Completely positive linear maps on complex matrices
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- Findingreasoning
If {u_i*} is linearly independent, then {V_i} is linearly independent.
“For some improvement, we may require {u:} to be linearly independent, then {Vi} must be linearly independent too.”
Quote in the paperSubstantial claim - Findingcitation
A linear map Φ: M_m → M_n is hermitian-preserving iff it admits an expression Φ(A)=Σ ε_i V_i* A V_i where ε_i=±1 and V_i are n×m matrices.
“Q : !JXn,+9Xm is hermitian-preserving iff @ admits an expression @(A) = Zei V;CAV, where ei = + 1, Vi are n X m matrices.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [7, p. 134, Theorem 2.11]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [5, p. 259, Theorem 21]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Findingreasoning
A linear map Φ: M_m → M_n is hermitian-preserving iff (Φ(E_ik))_{jk} is hermitian.
“For a linear map a: 9JIm,-+Izll,, it is obvious that Q is hermitian-preserving iff (@(Eik))jk is hermitian.”
Quote in the paperNot a substantial claim - Findingreasoning
A linear map Φ: M_m → M_n is completely positive iff the block matrix (Φ(E_jk))_{1≤j,k≤n} is positive.
“THEOREM 2. Let Cp be a linear map from %!, to %Q,,,.Then @ is completely positive iff (@(Ejk))lG i,k9n is positiue.”
Quote in the paperSubstantial claim - Findingreasoning
Let {W_p} be n×m matrices. Φ has the expression Φ(A)=Σ W_p* A W_p iff there exists an isometric Γ'×Γ matrix (β_{pi}) such that W_p=Σ_i β_{pi} V_i for all p.
“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.”
Quote in the paperSubstantial claim
- Findingnone given
Let Φ: M_m → M_n. Then Φ is extreme in CP[M_m,M_n;K] iff Φ admits an expression Φ(A)=Σ V_i* A V_i for all A in M_m, where V_i are n×m matrices, Σ V_i* V_i = K, and {V_i* V_j}_{ij} is a linearly independent set.
“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.”
Quote in the paperNot a substantial claim - Findingreasoning
A linear map Φ: M_m → M_n is completely positive iff there exist n×m matrices V_i such that Φ(A)=Σ V_i* A V_i for all A in M_m.
“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.”
Quote in the paperSubstantial claim- Rests on an assumption: Every positive matrix can be expressed as a sum of rank-one positive matrices (vectors).
- Findingreasoning
If Φ: M_2 → M_n is positive, then there exist n×2 matrices V_i such that Φ(A)=Σ V_i* A V_i for all 2×2 symmetric A.
“THEOREM 7 . Zf @:!!JL&lR,,, is positive, then there exist 2Xm matrices y such that Q(A) =ZiVi*AVi for all 2X2 symmetric A.”
Quote in the paperSubstantial claim - Premisecitation
If an m×m-coefficient quadratic form F(s,t)=B1 s^2 + B2 st + B3 t^2 with real indeterminates s,t is positive for all s,t, then there exist k×m matrices C,D such that F(s,t)=(Cs+Dt)*(Cs+Dt).
“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.)”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [3,Theorem 21]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [6, Appendix III]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Premisenone given
For each n×m matrix V, the map A ↦ V* A V from M_m to M_n is completely positive.
“For each n urn matrix V, it is evident that the map: %!,-+m,,, with A + V* A V is completely positive.”
Quote in the paperSubstantial claim - Interpretationreasoning
If Φ is extreme in CP[M_m,M_n;K], then l can be reduced to ≤ m.
“In case Cp is extreme in CP[9JI,,,%!Sl,;K], 1 can be reduced to < m.”
Quote in the paperSubstantial claim
- May combine two claims. 6
- Classified as probably not a substantial claim. 1
May combine two claims.
FindingreasoningThe class of hermitian-preserving maps is a partially ordered vector space over the reals, and the class of completely positive linear maps is its positive cone.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningIf Φ: M_m → M_n is completely positive, it can be written as Φ(A)=Σ V_i* A V_i where {V_i} is a linearly independent class of n×m matrices; hence l ≤ nm.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningMoreover, if {W_p} is also a linearly independent set, then Γ'=Γ and (β_{pi}) is unitary.
“Moreover, if { W,}; is also a linearly independent set, then F ' = I', and ( ppJpi is unitary.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningIn the proof of Theorem 1, the expression (Φ(E_jk))^+ = Σ u_i* u_i is not unique, hence {V_i} is not uniquely determined.
“In the proof of Theorem 1, the expression (@(Ejk))+ = Xi”* q is not unique, hence ( Vi} is not uniquely determined.”
Quote in the paperSubstantial claimMay combine two claims.
PremisereasoningCP[M_m,M_n;K] is convex and is the convex hull of its extreme points.
“It is evident that CP[!&, iDi!,; K] is a convex set, hence it is the convex hull of its extreme points.”
Quote in the paperSubstantial claimMay combine two claims.
Premisenone givenEach linear map Φ: M_m → M_n is determined by its values on E_ik; hence Φ is determined by the single element (Φ(E_jk))_{j,k}.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningIn particular, if m=1, Φ is a pure state (an extreme identity-preserving positive functional) on M_n iff Φ is a vector state (there exists a unit vector u such that Φ(A)=u* A u for all A).
“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).”
Quote in the paperNot a substantial claim
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 trace norm of the Choi-Jamiołkowski matrix of E(t,t1) equals 1 if E(t,t1) is completely positive and is greater than 1 otherwise.
“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]
Theorem 2 (in the copy’s Abstract section; page not provided) states that a map is completely positive if and only if its associated matrix is positive. The copy does not state the claimed trace-norm criterion or establish the normalization needed for a threshold of 1. (LLM, conf=0.86)
Experimental — accuracy not yet measuredCompletely positive maps are easier to characterize than merely positive maps and have a Kraus representation.
“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]
For a Hilbert space with finite dimension N_S = dim H_S, complete positivity is equivalent to N_S-positivity.
“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]
Choi's theorem states that a linear map on Mn is completely positive exactly when its amplification by the identity on Mn is positivity-preserving, equivalently when the block matrix (20) is positive semidefinite.
“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]
The minimal number of Kraus operators needed to represent a completely positive map in form (1) equals rank[β( )].
“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]
A stochastic map is an extreme point of S exactly when it can be written in Kraus form with the identity trace-preserving condition and the set of products Aj Ak† is linearly independent.
“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]
Gottesman showed 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 Shor for CSS codes, generalized to qupits.
“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]
The qPCA method for states extends to quantum processes through the Choi-Jamiolkowski state.
“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]
qPCA can build the eigenvectors for the dominant eigenvalues of the Choi-Jamiolkowski state, and the resulting spectral decomposition captures many key channel properties.
“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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“The Choi representation of maps is so-named for Choi’s 1975 paper [50] characterizing completely positive maps (as represented by the equivalence of statements 1 and 3 in Theorem 2.”
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Entanglement Breaking Channels
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“E) Υ ◦ Φ is completely positive for all positivity preserving maps Υ.”
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“This is easiest to do making use of the Choi-Jamiolkowski isomorphism (Choi, 1975; Jamiołkowski, 1972) which states that with each quantum channel, Λ : L(Hin) → L(Hout), we can associate a positive operator ΩΛ ∈ L(Hout ⊗ Hin), so that ΩΛ = (Λ ⊗ I)(|I〉〈I|), where |I〉 = ∑dim(Hin) i=1 |i〉|i〉 is a…”
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MethodDagger Compact Closed Categories and Completely Positive Maps
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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.”
MethodQuantum Mechanics as Quantum Information (and only a little more)Preprint
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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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“Further, any map N that preserves Hermiticity can be written as the difference of two completely positive maps N = N 1−N 2 [82].”
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“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 .”
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“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.”
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Shabani, A., & Lidar, D. A. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.100402 · Cited by 309 · see citing works
Quantum Machine Learning: What Quantum Computing Means to Data Mining
Wittek, P. (2014) · Cited by 319 · see citing works
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 · see citing works
“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 311 · see citing works
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 295 · see citing works
Wolf, M. M., & Cirac, J. I. (2008) · Communications in Mathematical Physics · doi:10.1007/s00220-008-0411-y · Cited by 288 · see citing works
Smolin, J. A., Gambetta, J., & Smith, G. (2012) · Physical Review Letters · doi:10.1103/physrevlett.108.070502 · Cited by 271 · see citing works
“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 · see citing works
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 271 · see citing works
“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 259 · see citing works
“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 · see citing works
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 · see citing works
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 · see citing works
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 251 · see citing works
Geometric perspective on quantum parameter estimation
Sidhu, J. S., & Kok, P. (2020) · AVS Quantum Science · doi:10.1116/1.5119961 · Cited by 253 · see citing works
Quantum Data HidingPreprint
Divincenzo, D. P., Leung, D., & Terhal, B. M. (2001) · Cited by 282 · see citing works
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 · see citing works
“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].”
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 221 · see citing works
“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 · see citing works
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 · see citing works
Two-photon interference: the Hong–Ou–Mandel effect
Bouchard, F., Sit, A., Zhang, Y., Fickler, R., Miatto, F. M., Yao, Y., Sciarrino, F., & Karimi, E. (2020) · Reports on Progress in Physics · doi:10.1088/1361-6633/abcd7a · Cited by 217 · see citing works
“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.”
BackgroundCompletely 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 · see citing works
“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 216 · see citing works
“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 214 · see citing works
“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 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
“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 · see citing works
“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 · see citing works
“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 201 · see citing works
“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., et al. (2015) · New Journal of Physics · doi:10.1088/1367-2630/17/10/102001 · Cited by 197 · see citing works
“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 · see citing works
“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 212 · see citing works
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 184 · see citing works
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 · see citing works
Everything you always wanted to know about LOCC (but were afraid to askPreprint
Chitambar, E., Leung, D., Ozols, M., & Winter, A. (2016) · Cited by 182 · see citing works
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 · see citing works
Fundamental properties of Tsallis relative entropy
Furuichi, S., Yanagi, K., & Kuriyama, K. (2004) · Journal of Mathematical Physics · doi:10.1063/1.1805729 · Cited by 178 · see citing works
“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 · see citing works
“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 176 · see citing works
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 170 · see citing works
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 173 · see citing works
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 177 · see citing works
“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 183 · see citing works
Havel, T. F. (2003) · Journal of Mathematical Physics · doi:10.1063/1.1518555 · Cited by 169 · see citing works
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 · see citing works
“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 200 · see citing works
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 · see citing works
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 · see citing works
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 · see citing works
“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).”
BackgroundInfluentialLagrangian representation for fermionic linear optics
Bravyi, S. (2005) · Quantum Information and Computation · doi:10.26421/qic5.3-3 · Cited by 162 · see citing works
- 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 › Computer Science › Artificial Intelligence › 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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