovery. The Unitary Recovery Principle itself is a clean, dimension-independent bound that may find independent use in channel calibration and quantum control. The additional results on optimal families, SDP computability, robustness, and composition give practical tools for designing and certifying error-correction protocols. The three-qubit example concretely illustrates that fidelity-optimal recovery can leave significant operational error, and that a single final unitary suffices to close that gap.

Major objections (3)

[Abstract and main text (Eq. (3))] The central equality inf_D(RN) = 1 - sup_F(RN) is proved under the condition that the decoder family C is closed under appending final logical unitaries (Ad_U R in C for every R in C and logical U). This condition is stated and applied to the unrestricted CPTP family, which indeed satisfies it. The paper correctly notes that families not closed under output unitaries do not automatically enjoy the equality. This is a clear and proper handling of the scope; no revision is required.
[End Matter, Eq. (15) and surrounding text] The proof that the optimum over final unitaries coincides with the maximum over coefficient vectors c of s_min(B_c)^2 uses the fact that for any U with F(Ad_U Phi) > 0 there exists a normalized c with s_min(B_c) >= sqrt(F(Ad_U Phi)). The reverse direction (any c gives a U_c with D <= 1 - s_min^2) relies on the centered-Kraus argument. The argument is sound. However, the step where the optimal U is said to satisfy F(Ad_U Phi) = q* and D(Ad_U Phi) = 1 - q* uses the fidelity-diamond equality for that specific U. The paper should make explicit that this equality holds for the polar-corrected U_c only after the SDP relaxation is proven tight (Gamma > 1/2), not for general c. The text currently reads as if every c with B_c positive yields D = 1 - q(c), which need not hold unless the centered-Kraus bound is actually tight for that particular B_c. Please clarify in the main text that the equality D(Ad_U Phi) = 1 - F(Ad_U Phi) for a specific unitary gate is only guaranteed when the gate is certified by the SDP or when the channel's optimal coherent amplitude has the property that the centered-Kraus bound saturates (which is not generally true for arbitrary B_c). The current wording in the paragraph following Eq. (15) could be misinterpreted.
[Section on 'Computing the best final unitary' (Eq. (9))] The SDP is claimed to determine the optimal final unitary exactly whenever Gamma > 1/2. The proof in End Matter uses a spectral rounding argument that requires t > 1/2 to force every optimal tau to be rank one. This is correct. However, the paper should state more prominently that the SDP variable tau has dimension r <= d^2, and that the choice of r affects the tightness: if the Kraus rank of the channel exceeds d^2, then r must be at least that rank for the relaxation to be faithful. The text mentions 'one may choose r <= d^2' but does not explain how to pick r. Since the SDP is over r x r matrices but uses the same Kraus operators A_j, a reader may think that the r dimension is independent of the channel's Kraus rank. Please add a brief note clarifying that r is the number of Kraus operators of the fixed decoded channel, and that the SDP formulation in Eq. (9) requires summing over all Kraus indices. This is already implicit in the notation tau_{ij} but would benefit from an explicit statement.

Minor objections (9)

[Abstract (line 12)] The phrase 'with a retained noise record, we determine the optimal error and characterize all optimal decoders subject to a Kraus-rank limit for each outcome' is vague. Please specify that this applies to instrument-decoder scenarios where the environment or classical side-information is kept, and that the rank constraint limits the number of Kraus operators per outcome branch. The full details are in the supplement, but a brief orientation in the abstract would help.
[Main text after Eq. (2)] The sentence 'Applying Eq. (2) to Phi = R N gives inf_{R in C} D(RN) = 1 - sup_{R in C} F(RN)' should note that the argument uses the lower bound D >= 1 - F and the fact that for any R in C, the corrected channel Ad_U (R N) belongs to C because C is closed under appending unitaries. This is clear from context, but a short reminder would prevent confusion about where the closure property is used.
[End Matter, Eq. (18)] The diamond-norm representation D(Psi) = inf_{||gamma||_2 < 1} ||I - sum gamma_j C_j||^2 / (1 - ||gamma||_2^2) is a key technical tool. The derivation in the supplement (Section II) is rigorous, but the main text states it without proof. Since it is not standard in the literature, it would be helpful to include a brief sketch or at least a clear citation to the supplement at the point where Eq. (18) is first used (first paragraph of 'Optimal families and their stability').
[End Matter, Eq. (15) (second minor comment)] The claim 'q* := max_{||c||_2=1} q(c) = max_U F(Ad_U Phi) = 1 - min_U D(Ad_U Phi)' is justified by the two directions described. However, the second equality (max fidelity = 1 - min diamond) relies on the fact that for the particular U that achieves min D, one can prove F(Ad_U Phi) = 1 - D(Ad_U Phi). The text states 'Conversely, applying Eq. (4) to any Ad_U Phi of positive fidelity gives a normalized c with q(c) >= F(Ad_U Phi); at zero fidelity this inequality is automatic.' This establishes only that max_U F(Ad_U Phi) <= max_c q(c). The reverse inequality follows because for any c, the channel Ad_{U_c} Phi has fidelity at least q(c) by the positive-Kraus bound. The text states this correctly. No change needed, but a reader might be helped by an explicit 'Thus' sentence.
[Three-qubit conditional-gate calibration] The example uses a unitary correction U_alpha acting only on the data output, as stated. The optimal recovery is claimed to be a 'doubly controlled target rotation' that achieves the optimum over all CPTP recoveries. The proof (End Matter and supplement) is given. However, the claim that 'The optimal recovery is a doubly controlled target rotation' might be read as saying that the optimal decoder itself (not just the final unitary) is of that form. In fact, the decoder is a CPTP map consisting of syndrome detection followed by that unitary; the text and supplement make this clear, but the short sentence in the main text (just above Eq. (13)) could be rephrased as 'The optimal correction is achieved by the doubly controlled rotation U_alpha* applied after a fixed decoder' to avoid ambiguity.
[Optimal families and their stability (Eq. (10))] The sharp bounds for gate distance d(U,U_c) are stated as <= sqrt(epsilon) for fidelity optimality and <= epsilon sqrt(3+epsilon) / 2 for operational optimality. The main text gives 'epsilon sqrt(3+epsilon) /2' which is ambiguous due to missing parentheses. Please write 'epsilon sqrt(3+epsilon) / 2' or use clear fractional notation.
[References] Reference [17] (H. Hao, 'Investigations on automorphism groups of quantum stabilizer codes') is listed as 'Unpublished manuscript, Sec. 2.2.5 (n.d.)'. This is acceptable as an unpublished source, but the paper uses it as the specification of an [[8,2,3]] code in the supplement (Section XI). If possible, please provide a more accessible reference (e.g., a published paper or a stable arXiv URL).
[General typography] The notation 'Ad_U' is sometimes written as 'AdU' (without underscore) in the main text, e.g., in the line 'For a channel Phi, write AdU (X) = U X U^dagger'. Consistency would improve readability.
[Supplement, Eq. (S1)] The environmental criterion D* = 1 - max_{omega_E} F_{root}(hat{N}, C_{omega_E})^2 uses F_{root} (unsquared fidelity). The sentence says 'Beny and Oreshkov's complementary-channel duality, Corollary 5 of Ref. [1], identifies the maximized squared fidelity with the optimal recovery fidelity.' This is correct. However, the letter's recovery equality (Eq. (3)) is used to convert the fidelity optimum to diamond error. The supplement should explicitly state that D* = 1 - (max_{omega_E} F_{root})^2 follows from the main theorem, not from Beny-Oreshkov alone. The current text could be read as if Beny-Oreshkov already gave the diamond-optimum equality. Please clarify.

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The paper

The paper claims

For every noise channel, worst-input entanglement infidelity and operational error have the same minimum over all recovery operations, a result proved by a single final unitary correction.

The reading

This paper proves that for every finite-dimensional noise channel and fixed encoding, the minimum operational error (normalized diamond distance) over all recovery operations is exactly one minus the maximum worst-input entanglement fidelity. The proof constructs a single final unitary that turns any recovery into one whose operational error is no larger than the original worst-input entanglement infidelity. This closes a gap left open in prior work and shows that fidelity optimization directly determines the best achievable worst-case measurement disturbance, without requiring shared randomness or additional resources. The result applies to any decoder family closed under appending logical unitaries, including all completely positive trace-preserving maps.

What carries the argument

The key machinery is the polar decomposition of the weighted Kraus amplitude T = Σ_j ar{α}_j A_j, where α_j are traces of Kraus operators against a worst-input state. The worst-input optimality condition forces Re T ⪰ f I (with f the fidelity), making T invertible and yielding a unique polar correction U. The proof then centers the corrected Kraus operators around the positive amplitude K = |T|/√f, leading to a centered Gram matrix H whose operator norm is bounded by 1 − f, which directly gives the diamond norm bound. For the optimal final unitary problem, a semidefinite program (SDP) relaxes the rank-one constraint on the coherent weight matrix τ, and exactness is certified whenever the optimum exceeds 1/2.

Extended reading notes

Core claim

The central discovery is the Unitary Recovery Principle: for any channel Φ there exists an output unitary U satisfying 1 − F(Ad_U Φ) ≤ D(Ad_U Φ) ≤ 1 − F(Φ) ≤ D(Φ). Here F is worst-input entanglement fidelity and D is normalized diamond distance. This dimension-independent chain implies that the minimum operational error over any unitary-closed recovery family equals one minus the maximum worst-input entanglement fidelity. Consequently, a recovery that maximizes fidelity can be made operationally optimal by appending a suitably chosen final unitary, and fidelity optimization suffices to determine the ultimate error floor.

Load-bearing premise

The equality holds for any decoder family that is closed under appending an arbitrary final logical unitary; this structural assumption is necessary for the proof that fidelity optimization determines operational error.

Editorial extensions

If this is right

Optimizing worst-input entanglement fidelity over any unitary-closed recovery family automatically minimizes the operational diamond error, so fidelity-based benchmarks are sufficient for predicting worst-case measurement errors in quantum memories.
Existing approximate error-correction schemes (e.g., Bény–Oreshkov) can be upgraded to operationally optimal by adding one final unitary correction, achieving a factor-four approximation in diamond distance.
For a fixed decoded channel, the optimal final unitary can be found via a single SDP whenever the minimum error is below 1/2, providing an efficient calibration tool.
The polar correction exhibits quadratic sensitivity to residual unitary errors, while arbitrary operationally optimal gates can have only s^{4/3} sensitivity, suggesting polar corrections are preferable in practice.
Product channels with local errors below 1/2 can be corrected globally by independent polar corrections, even against entangled inputs and a reference, preserving optimality.

Reading between the lines

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

The requirement that the decoder family be closed under appending final logical unitaries is mild for most practical settings (all CPTP maps, for example), but it excludes families with fixed circuit structure that cannot absorb a final unitary. In those cases the equality may fail, and more involved optimization would be needed.
The rank-one certificate in the SDP suggests a deeper connection between the coherent amplitude matrix and the complementary channel's spectrum; this may inspire new algorithms for joint encoder–decoder optimization.
The gap between average-gate-fidelity optimization and worst-case operational error illustrated by the three-qubit example warns against using average fidelity as a proxy for worst-case performance in quantum error correction.
The polynomial hierarchy for rank-constrained recovery (Section VII of the supplemental material) provides a practical way to compute near-optimal recoveries when the direct SDP is not exact, with certified error bounds.

Original abstract

Even a recovery that maximizes worst-input entanglement fidelity can retain avoidable errors in subsequent measurements. Entanglement fidelity is the probability of passing a test of whether the state and its entanglement were preserved; infidelity is the probability of failing it. Operational error is the largest possible change in any measurement probability, quantified by the normalized diamond distance. For every finite-dimensional noise channel and fixed encoding, we prove that operational error and worst-input entanglement infidelity have the same minimum over all recovery operations. The proof constructs one final unitary (a reversible quantum operation) that makes any recovery's operational error no greater than its original worst-input entanglement infidelity. With a retained noise record, we determine the optimal error and characterize all optimal decoders subject to a Kraus-rank limit for each outcome. For a fixed decoded channel, a semidefinite program finds the optimal final unitary in the low-error regime. We characterize all optimal final corrections and show that, at a fixed error floor $0<\epsilon<1$, the operational optimizers' sensitivity to residual unitary errors of size $s$ is at worst of order $s^{4/3}$, while the canonical polar correction has quadratic sensitivity.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.46 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.77 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Even a recovery that maximizes worst-input entanglement fidelity can retain avoidable errors in subsequent measurements. Entanglement fidelity is the probability of passing a test of whether the state and its entanglement were preserved; infidelity is the probability of failing it. Operational error is the largest possible change in any measurement probability, quantified by the normalized diamond distance. For every finite-dimensional noise channel and fixed encoding, we prove that operational error and worst-input entanglement infidelity have the same minimum over all recovery operations. The proof constructs one final unitary (a reversible quantum operation) that makes any recovery's operational error no greater than its original worst-input entanglement infidelity. With a retained noise record, we determine the optimal error and characterize all optimal decoders subject to a Kraus-rank limit for each outcome. For a fixed decoded channel, a semidefinite program finds the optimal final unitary in the low-error regime. We characterize all optimal final corrections and show that, at a fixed error floor $0<\epsilon<1$, the operational optimizers' sensitivity to residual unitary errors of size $s$ is at worst of order $s^{4/3}$, while the canonical polar correction has quadratic sensitivity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Even a recovery that maximizes worst-input entanglement fidelity can retain avoidable errors in subsequent measurements.	gap 0.48	0.25	0.14	hedged possibility 0.58
s1	Entanglement fidelity is the probability of passing a test of whether the state and its entanglement were preserved; infidelity is the probability of failing it.	background 1.00	0.08	0.03	asserted 1.00
s2	Operational error is the largest possible change in any measurement probability, quantified by the normalized diamond distance.	background 0.99	0.16	0.04	asserted 1.00
s3	For every finite-dimensional noise channel and fixed encoding, we prove that operational error and worst-input entanglement infidelity have the same minimum over all recovery operations.	result 0.94	0.95	0.84	asserted 1.00
s4	The proof constructs one final unitary (a reversible quantum operation) that makes any recovery's operational error no greater than its original worst-input entanglement infidelity.	method 0.73	0.94	0.24	asserted 1.00
s5	With a retained noise record, we determine the optimal error and characterize all optimal decoders subject to a Kraus-rank limit for each outcome.	contribution 0.77	0.94	0.07	asserted 1.00
s6	For a fixed decoded channel, a semidefinite program finds the optimal final unitary in the low-error regime.	method 0.97	0.88	0.03	asserted 1.00
s7	We characterize all optimal final corrections and show that, at a fixed error floor $0<\epsilon<1$, the operational optimizers' sensitivity to residual unitary errors of size $s$ is at worst of order $s^{4/3}$, while the canonical polar correction has quadratic sensitivity.	result 0.98	0.94	0.15	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
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levels run

P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Three-qubit conditional-gate calibration. (a) Conditional noise Φ, followed by output correction

Figures reproduced from arXiv:2611.07344 by the authors.

References

Works cited (20)

[17]

H. Hao, Investigations on automorphism groups of quan- tum stabilizer codes, Unpublished manuscript, Sec. 2.2.5 (n.d.). 6 END MA TTER Independence of the minimizing state At a worst input the vectorα j = Tr(ρ0Aj) has squared normf. If two minimizing states gave different vectors α, β, their midpoint would have score∥(α+β)/2∥ 22 = f− ∥α−β∥ 22 /4< f. Hence ...

View on Pith
[1]

B. Schumacher, Sending entanglement through noisy quantum channels, Physical Review A54, 2614 (1996), arXiv:quant-ph/9604023

View on Pith Pith review arXiv 1996
[2]

C. B´ eny and O. Oreshkov, General conditions for ap- proximate quantum error correction and near-optimal re- covery channels, Physical Review Letters104, 120501 (2010), arXiv:0907.5391

View on Pith Pith review arXiv 2010
[4]

C. A. Fuchs and J. van de Graaf, Cryptographic dis- tinguishability measures for quantum-mechanical states, IEEE Transactions on Information Theory45, 1216 (1999), arXiv:quant-ph/9712042

View on Pith Pith review arXiv 1999
[5]

D. Kretschmann, D. Schlingemann, and R. F. Werner, The information-disturbance tradeoff and the continu- ity of Stinespring’s representation, IEEE Transactions on Information Theory54, 1708 (2008), arXiv:quant- ph/0605009

View on Pith 2008
[6]

R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia, Comparing experiments to the fault-tolerance threshold, Physical Review Letters117, 170502 (2016), arXiv:1510.05653

View on Pith Pith review arXiv 2016
[7]

T. Holdsworth, V. Singh, and M. M. Wilde, Quan- tifying the performance of approximate teleportation and quantum error correction via symmetric 2-PPT- extendible channels, Physical Review A107, 012428 (2023), arXiv:2207.06931

View on Pith Pith review arXiv 2023
[8]

J. Watrous,The Theory of Quantum Information(Cam- bridge University Press, 2018)

View on Pith 2018
[9]

Supplemental Material (2026), finite error certificates, lo- cal readout, the average-fidelity comparison, the isomet- ric extension, the all-error hierarchy and the standard- code control

View on Pith 2026
[10]

C.-H. F. Fung and H. F. Chau, Relation between phys- ical time-energy cost of a quantum process and its in- formation fidelity, Physical Review A90, 022333 (2014), arXiv:1404.3309

View on Pith Pith review arXiv 2014
[12]

J. Skanes-Norman,On Realistic Errors in Quantum Computers, M.Math. thesis, University of Waterloo (2021)

View on Pith 2021
[14]

T. Lee and R. Mittal, Product theorems via semidefinite programming (2008), arXiv:0803.4206 [cs.CC]

View on Pith Pith review arXiv 2008
[18]

Scaling (β, z) to the boundary cannot increase the objective; settingγ=β/zproves Eq. (S7) forD >0. IfD= 0, all Kraus matrices are scalar multiples ofI; aligned coefficients tending to the cone boundary give the stated infimum. No full-rank assumption on an optimal input is required. III. COMPLETE OPTIMAL F AMILIES AND SHARP GA TE RADII Let Φ(X) = Pj AjXA ...

View on Pith
[19]

C. B´ eny and O. Oreshkov, Approximate simulation of quantum channels, Physical Review A 84, 022333 (2011), arXiv:1103.0649

View on Pith Pith review arXiv 2011
[20]

J. Watrous,The Theory of Quantum Information(Cambridge University Press, 2018)

View on Pith 2018
[21]

X.-D. Yu, T. Simnacher, H. C. Nguyen, and O. G¨ uhne, Quantum-inspired hierarchy for rank- constrained optimization, PRX Quantum3, 010340 (2022), arXiv:2012.00554

View on Pith Pith review arXiv 2022
[22]

M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A303, 249 (2002), arXiv:quant-ph/0205035

View on Pith Pith review arXiv 2002
[23]

A. Carignan-Dugas, M. Alexander, and J. Emerson, A polar decomposition for quantum chan- nels (with applications to bounding error propagation in quantum circuits), Quantum3, 173 (2019), arXiv:1904.08897

View on Pith Pith review arXiv 2019
[24]

H. Hao, Investigations on automorphism groups of quantum stabilizer codes, Unpublished manuscript, Sec. 2.2.5 (n.d.)

View on Pith
[25]

D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Physical Review A54, 1862 (1996), arXiv:quant-ph/9604038. 27

View on Pith Pith review arXiv 1996

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [17]

H. Hao, Investigations on automorphism groups of quan- tum stabilizer codes, Unpublished manuscript, Sec. 2.2.5 (n.d.). 6 END MA TTER Independence of the minimizing state At a worst input the vectorα j = Tr(ρ0Aj) has squared normf. If two minimizing states gave different vectors α, β, their midpoint would have score∥(α+β)/2∥ 22 = f− ∥α−β∥ 22 /4< f. Hence ...

View on Pith

[2] [1]

B. Schumacher, Sending entanglement through noisy quantum channels, Physical Review A54, 2614 (1996), arXiv:quant-ph/9604023

View on Pith Pith review arXiv 1996

[3] [2]

C. B´ eny and O. Oreshkov, General conditions for ap- proximate quantum error correction and near-optimal re- covery channels, Physical Review Letters104, 120501 (2010), arXiv:0907.5391

View on Pith Pith review arXiv 2010

[4] [4]

C. A. Fuchs and J. van de Graaf, Cryptographic dis- tinguishability measures for quantum-mechanical states, IEEE Transactions on Information Theory45, 1216 (1999), arXiv:quant-ph/9712042

View on Pith Pith review arXiv 1999

[5] [5]

D. Kretschmann, D. Schlingemann, and R. F. Werner, The information-disturbance tradeoff and the continu- ity of Stinespring’s representation, IEEE Transactions on Information Theory54, 1708 (2008), arXiv:quant- ph/0605009

View on Pith 2008

[6] [6]

R. Kueng, D. M. Long, A. C. Doherty, and S. T. Flammia, Comparing experiments to the fault-tolerance threshold, Physical Review Letters117, 170502 (2016), arXiv:1510.05653

View on Pith Pith review arXiv 2016

[7] [7]

T. Holdsworth, V. Singh, and M. M. Wilde, Quan- tifying the performance of approximate teleportation and quantum error correction via symmetric 2-PPT- extendible channels, Physical Review A107, 012428 (2023), arXiv:2207.06931

View on Pith Pith review arXiv 2023

[8] [8]

J. Watrous,The Theory of Quantum Information(Cam- bridge University Press, 2018)

View on Pith 2018

[9] [9]

Supplemental Material (2026), finite error certificates, lo- cal readout, the average-fidelity comparison, the isomet- ric extension, the all-error hierarchy and the standard- code control

View on Pith 2026

[10] [10]

C.-H. F. Fung and H. F. Chau, Relation between phys- ical time-energy cost of a quantum process and its in- formation fidelity, Physical Review A90, 022333 (2014), arXiv:1404.3309

View on Pith Pith review arXiv 2014

[11] [12]

J. Skanes-Norman,On Realistic Errors in Quantum Computers, M.Math. thesis, University of Waterloo (2021)

View on Pith 2021

[12] [14]

T. Lee and R. Mittal, Product theorems via semidefinite programming (2008), arXiv:0803.4206 [cs.CC]

View on Pith Pith review arXiv 2008

[13] [18]

Scaling (β, z) to the boundary cannot increase the objective; settingγ=β/zproves Eq. (S7) forD >0. IfD= 0, all Kraus matrices are scalar multiples ofI; aligned coefficients tending to the cone boundary give the stated infimum. No full-rank assumption on an optimal input is required. III. COMPLETE OPTIMAL F AMILIES AND SHARP GA TE RADII Let Φ(X) = Pj AjXA ...

View on Pith

[14] [19]

C. B´ eny and O. Oreshkov, Approximate simulation of quantum channels, Physical Review A 84, 022333 (2011), arXiv:1103.0649

View on Pith Pith review arXiv 2011

[15] [20]

J. Watrous,The Theory of Quantum Information(Cambridge University Press, 2018)

View on Pith 2018

[16] [21]

X.-D. Yu, T. Simnacher, H. C. Nguyen, and O. G¨ uhne, Quantum-inspired hierarchy for rank- constrained optimization, PRX Quantum3, 010340 (2022), arXiv:2012.00554

View on Pith Pith review arXiv 2022

[17] [22]

M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A303, 249 (2002), arXiv:quant-ph/0205035

View on Pith Pith review arXiv 2002

[18] [23]

A. Carignan-Dugas, M. Alexander, and J. Emerson, A polar decomposition for quantum chan- nels (with applications to bounding error propagation in quantum circuits), Quantum3, 173 (2019), arXiv:1904.08897

View on Pith Pith review arXiv 2019

[19] [24]

H. Hao, Investigations on automorphism groups of quantum stabilizer codes, Unpublished manuscript, Sec. 2.2.5 (n.d.)

View on Pith

[20] [25]

D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Physical Review A54, 1862 (1996), arXiv:quant-ph/9604038. 27

View on Pith Pith review arXiv 1996