The DOI matches its Crossref record; 36 of the 36 references it cites were found in a registry or an index; none of the 22 sources its claims cite is checked for support yet.
An analysis of completely-positive trace-preserving maps on M2
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“We give a useful new characterization of the set of all completely positive, trace-preserving maps : M 2 → M 2 from which one can easily check any trace-preserving map for complete positivity.”
§ An analysis of completely positive trace-preserving maps on M 2 > Abstract
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“Conditions on the diagonal elements and determinant of I -R † R suffice to insure R is a contraction.”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“The reformulation of the conditions of Theorem 4 as ( 15) and ( 16) implies that when is in the closure of the extreme points, the matrix in (3) can be written in the useful form”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.4. Trigonometric parameterization
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“The requirement that the diagonal elements of R † R and R R † must be 1 then implies that”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“Theorem 4. A map belongs to the closure of the set of extreme points of S if and only if it can be reduced to the form (3) so that at most one t k is non-zero and (with the convention this is t 3 ) (λ 1 ± λ 2 ) 2 = (1 ± λ 3 ) 2 -t 2 3 .”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“When |t 3 | + |λ 3 | = 1 no solution to (6) exists unless t 1 = t 2 = 0 and either λ 1 = λ 2 or λ 1 = -λ 2 .”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“Moreover, as will be shown in Section 2.3, that “equivalence” preserves the property of being a generalized extreme point of S.”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“We will see that for non-unital maps the extreme points correspond to those maps for which the translation allows the ellipsoid to touch the boundary of the sphere at two points”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
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“Moreover, is extreme unless t 3 = 0 and |λ 3 | < 1.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“Thus, in particular, a necessary condition for to be an extreme point of S is that it is equivalent [in the sense of (4)] to a map of the form (3) for which the corresponding R is unitary.”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“If λ 3 = 0 and |t 3 | + |λ 3 | = 1, then |t 3 | = 1 in which case will not be completely positive unless t 1 = t 2 = λ 1 = λ 2 = 0; then it is consistent and sufficient to interpret”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“Note that a trace-preserving map is positivity-preserving if and only if it maps the Bloch sphere into the “Bloch ball”, defined as the Bloch sphere plus its interior.”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
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“Adding and subtracting conditions (14) yields the equivalent conditions λ 3 = λ 1 λ 2 , (15 ) t 2 3 = (1 -λ 2 1 )(1 -λ 2 2 ).”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“This implies that the Algoet-Fujiwara condition [1] (λ 1 ± λ 2 ) 2 (1 ± λ 3 ) 2 -t 2 3 (11 )”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
What the sources say
- Not checked yetcites [1] A. Fujiwara, P. Algoet, Affine parameterization of completely positive maps on a matrix algebra, preprint; A. Fujiwara, P. Algoet, One-to-one parameterization of quantum channels, Phys. Rev. A 59 (1999) 3290–3294 source
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“The image of the Bloch sphere of pure states under a map of the form (3) is the ellipsoid”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
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“and give a useful parameterization after reduction to a certain canonical form.”
§ An analysis of completely positive trace-preserving maps on M 2 > Abstract
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“In addition to a condition on the diagonal, the requirement R † R I also implies that det(I -R † R ) 0. This leads to the condition”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“Moreover, the capacity is, in general, achieved neither with a pair of orthogonal input states, which would yield h(sin u sin v)h( 1sin 2 u cos 2 v), nor with a pair of minimal entropy states, which would yield h(sin u/sin v), where”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.4. Channel capacity
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“In fact, the non-zero Kraus operator is simply one of {I, σ x , σ y , σ z } and accordingly, either none or exactly two of the λ k can be negative.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“The expression log 2h(sin u) is the capacity of the unshifted channel.In this case, it is also the classical Shannon capacity.”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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“For non-unital maps, not only do two extreme points suffice, but they can be chosen so that an arbitrary non-unital map is the”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“In this case, is unitary and obviously extreme, and λ k = ±1 for k = 1, 2, 3.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“Similarly, Lemma 10 shows that (iv) ⇔ (v).”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
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“Although this tetrahedron lies in a three-dimensional space in a fixed basis, an arbitrary bistochastic map requires nine parameters that could be chosen either as the nine elements of the real 3 × 3 matrix T, or as the three λ k and two sets of Euler angles corresponding to the two rotations required to reduce to diagonal form.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
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“This is possible if, in addition, λ 3 and λ 1 λ 2 have the same sign, and t 1 = t 2 = 0.Under these conditions R , is always a contraction, but is unitary only for |λ 3 | = µ 2 .”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“Concerning the other conditions in the general case, Lemma 8 says that (ii) ⇔ rank β( ) 2, while it follows from (25) that rank β( ) = rank β( ); thus (ii) ⇔ (v).”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
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“If a bistochastic map is written in diagonal form, then it has a unique decomposition as a convex combination of the four unitary maps corresponding to the corners of a tetrahedron; however, it can also be written non-uniquely as a convex combination of two maps on the”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“After some straightforward trigonometry, we find |x| 2 =cos 2 θ cos 2 u + sin 2 θ cos 2 u cos 2 v + sin 2 u sin 2 v + 2 sin θ [sin u sin v cos u cos v] =1 -[sin u cos v -sin θ cos u sin v] 2(37) so that |x| = 1 if (and only if) sin θ = tan u/tan v.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“If R is unitary, but does not have the special form (7), it will still define a stochastic map, albeit not one of the”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“In a fixed basis, an arbitrary bistochastic map can be written (uniquely) as a convex combination of four true extreme points, or non-uniquely as a convex combination of two quasi-extreme points on the”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
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“We first note that the fact that W is diagonal implies A † 1 A 1 = V † 2 W † DW V 2 = V † 2 DV 2 . Thus A † 1 A 1 + A † 2 A 2 = V † 2 DV 2 + V † 2 (I -D)V 2 = I so that is a bistochastic map.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“These maps have two non-zero Kraus operators and hence, the form s j + (1s) k , where j, k ∈ {0, 1, 2, 3} and j (ρ) = σ j ρσ j (and σ 0 = I ).”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“However, unlike the case with µ > 0 such maps cannot be translated to the boundary; in fact, for µ = - 1 3 , one must have t = 0 so that no translation is possible.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“If exactly one of {sin v, sin u} is zero, then it follows from (17) that t 3 = sin v sin u = 0, and is unital.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“The conditions cos u = cos ω cos θ , sin v = sin u sin ω follow from (38) and imply that the solution is unique.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.2. Images of stochastic maps
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- The authors state this as an assumption: For solving the inverse problem, the authors assume 0 < ω < θ < 1/2 and seek solutions with 0 < u < v < 1/2, corresponding to λ1 > λ2 > 0.
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“Example (2a) includes maps with λ 1 = λ 2 = µ, λ 3 = 2µ -1, which may seem to contradict Example (1b).However, the condition 2µ -1 = λ 3 > λ 2 1 = µ 2 is never satisfied since if it would imply -µ 2 + 2µ -1 = -(1µ) 2 > 0. Hence the conditions for these two examples do not overlap.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“The former can be translated in any direction with |t| 2/3; however, the latter cannot be translated at all without losing complete positivity.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“If both of {sin v, sin u} are zero, then u, v are both of the form n for some integer n.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“For 0 < µ < 1 2 translation is possible, but clearly never reaches the boundary of pure states.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“It is easy to see that for extreme points of the types (II) and (III), C Holv ( ) = log 2 so that the capacity attains its maximal value, and for the completely noisy channel in Case (IC), C Holv ( ) = 0.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.4. Channel capacity
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“If D = µI , then both µ -1/2 A 1 and (1µ) -1/2 A 2 are unitary so that is a convex combination of unitary maps.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“(D) is an ellipsoid whose major axis has length one.Thus, (D) has one pair of orthogonal states on the unit sphere, which are also fixed points.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“This curve forms the boundary of Fig. 3.”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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“In place of the remained two edges there appear two segments (still on the surface of the”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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In the paper
“One can easily check that this yields a map of type (IC) with u = 2x -2 , v = 1 2 but the Kraus operators do not have the form (18).”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
plaindirect- Asserted: passes — The authors put it forward as true.
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In the 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”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 5.
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“This parameterization was also obtained independently by Rieffel and Zalka [21] who, like Niu and Griffiths, considered maps that arise via interactions between a pair of qubits”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.4. Trigonometric parameterization
What the sources say
- Not checked yetcites [21] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, MA (in press)
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“A similar parameterization was obtained by Niu and Griffiths [18] in their work on two-bit copying devices.”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.4. Trigonometric parameterization
What the sources say
- Not checked yetcites [18] Lieb, 1974, Some operator inequalities of the Schwarz type, Adv. Math. source
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In the paper
“For bistochastic maps, the extreme points are known [1,11] to consist of the maps that conjugate by a unitary matrix and in the λ k”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
What the sources say
- Not checked yetcites [1] A. Fujiwara, P. Algoet, Affine parameterization of completely positive maps on a matrix algebra, preprint; A. Fujiwara, P. Algoet, One-to-one parameterization of quantum channels, Phys. Rev. A 59 (1999) 3290–3294 source
- Not checked yetcites [11] Horn, 1991
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In the paper
“However, Terhal et al. [22] showed that this is false and another counterexample was obtained in [21].”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.4. Trigonometric parameterization
What the sources say
- Not checked yetcites [22] Niu, Griffiths, Two qubit copying machine for economical quantum eavesdropping, quant-ph/9810008
- Not checked yetcites [21] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, MA (in press)
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In the paper
“When has the form (17) this can only happen when λ 3 is degenerate [i.e., equal to λ 1 or λ 2 ].However, this is precluded by (15) unless λ 3 = 0, ±1.In the latter case we must have a unital channel with all t k = 0. Thus, two non-zero t k occur only in channels that are so noisy that at least two λ k = 0.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“This behavior was observed first by Fuchs for a non-extreme channel [5]; subsequently, the amplitude-damping channel was studied by Schumacher and Westmoreland [23].”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.4. Channel capacity
What the sources say
- Not checked yetcites [5] Choi, 1974, A schwarz inequality for positive linear maps on C* algebras, Illinois J. Math. source
- Not checked yetcites [23] V.I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman, Harlow, Wiley, New York, 1986
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Holevo [7,8] introduced such channels and showed that they suffice to demonstrate the”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
What the sources say
- Not checked yetcites [7] Haag, 1964, An algebraic approach to quantum field theory, J. Math. Phys. source
- Not checked yetcites [8] Holevo, 1979, On the capacity of quantum communication channel, Probl. Peredachi Inform.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“As the changes of basis rotate the tetrahedron (considered as a set of diagonal 3 × 3 matrices in R 9 ), we obtain a much larger set so that (by Theorem 14) any non-extreme map can be written as the midpoint of two”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Geometric considerations and the arguments leading to (38) imply that this will be possible only if | sin θ| > | sin ω| and |λ 1 | > |λ 2 |.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.2. Images of stochastic maps
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Since every ∈ S is equivalent in the sense of (4) (i.e., after changes of bases in the two-dimensional Hilbert spaces corresponding to the domain and the range of ) to a map of the form (3), this describes in principle all extreme points of S and all points in the closure of the set of extreme points.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In both cases the work was numerical, but suggests that this situation is generic for channels that are translated orthogonal to the major axis of the ellipse.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.4. Channel capacity
What the sources say
- Not checked yetcites [5] Choi, 1974, A schwarz inequality for positive linear maps on C* algebras, Illinois J. Math. source
- Not checked yetcites [23] V.I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman, Harlow, Wiley, New York, 1986
hedgedcorrelational- 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
“Holevo called channels of the form (IC)”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The authors state this as an assumption.
For describing the subclasses of extreme points, the authors assume |λ1| ≤ |λ2|, noting that permutations of indices cover all maps of form (3).
“We make the additional assumption that |λ 1 | |λ 2 | since, in any case, all permutations of indices must be considered to obtain the extreme points of all maps of the form (3).”
The authors state this as an assumption.
For solving the inverse problem, the authors assume 0 < ω < θ < 1/2 and seek solutions with 0 < u < v < 1/2, corresponding to λ1 > λ2 > 0.
“For simplicity, we assume 0 < ω < θ < 1 2 [corresponding to an upward translation and u ∈ (0, )] and seek solutions satisfying 0 < u < v < 1 2 (corresponding to λ 1 > λ 2 > 0).”
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In the citing paper
“Interestingly, for all η this channel is extremal48, which means that the CS is not applicable.”
From The elusive Heisenberg limit in quantum-enhanced metrology (2012) · cites this paper as [48]
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Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
“Thus, (D) has one pair of orthogonal states on the unit sphere, which are also fixed points.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.4. Channel capacity
“The extreme points of the tetrahedron are [1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1].The edges connecting any two of these extreme points have the form [±1, s, ±s] up to permutation and correspond to quasi-extreme points of type (III).”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
“Because the set {I, σ x , σ y , σ z } forms an orthonormal basis (with respect to the Hilbert-Schmidt inner product) for M 2”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
“the Hilbert-Schmidt inner product A, B = TrA † B”
Citations
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Metadata
- Authors
- Beth Ruskai, M., Szarek, S., & Werner, E.
- Affiliations
- Université de Lille (FR)
- ORCID
- Werner, Elisabeth
- Venue
- Linear Algebra and its Applications
- Publisher
- Elsevier BV
- Published
- 2002-05
- Volume
- 347
- Issue
- 1-3
- Pages
- 159-187
- ISSN
- 0024-3795 (ISSN-L)
- OpenAlex
- W2160212841
- Semantic Scholar
- b5eb07634f6099ceec94304193355fa8b3fe6acd
- Cited by
- 366
- References
- 29
- Citations by year
- 2026: 20 · 2025: 15 · 2024: 17 · 2023: 22 · 2022: 10 · 2021: 14 · 2020: 20 · 2019: 20 · 2018: 26 · 2017: 10 · 2016: 16 · 2015: 14 · 2014: 13 · 2013: 17 · 2012: 13
- FWCI
- 13.77
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Linear Algebra and its Applications (publishedVersion)
- arXiv (Cornell University) (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9706981, 0074566)
- National Security Agency
- Agricultural Research Development Agency
- Technische Universität Berlin
- Army Research Office
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- completely positive maps, trace-preserving maps, extreme points, stochastic maps, Bloch sphere, convex combination, canonical form, quantum channels
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