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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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- Rests on an assumption: Every positive matrix can be expressed as a sum of rank-one positive matrices (vectors).
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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
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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.
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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.
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- 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.
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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.
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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.
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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.
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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.
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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.
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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.
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- 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]
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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
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