The DOI matches its Crossref record; 12 of the 12 references it cites were found in a registry or an index; none of the 4 sources its claims cite is checked for support yet.
Completely positive linear maps on complex matrices
Open accessOverview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
14 No basis given · 2 Sources not found · 2 Own interpretation
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For some improvement, we may require {u:} to be linearly independent, then {Vi} must be linearly independent too.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Q : !JXn,+9Xm is hermitian-preserving iff @ admits an expression @(A) = Zei V;CAV, where ei = + 1, Vi are n X m matrices.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
What the sources say
- Not matchedcites [7, p. 134, Theorem 2.11]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [5, p. 259, Theorem 21]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For a linear map a: 9JIm,-+Izll,, it is obvious that Q is hermitian-preserving iff (@(Eik))jk is hermitian.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“THEOREM 2. Let Cp be a linear map from %!, to %Q,,,.Then @ is completely positive iff (@(Ejk))lG i,k9n is positiue.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let { W,}; be a class of n X m matrices, then Q, has the expression @(A) = X;W:AWp iff there exists an isometric I! ' x E matrix (@Pi, such that Wp = Zipppi Vi for all p.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let Q,: 9.Rm,41Qm,.Then ip is extreme in CP[ZRn, VA!,; K] iff @ admits an expression @(A)=ZiVFAVi for all A in 9X,,, where y are n X m matrices, Xi VF Vi = K, and { y vi} ii is a linearly independent set.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“THEOREM 1 . Let Q, : !J.R,+!&,,.Then CD is completely positive iff Q is of the form ia = ZiVTAV for all A in 92, where Vi are n X m matrices.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Rests on an assumption: Every positive matrix can be expressed as a sum of rank-one positive matrices (vectors).
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“THEOREM 7 . Zf @:!!JL&lR,,, is positive, then there exist 2Xm matrices y such that Q(A) =ZiVi*AVi for all 2X2 symmetric A.”
§ Completely Positive Linear Maps on Complex Matrices > Additional information
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Let F be an mm,-coefficient quadratic form F(s, t) = B,?+ B,st + B,t2 with real indeterminates s, t.If F(s, t) is positive for all s, t, then there exist k X m matrices C, D, such that F(s, t) = (Cs + Dt)*(Cs + Dt).(k is a certain integer.)”
§ Completely Positive Linear Maps on Complex Matrices > Additional information
What the sources say
- Not matchedcites [3,Theorem 21]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [6, Appendix III]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For each n urn matrix V, it is evident that the map: %!,-+m,,, with A + V* A V is completely positive.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- It may state two separate claims as one. 6
- Probably not a substantial claim. 1
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Endowed with the natural ordering induced by !TQ,(!lP,J, the class of hermitian-preserving maps is a partially ordered vector space over the reals, while the class of completely positive linear maps is just the positive cone.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Suppose a: YJ?”+%P, is completely positive.From Remark 4, we can write 0 in the form O(A) = Z,!VTAVi where { Vi} f is a class of linearly independent n X m matrices; hence e < nm.”
§ Completely Positive Linear Maps on Complex Matrices > Additional information
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Moreover, if { W,}; is also a linearly independent set, then F ' = I', and ( ppJpi is unitary.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In the proof of Theorem 1, the expression (@(Ejk))+ = Xi”* q is not unique, hence ( Vi} is not uniquely determined.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is evident that CP[!&, iDi!,; K] is a convex set, hence it is the convex hull of its extreme points.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Each linear map Q, : !IR2,-+'171, is determined by its values on Eik.Hence ip is completely determined by the single element (@(Ejk))l < i, k G n E 9JIm, (TRJ.”
§ Completely Positive Linear Maps on Complex Matrices > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In particular, if m = I, we obtain the well-known fact that @ is a 'pure state' (an extreme identity-preserving positive functional) on YJ$, iff @ is a 'vector state' (i.e., there exists a unit vector u such that Q(A) = v*Av for all A).”
§ Completely Positive Linear Maps on Complex Matrices > Additional information
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper relies on this without support.
Every positive matrix can be expressed as a sum of rank-one positive matrices (vectors).
“As (E&< i,kGn is positive, so P(EjJ)r< j,k(n E !I&, (!l.R,J is positive; thus there exist vectors vi” (regarded as nm x 1 matrices) such that (Q(Ejk))jk =Z,uFvi.”
The paper relies on this without support.
Theorem 7 is not valid for higher order matrices.
“We remark that Theorem 7 is not valid for higher order matrices.This is just because the quoted result for an mm,-coefficient quadratic form cannot be generalized to the case of more than 2 real indeterminates, as will be shown in a forthcoming paper.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“E (t,t 1 ) ⊗ 1 1 (| |) 1 = 1 iff E (t,t 1 ) is CP, > 1 otherwise.”
From Quantum non-Markovianity: characterization, quantification and detection (2014) · cites this paper as [19]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Complete positive maps are much easier to characterize than maps that are merely positive [19,20]; they admit the so-called Kraus representation,”
From Quantum non-Markovianity: characterization, quantification and detection (2014) · cites this paper as [19]
The check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“For this, typical methods used to generate hybrid entangled states using q-plates [46] and static meta-surfaces [47,48] will not suffice, hence we make use of an all-digital SPC approach [49]”
From Non-local skyrmions as topologically resilient quantum entangled states of light (2024) · cites this paper as [48]
Experimental — accuracy not yet measuredIn the citing paper
“for a Hilbert space with finite dimension N S = dim H S complete positivity is equivalent to N S -positivity [13].”
From Foundations and measures of quantum non-Markovianity (2012) · cites this paper as [13]
In the citing paper
“Theorem 5. A linear map : M n → M n is completely positive if and only if ⊗ I M n is positivity-preserving on M n ⊗ M n or equivalently, if and only if the matrix”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [4]
In the citing paper
“Lemma 10.The minimal number of Kraus operators A k needed to represent a completely positive map in the form (1) is rank[β( )].”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [4]
In the citing paper
“Lemma 9. A stochastic map is an extreme point of S if and only if it can be written in the form (1) (necessarily with k A k A † k = I ), so that the set of matrices {A j A † k } is linearly independent.”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [4]
In the citing paper
“After the results were first published, it was shown by Gottesman[32] that fault tolerant procedures for the universal set of gates introduced by Shor can be constructed for any stabilizer code, using basically the same technique as presented by Shor for the CSS codes, a result which was generalized to qupits[33].”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [32]
In the citing paper
“qPCA for quantum states can be extended to quantum processes by using the Choi-Jamiolkowski state (1/d) ij |i j| ⊗ S(|i j|) for a completely positive map S (ref.22).”
From Quantum principal component analysis (2014) · cites this paper as [22]
In the citing paper
“qPCA can then be used to construct the eigenvectors corresponding to the dominant eigenvalues of this state: the resulting spectral decomposition in turn encapsulates many of the most important properties of the channel 22 .”
From Quantum principal component analysis (2014) · cites this paper as [22]
Citations
Cites
Arveson, W. (1969) · Acta Mathematica · doi:10.1007/bf02392388 · Cited by 1,107
Positive functions on 𝐶*-algebras
Stinespring, W. F. (1955) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1955-0069403-4 · Cited by 1,085
Positive Functions on C ∗ -Algebras
Stinespring, W. F. (1955) · Proceedings of the American Mathematical Society · doi:10.2307/2032342 · Cited by 753
Positive Linear Maps of Operator Algebras
Størmer, E. (2012) · Springer monographs in mathematics · doi:10.1007/978-3-642-34369-8 · Cited by 428
Arveson, W. (1972) · Acta Mathematica · doi:10.1007/bf02392166 · Cited by 382
Positive linear maps of operator algebras
Størmer, E. (1963) · Acta Mathematica · doi:10.1007/bf02391860 · Cited by 354
Positive Linear Maps on C*-Algebras
Choi, M.-D. (1972) · Canadian Journal of Mathematics · doi:10.4153/cjm-1972-044-5 · Cited by 239
Self-citationLinear transformations which preserve hermitian and positive semidefinite operators
de Pillis, J. (1967) · Pacific Journal of Mathematics · doi:10.2140/pjm.1967.23.129 · Cited by 188
On subalgebras of $C^*$-algebras
Arveson, W. (1969) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1969-12293-7 · Cited by 144
Synthesis of Finite Passive n-Ports with Prescribed Positive Real Matrices of Several Variables
Koga, T. (1968) · IEEE Transactions on Circuit Theory · doi:10.1109/tct.1968.1082780 · Cited by 91
Cited by
Sorted by citations
Items 301–400 of 2,678
Experimental deterministic correction of qubit loss
Stricker, R., Vodola, D., Erhard, A., Postler, L., Meth, M., Ringbauer, M., Schindler, P., Monz, T., Müller, M., & Blatt, R. (2020) · Nature · doi:10.1038/s41586-020-2667-0 · Cited by 62
Geometric perspective on quantum parameter estimation
Sidhu, J. S., & Kok, P. (2020) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · Cited by 58
Probabilistic theories: What is special about Quantum Mechanics?
D’Ariano, G. M. (2010) · Cambridge University Press eBooks · doi:10.1017/cbo9780511676550.007 · Cited by 59
THE STRUCTURES OF STATE SPACE CONCERNING QUANTUM DYNAMICAL SEMIGROUPS
BAUMGARTNER, B., & Narnhofer, H. (2012) · Reviews in Mathematical Physics · doi:10.1142/s0129055x12500018 · Cited by 75
Structured near-optimal channel-adapted quantum error correction
Fletcher, A. S., Shor, P. W., & Win, M. Z. (2008) · Physical Review A · doi:10.1103/physreva.77.012320 · Cited by 61
Quantum channel construction with circuit quantum electrodynamics
Shen, C., Noh, K., Albert, V. V., Krastanov, S., Devoret, M., Schoelkopf, R. J., Girvin, S. M., & Jiang, L. (2017) · Physical review. B./Physical review. B · doi:10.1103/physrevb.95.134501 · Cited by 62
On a characterization of positive maps
Majewski, W. A., & Marciniak, M. (2001) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/34/29/308 · Cited by 61
Noisy Approach to Intrinsically Mixed-State Topological Order
Sohal, R., & Prem, A. (2025) · PRX Quantum · doi:10.1103/prxquantum.6.010313 · Cited by 89
Gate fidelity fluctuations and quantum process invariants
Magesan, E., Blume-Kohout, R., & Emerson, J. (2011) · Physical Review A · doi:10.1103/physreva.84.012309 · Cited by 60
Entangled states with positive partial transposes arising from indecomposable positive linear maps
Ha, K.-C., Kye, S.-H., & Park, Y. S. (2003) · Physics Letters A · doi:10.1016/s0375-9601(03)00733-3 · Cited by 60
Separable states and positive maps
Størmer, E. (2008) · Journal of Functional Analysis · doi:10.1016/j.jfa.2007.12.017 · Cited by 60
Theoretical framework for higher-order quantum theory
Bisio, A., & Perinotti, P. (2019) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2018.0706 · Cited by 60
Quantum process tomography on vibrational states of atoms in an optical lattice
Myrskog, S. H., Fox, J. K., Mitchell, M. W., & Steinberg, A. M. (2005) · Physical Review A · doi:10.1103/physreva.72.013615 · Cited by 64
Method to Find Quantum Noiseless Subsystems
Choi, M.-D., & Kribs, D. W. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.050501 · Cited by 64
Self-citationPrinciples of Quantum Communication Theory: A Modern ApproachPreprint
Khatri, S., Lami, L., & Wilde, M. M. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2011.04672 · Cited by 297
An Asymptotic Property of Factorizable Completely Positive Maps and the Connes Embedding Problem
Haagerup, U., & Musat, M. (2015) · Communications in Mathematical Physics · doi:10.1007/s00220-015-2325-9 · Cited by 59
Can a quantum state over time resemble a quantum state at a single time?
Horsman, D., Heunen, C., Pusey, M. F., Barrett, J., & Spekkens, R. W. (2017) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2017.0395 · Cited by 62
Quantum Circuits with Classical Versus Quantum Control of Causal Order
Wechs, J., Dourdent, H., Abbott, A. A., & Branciard, C. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.030335 · Cited by 101
SINGULAR VALUE DECOMPOSITION AND MATRIX REORDERINGS IN QUANTUM INFORMATION THEORY
Miszczak, J. A. (2011) · International Journal of Modern Physics C · doi:10.1142/s0129183111016683 · Cited by 60
Control landscapes for two-level open quantum systems
Pechen, A. N., Прохоренко, Д. В., Wu, R.-B., & RABITZ, H. A. (2008) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/41/4/045205 · Cited by 58
Environment-assisted quantum state restoration via weak measurements
Wang, K. C. P., Zhao, X., & Yu, T. (2014) · Physical Review A · doi:10.1103/physreva.89.042320 · Cited by 59
Fleming, C. H., & Hu, B. L. (2012) · Annals of Physics · doi:10.1016/j.aop.2011.12.006 · Cited by 59
Atomic Positive Linear Maps in Matrix Algebras
Ha, K.-C. (1998) · Publications of the Research Institute for Mathematical Sciences · doi:10.2977/prims/1195144425 · Cited by 59
Statistical Methods for Quantum State Verification and Fidelity Estimation
Yu, X., Shang, J., & Gühne, O. (2022) · Advanced Quantum Technologies · doi:10.1002/qute.202100126 · Cited by 58
Cerf, N. J., & Fiurášek, J. (2006) · Progress in optics · doi:10.1016/s0079-6638(06)49006-5 · Cited by 56
Tomography increases key rates of quantum-key-distribution protocols
Watanabe, S., Matsumoto, R., & Uyematsu, T. (2008) · Physical Review A · doi:10.1103/physreva.78.042316 · Cited by 56
Operator dynamics in Lindbladian SYK: a Krylov complexity perspective
Bhattacharjee, B., Nandy, P., & Pathak, T. (2024) · Journal of High Energy Physics · doi:10.1007/jhep01(2024)094 · Cited by 57
Quantum mechanics without state vectors
Weinberg, S. (2014) · Physical Review A · doi:10.1103/physreva.90.042102 · Cited by 57
Quantum circuit cutting with maximum-likelihood tomography
Perlin, M. A., Saleem, Z. H., Suchara, M., & Osborn, J. C. (2021) · npj Quantum Information · doi:10.1038/s41534-021-00390-6 · Cited by 72
Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage
Gao, X., Kalinowski, M., Chou, C.-N., Lukin, M. D., Barak, B., & Choi, S. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.010334 · Cited by 89
Tensor networks and graphical calculus for open quantum systems
Wood, C. J., Biamonte, J., & Cory, D. G. (2015) · Quantum Information and Computation · doi:10.26421/qic15.9-10-3 · Cited by 102
Channel capacity enhancement with indefinite causal order
Loizeau, N., & Grinbaum, A. (2020) · Physical Review A · doi:10.1103/physreva.101.012340 · Cited by 56
Yoshida, T., Kudo, K., Katsura, H., & Hatsugai, Y. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033428 · Cited by 61
Optimal estimation of one-parameter quantum channels
Sarovar, M., & Milburn, G. J. (2006) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/39/26/015 · Cited by 57
Classical shadows for quantum process tomography on near-term quantum computers
Levy, R., Luo, D., & Clark, B. K. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.013029 · Cited by 94
Squashing model for detectors and applications to quantum-key-distribution protocols
Gittsovich, O., Beaudry, N. J., Narasimhachar, V., et al. (2014) · Physical Review A · doi:10.1103/physreva.89.012325 · Cited by 55
Shrapnel, S., Costa, F., & Milburn, G. J. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aabe12 · Cited by 72
Non-Markovian dephasing and depolarizing channels
Utagi, S., Srikanth, R., & Banerjee, S. (2018) · Physical Review A · doi:10.1103/physreva.98.032328 · Cited by 56
Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction
Holbrook, J., Kribs, D. W., & Laflamme, R. (2003) · Quantum Information Processing · doi:10.1023/b:qinp.0000022737.53723.b4 · Cited by 58
Classical interventions in quantum systems. I. The measuring process
Peres, A. (2000) · Physical Review A · doi:10.1103/physreva.61.022116 · Cited by 60
Quantum advantages for Pauli channel estimation
Chen, S., Zhou, S., Seif, A., & Jiang, L. (2022) · Physical Review A · doi:10.1103/physreva.105.032435 · Cited by 71
Fast Quantum State Discrimination with Nonlinear Positive Trace‐Preserving Channels
Geller, M. R. (2023) · Advanced Quantum Technologies · doi:10.1002/qute.202200156 · Cited by 56
Precision-Guaranteed Quantum Tomography
Sugiyama, T., Turner, P. S., & Murao, M. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.160406 · Cited by 54
Experimental Implementation of the Optimal Linear-Optical Controlled Phase Gate
Lemr, K., Černoch, A., Soubusta, J., Kieling, K., Eisert, J., & Dušek, M. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.013602 · Cited by 54
Economical quantum cloning in any dimension
Durt, T., Fiurášek, J., & Cerf, N. J. (2005) · Physical Review A · doi:10.1103/physreva.72.052322 · Cited by 54
Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage
Gao, X., Kalinowski, M., Chou, C.-N., Lukin, M. D., Barak, B., & Choi, S. (2021) · arXiv · doi:10.48550/arxiv.2112.01657 · Cited by 54
Shadow process tomography of quantum channels
Kunjummen, J., Tran, M. C., Carney, D., et al. (2023) · Physical Review A · doi:10.1103/physreva.107.042403 · Cited by 54
Decoding algorithms for surface codes
deMarti iOlius, A., Fuentes, P., Orús, R., Crespo, P. M., & Etxezarreta Martinez, J. (2024) · Quantum · doi:10.22331/q-2024-10-10-1498 · Cited by 60
Non-Markovian Quantum Mpemba Effect
Strachan, D. J., Purkayastha, A., & Clark, S. R. L. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.220403 · Cited by 86
Information geometry of Gaussian channels
Monràs, A., & Illuminati, F. (2010) · Physical Review A · doi:10.1103/physreva.81.062326 · Cited by 52
Piani, M., & Mora, C. E. (2007) · Physical Review A · doi:10.1103/physreva.75.012305 · Cited by 51
Operational quantum theory without predefined time
Oreshkov, O., & Cerf, N. J. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/7/073037 · Cited by 65
Continuous-variable steering and incompatibility via state-channel duality
Kiukas, J., Budroni, C., Uola, R., & Pellonpää, J.-P. (2017) · Physical Review A · doi:10.1103/physreva.96.042331 · Cited by 55
Unextendible maximally entangled bases
Bravyi, S., & Smolin, J. A. (2011) · Physical Review A · doi:10.1103/physreva.84.042306 · Cited by 52
On quantum advantage in dense coding
Horodecki, M., & Piani, M. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/45/10/105306 · Cited by 51
Non-Asymptotic Entanglement Distillation
Fang, K., Wang, X., Tomamichel, M., & Duan, R. (2019) · IEEE Transactions on Information Theory · doi:10.1109/tit.2019.2914688 · Cited by 52
Experimental phase-covariant cloning of polarization states of single photons
Černoch, A., Bartůšková, L., Soubusta, J., Ježek, M., Fiurášek, J., & Dušek, M. (2006) · Physical Review A · doi:10.1103/physreva.74.042327 · Cited by 52
On quantum operations as quantum states
Arrighi, P. J., & Patricot, C. (2003) · Annals of Physics · doi:10.1016/j.aop.2003.11.005 · Cited by 51
From Time-Reversal Symmetry to Quantum Bayes’ Rules
Parzygnat, A. J., & Fullwood, J. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.020334 · Cited by 78
Quantum-coherent mixtures of causal relations
MacLean, J.-P. W., Ried, K., Spekkens, R. W., & Resch, K. J. (2017) · Nature Communications · doi:10.1038/ncomms15149 · Cited by 52
A Class of Linear Positive Maps in Matrix Algebras
Kossakowski, A. (2003) · Open Systems & Information Dynamics · doi:10.1023/a:1025101606680 · Cited by 51
Squeezing the limit: quantum benchmarks for the teleportation and storage of squeezed states
Owari, M., Plenio, M. B., Polzik, E. S., Serafini, A., & Wolf, M. M. (2008) · New Journal of Physics · doi:10.1088/1367-2630/10/11/113014 · Cited by 56
Optimal Counterfeiting Attacks and Generalizations for Wiesner’s Quantum Money
Molina, A., Vidick, T., & Watrous, J. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-35656-8_4 · Cited by 52
On the Choi-Jamiolkowski Correspondence in Infinite DimensionsPreprint
Holevo, A. S. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1004.0196 · Cited by 51
Capacities of repeater-assisted quantum communicationsPreprint
Pirandola, S. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1601.00966 · Cited by 128
MacCormack, I., Tan, M. T., Kudler-Flam, J., & Ryu, S. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.214202 · Cited by 55
Open-system approach to nonequilibrium quantum thermodynamics at arbitrary coupling
Colla, A., & Breuer, H. (2022) · Physical Review A · doi:10.1103/physreva.105.052216 · Cited by 54
Measuring non-Markovianity based on local quantum uncertainty
He, Z., Yao, C., Wang, Q., & Zou, J. (2014) · Physical Review A · doi:10.1103/physreva.90.042101 · Cited by 51
Non-Gaussian operations on bosonic modes of light: Photon-added Gaussian channels
Sabapathy, K. K., & Winter, A. (2017) · Physical Review A · doi:10.1103/physreva.95.062309 · Cited by 49
Operator Scaling: Theory and Applications
Garg, A., Gurvits, L., Oliveira, R., & Wigderson, A. (2019) · Foundations of Computational Mathematics · doi:10.1007/s10208-019-09417-z · Cited by 63
Is there a Floquet Lindbladian?
Schnell, A., Eckardt, A., & Denisov, S. (2020) · Physical review. B./Physical review. B · doi:10.1103/physrevb.101.100301 · Cited by 50
Probabilistic exact universal quantum circuits for transforming unitary operations
Quintino, M. T., Dong, Q., Shimbo, A., Soeda, A., & Murao, M. (2019) · Physical Review A · doi:10.1103/physreva.100.062339 · Cited by 49
Accardi, L., & Fidaleo, F. (2005) · Annali di Matematica Pura ed Applicata (1923 -) · doi:10.1007/s10231-004-0118-4 · Cited by 49
Presence and Absence of Barren Plateaus in Tensor-Network Based Machine Learning
Liu, Z., Yu, L.-W., Duan, L., & Deng, D.-L. (2022) · Physical Review Letters · doi:10.1103/physrevlett.129.270501 · Cited by 56
Quantum complexity in gravity, quantum field theory, and quantum information science
Baiguera, S., Balasubramanian, V., Caputa, P., Chapman, S., Haferkamp, J., Heller, M. P., & Halpern, N. Y. (2025) · Physics Reports · doi:10.1016/j.physrep.2025.11.001 · Cited by 57
Conjugate degradability and the quantum capacity of cloning channels
Brádler, K., Dutil, N., Hayden, P. M., & Muhammad, A. (2010) · Journal of Mathematical Physics · doi:10.1063/1.3449555 · Cited by 51
Practical adaptive quantum tomography
Granade, C., Ferrie, C., & Flammia, S. T. (2017) · New Journal of Physics · doi:10.1088/1367-2630/aa8fe6 · Cited by 60
Korzekwa, K., Czachórski, S., Puchała, Z., & Życzkowski, K. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aaaff3 · Cited by 51
Temporal self-similarity of quantum dynamical maps as a concept of memorylessness
Utagi, S., Srikanth, R., & Banerjee, S. (2020) · Scientific Reports · doi:10.1038/s41598-020-72211-3 · Cited by 51
Quantum Simulation of Dissipative Collective Effects on Noisy Quantum Computers
Cattaneo, M., Rossi, M. A. C., García-Pérez, G., Zambrini, R., & Maniscalco, S. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.010324 · Cited by 54
Quantum computing with and for many-body physics
Ayral, T., Besserve, P., Denis Lacroix, A., & Guzman, E. A. R. (2023) · The European Physical Journal A · doi:10.1140/epja/s10050-023-01141-1 · Cited by 71
Entanglement-annihilating and entanglement-breaking channels
Moravčíková, L., & Ziman, M. (2010) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/43/27/275306 · Cited by 53
Practical adaptive quantum tomography
Granade, C., Ferrie, C., & Flammia, S. T. (2017) · OPUS - Open Publications of UTS Scholars (University of Technology Sydney) · Cited by 47
Measurement-induced purification in large- N hybrid Brownian circuits
Bentsen, G. S., Sahu, S., & Swingle, B. G. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.094304 · Cited by 49
Local copying and local discrimination as a study for nonlocality of a set of states
Owari, M., & Hayashi, M. (2006) · Physical Review A · doi:10.1103/physreva.74.032108 · Cited by 46
A survey of canonical forms and invariants for unitary similarity
Shapiro, H. M. (1991) · Linear Algebra and its Applications · doi:10.1016/0024-3795(91)90232-l · Cited by 48
Operator-Valued Measures, Dilations, and the Theory of FramesPreprint
Han, D., Larson, D. R., Liu, B., et al. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1110.5833 · Cited by 48
Work Extraction Processes from Noisy Quantum Batteries: The Role of Nonlocal Resources
Tirone, S., Salvia, R., Chessa, S., et al. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.060402 · Cited by 47
Dilation theory in finite dimensions: The possible, the impossible and the unknown
Levy, E., & Shalit, O. M. (2014) · Rocky Mountain Journal of Mathematics · doi:10.1216/rmj-2014-44-1-203 · Cited by 57
Quantum Capacity under Adversarial Quantum Noise: Arbitrarily Varying Quantum Channels
Ahlswede, R., Bjelaković, I., Boche, H., & Nötzel, J. (2012) · Communications in Mathematical Physics · doi:10.1007/s00220-012-1613-x · Cited by 61
Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling
Garg, A., Gurvits, L., Oliveira, R., et al. (2018) · Geometric and Functional Analysis · doi:10.1007/s00039-018-0434-2 · Cited by 47
Conformal field theory and the web of quantum chaos diagnostics
Kudler-Flam, J., Nie, L., & Ryu, S. (2020) · Journal of High Energy Physics · doi:10.1007/jhep01(2020)175 · Cited by 67
Incompatibility in quantum parameter estimation
Belliardo, F., & Giovannetti, V. (2021) · New Journal of Physics · doi:10.1088/1367-2630/ac04ca · Cited by 54
Evolution in quantum causal histories
Hawkins, E., Markopoulou, F., & Sahlmann, H. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/16/320 · Cited by 46
Flowing the Information from Shannon to Fisher: Towards the Fundamental Tradeoff in ISAC
Xiong, Y., Liu, F., Cui, Y., Yuan, W., & Han, T. X. (2022) · Globecom · doi:10.1109/globecom48099.2022.10001144 · Cited by 46
Constructing optimal entanglement witnesses
Chruściński, D., Pytel, J., & Sarbicki, G. (2009) · Physical Review A · doi:10.1103/physreva.80.062314 · Cited by 46
Restricted numerical range: A versatile tool in the theory of quantum information
Gawron, P., Puchała, Z., Miszczak, J. A., Skowronek, Ł., & Życzkowski, K. (2010) · Journal of Mathematical Physics · doi:10.1063/1.3496901 · Cited by 49
Davies maps for qubits and qutrits
Roga, W., Fannes, M., & Życzkowski, K. (2010) · Reports on Mathematical Physics · doi:10.1016/s0034-4877(11)00003-6 · Cited by 46
Two-Message Quantum Interactive Proofs Are in PSPACE
Jain, R., Upadhyay, S., & Watrous, J. (2009) · Annual Symposium on Foundations of Computer Science · doi:10.1109/focs.2009.30 · Cited by 57
High-Dimensional Entanglement in States with Positive Partial Transposition
Huber, M., Lami, L., Lancien, C., & Müller‐Hermes, A. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.200503 · Cited by 46
Metadata
- Authors
- Choi, M.-D.
- Affiliations
- University of California, Berkeley (US)
- Corresponding author
- Choi, Man-Duen
- Venue
- Linear Algebra and its Applications
- Publisher
- Elsevier BV
- Published
- 1975-06
- Volume
- 10
- Issue
- 3
- Pages
- 285-290
- ISSN
- 0024-3795 (ISSN-L)
- OpenAlex
- W2116102644
- Cited by
- 2,659
- References
- 12
- Citations by year
- 2026: 122 · 2025: 216 · 2024: 189 · 2023: 205 · 2022: 161 · 2021: 167 · 2020: 132 · 2019: 152 · 2018: 112 · 2017: 98 · 2016: 95 · 2015: 100 · 2014: 93 · 2013: 90 · 2012: 97
- FWCI
- 4.55
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Linear Algebra and its Applications (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Keywords
- complex matrices, matrix algebra, positive linear maps, operator algebras
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-11, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-11 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
