### Archive

Posts Tagged ‘Quantum Information Theory’

## Separability-Preserving Operators in Entanglement Theory

June 14th, 2011

One of the key concepts in quantum information theory is the difference between separable states and entangled states. A pure quantum state (that is, a unit vector) v ∈ CnCn is said to be separable if it can be written as v = a ⊗ b for some a,b ∈ Cn; otherwise v is called entangled. In this post we will investigate what operators preserve the set of separable pure states, as well as what operators entangle all separable pure states.

### Separable Pure State Preservers and Entangling Gates

In the design of quantum algorithms, entangling gates play a very important role. Entangling gates are unitary operators that are able to generate entanglement. A bit more specifically, a unitary operator U ∈ Mn ⊗ Mn (where Mn is the space of n × n complex matrices) is called an entangling gate if there exists a separable pure state v = a ⊗ b ∈ CnCn such that Uv is entangled. Conversely, we will say that a unitary operator U preserves separability if Uv is separable whenever v is separable.

In order to answer the question of what unitaries preserve separability, it is instructive to consider some simple examples (this is often a useful way to formulate conjectures regarding preserver problems). For example, it is clear that if U = A ⊗ B for some unitary operators A, B ∈ Mn, then U preserves separability (because U(a ⊗ b) = Aa ⊗ Bb is separable). Another example of a unitary operator that preserves separability is the swap (or flip) operator S defined on separable states by S(a ⊗ b) = b ⊗ a (the action of S on the rest of CnCn is determined by extending linearly). It turns out that these are essentially the only operators that preserve separability [1,2,3]:

Theorem 1. Let U ∈ Mn ⊗ Mn be a unitary operator. Then U preserves separability (i.e., U is not an entangling gate) if and only if there exist unitary operators A, B ∈ Mn such that either U = A ⊗ B or U = S(A ⊗ B).

As we already saw, the “if” direction of the above result is trivial – the meat and potatoes of the theorem comes from the “only if” direction (as is typically the case with results about linear preservers). Theorem 1 was first proved in [1] essentially by case analysis and checking the action of a separability-preserving unitary on a basis of CnCn, and was subsequently re-proved using similar techniques (but with different motivations and connections) in [2]. The result was proved in [3] by using the vector-operator isomorphism and the fact that a linear map Φ : Mn → Mn preserves the set of rank-1 operators if and only if there exist A, B ∈ Mn such that either Φ(X) ≡ AXB or Φ(X) ≡ AXtB [4].

Theorem 1 also follows as a simple corollary of several related results that have recently been proved in [5,6]. A version of Theorem 1 for multipartite systems (i.e., systems that are the tensor product of more than two copies of Cn) can be found in [3] and [7].

### Universal Entangling Gates

A universal entangling gate is, as its name suggests, a stronger form of an entangling gate – it is a unitary operator U such that U(a ⊗ b) is entangled for all a, b ∈ Cn (contrast this with entangling gates, which require only that U(a ⊗ b) is entangled for some a, b ∈ Cn). The structure of universal entangling gates is much less well-understood than that of entangling gates, though we can still at least say when they exist.

It is not difficult to convince yourself that universal entangling gates can’t exist in small dimensions. Let’s begin by supposing n = 2. The set of pure states in C2C2 can be regarded as a 7-dimensional real manifold (7 = 2 × (n × n) – 1, where we subtract one because pure states all have unit length), while the set of separable pure states in C2C2 can be regarded as a 5-dimensional real manifold (5 = (2 × n – 1) + (2 × n – 1) – 1, where the final one is subtracted because the overall phase of the first system relative to the second system is irrelevant). Thus, if U ∈ M2 ⊗ M2 were a universal entangler, it would have to send a 5-dimensional manifold into the 7 – 5 = 2 remaining dimensions of the space, which seems unlikely. Similarly, if n = 3 and U ∈ M3 ⊗ M3 were a universal entangler, it would have to send a 9-dimensional manifold into the 17 – 9 = 8 remaining dimensions of the space, which also seems unlikely.

Indeed, this type of argument was made rigorous via methods of algebraic geometry in [8], where the following result was proved:

Theorem 2. There exists a universal entangling gate in Mn ⊗ Mn if and only if n ≥ 4.

Despite knowing when universal entangling gates exist, we still don’t have a characterization of such operators, nor do we even have many explicit examples (does anyone have an explicit example for 3 ⊗ 4 or 4 ⊗ 4 systems?). Similar techniques to those used in the proof of Theorem 2 should also shed light on when universal entangling gates exist in multipartite systems Mn1 ⊗ Mn2 ⊗ … ⊗ Mnk, but to my knowledge this calculation has not been explicitly carried out.

References:

1. M. Marcus and B. N. Moyls, Transformations on tensor product spaces. Pacific Journal of Mathematics 9, 1215–1221 (1959).
2. F. Hulpke, U. V. Poulsen, A. Sanpera, A. Sen De, U. Sen, and M. Lewenstein, Unitarity as preservation of entropy and entanglement in quantum systems. Foundations of Physics 36, 477–499 (2006). E-print: arXiv:quant-ph/0407118
3. N. Johnston, Characterizing Operations Preserving Separability Measures via Linear Preserver Problems. To appear in Linear and Multilinear Algebra (2011). E-print: arXiv:1008.3633 [quant-ph]
4. L. Beasley, Linear operators on matrices: the invariance of rank k matrices. Linear Algebra and its Applications 107, 161–167 (1988).
5. E. Alfsen and F. Shultz, Unique decompositions, faces, and automorphisms of separable states. Journal of Mathematical Physics 51, 052201 (2010). E-print: arXiv:0906.1761 [math.OA]
6. S. Friedland, C.-K. Li, Y.-T. Poon, and N.-S. Sze, The automorphism group of separable states in quantum information theory. Journal of Mathematical Physics 52, 042203 (2011). E-print: arXiv:1012.4221 [quant-ph]
7. R. Westwick, Transformations on tensor spaces. Pacific Journal of Mathematics 23, 613–620 (1967).
8. J. Chen, R. Duan, Z. Ji, M. Ying, J. Yu, Existence of Universal Entangler. Journal of Mathematical Physics 49, 012103 (2008). E-print: arXiv:0704.1473 [quant-ph]

## The Other Superoperator Isomorphism

November 20th, 2009

A few months ago, I spent two posts describing the Choi-Jamiolkowski isomorphism between linear operators from Mn to Mm (often referred to as “superoperators“) and linear operators living in the space Mn ⊗ Mm. However, there is another isomorphism between superoperators and regular operators — one that I’m not sure of any name for but which has just as many interesting properties.

Recall from Section 1 of this post that any superoperator Φ can be written as

for some operators {Ai} and {Bi}. The isomorphism that I am going to focus on in this post is the one given by associating Φ with the operator

The main reason that MΦ can be so useful is that it retains the operator structure of Φ. In particular, if you define vec(X) to be the vectorization of the operator X, then

In other words, if you treat X as a vector, then MΦ is the operator describing the action of Φ on X. From this it becomes simple to compute some basic quantities describing Φ. For example, the induced Frobenius norm,

is equal to the standard operator norm of MΦ. If n = m then we can define the eigenvalues {λ} and the eigenmatrices {V} of Φ in the obvious way via

Then the eigenvalues of Φ are exactly the eigenvalues of MΦ, and the corresponding eigenvectors of MΦ are the vectorizations of the eigenmatrices of Φ. It is similarly easy to check whether Φ is invertible (by checking whether or not det(MΦ) = 0), find the inverse if it exists, or find the nullspace (and a pseudoinverse) if it doesn’t.

Finally, here’s a question for the interested reader to think about: why is the transpose required on the Bi operators for this isomorphism to make sense? That is, why can we not define an isomorphism between Φ and the operator

## Approximating the Distribution of Schmidt Vector Norms

November 6th, 2009

Recently, a family of vector norms [1,2] have been introduced in quantum information theory that are useful for helping classify entanglement of quantum states. In particular, the Schmidt vector k-norm of a vector v ∈ CnCn, for an integer 1 ≤ k ≤ n, is defined by

In the above definition, SR(w) refers to the Schmidt rank of the vector w and so these norms are in some ways like a measure of entanglement for pure state vectors. One of the results of [2] shows how to compute these norms efficiently, so with that in mind we can perform all sorts of fun numerical analysis on them. Analytic results are provided in the paper, so I’ll provide more hand-wavey stuff and pictures here. In particular, let’s look at what the distributions of the Schmidt vector norms look like.

Figure 1: The distribution of the Schmidt 1 and 2 vector norms in (3 ⊗ 3)-dimensional space

Figure 1 shows the distributions of the Schmidt 1 and 2 norms of unit vectors distributed according to the Haar measure in C3C3, based on 5×105 vectors generated randomly via MATLAB. Note that the Schmidt 3-norm just equals the standard Euclidean norm so it always equals 1 and is thus not shown. Figures 2 and 3 show similar distributions in C4C4 and C5C5.

Figure 2: The distribution of the Schmidt 1, 2, and 3 vector norms in (4 ⊗ 4)-dimensional space

Figure 3: The distribution of the Schmidt 1, 2, 3, and 4 vector norms in (5 ⊗ 5)-dimensional space

The following table shows various basic statistics about the above distributions. I suppose the natural next step is to ask whether or not we can analytically determine the distribution of the Schmidt vector norms. Since these norms are essentially just the singular values of an operator that is associated with the vector, it seems like this might even already be a (partially) solved problem, since many results are known about the distribution of the singular values of random matrices. The difficulty comes in trying to interpret the Haar measure (or any other natural measure on pure states, such as the Hilbert-Schmidt measure) on the associated operators.

Space k Mean Median Std. Dev.
C3C3 1 0.8494 0.8516 0.0554
2 0.9811 0.9860 0.0171
C4C4 1 0.7799 0.7792 0.0501
2 0.9411 0.9435 0.0247
3 0.9921 0.9943 0.0074
C5C5 1 0.7240 0.7225 0.0444
2 0.8976 0.8987 0.0268
3 0.9707 0.9722 0.0129
4 0.9960 0.9971 0.0039

References:

1. D. Chruscinski, A. Kossakowski, G. Sarbicki, Spectral conditions for entanglement witnesses vs. bound entanglement, Phys. Rev A 80, 042314 (2009). arXiv:0908.1846v2 [quant-ph]
2. N. Johnston and D. W. Kribs, A Family of Norms With Applications in Quantum Information Theory. Journal of Mathematical Physics 51, 082202 (2010). arXiv:0909.3907 [quant-ph]

## The Equivalences of the Choi-Jamiolkowski Isomorphism (Part II)

October 23rd, 2009

This is a continuation of this post.

In part 1, we learned about hermicity-preserving linear maps, positive maps, k-positive maps, and completely positive maps. Now let’s see what other types of linear maps have interesting equivalences through the Choi-Jamiolkowski isomorphism. Recall that the notation CΦ is used to represent the Choi matrix of the linear map Φ.

### 6. Entanglement Breaking Maps / Separable Quantum States

An entanglement breaking map is defined as a completely positive map Φ with the property that (idn ⊗ Φ)(ρ) is a separable quantum state whenever ρ is a quantum state (i.e., a density operator). A separable quantum state σ is one that can be written in the form

where {pi} forms a probability distribution (i.e., pi ≥ 0 for all i and the pi‘s sum to 1) and each σi and τi is a density operator. It turns out that the Choi-Jamiolkowski equivalence for entanglement-breaking maps is very natural — Φ is entanglement breaking if and only if CΦ is separable. Because it is known that determining whether or not a given state is separable is NP-HARD [1], it follows that determining whether or not a given linear map is entanglement breaking is also NP-HARD. Nonetheless, there are several nice characterizations of entanglement breaking maps. For example, Φ is entanglement breaking if and only if it can be written in the form

where each operator Ai has rank 1 (recall from Section 4 of the previous post that every completely positive map can be written in this form for some operators Ai — the rank 1 condition is what makes the map entanglement breaking). For more properties of entanglement breaking maps, the interested reader is encouraged to read [2].

### 7. k-Partially Entanglement Breaking Maps / Quantum States with Schmidt Number at Most k

The natural generalization of entanglement breaking maps are k-partially entanglement breaking maps, which are completely positive maps Φ with the property that (idn ⊗ Φ)(ρ) always has Schmidt number [3] at most k for any density operator ρ. Recall that an operator has Schmidt number 1 if and only if it is separable, so the k = 1 case recovers exactly the entanglement breaking maps of Section 6. The set of operators associated with the k-partially entanglement breaking maps via the Choi-Jamiolkowski isomorphism are exactly what we would expect: the operators with Schmidt number no larger than k. In fact, pretty much all of the properties of entanglement breaking maps generalize in a completely natural way to this situation. For example, a map is k-partially entanglement breaking if and only if it can be written in the form

where each operator Ai has rank no greater than k. For more information about k-partially entanglement breaking maps, the interested reader is pointed to [4]. Additionally, there is an interesting geometric relationship between k-positive maps (see Section 5 of the previous post) and k-partially entanglement breaking maps that is explored in this note and in [5].

### 8. Unital Maps / Operators with Left Partial Trace Equal to Identity

A linear map Φ is said to be unital if it sends the identity operator to the identity operator — that is, if Φ(In) = Im. It is a simple exercise in linear algebra to show that Φ is unital if and only if

where Tr1 denotes the partial trace over the first subsystem. In fact, it is not difficult to show that Tr1(CΦ) always equals exactly Φ(In).

### 9. Trace-Preserving Maps / Operators with Right Partial Trace Equal to Identity

In quantum information theory, maps that are trace-preserving (i.e., maps Φ such that Tr(Φ(X)) = Tr(X) for every operator X ∈ Mn) are of particular interest because quantum channels are modeled by completely positive trace-preserving maps (see Section 4 of the previous post to learn about completely positive maps). Well, some simple linear algebra shows that the map Φ is trace-preserving if and only if

where Tr2 denotes the partial trace over the second subsystem. The reason for the close relationship between this property and the property of Section 8 is that unital maps and trace-preserving maps are dual to each other in the Hilbert-Schmidt inner product.

### 10. Completely Co-Positive Maps / Positive Partial Transpose Operators

A map Φ such that T○Φ is completely positive, where T represents the transpose map, is called a completely co-positive map. Thanks to Section 4 of the previous post, we know that Φ is completely co-positive if and only if the Choi matrix of T○Φ is positive semi-definite. Another way of saying this is that

This condition says that the operator CΦ has positive partial transpose (or PPT), a property that is of great interest in quantum information theory because of its connection with the problem of determining whether or not a given quantum state is separable. In particular, any quantum state that is separable must have positive partial transpose (a condition that has become known as the Peres-Horodecki criterion). If n = 2 and m ≤ 3, then the converse is also true: any PPT state is necessarily separable [6]. It follows via our equivalences of Sections 4 and 6 that any entanglement breaking map is necessarily completely co-positive. Conversely, if n = 2 and m ≤ 3 then any map that is both completely positive and completely co-positive must be entanglement breaking.

### 11. Entanglement Binding Maps / Bound Entangled States

A bound entangled state is a state that is entangled (i.e., not separable) yet can not be transformed via local operations and classical communication to a pure maximally entangled state. In other words, they are entangled but have zero distillable entanglement. Currently, the only states that are known to be bound entangled are states with positive partial transpose — it is an open question whether or not other such states exist.

An entanglement binding map [7] is a completely positive map Φ such that (idn ⊗ Φ)(ρ) is bound entangled for any quantum state ρ. It turns out that a map is entanglement binding if and only if its Choi matrix CΦ is bound entangled. Thus, via the result of Section 10 we see that a map is entanglement binding if it is both completely positive and completely co-positive. It is currently unknown if there exist other entanglement binding maps.

References:

1. L. Gurvits, Classical deterministic complexity of Edmonds’ Problem and quantum entanglement, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, 10-19 (2003). arXiv:quant-ph/0303055v1
2. M. Horodecki, P. W. Shor, M. B. Ruskai, General Entanglement Breaking Channels, Rev. Math. Phys 15, 629–641 (2003). arXiv:quant-ph/0302031v2
3. B. Terhal, P. Horodecki, A Schmidt number for density matrices, Phys. Rev. A Rapid Communications Vol. 61, 040301 (2000). arXiv:quant-ph/9911117v4
4. D. Chruscinski, A. Kossakowski, On partially entanglement breaking channels, Open Sys. Information Dyn. 13, 17–26 (2006). arXiv:quant-ph/0511244v1
5. L. Skowronek, E. Stormer, K. Zyczkowski, Cones of positive maps and their duality relations, J. Math. Phys. 50, 062106 (2009). arXiv:0902.4877v1 [quant-ph]
6. M. Horodecki, P. Horodecki, R. Horodecki, Separability of Mixed States: Necessary and Sufficient Conditions, Physics Letters A 223, 1–8 (1996). arXiv:quant-ph/9605038v2
7. P. Horodecki, M. Horodecki, R. Horodecki, Binding entanglement channels, J.Mod.Opt. 47, 347–354 (2000). arXiv:quant-ph/9904092v1

## The Equivalences of the Choi-Jamiolkowski Isomorphism (Part I)

October 16th, 2009

The Choi-Jamiolkowski isomorphism is an isomorphism between linear maps from Mn to Mm and operators living in the tensor product space Mn ⊗ Mm. Given any linear map Φ : Mn → Mm, we can define the Choi matrix of Φ to be

It turns out that this association between Φ and CΦ defines an isomorphism, which has become known as the Choi-Jamiolkowski isomorphism. Because much is already known about linear operators, the Choi-Jamiolkowski isomorphism provides a simple way of studying linear maps on operators — just study the associated linear operators instead. Thus, since there does not seem to be a list compiled anywhere of all of the known associations through this isomorphism, I figure I might as well start one here. I’m planning on this being a two-parter post because there’s a lot to be said.

### 1. All Linear Maps / All Operators

By the very fact that we’re talking about an isomorphism, it follows that the set of all linear maps from Mn to Mm corresponds to the set of all linear operators in Mn ⊗ Mm. One can then use the singular value decomposition on the Choi matrix of the linear map Φ to see that we can find sets of operators {Ai} and {Bi} such that

To construct the operators Ai and Bi, simply reshape the left singular vectors and right singular vectors of the Choi matrix and multiply the Ai operators by the corresponding singular values. An alternative (and much more mathematically-heavy) method of proving this representation of Φ is to use the Generalized Stinespring Dilation Theorem [1, Theorem 8.4].

### 2. Hermicity-Preserving Maps / Hermitian Operators

The set of Hermicity-Preserving linear maps (that is, maps Φ such that Φ(X) is Hermitian whenever X is Hermitian) corresponds to the set of Hermitian operators. By using the spectral decomposition theorem on CΦ and recalling that Hermitian operators have real eigenvalues, it follows that there are real constants {λi} such that

Again, the trick is to construct each Ai so that the vectorization of Ai is the ith eigenvector of CΦ and λi is the corresponding eigenvalue. Because every Hermitian operator can be written as the difference of two positive semidefinite operators, it is a simple corollary that every Hermicity-Preserving Map can be written as the difference of two completely positive linear maps — this will become more clear after Section 4. It is also clear that we can absorb the magnitude of the constant λi into the operator Ai, so we can write any Hermicity-preserving linear map in the form above, where each λi = ±1.

### 3. Positive Maps / Block Positive Operators

A linear map Φ is said to be positive if Φ(X) is positive semidefinite whenever X is positive semidefinite. A useful characterization of these maps is still out of reach and is currently a very active area of research in quantum information science and operator theory. The associated operators CΦ are those that satisfy

In terms of quantum information, these operators are positive on separable states. In the world of operator theory, these operators are usually referred to as block positive operators. As of yet we do not have a quick deterministic method of testing whether or not an operator is block positive (and thus we do not have a quick deterministic way of testing whether or not a linear map is positive).

### 4. Completely Positive Maps / Positive Semidefinite Operators

The most famous class of linear maps in quantum information science, completely positive maps are maps Φ such that (idk ⊗ Φ) is a positive map for any natural number k. That is, even if there is an ancillary system of arbitrary dimension, the map still preserves positivity. These maps were characterized in terms of their Choi matrix in the early ’70s [2], and it turns out that Φ is completely positive if and only if CΦ is positive semidefinite. It follows from the spectral decomposition theorem (much like in Section 2) that Φ can be written as

Again, the Ai operators (which are known as Kraus operators) are obtained by reshaping the eigenvectors of CΦ. It also follows (and was proved by Choi) that Φ is completely positive if and only if (idn ⊗ Φ) is positive. Also note that, as there exists an orthonormal basis of eigenvectors of CΦ, the Ai operators can be constructed so that Tr(Ai*Aj) = δij, the Kronecker delta. An alternative method of deriving the representation of Φ(X) is to use the Stinespring Dilation Theorem [1, Theorem 4.1] of operator theory.

### 5. k-Positive Maps / k-Block Positive Operators

Interpolating between the situations of Section 3 and Section 4 are k-positive maps. A map is said to be k-positive if (idk ⊗ Φ) is a positive map. Thus, complete positivity of a map Φ is equivalent to Φ being k-positive for all natural numbers k, which is equivalent to Φ being n-positive. Positivity of Φ is the same as 1-positivity of Φ. Since we don’t even have effective methods for determining positivity of linear maps, it makes sense that we don’t have effective methods for determining k-positivity of linear maps, so they are still a fairly active area of research. It is known that Φ is k-positive if and only if

Operators of this type are referred to as k-block positive operators, and SR(x) denotes the Schmidt rank of the vector x. Because a vector has Schmidt rank 1 if and only if it is separable, it follows that this condition reduces to the condition that we saw in Section 3 for positive maps in the k = 1 case. Similarly, since all vectors have Schmidt rank less than or equal to n, it follows that Φ is n-positive if and only if CΦ is positive semidefinite, which we saw in Section 4.

Update [October 23, 2009]: Part II of this post is now online.

References:

1. V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2003.
2. M.-D. Choi, Completely Positive Linear Maps on Complex Matrices, Lin. Alg. Appl, 285-290 (1975).

## An Introduction to Schmidt Norms

October 2nd, 2009

In [1], a family of matrix norms (called Schmidt norms) are studied and some of their uses in quantum information theory are explored. The interested reader is of course welcome to read the results presented in that paper, but for the more casual reader I present here one very crucial preliminary, the Schmidt decomposition theorem, and a proof that the Schmidt norms actually are (as their name suggests) norms.

### Schmidt Decomposition Theorem

The Schmidt decomposition theorem says that any complex vector vCnCn can be written as

where k ≤ n, {αj} ⊆ R is a family of non-negative real scalars, and {ej}, {fj} ⊆ Cn are two orthonormal sets of vectors. I won’t prove the theorem here — a proof can be found on its Wikipedia page (it’s basically the singular value decomposition in disguise). For our purposes the most important thing to realize is that, for some vectors v, we can write v in its Schmidt decomposition with k < n. The least k such that v can be written in the form above is called the Schmidt rank of v, and we denote it by SR(v). Every vector v has SR(v) ≤ n.

### Schmidt Matrix Norms

The Schmidt k-norm of a matrix X ∈ Mn is defined to be

That might look like a horribly complex definition upon first glance, but it’s not so hard to get your head around when you realize that the Schmidt k-norm for k = n is simply the standard operator norm of X. It is clear then that the Schmidt k-norm for k < n must be a smaller quantity. Indeed, from a quantum information perspective, the norm measures how much the operator represented by X can stretch pure states that “aren’t very entangled.” The interested reader can learn about the various properties and applications of these norms in [1] — what I present here is simply a proof that the Schmidt k-norm is indeed a norm (since this is not explicitly done in the paper).

Proof that the Schmidt k-norm is a norm. It is clear from the definition that the absolute value of a constant pulls out of the Schmidt norms and that the Schmidt norms satisfy the triangle inequality. The only challenging property of the norm to verify is that the Schmidt norm of X being zero implies X = 0.

To prove this, assume that we are in the k = 1 case (if we can show that this property holds for k = 1, it immediately follows that the same property must hold for k > 1). Then recall that we can write X as the sum of elementary tensors, so we can write

Furthermore, we may write X in this way using matrices Bj that are linearly independent (see, for example, Proposition 24 of [2], or simply note that you could choose them to be a family of matrix units). Thus, if the Schmidt 1-norm of X equals zero, then it follows that for any v1, v2, w1, and w2:

Since this holds for any v2 and w2, it follows that

Because we chose the Bj matrices to be linearly independent, it follows that cj = 0 for all j. By referring back to the definition of cj, we see that this then implies Aj = 0 for all j, so X = 0 as desired. QED.

References:

1. N. Johnston and D. W. Kribs, A family of norms with applications in quantum information theory. Journal of Mathematical Physics 51, 082202 (2010). arXiv:0909.3907 [quant-ph]
2. Johnston, N., Kribs, D. W., and Paulsen, V., Computing stabilized norms for quantum operations. Quantum Information & Computation 9 1 & 2, 16-35 (2009). arXiv:0711.3636v1 [quant-ph]