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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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“However, not all ellipsoids contained in the “Bloch ball” correspond to images of a stochastic map .”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
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“Theorem 3. Let be a stochastic map induced by a matrix T of the form (3). Then belongs to the closure of the set of extreme points of S if and only if the matrix R [as given by (7) or (8) above] is unitary.”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“For now, we emphasize that the interesting new class of extreme points are those that also satisfy 1 > |λ 1 | > |λ 2 | > |λ 3 | > 0.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“Since a cyclic permutation of indices corresponds to a rotation in R 3 which can be incorporated into the maps U and V in (4), stating conditions (for membership of S or for any form of extremality) in a form in which, say, λ 3 and t 3 play a special role does not lead to any real loss of generality.”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“If ∈ S, then it is unitarily equivalent, in the sense of (4), to a map of the form (3).”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“Theorem 1. A map given by (3) for which |t 3 | + |λ 3 | 1 is completely positive if and only if the equation”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 1.
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“Moreover, it follows that a necessary condition for complete positivity of a map of the form (3) is (λ 1 ± λ 2 ) 2 (1 ± λ 3 ) 2 .”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“and show that any stochastic map on M 2 can be written as a convex combination of two “generalized” extreme points.”
§ An analysis of completely positive trace-preserving maps on M 2 > Abstract
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“Although not obvious from this analysis, (12) holds if and only if it is valid for any permutation of the parameters λ k .”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“We also determine explicitly all extreme points of this set”
§ An analysis of completely positive trace-preserving maps on M 2 > Abstract
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“However, it should be noted that the requirement t 1 = t 2 = 0 is equivalent to requiring that R is skew diagonal.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“Thus, is in the closure of the extreme points of S if and only if, up to a permutation of indices, equality holds in ( 9), ( 10) and ( 13).”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“This allows a detailed examination of an important class of non-unital extreme points that can be characterized as having exactly two images on the Bloch sphere.”
§ An analysis of completely positive trace-preserving maps on M 2 > Abstract
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“We conclude this summary by noting that we have obtained three equivalent sets of conditions on the closure of the set of extreme points of S after appropriate reduction to the “diagonal” form (3).”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“For such maps, it is easy to verify that a necessary condition for to be positivity-preserving is that |t k | + |λ k | 1 for k = 1, 2, 3.”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.3. Summary of results
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“When |t k | + |λ k | < 1 (6) has the unique solution R =”
§ An analysis of completely positive trace-preserving maps on M 2 > Remark.
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“It may be worth noting that when one of the λ k = 0, (15) implies that at least two λ k are zero.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 4.
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“It is straightforward to verify that in this parameterization can be realized with the Kraus operators”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.4. Trigonometric parameterization
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“In a fixed basis, the convex set of stochastic maps with t = 0 forms a tetrahedron (see Fig. 2) that we can describe using the λ k in (3) which we write as”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
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“one can see that, depending on the sign of λ 3 one of the pair inequalities in (39) holds.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“When t 3 = 0, the linearity in λ 3 yields a line segment so that we no longer have”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
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“Theorem 16. If is an extreme point of S, then at least one point in the image (D) is a pure state.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“In the preceding section we showed that, in the special case when is represented by a matrix T of the form (3), the first three conditions are equivalent.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
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“Maps of type (I) and (II) correspond to the non-unital and unital extreme points of S, respectively.”
§ 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 eigenvalues [ 1 3 , 1 3 , 1 3 ] and [ 1 3 , 1 3 , -1 3 ] both yield the same ellipsoid (actually, a sphere of radius 1 3 ).”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“If R is a contraction, its singular value decomposition can be written in the form R =V cos θ 1 0 0 cosθ 2 W † = 1 2 V e iθ 1 0 0 e iθ 2 W † + 1 2 V e -iθ 1 0 0 e -iθ 2 W † , (36 )”
§ An analysis of completely positive trace-preserving maps on M 2 > Lemma 15.
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“A general extreme point is then the composition of a map of type (I) or (II) with unitary maps, as in (4).”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“In general, however, it is not possible to translate the image ellipsoid to the boundary of the unit ball.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“In particular, (25) shows that β( ) is positive semi-definite if and only if β( ) is.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 5.
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“Lemma 10.The minimal number of Kraus operators A k needed to represent a completely positive map in the form (1) is rank[β( )].”
§ An analysis of completely positive trace-preserving maps on M 2 > Lemma 9.
What the sources say
- Not checked yetcites [4] Choi, 1975, Completely positive linear maps on complex matrices, Linear Algebra Appl. source
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“However, it is readily verified that such a is a convex combination of two extreme maps of type (II).”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“Theorem 14. Any stochastic map on M 2 can be written as the convex combination of two maps in the closure of the set of extreme points.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 14.
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“For each pair (j, k) we obtain a line between two of the extreme points that form the tetrahedron of bistochastic maps.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“Theorem 13. A stochastic map that, when written in the form (1), requires two Kraus operators A k but not more, is either an extreme point of S or bistochastic.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“Lemma 8. Let M ∈ M 4 with M 0. Then M admits a factorization (29) with R unitary if and only if rank(M) 2.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 7.
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“When t 3 / = 0, the corners depend non-linearly on λ 3 , yielding a curve of true extreme points.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
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“If (D) contains two or more pure states, it must be in the closure of the extreme points of S.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“Moreover, if the intersection of (D) with the Bloch sphere consists of exactly two non-orthogonal pure states, then must be a non-unital extreme point;”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“Indeed, if = α 1 + (1α) 2 with 0 < α < 1 is not extreme and has an image point x = (u) on the boundary of the Bloch sphere, then it follows immediately from the strict convexity of the Euclidean unit ball that x = (u) = 1 (u) = 2 (u).”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“On the contrary, there are many situations in which the translation is severely limited despite contraction of the ellipsoid.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“Such a translation would have to be representable in the form 1 0 t I , which does not satisfy the conditions for complete positivity.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“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 , the operators in (32) are linearly independent if and only if sin v sin u / = 0.”
§ 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 addition, Lemma 11 above implies that (v) is invariant under a change of basis.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
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“(b) A map with λ 1 = λ 2 = λ 3 = µ < 0 is not completely positive unless |µ| 1 3 .”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“In particular, maps the pair of states corresponding to (± cos θ, 0, sin θ) on the Bloch sphere to the pair of states corresponding to (± cos ω, 0, sin ω) when cos ω = cos θ cos u = cos 2 u sin 2 v -sin 2 u cos 2 v sin v , (38 ) sin ω = sin u sin v + sin θ cos u cos v = sin u sin v .”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“Lemma 15. Any contraction in M 2 can be written as the convex combination of two unitary matrices.”
§ An analysis of completely positive trace-preserving maps on M 2 > Lemma 15.
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“Theorem 7. A map is a generalized extreme point of S if and only if the corresponding matrix M = β( ) defined via (23) is of the form”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 7.
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“A non-extreme point can have an image point on the boundary of the Bloch sphere only if all its extreme components have the same pre-image for that point.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“Lemma 11.Let be a positive map on M 2 and U, V unitary.Then = U • D • V if and only if ( ⊗ I )(B 0 ) = [U ⊗ V T ] [( D ⊗ I )(B 0 )] [U † ⊗ V ], (34 )”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.3. Invariance of conditions under change of basis
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“By Lemma 9, is extreme if and only if the set {A 1 A † 1 , A 2 A † 2 , A 1 A † 2 , A 2 A † 1 } is linearly independent.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“(c) When λ 1 = λ 2 = µ, λ 3 = 0 which implies |µ| 1 2 and the image of is a circle in a plane parallel to the equator.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“We consider a pure state of the form ρ = 1 2 [I + w• ] with w = (± cos θ, 0, sin θ) so that (ρ) = 1 2 [I + x• ] with x = (± cos θ cos u, 0, sin u sin v + sin θ cos u cos v).”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“By contrast, the non-unital situation (I) is more interesting because it includes channels for which capacity (42) is strictly bigger than the classical Shannon capacity, i.e., the capacity for communication restricted to product input and measurements.Such channels demonstrate a definite quantum advantage.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.4. Channel capacity
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“Thus, we have verified that unitarity of R implies (14) in all cases.”
§ 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 general, it is not possible to write a non-unital map as the composition of a unital map (which contracts the Bloch sphere to an ellipsoid) and a translation.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“Thus, when µ = 1 2 no translation the image (D) is possible despite the fact that the ellipsoid has shrunk to a flat disk of radius 1 2 .”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“Moreover, the implication (i) ⇒ (ii) holds in full generality.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
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“However, even when equality holds with only one sign, (11) implies that 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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“It is worth pointing out that Case (IC) is the only situation in which one can have extreme points with more than one t k non-zero.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.1. Types of extreme points
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“while (D) contains the entire Bloch sphere if and only if it is a unital extreme point.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“We now show that this unitarity implies condition (14) in Theorem 4.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“Note that in both cases we have proven the somewhat stronger result that the convex combination can be chosen to be a midpoint.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 14.
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“The cases t = ( √ 1α 2 , 0, 0) and t = (0, √ 1α 2 , 0) are similar.The only difference is the orientation and displacement, e.g., whether the point [1, 1, 1] is replaced by [α 2 , α, α] or to [α, α 2 , α].”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.3. Geometry of stochastic maps
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“Conversely, one can easily check that under the hypotheses t 1 = t 2 = 0 and ( 14), R as given by ( 7) or ( 8) is always unitary.”
§ 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 argument presented thus far yields the extreme points of S that are represented by a matrix T of the form (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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“One can readily verify that this set is linearly independent unless D is a multiple of the identity or W is diagonal.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“(a) If (λ 1 ± λ 2 ) 2 = (1 ± λ 3 ) 2 holds with exactly one sign, then λ 1 , λ 2 , λ 3 lie on the surface of the tetrahedron defined by (II), but on the interior of one of the four faces.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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“This is equivalent to linear independence of the set D, I -D, √ DW √ I -D, √ I -DW † √ D , where W = V 1 V † 2 .”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
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“In this case one and only one of the two Kraus operators given by ( 18) is non-zero and hence, unitary.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
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“However, the condition |t 3 | = (1 ± λ 3 ) implies that the North or South pole is a fixed point.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
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“It follows from (17) that the extreme points are given by the curve [cos u, cos v, cos u cos v] subject to the constraint sin u sin v = t 3 .”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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“The four points above split the curve into four pieces, corresponding to four of the six edges of the tetrahedron.”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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“Letting α 2 = 1t 3 , we see that the curve (41) passes through the four points [α, α, α 2 ], [-α, -α, α 2 ], [α, -α, -α 2 ], [-α, α, -α 2 ] when sin u = ± sin v.”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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“Unlike the case t 3 = 0 these lines are not extreme points, even in a generalized sense.”
§ 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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“This can be written in parametric form as a pair of curves cos u, ± cos sin -1 t 3 sin u , ± cos u cos sin -1 t 3 sin u (41) with sin -1 (t 3 ) u sin -1 (t 3 ).”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
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In the paper
“It was shown in [1,11] that the λ k 's are limited by the inequalities (λ 1 ± λ 2 ) 2 (1 ± λ 3 ) 2 .”
§ 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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“King and Ruskai [11] showed that any such map can be reduced, via changes of basis in C 2 , to the form”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
What the sources say
- Not checked yetcites King and Ruskai [11]
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“We are interested in n = 2, in which case this condition is equivalent [10] to ( ⊗ I ) (B 0 ) 0,”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 5.
What the sources say
- Not checked yetcites [10] Horn, 1985
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The crucial point about the reduction (4) is that is completely positive if and only if t, is.”
§ An analysis of completely positive trace-preserving maps on M 2 > 1.2. Notation
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“Moreover, Niu and Griffiths's construction showed that taking the partial trace T 1 rather than T 2 is equivalent to switching sin and cos in (17).”
§ 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
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“It is tempting to think of a stochastic map as the composition of a rotation, contraction (to an ellipsoid), translation, and another rotation.However, this is not accurate in the sense that the individual maps in this process will no longer be stochastic.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Hence, even when |λ 3 | > µ 2 so that is not extreme, the ellipsoid is shifted to the boundary in a manner analogous to an amplitude damping channel.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 16.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“One might think that a map of the form (3) with all three t k non-zero would require three extreme points.However, this is not the case.In general, two maps in the set of generalized points suffice.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 13.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Lemma 9. A stochastic map is an extreme point of S if and only if it can be written in the form (1) (necessarily with k A k A † k = I ), so that the set of matrices {A j A † k } is linearly independent.”
§ An analysis of completely positive trace-preserving maps on M 2 > Lemma 9.
What the sources say
- Not checked yetcites [4] Choi, 1975, Completely positive linear maps on complex matrices, Linear Algebra Appl. source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Hence, it follows that when two distinct extreme points have a common pair of points on the image of the Bloch sphere, their pre-images must be distinct.”
§ An analysis of completely positive trace-preserving maps on M 2 > 3.2. Images of stochastic maps
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus, by Lemma 9, the corresponding stochastic map is extreme if both of {sin v, sin u} are non-zero or, equivalently, t 3 / = 0 and all |λ k | < 1.”
§ 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 the paper
“Example (2b) illustrates that the ellipsoid picture, while useful, is incomplete.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“The discussion at the beginning of Section 3.1 suggests that, roughly speaking, extreme points correspond to maps for which two (or more) points in the image (D) lie on the Bloch sphere as shown in Fig. 1.However, this statement is correct only if we interpret the single point at a pole as a pair of degenerate images when |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
“It thus remains to prove that (ii) ⇒ (iii) ⇒ (i) in full generality rather than just for maps of the form (3).But this follows from the facts that every stochastic map can be reduced to the form (3) via changes of bases (4) and that all these conditions are invariant under such changes of bases.”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus, one can find maps with all |λ k | < 1 for which the image (D) is an ellipsoid strictly contained within the unit ball but no translation in any direction is possible.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2. Example of maps with limited translation
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In the paper
“Hence maps for which exactly one of {sin v, sin u} is zero cannot be extreme.”
§ An analysis of completely positive trace-preserving maps on M 2 > 2.2. Proof of results in Section 1.3
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus, as t 3 moves away from zero, the corners of the tetrahedron are replaced by these points as shown in Fig. 3 which might be described as an asymmetric rounded tetrahedron.”
§ An analysis of completely positive trace-preserving maps on M 2 > Additional information
hedgeddirect- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 38
- Probably not a substantial claim. 9
- The quote only partly states this claim. 6
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
“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.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“and give a useful parameterization after reduction to a certain canonical form.”
§ An analysis of completely positive trace-preserving maps on M 2 > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Similarly, Lemma 10 shows that (iv) ⇔ (v).”
§ An analysis of completely positive trace-preserving maps on M 2 > Theorem 12.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
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
“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
hedged- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- 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.
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
“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
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For 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
boostedSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“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
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(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
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This curve forms the boundary of Fig. 3.”
§ 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 paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In 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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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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None of the sources cited for this claim have been checked yet.
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.
What the sources say
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
plaincorrelational- 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.
None of the sources cited for this claim have been checked yet.
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
boostedcorrelational- 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. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
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)
boosted- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
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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- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The 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).”
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?
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]
Self-check
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
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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