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
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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
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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
Cites
Horn, R. A., & Johnson, C. R. (1985) · Cited by 22,203
Quantum Computation and Quantum Information
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895
Horn, R. A., & Johnson, C. R. (1991) · Cambridge University Press eBooks · doi:10.1017/cbo9780511840371 · Cited by 9,324
Lloyd, S. (1996) · Science · doi:10.1126/science.273.5278.1073 · Cited by 3,088
Completely positive linear maps on complex matrices
Choi, M.-D. (1975) · Linear Algebra and its Applications · doi:10.1016/0024-3795(75)90075-0 · Cited by 2,659
C-Algebras and Their Automorphism Groups
Pedersen, G. K. (1979) · Cited by 2,423
States, Effects, and Operations Fundamental Notions of Quantum Theory
(1983) · Lecture Notes in Physics · doi:10.1007/3-540-12732-1 · Cited by 1,565
An algebraic approach to quantum field theory
Haag, R., & Kastler, D. (1964) · Journal of Mathematical Physics · doi:10.1063/1.1704187 · Cited by 1,250
General state changes in quantum theory
Kraus, K. (1971) · Annals of Physics · doi:10.1016/0003-4916(71)90108-4 · Cited by 777
Completely positive maps and entropy inequalities
Lindblad, G. (1975) · Communications in Mathematical Physics · doi:10.1007/bf01609396 · Cited by 666
Cited by
Sorted by citations
The elusive Heisenberg limit in quantum-enhanced metrology
Demkowicz-Dobrzański, R., Kołodyński, J., & Guţă, M. (2012) · Nature Communications · doi:10.1038/ncomms2067 · Cited by 773
Entanglement Breaking Channels
Horodecki, M., Shor, P. W., & Ruskai, M. B. (2003) · Reviews in Mathematical Physics · doi:10.1142/s0129055x03001709 · Cited by 603
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
A Survey on Quantum Channel Capacities
Gyongyosi, L., Imre, S., & Nguyen, H. V. (2018) · IEEE Communications Surveys & Tutorials · doi:10.1109/comst.2017.2786748 · Cited by 337
Wolf, M. M., & Cirac, J. I. (2008) · Communications in Mathematical Physics · doi:10.1007/s00220-008-0411-y · Cited by 288
A Survey on Quantum Channel Capacities
Gyöngyösi, L., Imre, S., & Nguyen, H. V. (2018) · Surrey Research Insight Open Access (The University of Surrey) · Cited by 299
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Verstraete, F., Cirac, J. I., Latorre, J. I., Rico, E., & Wolf, M. M. (2005) · Physical Review Letters · doi:10.1103/physrevlett.94.140601 · Cited by 191
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Smirne, A., Kołodyński, J., Huelga, S. F., & Demkowicz-Dobrzański, R. (2016) · Physical Review Letters · doi:10.1103/physrevlett.116.120801 · Cited by 183
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
Randomized benchmarking with confidence
Wallman, J. J., & Flammia, S. T. (2014) · New Journal of Physics · doi:10.1088/1367-2630/16/10/103032 · Cited by 174
On Duality between Quantum Maps and Quantum States
Życzkowski, K., & Bengtsson, I. (2004) · Open Systems & Information Dynamics · doi:10.1023/b:opsy.0000024753.05661.c2 · Cited by 160
Hayashi, M., Iwama, K., Nishimura, H., Raymond, R., & Yamashita, S. (2007) · Springer eBooks · doi:10.1007/978-3-540-70918-3_52 · Cited by 150
Negativity of quantumness and its interpretations
Nakano, T., Piani, M., & Adesso, G. (2013) · Physical Review A · doi:10.1103/physreva.88.012117 · Cited by 133
Qubit Entanglement Breaking Channels
Ruskai, M. B. (2003) · Reviews in Mathematical Physics · doi:10.1142/s0129055x03001710 · Cited by 127
General Framework for Randomized Benchmarking
Helsen, J., Roth, I., Onorati, E., Werner, A. H., & Eisert, J. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.020357 · Cited by 179
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Kimmel, S., Silva, M. P. D., Ryan, C. A., Johnson, B., & Ohki, T. (2014) · Physical Review X · doi:10.1103/physrevx.4.011050 · Cited by 127
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Apollaro, T. J. G., Di Franco, C., Plastina, F., & Paternostro, M. (2011) · Physical Review A · doi:10.1103/physreva.83.032103 · Cited by 106
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Qubit Entanglement Breaking ChannelsPreprint
Ruskai, M. B. (2003) · Cited by 87
Private classical capacity with a symmetric side channel and its application to quantum cryptography
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Filippov, S. N., Piilo, J., Maniscalco, S., & Ziman, M. (2017) · Physical Review A · doi:10.1103/physreva.96.032111 · Cited by 84
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Beale, S. J., Wallman, J. J., Gutiérrez, M., Brown, K. R., & Laflamme, R. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.190501 · Cited by 84
The structure of degradable quantum channelsPreprint
Cubitt, T. S., Ruskai, M. B., & Smith, G. (2008) · Cited by 90
Quantum process tomography and Linblad estimation of a solid state qubit
Howard, M., Gaebel, T., Jelezko, F., Twamley, J., Wittmann, C., & Wrachtrup, J. (2006) · CERN Document Server (European Organization for Nuclear Research) · Cited by 75
Uhlmann, A. (2010) · Entropy · doi:10.3390/e12071799 · Cited by 70
Orbits of quantum states and geometry of Bloch vectors forN-level systems
Schirmer, S. G., Zhang, T., & Leahy, J. V. (2004) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/37/4/022 · Cited by 66
Exact performance of concatenated quantum codes
Rahn, B., Doherty, A. C., & Mabuchi, H. (2002) · Physical Review A · doi:10.1103/physreva.66.032304 · Cited by 64
Quantum Markov channels for qubits
Daffer, S., Wódkiewicz, K., & McIver, J. K. (2003) · Physical Review A · doi:10.1103/physreva.67.062312 · Cited by 64
Modeling quantum noise for efficient testing of fault-tolerant circuits
Magesan, E., Puzzuoli, D., Granade, C., & Cory, D. G. (2013) · Physical Review A · doi:10.1103/physreva.87.012324 · Cited by 69
Unitary quantum gates, perfect entanglers, and unistochastic maps
Musz, M., Kuś, M., & Życzkowski, K. (2013) · Physical Review A · doi:10.1103/physreva.87.022111 · Cited by 58
Concurrence versus purity: Influence of local channels on Bell states of two qubits
Ziman, M., & Bužek, V. (2005) · Physical Review A · doi:10.1103/physreva.72.052325 · Cited by 57
Solovay-Kitaev Decomposition Strategy for Single-Qubit Channels
Wang, D.-S., Berry, D. W., de Oliveira, M. C., & Sanders, B. C. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.130504 · Cited by 57
Generating random quantum channels
Kukulski, R., Nechita, I., Pawela, Ł., Puchała, Z., & Życzkowski, K. (2021) · Journal of Mathematical Physics · doi:10.1063/5.0038838 · Cited by 64
Maximization of capacity and lp norms for some product channels
King, C. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1433943 · Cited by 52
Gross, D., & Eisert, J. (2010) · Physical Review A · doi:10.1103/physreva.82.040303 · Cited by 51
Korzekwa, K., Czachórski, S., Puchała, Z., & Życzkowski, K. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aaaff3 · Cited by 51
Conjugate degradability and the quantum capacity of cloning channels
Brádler, K., Dutil, N., Hayden, P. M., & Muhammad, A. (2010) · Journal of Mathematical Physics · doi:10.1063/1.3449555 · Cited by 51
Hayashi, M., Imai, H., Matsumoto, K., Ruskai, M. B., & Shimono, T. (2005) · Quantum Information and Computation · doi:10.26421/qic5.1-2 · Cited by 48
All classes of informationally complete symmetric measurements in finite dimensions
Siudzińska, K. (2022) · Physical Review A · doi:10.1103/physreva.105.042209 · Cited by 47
A Taxonomy of Small Markovian Errors
Blume-Kohout, R., Silva, M. P. D., Nielsen, E., Proctor, T. J., Rudinger, K., Sarovar, M., & Young, K. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.020335 · Cited by 53
Efficient universal quantum channel simulation in IBM’s cloud quantum computer
Wei, S.-J., Xin, T., & Long, G. (2018) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-017-9181-9 · Cited by 46
Quantum simulation of quantum channels in nuclear magnetic resonance
Xin, T., Wei, S.-J., Pedernales, J. S., Solano, E., & Long, G. (2017) · Physical Review A · doi:10.1103/physreva.96.062303 · Cited by 45
Pauli diagonal channels constant on axes
Nathanson, M., & Ruskai, M. B. (2007) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/40/28/s22 · Cited by 49
Local two-qubit entanglement-annihilating channels
Filippov, S. N., Rybár, T., & Ziman, M. (2012) · Physical Review A · doi:10.1103/physreva.85.012303 · Cited by 44
Estimating the Coherence of Noise in Quantum Control of a Solid-State Qubit
Feng, G., Wallman, J. J., Buonacorsi, B., Cho, F. H., Park, D. K., Xin, T., Lu, D., Baugh, J., & Laflamme, R. (2016) · Physical Review Letters · doi:10.1103/physrevlett.117.260501 · Cited by 50
Generalized Pauli channels and a class of non-Markovian quantum evolution
Chruściński, D., & Siudzińska, K. (2016) · Physical Review A · doi:10.1103/physreva.94.022118 · Cited by 44
Phase Covariant Qubit Dynamics and Divisibility
Filippov, S. N., Glinov, A. N., & Leppäjärvi, L. (2020) · Lobachevskii Journal of Mathematics · doi:10.1134/s1995080220040095 · Cited by 43
Complete positivity of a spin- 1∕2 master equation with memory
Maniscalco, S. (2007) · Physical Review A · doi:10.1103/physreva.75.062103 · Cited by 43
Geometry of quantum correlations in space-time
Zhao, Z., Pisarczyk, R., Thompson, J. E., Gu, M., Vedral, V., & Fitzsimons, J. F. (2018) · Physical Review A · doi:10.1103/physreva.98.052312 · Cited by 42
Divisibility of qubit channels and dynamical maps
Davalos, D., Ziman, M., & Pineda, C. (2019) · Quantum · doi:10.22331/q-2019-05-20-144 · Cited by 38
A universal set of qubit quantum channels
Braun, D., Giraud, O., Nechita, I., Pellegrini, C., & Žnidarič, M. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/47/13/135302 · Cited by 42
A small quantum computer is needed to optimize fault-tolerant protocols
Iyer, P., & Poulin, D. (2018) · Quantum Science and Technology · doi:10.1088/2058-9565/aab73c · Cited by 34
Simulation of single-qubit open quantum systems
Sweke, R., Sinayskiy, I., & Petruccione, F. (2014) · Physical Review A · doi:10.1103/physreva.90.022331 · Cited by 33
Separability of entangled qutrits in noisy channels
Chęcińska, A., & Wódkiewicz, K. (2007) · Physical Review A · doi:10.1103/physreva.76.052306 · Cited by 34
Extending additivity from symmetric to asymmetric channels
Fukuda, M. (2005) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/38/45/l02 · Cited by 34
Quantum Channel Capacities With Passive Environment Assistance
Karumanchi, S., Mancini, S., Winter, A., & Yang, D. (2016) · IEEE Transactions on Information Theory · doi:10.1109/tit.2016.2522192 · Cited by 35
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Carignan-Dugas, A., Boone, K., Wallman, J. J., & Emerson, J. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aadcc7 · Cited by 33
Pauli Diagonal Channels Constant on AxesPreprint
Nathanson, M., & Ruskai, M. B. (2006) · Cited by 38
Szarek, S. J., Werner, E. M., & Życzkowski, K. (2008) · Journal of Mathematical Physics · doi:10.1063/1.2841325 · Cited by 31
Self-citationHypercontractivity for Semigroups of Unital Qubit Channels
King, C. (2014) · Communications in Mathematical Physics · doi:10.1007/s00220-014-1982-4 · Cited by 30
Quantum universality by state distillationPreprint
Reichardt, B. W. (2009) · doi:10.5555/2012098.2012105 · Cited by 30
Quantum circuit design for accurate simulation of qudit channels
Wang, D., & Sanders, B. C. (2015) · New Journal of Physics · doi:10.1088/1367-2630/17/4/043004 · Cited by 26
Simulating and Optimising Quantum Thermometry Using Single Photons
Tham, W. K., Ferretti, H., Sadashivan, A. V., & Steinberg, A. M. (2016) · Scientific Reports · doi:10.1038/srep38822 · Cited by 28
Lankinen, J., Lyyra, H., Sokolov, B., Teittinen, J., Ziaei, B., & Maniscalco, S. (2016) · Physical Review A · doi:10.1103/physreva.93.052103 · Cited by 29
Uhlmann, A. (2010) · Cited by 29
The additivity problem in quantum information theory
Holevo, A. S. (2007) · Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 · doi:10.4171/022-3/49 · Cited by 29
Error-detection-based quantum fault tolerance against discrete Pauli noisePreprint
Reichardt, B. W. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0612004 · Cited by 34
Linear optics and quantum maps
Aiello, A., Puentes, G., & Woerdman, J. P. (2007) · Physical Review A · doi:10.1103/physreva.76.032323 · Cited by 28
Datta, C., Sazim, S., Pati, A. K., & Agrawal, P. (2018) · Annals of Physics · doi:10.1016/j.aop.2018.08.014 · Cited by 29
Multiplicativity properties of entrywise positive maps
King, C., Nathanson, M., & Ruskai, M. B. (2005) · Linear Algebra and its Applications · doi:10.1016/j.laa.2005.02.035 · Cited by 28
Conditions for multiplicativity of maximal ℓp-norms of channels for fixed integer p
Giovannetti, V., Lloyd, S., & Ruskai, M. B. (2005) · Journal of Mathematical Physics · doi:10.1063/1.1862094 · Cited by 28
ENTROPIC CHARACTERIZATION OF QUANTUM OPERATIONS
Roga, W., Życzkowski, K., & Fannes, M. (2011) · International Journal of Quantum Information · doi:10.1142/s0219749911007794 · Cited by 27
Comments on multiplicativity of maximal p-norms when p=2
King, C., & Ruskai, M. B. (2004) · Quantum Information and Computation · doi:10.26421/qic4.6-7-9 · Cited by 27
Can Communication Power of Separable Correlations Exceed That of Entanglement Resource?
Horodecki, P., Tuziemski, J., Mazurek, P., & Horodecki, R. (2014) · Physical Review Letters · doi:10.1103/physrevlett.112.140507 · Cited by 26
Bipartite entanglement-annihilating maps: Necessary and sufficient conditions
Filippov, S. N., & Ziman, M. (2013) · Physical Review A · doi:10.1103/physreva.88.032316 · Cited by 26
Maximal output purity and capacity for asymmetric unital qudit channels
Datta, N., & Ruskai, M. B. (2005) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/38/45/005 · Cited by 26
How often is a random quantum statek-entangled?
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Self-citationHelsen, J., Battistel, F., & Terhal, B. M. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0189-0 · Cited by 29
Contraction coefficients for noisy quantum channels
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Holevo, A. S. (2006) · Russian Mathematical Surveys · doi:10.1070/rm2006v061n02abeh004313 · Cited by 25
Geometry of color perception. Part 2: perceived colors from real quantum states and Hering’s rebit
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The Dobrushin ergodicity coefficient and the ergodicity of noncommutative Markov chains
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Complementarity between entanglement-assisted and quantum distributed random access code
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Logarithmic Sobolev inequalities in non-commutative algebras
Carbone, R., & Martinelli, A. (2015) · Infinite Dimensional Analysis Quantum Probability and Related Topics · doi:10.1142/s0219025715500113 · Cited by 24
Lower bounds on the quantum capacity and highest error exponent of general memoryless channels
Hamada, M. (2002) · IEEE Transactions on Information Theory · doi:10.1109/tit.2002.801470 · Cited by 27
Channel covariance, twirling, contraction and some upper bounds on the quantum capacity
Ouyang, Y. (2014) · Quantum Information and Computation · doi:10.26421/qic14.11-12-2 · Cited by 26
Entropic trade-off relations for quantum operations
Roga, W., Puchała, Z., Rudnicki, Ł., & Życzkowski, K. (2013) · Physical Review A · doi:10.1103/physreva.87.032308 · Cited by 24
Avron, J. E., Bisker, G., & Kenneth, O. (2007) · Journal of Mathematical Physics · doi:10.1063/1.2795217 · Cited by 22
Maximal output purity and capacity for asymmetric unital qudit channelsPreprint
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Non-locality breaking qubit channels: the case for CHSH inequality
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State retrieval beyond Bayes' retrodiction
Surace, J., & Scandi, M. (2023) · Quantum · doi:10.22331/q-2023-04-27-990 · Cited by 22
Circuits of space and time quantum channels
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Detecting Lower Bounds to Quantum Channel Capacities
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Channel kets, entangled states, and the location of quantum information
Griffiths, R. B. (2005) · Physical Review A · doi:10.1103/physreva.71.042337 · Cited by 24
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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