An analysis of completely-positive trace-preserving maps on M2

The paper provides a new characterization of completely positive trace-preserving maps on M2 that makes it easy to test any trace-preserving map for complete positivity.

“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.”

Conditions on the diagonal elements and determinant of I - R†R are sufficient to ensure that R is a contraction.

“Conditions on the diagonal elements and determinant of I -R † R suffice to insure R is a contraction.”

When a map is in the closure of the extreme points, its matrix in (3) can be written in the trigonometric form (17) with u in [0,2π) and v in [0,π).

“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”

Requiring the diagonal elements of R†R and RR† to be 1 implies inequalities (9) and (10).

“The requirement that the diagonal elements of R † R and R R † must be 1 then implies that”

A map belongs to the closure of the extreme points of S exactly when it can be reduced to form (3) with at most one tk nonzero and, taking that index as 3, (λ1 ± λ2)^2 = (1 ± λ3)^2 - t3^2.

“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 .”

When |t3| + |λ3| = 1, equation (6) has no solution unless t1 = t2 = 0 and either λ1 = λ2 or λ1 = -λ2.

“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 .”

The equivalence in (4) preserves the property of being a generalized extreme point of S.

“Moreover, as will be shown in Section 2.3, that “equivalence” preserves the property of being a generalized extreme point of S.”

For non-unital maps, the extreme points are those for which the translated ellipsoid touches the boundary of the Bloch sphere at two points, with degenerate cases treated as two coincident points.

“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”

Such a map is extreme unless t3 = 0 and |λ3| < 1.

“Moreover, is extreme unless t 3 = 0 and |λ 3 | < 1.”

A necessary condition for a map to be an extreme point of S is that it is equivalent in the sense of (4) to a map of the form (3) whose corresponding R is unitary.

“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.”

Not every ellipsoid contained in the Bloch ball is the image of a stochastic map.

“However, not all ellipsoids contained in the “Bloch ball” correspond to images of a stochastic map .”

A stochastic map induced by a matrix T of the form (3) belongs to the closure of the extreme points of S exactly when the matrix R given by (7) or (8) is unitary.

“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.”

The interesting new class of extreme points consists of those satisfying 1 > |λ1| > |λ2| > |λ3| > 0.

“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.”

Because a cyclic permutation of indices corresponds to a rotation in R3 that can be absorbed into U and V in (4), treating λ3 and t3 as special does not lose generality.

“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.”

Every map in S is unitarily equivalent, in the sense of (4), to a map of the form (3).

“If ∈ S, then it is unitarily equivalent, in the sense of (4), to a map of the form (3).”

A map of the form (3) with |t3| + |λ3| ≤ 1 is completely positive exactly when equation (6) has a contraction solution R.

“Theorem 1. A map given by (3) for which |t 3 | + |λ 3 | 1 is completely positive if and only if the equation”

A necessary condition for complete positivity of a map of the form (3) is (λ1 ± λ2)^2 ≤ (1 ± λ3)^2.

“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 .”

Every stochastic map on M2 can be expressed as a convex combination of two generalized extreme points.

“and show that any stochastic map on M 2 can be written as a convex combination of two “generalized” extreme points.”

Condition (12) holds exactly when it remains valid under any permutation of the parameters λk.

“Although not obvious from this analysis, (12) holds if and only if it is valid for any permutation of the parameters λ k .”

All extreme points of the set of completely positive trace-preserving maps on M2 are determined explicitly.

“We also determine explicitly all extreme points of this set”

The condition t1 = t2 = 0 is equivalent to R being skew diagonal.

“However, it should be noted that the requirement t 1 = t 2 = 0 is equivalent to requiring that R is skew diagonal.”

A map lies in the closure of the extreme points of S exactly when, up to a permutation of indices, equality holds in (9), (10), and (13).

“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 important class of non-unital extreme points is characterized by having exactly two images on the Bloch sphere.

“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.”

After reduction to the diagonal form (3), there are three equivalent sets of conditions for the closure of the extreme points of S.

“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).”

For maps of the form (3), a necessary condition for positivity preservation is |tk| + |λk| ≤ 1 for each k = 1,2,3.

“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.”

When |tk| + |λk| < 1 for all k, equation (6) has the unique solution R given by (7).

“When |t k | + |λ k | < 1 (6) has the unique solution R =”

If one of the λk is zero, condition (15) implies that at least two of the λk are zero.

“It may be worth noting that when one of the λ k = 0, (15) implies that at least two λ k are zero.”

In the trigonometric parameterization, the map can be realized with the Kraus operators A± given by (18).

“It is straightforward to verify that in this parameterization can be realized with the Kraus operators”

In a fixed basis, the convex set of stochastic maps with t = 0 forms a tetrahedron.

“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”

Depending on the sign of λ3, one of the pair inequalities in (39) holds.

“one can see that, depending on the sign of λ 3 one of the pair inequalities in (39) holds.”

When t3 = 0, the linearity in λ3 yields a line segment, so there are no longer true extreme points.

“When t 3 = 0, the linearity in λ 3 yields a line segment so that we no longer have”

Theorem 16: If a map is an extreme point of S, then at least one point in its image (D) is a pure state.

“Theorem 16. If is an extreme point of S, then at least one point in the image (D) is a pure state.”

In the special case where the map is represented by a matrix T of form (3), the first three conditions of Theorem 12 are equivalent.

“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.”

Maps of type (I) and (II) correspond respectively to the non-unital and unital extreme points of S.

“Maps of type (I) and (II) correspond to the non-unital and unital extreme points of S, respectively.”

The eigenvalues [1/3, 1/3, 1/3] and [1/3, 1/3, -1/3] both yield the same ellipsoid, a sphere of radius 1/3.

“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 ).”

If R is a contraction, its singular value decomposition can be written in the form of equation (36).

“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 )”

A general extreme point is the composition of a map of type (I) or (II) with unitary maps, as in (4).

“A general extreme point is then the composition of a map of type (I) or (II) with unitary maps, as in (4).”

In general, it is not possible to translate the image ellipsoid to the boundary of the unit ball.

“In general, however, it is not possible to translate the image ellipsoid to the boundary of the unit ball.”

The matrix β( ) is positive semidefinite exactly when β( ) is positive semidefinite.

“In particular, (25) shows that β( ) is positive semi-definite if and only if β( ) is.”

The minimal number of Kraus operators needed to represent a completely positive map in form (1) equals rank[β( )].

“Lemma 10.The minimal number of Kraus operators A k needed to represent a completely positive map in the form (1) is rank[β( )].”

• cites [4] Choi, 1975, Completely positive linear maps on complex matrices, Linear Algebra Appl. sourceNot checked yet

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Such a map is a convex combination of two extreme maps of type (II).

“However, it is readily verified that such a is a convex combination of two extreme maps of type (II).”

Theorem 14: Any stochastic map on M2 can be written as the convex combination of two maps in the closure of the extreme-point set.

“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.”

For each pair (j,k), one obtains a line between two extreme points that form the tetrahedron of bistochastic maps.

“For each pair (j, k) we obtain a line between two of the extreme points that form the tetrahedron of bistochastic maps.”

Theorem 13: A stochastic map that requires two Kraus operators but not more is either an extreme point of S or bistochastic.

“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.”

For a positive semidefinite matrix M in M4, M admits a factorization (29) with R unitary exactly when rank(M) ≤ 2.

“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.”

When t3 is non-zero, the corners depend non-linearly on λ3, yielding a curve of true extreme points.

“When t 3 / = 0, the corners depend non-linearly on λ 3 , yielding a curve of true extreme points.”

If (D) contains two or more pure states, then the map must be in the closure of the extreme points of S.

“If (D) contains two or more pure states, it must be in the closure of the extreme points of S.”

If the intersection of (D) with the Bloch sphere consists of exactly two non-orthogonal pure states, then the map must be a non-unital extreme point.

“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;”

If = α1 + (1-α)2 with 0 < α < 1 is not extreme and has an image point x = (u) on the boundary, then x = 1(u) = 2(u).

“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).”

There are many situations in which the translation is severely limited despite contraction of the ellipsoid.

“On the contrary, there are many situations in which the translation is severely limited despite contraction of the ellipsoid.”

Such a translation would have to be representable in the form [[1,0],[t,I]], which does not satisfy the conditions for complete positivity.

“Such a translation would have to be representable in the form 1 0 t I , which does not satisfy the conditions for complete positivity.”

The operators in (32) are linearly independent if and only if sin v sin u is non-zero.

“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.”

Lemma 11 implies that condition (v) is invariant under a change of basis.

“In addition, Lemma 11 above implies that (v) is invariant under a change of basis.”

A map with λ1 = λ2 = λ3 = μ < 0 is not completely positive unless |μ| ≤ 1/3.

“(b) A map with λ 1 = λ 2 = λ 3 = µ < 0 is not completely positive unless |µ| 1 3 .”

The map sends the pair of states corresponding to (±cosθ, 0, sinθ) to the pair corresponding to (±cosω, 0, sinω) when equations (38) hold.

“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 .”

Lemma 15: Any contraction in M2 can be written as the convex combination of two unitary matrices.

“Lemma 15. Any contraction in M 2 can be written as the convex combination of two unitary matrices.”

A map is a generalized extreme point of S exactly when its corresponding matrix M = β( ) has the form (28) with 0 ≤ A ≤ I and U unitary.

“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”

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.

“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.”

Lemma 11: For a positive map on M2 and unitary U, V, the equality = U • D • V holds if and only if equation (34) holds.

“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 )”

By Lemma 9, the map is extreme if and only if the set {A1A1†, A2A2†, A1A2†, A2A1†} is linearly independent.

“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.”

When λ1 = λ2 = μ and λ3 = 0, which implies |μ| ≤ 1/2, the image of the map is a circle in a plane parallel to the equator.

“(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.”

For a pure state ρ = 1/2[I + w•σ] with w = (±cosθ, 0, sinθ), the image is 1/2[I + x•σ] with x = (±cosθ cos u, 0, sin u sin v + sinθ cos u cos v).

“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).”

The non-unital situation (I) includes channels for which the Holevo capacity is strictly larger than the classical Shannon capacity, demonstrating a definite quantum advantage.

“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.”

Unitarity of R implies condition (14) in every case.

“Thus, we have verified that unitarity of R implies (14) in all cases.”

In general, it is not possible to write a non-unital map as the composition of a unital map and a translation.

“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.”

When μ = 1/2, no translation of the image (D) is possible despite the ellipsoid having shrunk to a flat disk of radius 1/2.

“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 .”

The implication (i) ⇒ (ii) of Theorem 12 holds in full generality.

“Moreover, the implication (i) ⇒ (ii) holds in full generality.”

Even when equality holds with only one sign, (11) implies t = 0 so no translation is possible.

“However, even when equality holds with only one sign, (11) implies that t = 0 so that no translation is possible.”

Case (IC) is the only situation in which one can have extreme points with more than one t_k non-zero.

“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.”

(D) contains the entire Bloch sphere if and only if the map is a unital extreme point.

“while (D) contains the entire Bloch sphere if and only if it is a unital extreme point.”

The unitarity of R implies condition (14) in Theorem 4.

“We now show that this unitarity implies condition (14) in Theorem 4.”

The convex combination in Theorem 14 can be chosen to be a midpoint.

“Note that in both cases we have proven the somewhat stronger result that the convex combination can be chosen to be a midpoint.”

The cases t = (√(1-α^2), 0, 0) and t = (0, √(1-α^2), 0) are similar, differing only in orientation and displacement.

“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].”

If t1 = t2 = 0 and condition (14) holds, then R as given by (7) or (8) is always unitary.

“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.”

The argument yields the extreme points of S represented by a matrix T of form (3).

“The argument presented thus far yields the extreme points of S that are represented by a matrix T of the form (3).”

This set is linearly independent unless D is a multiple of the identity or W is diagonal.

“One can readily verify that this set is linearly independent unless D is a multiple of the identity or W is diagonal.”

If (λ1 ± λ2)^2 = (1 ± λ3)^2 holds with exactly one sign, then λ1, λ2, λ3 lie on the surface of the tetrahedron defined by (II), but on the interior of one of its four faces.

“(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.”

This is equivalent to linear independence of the set D, I-D, √D W √(I-D), √(I-D) W† √D, where W = V1 V2†.

“This is equivalent to linear independence of the set D, I -D, √ DW √ I -D, √ I -DW † √ D , where W = V 1 V † 2 .”

In the case where both sin v and sin u vanish, exactly one of the two Kraus operators in (18) is non-zero and hence unitary.

“In this case one and only one of the two Kraus operators given by ( 18) is non-zero and hence, unitary.”

The condition |t3| = (1 ± λ3) implies that the North or South pole is a fixed point.

“However, the condition |t 3 | = (1 ± λ 3 ) implies that the North or South pole is a fixed point.”

For non-zero t with t fixed, the extreme points are given by the curve [cos u, cos v, cos u cos v] subject to sin u sin v = t3.

“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 .”

The four points split the curve into four pieces corresponding to four of the six edges of the tetrahedron.

“The four points above split the curve into four pieces, corresponding to four of the six edges of the tetrahedron.”

Letting α^2 = 1 - t3, the curve (41) passes through the four points [α, α, α^2], [-α, -α, α^2], [α, -α, -α^2], and [-α, α, -α^2] when sin u = ± sin v.

“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.”

Unlike the case t3 = 0, these lines are not extreme points, even in a generalized sense.

“Unlike the case t 3 = 0 these lines are not extreme points, even in a generalized sense.”

This curve can be written in parametric form as a pair of curves (41) with sin^{-1}(t3) ≤ u ≤ π - sin^{-1}(t3).

“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 ).”

The parameters are constrained by the Algoet-Fujiwara inequalities (λ1 ± λ2)^2 ≤ (1 ± λ3)^2.

“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 .”

• cites [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 sourceNot checked yet

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• cites [11] Horn, 1991Not checked yet

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Any positivity-preserving map on M2 can be reduced by changes of basis to the diagonal form (3).

“King and Ruskai [11] showed that any such map can be reduced, via changes of basis in C 2 , to the form”

• cites King and Ruskai [11]Not checked yet

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For n = 2, Choi's condition is equivalent to (⊗ I)(B0) ≥ 0, where B0 projects onto the maximally entangled Bell state.

“We are interested in n = 2, in which case this condition is equivalent [10] to ( ⊗ I ) (B 0 ) 0,”

• cites [10] Horn, 1985Not checked yet

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Under the reduction (4), a map is completely positive exactly when its reduced map is completely positive.

“The crucial point about the reduction (4) is that is completely positive if and only if t, is.”

Niu and Griffiths's construction showed that taking the partial trace T1 instead of T2 is equivalent to swapping sin and cos in (17).

“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).”

• cites [18] Lieb, 1974, Some operator inequalities of the Schwarz type, Adv. Math. sourceNot checked yet

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It is not accurate to view a stochastic map as the composition of a rotation, a contraction to an ellipsoid, a translation, and another rotation, because the individual maps in that process are no longer stochastic.

“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.”

Even when |λ3| > μ^2 so the map is not extreme, the ellipsoid is shifted to the boundary in a manner analogous to an amplitude damping channel.

“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.”

A map of form (3) with all three t_k non-zero does not necessarily require three extreme points; in general two maps in the generalized-point set suffice.

“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.”

A stochastic map is an extreme point of S exactly when it can be written in Kraus form with the identity trace-preserving condition and the set of products Aj Ak† is linearly independent.

“Lemma 9. A stochastic map is an extreme point of S if and only if it can be written in the form (1) (necessarily with k A k A † k = I ), so that the set of matrices {A j A † k } is linearly independent.”

• cites [4] Choi, 1975, Completely positive linear maps on complex matrices, Linear Algebra Appl. sourceNot checked yet

  Not checked in the free run.

When two distinct extreme points have a common pair of points on the image of the Bloch sphere, their pre-images must be distinct.

“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.”

By Lemma 9, the corresponding stochastic map is extreme when both sin v and sin u are non-zero, equivalently when t3 is non-zero and all |λk| are less than 1.

“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.”

Example (2b) shows that the ellipsoid picture, while useful, is incomplete.

“Example (2b) illustrates that the ellipsoid picture, while useful, is incomplete.”

The statement that extreme points correspond to maps for which two or more image points lie on the Bloch sphere is correct only if a single point at a pole is interpreted as a pair of degenerate images when |λ1| = |λ2|.

“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 |.”

The implications (ii) ⇒ (iii) ⇒ (i) in full generality follow from reduction of every stochastic map to form (3) and invariance of the conditions under changes of bases.

“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.”

One can find maps with all |λk| < 1 for which the image (D) is an ellipsoid strictly contained within the unit ball but no translation in any direction is possible.

“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.”

Maps for which exactly one of sin v and sin u is zero cannot be extreme.

“Hence maps for which exactly one of {sin v, sin u} is zero cannot be extreme.”

As t3 moves away from zero, the corners of the tetrahedron are replaced by these points, forming an asymmetric rounded tetrahedron.

“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.”