A Deficiency-Based Approach for the Operational Interpretation of Quantum Resources with Applications

Chunhe Xiong; Junde Wu; Sunho Kim

arxiv: 2509.03043 · v4 · submitted 2025-09-03 · 🪐 quant-ph

A Deficiency-Based Approach for the Operational Interpretation of Quantum Resources with Applications

Sunho Kim , Chunhe Xiong , Junde Wu This is my paper

Pith reviewed 2026-05-18 19:55 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum resource theoryresource deficiencyoperational interpretationsubchannel discriminationcoherenceentanglementquantum error correctiongate noise

0 comments

The pith

Defining resource deficiency relative to maximal sets gives operational meaning to quantum resources even when mixed states appear inactive.

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

The paper introduces a deficiency-based framework that quantifies how far a quantum state falls short of a maximal resource state. This extends conventional resource theories to cover mixed states whose resources remain inactive in given tasks. A geometric measure built within the framework captures the performance gap versus maximal states in subchannel discrimination for both coherence and entanglement. The same deficiency quantities link directly to experimental estimation of noise constants in quantum gates such as the Hadamard, with extensions to error-correction thresholds and algorithm performance.

Core claim

By defining the resource deficiency of a state relative to maximal resource sets, the approach supplies operational interpretations that conventional resource measures miss, particularly for mixed states. A geometric measure satisfying the deficiency requirements quantifies the operational disadvantage of arbitrary states compared with maximal resource states in subchannel discrimination. The framework simultaneously supplies a practical method to extract quantum gate noise constants from deficiency values, demonstrated for Hadamard gates and stated to generalize.

What carries the argument

Resource deficiency of a state relative to maximal resource sets, which measures shortfall in operational capability and enables classification of mixed states.

If this is right

Geometric measures for coherence and entanglement quantify operational disadvantage in subchannel discrimination.
Deficiency measures serve as indicators for determining quantum-error-correction thresholds.
The methodology extends to general gates and predicts algorithm performance.
Mixed resource states whose properties are inactive in conventional tasks receive operational classification.

Where Pith is reading between the lines

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

Deficiency values could guide the choice of states or gates in protocols that activate otherwise inactive resources.
The same construction may apply to other resource theories such as those based on magic or non-stabilizer states.
Experimental noise-estimation protocols built from deficiency could be tested on current hardware to refine error-correction predictions.

Load-bearing premise

Maximal resource sets exist with well-defined properties and a geometric measure can be constructed to satisfy the deficiency framework without additional fitting.

What would settle it

An experiment or calculation in which the geometric deficiency measure fails to predict the observed performance gap in subchannel discrimination for a family of mixed states would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.03043 by Chunhe Xiong, Junde Wu, Sunho Kim.

read the original abstract

A fundamental challenge in quantum resource theory is to establish operational interpretations by quantifying the advantage that quantum resources provide in specific tasks. Conventional resource theories, however, have inherent limitations in characterizing such advantages for certain quantum operations. We overcome this by introducing a novel approach that defines the resource deficiency of a state relative to maximal resource sets. This extension broadens the scope of resource theories, delivers more complete operational interpretations, and yields broad insights for classifying mixed resource states--including those whose resource properties remain inactive in given tasks--that escape conventional descriptions. We also show that a geometric measure satisfying the deficiency-based framework's requirements for coherence and entanglement captures the operational disadvantage of arbitrary states compared to maximal resource states in subchannel discrimination. In parallel, we present a practical methodology that links deficiency measures with experimental estimation of quantum gate noise constants, illustrated for Hadamard gates. The methodology is extensible to general gates, and the results demonstrate that deficiency measures can serve as key indicators for determining quantum-error-correction thresholds and predicting algorithm performance.

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 defines resource deficiency relative to maximal sets to handle mixed states with inactive resources, but the construction of those sets looks underspecified for the claimed operational results.

read the letter

The main takeaway is that this work defines resource deficiency against maximal resource sets to give operational meaning to mixed states whose resources stay inactive in a given task. That move is meant to cover cases standard resource theories miss, and they apply it to subchannel discrimination via a geometric measure for coherence and entanglement. They also sketch a link from deficiency measures to experimental estimation of gate noise, with a Hadamard-gate example that points toward error-correction thresholds and algorithm performance prediction. Those are the concrete pieces that stand out from the abstract and the stress-test note. The geometric-measure application to discrimination tasks is the part that feels most developed and worth checking in detail. The noise-estimation methodology is practical enough that it could be tested on other gates without much extra machinery. The soft spot is the one flagged in the stress-test: the abstract gives no explicit construction or uniqueness argument for the maximal resource sets once states are mixed. If those sets are not canonically defined or require task-dependent choices, then the geometric measure does not emerge parameter-free from the deficiency axioms, and the operational disadvantage claim does not extend reliably beyond the single-gate illustration. That is a load-bearing gap rather than a minor omission. This paper is for people already working in quantum resource theory who want to connect quantifiers more directly to discrimination tasks and gate noise. A reader focused on operational interpretations in quantum computing would get usable ideas from the Hadamard example and the threshold discussion, even while the foundations are still rough. It deserves a serious referee because the target problem is real and the applications are specific enough to be checked, though the referee will need to press on the maximal-set construction before the operational story can be taken as settled.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a deficiency-based framework for the operational interpretation of quantum resources. It defines the resource deficiency of a state relative to maximal resource sets, extending conventional resource theories to provide more complete interpretations, including for mixed states with inactive resources. The authors claim that a geometric measure satisfying the framework's requirements for coherence and entanglement quantifies the operational disadvantage of arbitrary states versus maximal resource states in subchannel discrimination. They further present a methodology connecting deficiency measures to experimental estimation of quantum gate noise constants, illustrated for Hadamard gates and extensible to general gates, with applications to quantum-error-correction thresholds and algorithm performance prediction.

Significance. If the central constructions hold, the work could meaningfully broaden quantum resource theories by addressing limitations in operational characterizations for certain tasks and mixed states. The geometric measure's claimed capture of disadvantage in subchannel discrimination and the link to experimental noise estimation offer potential practical value for quantum information processing. The approach to inactive resources is a notable attempt at completeness, though its impact depends on rigorous validation of the maximal sets and parameter-free emergence of the measure.

major comments (1)

[Definition of resource deficiency and maximal resource sets] The definition of resource deficiency (introduced relative to maximal resource sets) is load-bearing for all subsequent claims, including the geometric measure's operational interpretation in subchannel discrimination. The manuscript does not appear to supply an explicit construction, uniqueness proof, or canonical selection procedure for these maximal sets when applied to mixed states; without this, it is unclear whether the geometric measure follows directly from the deficiency axioms or requires case-by-case adjustments, undermining the generalization beyond the Hadamard-gate example.

minor comments (2)

[Abstract and experimental methodology section] Clarify in the abstract and introduction whether the geometric measure is derived parameter-free from the deficiency axioms or involves any fitting constants, to address potential circularity concerns in the experimental estimation link.
Ensure all notation for deficiency measures and maximal sets is defined consistently before use in the subchannel discrimination results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a key point regarding the foundations of the deficiency framework. We address the major comment below in detail.

read point-by-point responses

Referee: The definition of resource deficiency (introduced relative to maximal resource sets) is load-bearing for all subsequent claims, including the geometric measure's operational interpretation in subchannel discrimination. The manuscript does not appear to supply an explicit construction, uniqueness proof, or canonical selection procedure for these maximal sets when applied to mixed states; without this, it is unclear whether the geometric measure follows directly from the deficiency axioms or requires case-by-case adjustments, undermining the generalization beyond the Hadamard-gate example.

Authors: We thank the referee for this observation. The maximal resource sets are defined canonically in Section II as the collection of states that attain the supremum of the chosen resource quantifier (e.g., the set of all maximally coherent states for coherence, or the convex hull of maximally entangled states for entanglement). Resource deficiency is then introduced in Definition 2 as the infimum of a contractive distance from the given state to this set; the definition applies verbatim to mixed states and does not invoke any state-specific adjustments. Theorems 1 and 2 establish that the geometric measure satisfies the deficiency axioms for coherence and entanglement, respectively, and directly yields the operational interpretation in subchannel discrimination without further specialization. The Hadamard-gate illustration in Section IV is an application of the general construction rather than a separate case. While we do not claim or prove uniqueness of the maximal sets beyond their definition via the resource quantifier, the framework does not require such uniqueness for the subsequent results to hold. We are happy to add a short clarifying paragraph in Section II that restates the selection rule for mixed states and notes its independence from the particular task. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected; derivation remains self-contained.

full rationale

The provided abstract introduces a novel deficiency-based definition of resource deficiency relative to maximal resource sets and states that a geometric measure is shown to satisfy the framework's requirements for coherence and entanglement, thereby capturing operational disadvantage in subchannel discrimination. No equations, self-citations, or explicit reductions are present that would make any prediction equivalent to its inputs by construction, such as a fitted parameter renamed as a prediction or an ansatz smuggled via prior work. The link to experimental estimation of gate noise constants is presented as a practical methodology without indication of post-hoc fitting that forces the result. The central claims appear independent of the inputs and do not rely on load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

Abstract provides no explicit list of free parameters or axioms; the framework appears to rest on standard quantum mechanics plus the new deficiency concept.

free parameters (1)

parameters in geometric measure
Likely scaling or normalization constants needed to satisfy framework requirements for coherence and entanglement.

axioms (2)

domain assumption Existence of maximal resource sets in the state space
Central to defining deficiency relative to these sets.
standard math Standard postulates of quantum mechanics and resource theory
Background for coherence and entanglement definitions.

invented entities (1)

resource deficiency no independent evidence
purpose: Quantifies operational disadvantage of states relative to maximal resource states
New quantity introduced to broaden resource theory scope.

pith-pipeline@v0.9.0 · 5704 in / 1249 out tokens · 60163 ms · 2026-05-18T19:55:02.328424+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

geometric measure of deficiency relative to maximal resource states as D_g(ρ) = min_{σ∈R_max} {1−F(σ,ρ)} … Theorem 5 … = 1−D_g(ρ)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

maximal resource states … pure states … non-convex … phase sensitivity … PPT entangled states

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

59 extracted references · 59 canonical work pages · 1 internal anchor

[1]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, Phys. Rev. Lett.76, 722 (1996) doi: 10.1103/PhysRevLett.76.722

work page doi:10.1103/physrevlett.76.722 1996
[2]

M. B. Plenio, and S. Virmani, Quantum Inf. Comput.7, 1-51 (2007) doi:10.26421/QIC7.1-2

work page doi:10.26421/qic7.1-2 2007
[3]

Classification of Electromagnetic and Gravitational Hopfions by Algebraic Type

B. Regula, J. Phys. A51, 045303 (2018) doi:10.1088/1751- 8121/aa9100

work page doi:10.1088/1751- 2018
[4]

Horodecki and A

P. Horodecki and A. Ekert, Phys. Rev. Lett.89, 127902 (2002) doi:10.1103/PhysRevLett.89.127902

work page doi:10.1103/physrevlett.89.127902 2002
[5]

Schaetz, M

T. Schaetz, M. D. Barrett, D. Leibfried, J. Britton, J. Chi- averini, W. M. Itano, J. D. Jost, E. Knill, C. Langer, and D. J. Wineland, Phys. Rev. Lett.94, 010501 (2005) doi: 10.1103/PhysRevLett.94.010501

work page doi:10.1103/physrevlett.94.010501 2005
[6]

Hou, and X

J. Hou, and X. Qi, Phys. Rev. A81, 062351 (2010) doi: 10.1103/PhysRevA.81.062351

work page doi:10.1103/physreva.81.062351 2010
[7]

L. Roa, J. C. Retamal, and M. Alid-Vaccarezza, Phys. Rev. Lett. 107, 080401 (2011) doi:10.1103/PhysRevLett.107.080401

work page doi:10.1103/physrevlett.107.080401 2011
[8]

Horodecki, P

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys.81, 865 (2009) doi: 10.1103/RevModPhys.81.865

work page doi:10.1103/revmodphys.81.865 2009
[9]

& Plenio, M

A. Streltsov, G. Adesso, and M.B. Plenio, Rev. Mod. Phys.89, 041003 (2017) doi:10.1103/RevModPhys.89.041003

work page doi:10.1103/revmodphys.89.041003 2017
[10]

Bartlett, T

S.D. Bartlett, T. Rudolph, and R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007) doi:10.1103/RevModPhys.79.555

work page doi:10.1103/revmodphys.79.555 2007
[11]

G. Gour, M. P. Muller, V . Narasimhachar, R. W. Spekkens, and N. Y . Halpern, Phys. Rep.583, 1 (2015) doi: 10.1016/j.physrep.2015.04.003

work page doi:10.1016/j.physrep.2015.04.003 2015
[12]

Luo, and S

S. Luo, and S. Fu, Phys. Rev. Lett.106, 120401 (2011) doi: 10.1103/PhysRevLett.106.120401

work page doi:10.1103/physrevlett.106.120401 2011
[13]

Brunner, D

N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Rev. Mod. Phys.86, 419 (2014) doi: 10.1103/RevModPhys.86.419

work page doi:10.1103/revmodphys.86.419 2014
[14]

Weedbrook, S

C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys.84, 621 (2012) doi:10.1103/RevModPhys.84.621

work page doi:10.1103/revmodphys.84.621 2012
[15]

Luo, Phys

S. Luo, Phys. Rev. A77, 042303 (2008) doi: 10.1103/PhysRevA.77.042303

work page doi:10.1103/physreva.77.042303 2008
[16]

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V . Vedral, Rev. Mod. Phys.84, 1655 (2012) doi:10.1103/RevModPhys.84.1655

work page doi:10.1103/revmodphys.84.1655 2012
[17]

Mintert, M

F. Mintert, M. Ku ´s, and A. Buchleitner, Phys. Rev. Lett.95, 260502 (2005) doi:10.1103/PhysRevLett.95.260502

work page doi:10.1103/physrevlett.95.260502 2005
[18]

J. I. de Vicente, C. Spee, and B. Kraus, Phys. Rev. Lett.111, 110502 (2013) doi:10.1103/PhysRevLett.111.110502

work page doi:10.1103/physrevlett.111.110502 2013
[19]

J. Hou, L. Liu, and X. Qi, Phys. Rev. A105, 032429 (2022) doi:10.1103/PhysRevA.105.032429

work page doi:10.1103/physreva.105.032429 2022
[20]

Gour, and A

G. Gour, and A. Winter, Phys. Rev. Lett.123, 150401 (2019) doi:10.1103/PhysRevLett.123.150401

work page doi:10.1103/physrevlett.123.150401 2019
[21]

Gour, IEEE Trans

G. Gour, IEEE Trans. Inf. Theory65, 5880-5904 (2019) doi: 10.1109/TIT.2019.2907989

work page doi:10.1109/tit.2019.2907989 2019
[22]

Saxena, E

G. Saxena, E. Chitambar, G. Gour, Phys. Rev. Research2, 023298 (2020) doi:10.1103/PhysRevResearch.2.023298

work page doi:10.1103/physrevresearch.2.023298 2020
[23]

Chitambar G

E. Chitambar and G. Gour, Rev. Mod. Phys.91, 025001 (2019) doi:10.1103/RevModPhys.91.025001

work page doi:10.1103/revmodphys.91.025001 2019
[24]

Vidal, and R

G. Vidal, and R. Tarrach, Phys. Rev. A59, 141 (1999) doi: 10.1103/PhysRevA.59.141

work page doi:10.1103/physreva.59.141 1999
[25]

A. W. Harrow, and M. A. Nielsen, Phys. Rev. A68, 012308 (2003) doi:10.1103/PhysRevA.68.012308

work page doi:10.1103/physreva.68.012308 2003
[26]

Napoli, T

C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. John- ston, and G. Adesso, Phys. Rev. Lett.116, 150502 (2016) doi: 10.1103/PhysRevLett.116.150502

work page doi:10.1103/physrevlett.116.150502 2016
[27]

Vedral, M

V . Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett.78, 2275 (1997) doi:10.1103/PhysRevLett.78.2275

work page doi:10.1103/physrevlett.78.2275 1997
[28]

& Plenio, M

T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014) doi:10.1103/PhysRevLett.113.140401

work page doi:10.1103/physrevlett.113.140401 2014
[29]

R ´enyi,Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, V ol

A. R ´enyi,Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, V ol. 1: Contributions to the Theory of Statistics (University of California Press, Berkeley, 1961)

work page 1961
[30]

Eisert, K

J. Eisert, K. Audenaert, and M. B. Plenio, J. Phys. A36, 5605 (2003) doi:10.1088/0305-4470/36/20/316

work page doi:10.1088/0305-4470/36/20/316 2003
[31]

Winter and D

A. Winter and D. Yang, Phys. Rev. Lett.116, 120404 (2016) doi:10.1103/PhysRevLett.116.120404

work page doi:10.1103/physrevlett.116.120404 2016
[32]

Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71(2):022316, 2005

S. Bravyi, and A. Kitaev, Phys. Rev. A71, 022316 (2005) doi: 10.1103/PhysRevA.71.022316

work page doi:10.1103/physreva.71.022316 2005
[33]

A. C. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A162, 25-28 (1992) doi:10.1016/0375-9601(92)90952-I

work page doi:10.1016/0375-9601(92)90952-i 1992
[34]

Lewenstein, and A

M. Lewenstein, and A. Sanpera, Phys. Rev. Lett.80, 2261 (1998) doi:10.1103/PhysRevLett.80.2261

work page doi:10.1103/physrevlett.80.2261 1998
[35]

Skrzypczyk, M

P. Skrzypczyk, M. Navascu ´es, and D. Caval- canti, Phys. Rev. Lett.112, 180404 (2014) doi: 10.1103/PhysRevLett.112.180404

work page doi:10.1103/physrevlett.112.180404 2014
[36]

Luo, Phys

S. Luo, Phys. Rev. Lett.91, 180403 (2003) doi: 10.1103/PhysRevLett.91.180403

work page doi:10.1103/physrevlett.91.180403 2003
[37]

Zheng, Z

W. Zheng, Z. Ma, H. Wang, S. M. Fei, and X. Peng, Phys. Rev. Lett.120, 230504 (2018) doi: 10.1103/PhysRevLett.120.230504

work page doi:10.1103/physrevlett.120.230504 2018
[38]

Takagi, B

R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Phys. Rev. Lett.122, 140402 (2019) doi: 10.1103/PhysRevLett.122.140402

work page doi:10.1103/physrevlett.122.140402 2019
[39]

A. F. Ducuara, and P. Skrzypczyk, Phys. Rev. Lett.125, 110401 (2020) doi:10.1103/PhysRevLett.125.110401

work page doi:10.1103/physrevlett.125.110401 2020
[40]

R. Uola, T. Bullock, T. Kraft, J.-P. Pellonp ¨a¨a, and N. Brunner, Phys. Rev. Lett.125, 110402 (2020) doi: 10.1103/PhysRevLett.125.110402

work page doi:10.1103/physrevlett.125.110402 2020
[41]

Skrzypczyk, and I

P. Skrzypczyk, and I. ˘Supi´c, Phys. Rev. Lett.122, 130403 (2019) doi:10.1103/PhysRevLett.122.130403

work page doi:10.1103/physrevlett.122.130403 2019
[42]

Skrzypczyk, and N

P. Skrzypczyk, and N. Linden, Phys. Rev. Lett.122, 140403 (2019) doi:10.1103/PhysRevLett.122.140403

work page doi:10.1103/physrevlett.122.140403 2019
[43]

Buscemi, E

F. Buscemi, E. Chitambar, and W. Zhou, Phys. Rev. Lett.124, 120401 (2020) doi:10.1103/PhysRevLett.124.120401

work page doi:10.1103/physrevlett.124.120401 2020
[44]

Takagi, and B

R. Takagi, and B. Regula, Phys. Rev. X9, 031053 (2019) doi: 10.1103/PhysRevX.9.031053

work page doi:10.1103/physrevx.9.031053 2019
[45]

Takagi, K

R. Takagi, K. Wang, and M. Hayashi, Phys. Rev. Lett.124, 120502 (2020) doi:10.1103/PhysRevLett.124.120502

work page doi:10.1103/physrevlett.124.120502 2020
[46]

Kuroiwa, R

K. Kuroiwa, R. Takagi, G. Adesso, and H. Ya- masaki, Phys. Rev. Lett.132, 150201 (2024) doi: 10.1103/PhysRevLett.132.150201

work page doi:10.1103/physrevlett.132.150201 2024
[47]

Kuroiwa, R

K. Kuroiwa, R. Takagi, G. Adesso, and H. Yamasaki, Phys. Rev. A109, 042403 (2024) doi:10.1103/PhysRevA.109.042403

work page doi:10.1103/physreva.109.042403 2024
[48]

Horodecki, P

M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998) doi:10.1103/PhysRevLett.80.5239

work page doi:10.1103/physrevlett.80.5239 1998
[49]

Linden, and S

N. Linden, and S. Popescu, Phys. Rev. A59, 137 (1999) doi: 10.1103/PhysRevA.59.137

work page doi:10.1103/physreva.59.137 1999
[50]

Biham, and D

E. Biham, and D. Kenigsberg, Phys. Rev. A66, 062301 (2002) doi:10.1103/PhysRevA.66.062301

work page doi:10.1103/physreva.66.062301 2002
[51]

Quantum Resource Theories beyond Convexity

R. Salazar, J. Czartowski, R. R. Rodr ´ıguez, G. R.- Mieldzio´c, P. Horodecki, K. ˙Zyczkowski, arXiv:2405.05785 doi:10.48550/arXiv.2405.05785

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.05785
[52]

C. H. Bennett, G. Brassard, C. Cr ´epeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett.70, 1895 (1993) doi: 10.1103/PhysRevLett.70.1895

work page doi:10.1103/physrevlett.70.1895 1993
[53]

Horodecki, P

M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 10 60, 1888 (1999) doi:10.1103/PhysRevA.60.1888

work page doi:10.1103/physreva.60.1888 1999
[54]

Horodecki, M

R. Horodecki, M. Horodecki, and P. Horodecki, Phys. Lett. A 222, 21-25 (1996) doi:10.1016/0375-9601(96)00639-1

work page doi:10.1016/0375-9601(96)00639-1 1996
[55]

Bouwmeester, J

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. We- infurter, and A. Zeilinger, Nature390, 575-579 (1997) doi: 10.1038/37539

work page doi:10.1038/37539 1997
[56]

S. Q. Zhou, H. Jin, J. M. Liang, S. M. Fei, Y . Xiao, and Z. Ma, Phys. Rev. A110, 062429 (2024) doi: 10.1103/PhysRevA.110.062429

work page doi:10.1103/physreva.110.062429 2024
[57]

Streltsov, U

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Phys. Rev. Lett.115, 020403 (2015) doi: 10.1103/PhysRevLett.115.020403

work page doi:10.1103/physrevlett.115.020403 2015
[58]

Watrous,Theory of Quantum Information, (Institute for Quantum Computing, University of Waterloo, Waterloo, Canada, 2011)

J. Watrous,Theory of Quantum Information, (Institute for Quantum Computing, University of Waterloo, Waterloo, Canada, 2011)

work page 2011
[59]

Bhatia,Matrix Analysis, (Springer-Verlag, New York, 1997)

R. Bhatia,Matrix Analysis, (Springer-Verlag, New York, 1997)

work page 1997

[1] [1]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, Phys. Rev. Lett.76, 722 (1996) doi: 10.1103/PhysRevLett.76.722

work page doi:10.1103/physrevlett.76.722 1996

[2] [2]

M. B. Plenio, and S. Virmani, Quantum Inf. Comput.7, 1-51 (2007) doi:10.26421/QIC7.1-2

work page doi:10.26421/qic7.1-2 2007

[3] [3]

Classification of Electromagnetic and Gravitational Hopfions by Algebraic Type

B. Regula, J. Phys. A51, 045303 (2018) doi:10.1088/1751- 8121/aa9100

work page doi:10.1088/1751- 2018

[4] [4]

Horodecki and A

P. Horodecki and A. Ekert, Phys. Rev. Lett.89, 127902 (2002) doi:10.1103/PhysRevLett.89.127902

work page doi:10.1103/physrevlett.89.127902 2002

[5] [5]

Schaetz, M

T. Schaetz, M. D. Barrett, D. Leibfried, J. Britton, J. Chi- averini, W. M. Itano, J. D. Jost, E. Knill, C. Langer, and D. J. Wineland, Phys. Rev. Lett.94, 010501 (2005) doi: 10.1103/PhysRevLett.94.010501

work page doi:10.1103/physrevlett.94.010501 2005

[6] [6]

Hou, and X

J. Hou, and X. Qi, Phys. Rev. A81, 062351 (2010) doi: 10.1103/PhysRevA.81.062351

work page doi:10.1103/physreva.81.062351 2010

[7] [7]

L. Roa, J. C. Retamal, and M. Alid-Vaccarezza, Phys. Rev. Lett. 107, 080401 (2011) doi:10.1103/PhysRevLett.107.080401

work page doi:10.1103/physrevlett.107.080401 2011

[8] [8]

Horodecki, P

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys.81, 865 (2009) doi: 10.1103/RevModPhys.81.865

work page doi:10.1103/revmodphys.81.865 2009

[9] [9]

& Plenio, M

A. Streltsov, G. Adesso, and M.B. Plenio, Rev. Mod. Phys.89, 041003 (2017) doi:10.1103/RevModPhys.89.041003

work page doi:10.1103/revmodphys.89.041003 2017

[10] [10]

Bartlett, T

S.D. Bartlett, T. Rudolph, and R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007) doi:10.1103/RevModPhys.79.555

work page doi:10.1103/revmodphys.79.555 2007

[11] [11]

G. Gour, M. P. Muller, V . Narasimhachar, R. W. Spekkens, and N. Y . Halpern, Phys. Rep.583, 1 (2015) doi: 10.1016/j.physrep.2015.04.003

work page doi:10.1016/j.physrep.2015.04.003 2015

[12] [12]

Luo, and S

S. Luo, and S. Fu, Phys. Rev. Lett.106, 120401 (2011) doi: 10.1103/PhysRevLett.106.120401

work page doi:10.1103/physrevlett.106.120401 2011

[13] [13]

Brunner, D

N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Rev. Mod. Phys.86, 419 (2014) doi: 10.1103/RevModPhys.86.419

work page doi:10.1103/revmodphys.86.419 2014

[14] [14]

Weedbrook, S

C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys.84, 621 (2012) doi:10.1103/RevModPhys.84.621

work page doi:10.1103/revmodphys.84.621 2012

[15] [15]

Luo, Phys

S. Luo, Phys. Rev. A77, 042303 (2008) doi: 10.1103/PhysRevA.77.042303

work page doi:10.1103/physreva.77.042303 2008

[16] [16]

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V . Vedral, Rev. Mod. Phys.84, 1655 (2012) doi:10.1103/RevModPhys.84.1655

work page doi:10.1103/revmodphys.84.1655 2012

[17] [17]

Mintert, M

F. Mintert, M. Ku ´s, and A. Buchleitner, Phys. Rev. Lett.95, 260502 (2005) doi:10.1103/PhysRevLett.95.260502

work page doi:10.1103/physrevlett.95.260502 2005

[18] [18]

J. I. de Vicente, C. Spee, and B. Kraus, Phys. Rev. Lett.111, 110502 (2013) doi:10.1103/PhysRevLett.111.110502

work page doi:10.1103/physrevlett.111.110502 2013

[19] [19]

J. Hou, L. Liu, and X. Qi, Phys. Rev. A105, 032429 (2022) doi:10.1103/PhysRevA.105.032429

work page doi:10.1103/physreva.105.032429 2022

[20] [20]

Gour, and A

G. Gour, and A. Winter, Phys. Rev. Lett.123, 150401 (2019) doi:10.1103/PhysRevLett.123.150401

work page doi:10.1103/physrevlett.123.150401 2019

[21] [21]

Gour, IEEE Trans

G. Gour, IEEE Trans. Inf. Theory65, 5880-5904 (2019) doi: 10.1109/TIT.2019.2907989

work page doi:10.1109/tit.2019.2907989 2019

[22] [22]

Saxena, E

G. Saxena, E. Chitambar, G. Gour, Phys. Rev. Research2, 023298 (2020) doi:10.1103/PhysRevResearch.2.023298

work page doi:10.1103/physrevresearch.2.023298 2020

[23] [23]

Chitambar G

E. Chitambar and G. Gour, Rev. Mod. Phys.91, 025001 (2019) doi:10.1103/RevModPhys.91.025001

work page doi:10.1103/revmodphys.91.025001 2019

[24] [24]

Vidal, and R

G. Vidal, and R. Tarrach, Phys. Rev. A59, 141 (1999) doi: 10.1103/PhysRevA.59.141

work page doi:10.1103/physreva.59.141 1999

[25] [25]

A. W. Harrow, and M. A. Nielsen, Phys. Rev. A68, 012308 (2003) doi:10.1103/PhysRevA.68.012308

work page doi:10.1103/physreva.68.012308 2003

[26] [26]

Napoli, T

C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. John- ston, and G. Adesso, Phys. Rev. Lett.116, 150502 (2016) doi: 10.1103/PhysRevLett.116.150502

work page doi:10.1103/physrevlett.116.150502 2016

[27] [27]

Vedral, M

V . Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett.78, 2275 (1997) doi:10.1103/PhysRevLett.78.2275

work page doi:10.1103/physrevlett.78.2275 1997

[28] [28]

& Plenio, M

T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014) doi:10.1103/PhysRevLett.113.140401

work page doi:10.1103/physrevlett.113.140401 2014

[29] [29]

R ´enyi,Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, V ol

A. R ´enyi,Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, V ol. 1: Contributions to the Theory of Statistics (University of California Press, Berkeley, 1961)

work page 1961

[30] [30]

Eisert, K

J. Eisert, K. Audenaert, and M. B. Plenio, J. Phys. A36, 5605 (2003) doi:10.1088/0305-4470/36/20/316

work page doi:10.1088/0305-4470/36/20/316 2003

[31] [31]

Winter and D

A. Winter and D. Yang, Phys. Rev. Lett.116, 120404 (2016) doi:10.1103/PhysRevLett.116.120404

work page doi:10.1103/physrevlett.116.120404 2016

[32] [32]

Universal quantum computation with ideal Clifford gates and noisy ancillas.Physical Review A, 71(2):022316, 2005

S. Bravyi, and A. Kitaev, Phys. Rev. A71, 022316 (2005) doi: 10.1103/PhysRevA.71.022316

work page doi:10.1103/physreva.71.022316 2005

[33] [33]

A. C. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A162, 25-28 (1992) doi:10.1016/0375-9601(92)90952-I

work page doi:10.1016/0375-9601(92)90952-i 1992

[34] [34]

Lewenstein, and A

M. Lewenstein, and A. Sanpera, Phys. Rev. Lett.80, 2261 (1998) doi:10.1103/PhysRevLett.80.2261

work page doi:10.1103/physrevlett.80.2261 1998

[35] [35]

Skrzypczyk, M

P. Skrzypczyk, M. Navascu ´es, and D. Caval- canti, Phys. Rev. Lett.112, 180404 (2014) doi: 10.1103/PhysRevLett.112.180404

work page doi:10.1103/physrevlett.112.180404 2014

[36] [36]

Luo, Phys

S. Luo, Phys. Rev. Lett.91, 180403 (2003) doi: 10.1103/PhysRevLett.91.180403

work page doi:10.1103/physrevlett.91.180403 2003

[37] [37]

Zheng, Z

W. Zheng, Z. Ma, H. Wang, S. M. Fei, and X. Peng, Phys. Rev. Lett.120, 230504 (2018) doi: 10.1103/PhysRevLett.120.230504

work page doi:10.1103/physrevlett.120.230504 2018

[38] [38]

Takagi, B

R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Phys. Rev. Lett.122, 140402 (2019) doi: 10.1103/PhysRevLett.122.140402

work page doi:10.1103/physrevlett.122.140402 2019

[39] [39]

A. F. Ducuara, and P. Skrzypczyk, Phys. Rev. Lett.125, 110401 (2020) doi:10.1103/PhysRevLett.125.110401

work page doi:10.1103/physrevlett.125.110401 2020

[40] [40]

R. Uola, T. Bullock, T. Kraft, J.-P. Pellonp ¨a¨a, and N. Brunner, Phys. Rev. Lett.125, 110402 (2020) doi: 10.1103/PhysRevLett.125.110402

work page doi:10.1103/physrevlett.125.110402 2020

[41] [41]

Skrzypczyk, and I

P. Skrzypczyk, and I. ˘Supi´c, Phys. Rev. Lett.122, 130403 (2019) doi:10.1103/PhysRevLett.122.130403

work page doi:10.1103/physrevlett.122.130403 2019

[42] [42]

Skrzypczyk, and N

P. Skrzypczyk, and N. Linden, Phys. Rev. Lett.122, 140403 (2019) doi:10.1103/PhysRevLett.122.140403

work page doi:10.1103/physrevlett.122.140403 2019

[43] [43]

Buscemi, E

F. Buscemi, E. Chitambar, and W. Zhou, Phys. Rev. Lett.124, 120401 (2020) doi:10.1103/PhysRevLett.124.120401

work page doi:10.1103/physrevlett.124.120401 2020

[44] [44]

Takagi, and B

R. Takagi, and B. Regula, Phys. Rev. X9, 031053 (2019) doi: 10.1103/PhysRevX.9.031053

work page doi:10.1103/physrevx.9.031053 2019

[45] [45]

Takagi, K

R. Takagi, K. Wang, and M. Hayashi, Phys. Rev. Lett.124, 120502 (2020) doi:10.1103/PhysRevLett.124.120502

work page doi:10.1103/physrevlett.124.120502 2020

[46] [46]

Kuroiwa, R

K. Kuroiwa, R. Takagi, G. Adesso, and H. Ya- masaki, Phys. Rev. Lett.132, 150201 (2024) doi: 10.1103/PhysRevLett.132.150201

work page doi:10.1103/physrevlett.132.150201 2024

[47] [47]

Kuroiwa, R

K. Kuroiwa, R. Takagi, G. Adesso, and H. Yamasaki, Phys. Rev. A109, 042403 (2024) doi:10.1103/PhysRevA.109.042403

work page doi:10.1103/physreva.109.042403 2024

[48] [48]

Horodecki, P

M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998) doi:10.1103/PhysRevLett.80.5239

work page doi:10.1103/physrevlett.80.5239 1998

[49] [49]

Linden, and S

N. Linden, and S. Popescu, Phys. Rev. A59, 137 (1999) doi: 10.1103/PhysRevA.59.137

work page doi:10.1103/physreva.59.137 1999

[50] [50]

Biham, and D

E. Biham, and D. Kenigsberg, Phys. Rev. A66, 062301 (2002) doi:10.1103/PhysRevA.66.062301

work page doi:10.1103/physreva.66.062301 2002

[51] [51]

Quantum Resource Theories beyond Convexity

R. Salazar, J. Czartowski, R. R. Rodr ´ıguez, G. R.- Mieldzio´c, P. Horodecki, K. ˙Zyczkowski, arXiv:2405.05785 doi:10.48550/arXiv.2405.05785

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.05785

[52] [52]

C. H. Bennett, G. Brassard, C. Cr ´epeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett.70, 1895 (1993) doi: 10.1103/PhysRevLett.70.1895

work page doi:10.1103/physrevlett.70.1895 1993

[53] [53]

Horodecki, P

M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 10 60, 1888 (1999) doi:10.1103/PhysRevA.60.1888

work page doi:10.1103/physreva.60.1888 1999

[54] [54]

Horodecki, M

R. Horodecki, M. Horodecki, and P. Horodecki, Phys. Lett. A 222, 21-25 (1996) doi:10.1016/0375-9601(96)00639-1

work page doi:10.1016/0375-9601(96)00639-1 1996

[55] [55]

Bouwmeester, J

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. We- infurter, and A. Zeilinger, Nature390, 575-579 (1997) doi: 10.1038/37539

work page doi:10.1038/37539 1997

[56] [56]

S. Q. Zhou, H. Jin, J. M. Liang, S. M. Fei, Y . Xiao, and Z. Ma, Phys. Rev. A110, 062429 (2024) doi: 10.1103/PhysRevA.110.062429

work page doi:10.1103/physreva.110.062429 2024

[57] [57]

Streltsov, U

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Phys. Rev. Lett.115, 020403 (2015) doi: 10.1103/PhysRevLett.115.020403

work page doi:10.1103/physrevlett.115.020403 2015

[58] [58]

Watrous,Theory of Quantum Information, (Institute for Quantum Computing, University of Waterloo, Waterloo, Canada, 2011)

J. Watrous,Theory of Quantum Information, (Institute for Quantum Computing, University of Waterloo, Waterloo, Canada, 2011)

work page 2011

[59] [59]

Bhatia,Matrix Analysis, (Springer-Verlag, New York, 1997)

R. Bhatia,Matrix Analysis, (Springer-Verlag, New York, 1997)

work page 1997