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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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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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]
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plaindirect- Asserted: passes — The authors put it forward as true.
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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.
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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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“is a faithful representation of Φ1 (known as the Choi representation, according to the Choi-Jamiołkowski isomorphism [50], [51]), where Eij is an M ×M matrix with its (i, j)-th entry equals to 1, while all other entries are zero.”
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“It has long been known that the trace preserving completely positive linear maps Φ over a D-dimensional vector space can be placed in a one-to-one correspondence with density operators on a D2-dimensional space via the relation [79, 99, 100] Υ = I ⊗ Φ(|ψME〉〈ψME|) (116)”
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Quantum channels and memory effects
Caruso, F., Giovannetti, V., Lupo, C., & Mancini, S. (2014) · Reviews of Modern Physics · doi:10.1103/revmodphys.86.1203 · Cited by 349
“The CPTP conditions are also equivalent to the possibility of expressing Φ in operator sum (or Kraus) representation (Choi, 1975; Kraus, 1971), Φ(ρQ) = ∑ j KjρQK † j , (8) with {Kj} being operators on HQ satisfying the normalization condition ∑ j K † jKj = 1 .”
BackgroundMatrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs
Scherer, C. W., & Hol, C. W. J. (2005) · Mathematical Programming · doi:10.1007/s10107-005-0684-2 · Cited by 327
“Since the trace operator as a mapping from Rp×p into R is completely positive [8], we obtain trp((Ip ⊗D)A(Ip ⊗D)) < 0.”
Transforming quantum operations: Quantum supermaps
Chiribella, G., D’Ariano, G. M., & Perinotti, P. (2008) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/83/30004 · Cited by 322
LMI Relaxations in Robust Control
Scherer, C. W. (2006) · European Journal of Control · doi:10.3166/ejc.12.3-29 · Cited by 319
Practical Characterization of Quantum Devices without Tomography
Silva, M. P. D., Landon-Cardinal, O., & Poulin, D. (2011) · Physical Review Letters · doi:10.1103/physrevlett.107.210404 · Cited by 340
Majorization, doubly stochastic matrices, and comparison of eigenvalues
Ando, T. (1989) · Linear Algebra and its Applications · doi:10.1016/0024-3795(89)90580-6 · Cited by 305
Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps
Shabani, A., & Lidar, D. A. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.100402 · Cited by 309
Quantum Machine Learning: What Quantum Computing Means to Data Mining
Wittek, P. (2014) · Cited by 319
Classical complexity and quantum entanglement
Gurvits, L. (2004) · Journal of Computer and System Sciences · doi:10.1016/j.jcss.2004.06.003 · Cited by 306
“This question is related to lower bounds on Cap(T ) provided that Choi's representation CH (T ) is an integer semidefinite matrix.”
BackgroundInfluentialNon-Gaussian Quantum States and Where to Find Them
Walschaers, M. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.030204 · Cited by 352
Self-consistent quantum process tomography
Merkel, S., Gambetta, J., Smolin, J. A., Poletto, S., Corcoles, A. D., Johnson, B., Ryan, C. A., & Steffen, M. (2013) · Physical Review A · doi:10.1103/physreva.87.062119 · Cited by 308
Wolf, M. M., & Cirac, J. I. (2008) · Communications in Mathematical Physics · doi:10.1007/s00220-008-0411-y · Cited by 288
Smolin, J. A., Gambetta, J., & Smith, G. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.070502 · Cited by 289
“Due to the Choi-Jamiolkowski isomorphism [12, 13] such a map can be represented by a density matrix.”
MethodThe entanglement of purification
Terhal, B. M., Horodecki, M., Leung, D., & Divincenzo, D. P. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1498001 · Cited by 270
Quantum Conditional Mutual Information and Approximate Markov Chains
Fawzi, O., & Renner, R. (2015) · Communications in Mathematical Physics · doi:10.1007/s00220-015-2466-x · Cited by 288
“Even though we do not need it here, it is worth pointing out that, by virtue of the Choi–Jamiołkowski isomorphism [12,34], the claim above can be rephrased in terms of completely positive trace-preserving maps.”
BackgroundExperimental verification of an indefinite causal order
Rubino, G., Rozema, L. A., Feix, A., Araújo, M., Zeuner, J., Procopio, L. M., Brukner, Č., & Walther, P. J. (2017) · Science Advances · doi:10.1126/sciadv.1602589 · Cited by 295
“With this in mind, the probability of obtaining the outcomesMa;x,M B b;y, and D ðoutÞ d , with the input state rðinÞ z ; can be written, using the Choi-Jamiołkowski isomorphism (16) (see the Supplementary Materials), as”
MethodPositive semidefinite biquadratic forms
Choi, M.-D. (1975) · Linear Algebra and its Applications · doi:10.1016/0024-3795(75)90058-0 · Cited by 258
Self-citationPerfect discrimination of no-signalling channels via quantum superposition of causal structures
Chiribella, G. (2012) · Physical Review A · doi:10.1103/physreva.86.040301 · Cited by 253
QuEST and High Performance Simulation of Quantum Computers
Jones, T., Brown, A., Bush, I., & Benjamin, S. C. (2019) · Scientific Reports · doi:10.1038/s41598-019-47174-9 · Cited by 275
Salles, A., de Melo, F., de Almeida, M. P., Hor-Meyll, M. O., Walborn, S. P., Ribeiro, P. H. S., & Davidovich, L. (2008) · Physical Review A · doi:10.1103/physreva.78.022322 · Cited by 256
Geometric perspective on quantum parameter estimation
Sidhu, J. S., & Kok, P. (2020) · AVS Quantum Science · doi:10.1116/1.5119961 · Cited by 282
Quantum Data HidingPreprint
Divincenzo, D. P., Leung, D., & Terhal, B. M. (2001) · Cited by 282
Strong NP-hardness of the quantum separability problem
Gharibian, S. (2010) · Quantum Information and Computation · doi:10.26421/qic10.3-4-11 · Cited by 266
“If and only if Φ is CP, then J(Φ) is positive semidefinite [39], and if and only if Φ is TP, then TrA(J(Φ)) = I/MN [16].”
BackgroundTwo-photon interference: the Hong–Ou–Mandel effect
Bouchard, F., Sit, A., Zhang, Y., Fickler, R., Miatto, F. M., Yao, Y., Sciarrino, F., & Karimi, E. (2021) · Reports on Progress in Physics · doi:10.1088/1361-6633/abcd7a · Cited by 232
“Furthermore, in light of the Choi–Jamiołkowski isomorphism which allows us to associate a bipartite state to a quantum channel [136, 137], generating the desired quantum states also has consequences on our ability to implement full quantum channels through the technique of gate teleportation.”
BackgroundQuantum discord and its allies: a review of recent progress
Bera, A., Das, T., Sadhukhan, D., Roy, S. S., Sen, A., & Sen, U. (2017) · Reports on Progress in Physics · doi:10.1088/1361-6633/aa872f · Cited by 229
“24 520 [592] Choi M D 1975 Completely positive linear maps on Complex Matrices Linear Algebra Appl.”
BackgroundFoundations and measures of quantum non-Markovianity
Breuer, H.-P. (2012) · Journal of Physics B: Atomic, Molecular and Optical Physics · doi:10.1088/0953-4075/45/15/154001 · Cited by 225
Quantum circuits for isometries
Iten, R., Colbeck, R., Kukuljan, I., Home, J., & Christandl, M. (2016) · Physical Review A · doi:10.1103/physreva.93.032318 · Cited by 218
Completely positive maps and classical correlations
Rodríguez-Rosario, C. A., Modi, K. K., Kuah, A.-M., Shaji, A., & Sudarshan, E. C. G. (2008) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/41/20/205301 · Cited by 212
“Any map that can be written in this form is completely positive [15, 16].”
BackgroundEntanglement witnesses: construction, analysis and classification
Chruściński, D., & Sarbicki, G. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/47/48/483001 · Cited by 231
“Choi maps Φ[1, 1, 0] and Φ[1, 0, 1] provided first examples of positive indecomposable maps [22].”
BackgroundInfluentialQuantum Common Causes and Quantum Causal Models
Allen, J.-M. A., Barrett, J., Horsman, D. C., Lee, C. M., & Spekkens, R. W. (2017) · Physical Review X · doi:10.1103/physrevx.7.031021 · Cited by 252
“An alternative way to express the channel EBjA is as an operator, using a variant of the Choi-Jamiołkowski isomorphism [19,20]:”
MethodFuller, A. H., Hartz, M., & Lupini, M. (2026) · Journal of Operator Theory · doi:10.7900/jot · Cited by 210
Non-Markovian Dynamical Maps: Numerical Processing of Open Quantum Trajectories
Cerrillo, J., & Cao, J. (2014) · Physical Review Letters · doi:10.1103/physrevlett.112.110401 · Cited by 205
Realization of quantum process tomography in NMR
Childs, A. M., Chuang, I. L., & Leung, D. W. (2001) · Physical Review A · doi:10.1103/physreva.64.012314 · Cited by 205
Non-Markovianity-Assisted Steady State Entanglement
Huelga, S. F., Rivas, Á., & Plenio, M. B. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.160402 · Cited by 203
The uniqueness theorem for entanglement measures
Donald, M. J., Horodecki, M., & Rudolph, O. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1495917 · Cited by 204
“Indeed, it might not even be required that the measure is defined except on pure states.”
BackgroundOn quantum fidelities and channel capacities
Barnum, H., Knill, E., & Nielsen, M. A. (2000) · IEEE Transactions on Information Theory · doi:10.1109/18.850671 · Cited by 192
“Hilbert space Hc, to the space B(Ho) of bounded linear operators on an output Hilbert space Ho [25],[26],[27].”
BackgroundTangent-space methods for uniform matrix product states
Vanderstraeten, L., Haegeman, J., & Verstraete, F. (2019) · SciPost Physics Lecture Notes · doi:10.21468/scipostphyslectnotes.7 · Cited by 233
“From its definition it follows that the transfer matrix is a completely positive map [9], where the MPS matrices As play the role of Kraus operators.”
BackgroundAn introductory review of the resource theory approach to thermodynamics
Lostaglio, M. (2019) · Reports on Progress in Physics · doi:10.1088/1361-6633/ab46e5 · Cited by 218
“The complete positivity of E is equivalent to the positivity of the ChoiJamiołkowski state J [E ] := E ⊗ I(Φ+), where Φ+ is the maximally entangled state (Φ+ = |φ+〉〈φ+|, |φ+〉 ∝ ∑ i |ii〉) [100, 101].”
Witnessing causal nonseparability
Araújo, M., Branciard, C., Costa, F., Feix, A., Giarmatzi, C., & Brukner, Č. (2015) · New Journal of Physics · doi:10.1088/1367-2630/17/10/102001 · Cited by 238
“According to Choi’s theorem [36], a linear map MA : AI → AO is CP if and only if its CJ matrix is positive semidefinite, MAI AO ≥ 0.”
BackgroundDanos, V., Kashefi, E., & Panangaden, P. (2007) · Journal of the ACM · doi:10.1145/1219092.1219096 · Cited by 195
“The important result in this regard is the Kraus representation theorem [Cho75].”
BackgroundNielsen, E., Gamble, J. K., Rudinger, K., Scholten, T. L., Young, K., & Blume-Kohout, R. (2021) · Quantum · doi:10.22331/q-2021-10-05-557 · Cited by 260
Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems
Werner, A. H., Jaschke, D., Silvi, P., Kliesch, M., Calarco, T., Eisert, J., & Montangero, S. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.237201 · Cited by 211
On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras
Landau, L., & Streater, R. F. (1993) · Linear Algebra and its Applications · doi:10.1016/0024-3795(93)90274-r · Cited by 189
Everything you always wanted to know about LOCC (but were afraid to askPreprint
Chitambar, E., Leung, D., Ozols, M., & Winter, A. (2016) · Cited by 229
Markovianity and non-Markovianity in quantum and classical systems
Vacchini, B., Smirne, A., Laine, E.-M., Piilo, J., & Breuer, H.-P. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/9/093004 · Cited by 198
Fundamental properties of Tsallis relative entropy
Furuichi, S., Yanagi, K., & Kuriyama, K. (2004) · Journal of Mathematical Physics · doi:10.1063/1.1805729 · Cited by 183
“If A and B are density operators, then the nonnegativity of the quantum Tsallis relative entropy follows from Theorem 3.3.”
BackgroundInfluentialQuantum Theory: Informational Foundations and Foils
(2016) · Fundamental Theories of Physics · doi:10.1007/978-94-017-7303-4 · Cited by 178
“In quantum theory, the state-transformation isomorphism concides with the Choi isomorphism [63].”
Completely positive post-Markovian master equation via a measurement approach
Shabani, A., & Lidar, D. A. (2005) · Physical Review A · doi:10.1103/physreva.71.020101 · Cited by 174
Quantum channels and their entropic characteristics
Holevo, A. S., & Giovannetti, V. (2012) · Reports on Progress in Physics · doi:10.1088/0034-4885/75/4/046001 · Cited by 178
Quantum dynamics as a physical resource
Nielsen, M. A., Dawson, C. M., Dodd, J. L., Gilchrist, A., Mortimer, D., Osborne, T. J., Bremner, M. J., Harrow, A. W., & Hines, A. P. (2003) · Physical Review A · doi:10.1103/physreva.67.052301 · Cited by 181
Quantum fidelity measures for mixed states
Liang, Y.-C., Yeh, Y.-H., Mendonça, P. E. M. F., Teh, R. Y., Reid, M. D., & Drummond, P. D. (2019) · Reports on Progress in Physics · doi:10.1088/1361-6633/ab1ca4 · Cited by 203
“At first, this seems rather different to density matrices, as discussed throughout this review Yet it is easy to show via the Choi-Jamiolkowski isomorphism [136, 137], that one can define a new quantum “density matrix” [22] on the enlarged Hilbert space of dimension d(2), such that ρE = P/d.”
BackgroundQuantum Stochastic Processes and Quantum non-Markovian Phenomena
Milz, S., & Modi, K. K. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.030201 · Cited by 212
Havel, T. F. (2003) · Journal of Mathematical Physics · doi:10.1063/1.1518555 · Cited by 169
Non-local skyrmions as topologically resilient quantum entangled states of light
Ornelas, P., Nape, I., de Mello Koch, R., & Forbes, A. (2024) · Nature Photonics · doi:10.1038/s41566-023-01360-4 · Cited by 167
Monotone Riemannian metrics and relative entropy on noncommutative probability spaces
Lesniewski, A., & Ruskai, M. B. (1999) · Journal of Mathematical Physics · doi:10.1063/1.533053 · Cited by 156
“General conditions can be obtained from the Stinespring representation [31] for completely positive maps or the subsequent work of Choi [ 8 ] and Kraus [14] who showed that φ is a completely positive if and only if there exist operators {Vk} with Vk : A1 → A2 such that”
BackgroundA Postquantum Theory of Classical Gravity?
Oppenheim, J. (2023) · Physical Review X · doi:10.1103/physrevx.13.041040 · Cited by 243
Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code
Noh, K., & Chamberland, C. (2020) · Physical Review A · doi:10.1103/physreva.101.012316 · Cited by 164
A fibre bundle over manifolds of quantum channels and its application to quantum statistics
Fujiwara, A., & Imai, H. (2008) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8113/41/25/255304 · Cited by 163
Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
Carlen, E. A., & Maas, J. (2017) · Journal of Functional Analysis · doi:10.1016/j.jfa.2017.05.003 · Cited by 165
“The Choi matrix of K is the element of Mn2(C) C(K ) = n∑ i,j=1 K (Ei,j)⊗ Ei,j , (A.8) viewed as the n× n block matrix with entries in Mn(C) whose i, j entry is K (Ei,j).”
BackgroundInfluential
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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