This is a continuation of this post.
Please read that post to learn what the Choi-Jamiolkowski isomorphism is.

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 C_Φ 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 (id_n ⊗ 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 {p_i} forms a probability distribution (i.e., p_i ≥ 0 for all i and the p_i‘s sum to 1) and each σ_i and τ_i 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 C_Φ 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 A_i 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 A_i — 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 (id_n ⊗ 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 A_i 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(I_n) = I_m. 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(C_Φ) always equals exactly Φ(I_n).

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 ∈ M_n) 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 C_Φ 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 (id_n ⊗ 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 C_Φ 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 Choi-Jamiolkowski isomorphism is an isomorphism between linear maps from M_n to M_m and operators living in the tensor product space M_n ⊗ M_m. Given any linear map Φ : M_n → M_m, we can define the Choi matrix of Φ to be

It turns out that this association between Φ and C_Φ 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 M_n to M_m corresponds to the set of all linear operators in M_n ⊗ M_m. 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 {A_i} and {B_i} such that

To construct the operators A_i and B_i, simply reshape the left singular vectors and right singular vectors of the Choi matrix and multiply the A_i 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 C_Φ and recalling that Hermitian operators have real eigenvalues, it follows that there are real constants {λ_i} such that

Again, the trick is to construct each A_i so that the vectorization of A_i is the i^th eigenvector of C_Φ and λ_i 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 λ_i into the operator A_i, so we can write any Hermicity-preserving linear map in the form above, where each λ_i = ±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 C_Φ 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 (id_k ⊗ 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 C_Φ is positive semidefinite. It follows from the spectral decomposition theorem (much like in Section 2) that Φ can be written as

Again, the A_i operators (which are known as Kraus operators) are obtained by reshaping the eigenvectors of C_Φ. It also follows (and was proved by Choi) that Φ is completely positive if and only if (id_n ⊗ Φ) is positive. Also note that, as there exists an orthonormal basis of eigenvectors of C_Φ, the A_i operators can be constructed so that Tr(A_i^*A_j) = δ_ij, 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 (id_k ⊗ 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 C_Φ 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).

A matrix norm on M_n is said to be weakly unitarily-invariant if conjugating a matrix by a unitary U does not change the norm. That is,

Many commonly-used matrix norms are weakly unitarily-invariant, including the operator norm, Frobenius norm, numerical radius, Ky Fan norms and Schatten p-norms. One might naturally wonder whether there are matrix norms that satisfy the slightly stronger property of similarity-invariance:

Upon first glance there doesn’t seem to be any reason why this shouldn’t be possible — one can look for simple examples that cause problems, but you’ll have trouble coming up with a matrix that causes problems if you restrict your attention to “nice” (i.e., normal) matrices. Nevertheless, we have the following lemma, which appeared as Exercise IV.4.1 in [1]:

Lemma (No Similarity-Invariant Norm). Let f : M_n → R be a function satisfying f(SXS^-1) = f(X) for all X,S ∈ M_n with S invertible. Then f is not a norm.

If you’re interested in the (very short and elementary) proof of this lemma, see the pdf attached below. I would be greatly interested in seeing a proof of this fact that relies less on the structure of matrices themselves. It seems as though there should be a more general result that characterizes when we can and can not find a norm on a given vector space that is invariant with respect to some given subgroup, or some such thing. Would anyone care to enlighten me?

Related Links:

• Lemma of the Month #4: No Similarity-Invariant Matrix Norm [pdf]

References:

1. R. Bhatia, Matrix analysis. Volume 169 of Graduate texts in mathematics (1997).

In matrix analysis, there are several different matrix norms that you might use depending on the context of your particular problem. If you are treating the matrix as an operator acting on a the complex vector space Cⁿ, then you would likely use the operator norm. If you are considering the matrix as a density operator (i.e., if you’re a quantum information nerd like me) then you might want to use the trace norm. If you just want something that’s easy to calculate, you might be better off going with the Frobenius norm. These are three of the most well-studied and well-used matrix norms, and they have one very important thing in common — they are unitarily invariant. That is, if X ∈ M_n, then

Unitarily-invariant norms are particularly “nice” in that they satisfy submultiplicativity as well as various other desirable properties. Here I will present two particular families of unitarily-invariant norms, briefly discuss some of their applications, and then define a family of norms that encompass all of the other norms mentioned in this post as special cases.

Before proceeding, recall that for any matrix X ∈ M_n we can define the absolute value |X| of X to be the positive matrix square root of X^*X. Then the singular values of X, s₁(X), s₂(X), …, s_n(X), are defined to be the eigenvalues of |X|. Throughout this post we will assume that the singular values are ordered from largest to smallest (this is pretty standard practice when dealing with singular values):

Ky Fan Norms

Given a natural number k such that 1 ≤ k ≤ n, the Ky Fan k-norm of a matrix X ∈ M_n is defined to be the sum of the k largest singular values of X:

While Ky Fan norms aren’t extremely well-known, they have applications is matrix theory as well as quantum information theory. For example, they have recently appeared in [1] as a tool for determining whether a linear map from M_n to M_m is k-positive, which is one of the difficult open problems in quantum information. If P_k ⊆ M_n denotes the space of rank-k orthogonal projections (i.e., matrices such that P² = P^* = P), then it is not difficult to show that

Several properties of these norms are obvious from the definition — for example, the Ky Fan k-norm is upper-bounded by the Ky Fan (k+1)-norm and each Ky Fan norm is unitarily-invariant. One property that isn’t immediately obvious, however, is the following very cool result:

Fan Dominance Theorem [2, Section IV.2]. Let X, Y ∈ M_n. Then

if and only if

Schatten Norms

Given a real number p ≥ 1, the Schatten p-norm of a matrix X ∈ M_n is defined to be the standard vector p-norm of the vector of singular values of X:

There are numerous applications of Schatten norms in quantum information theory. For example, they are used to define completely bounded norms for linear maps acting on matrices, which are probably the most important norms for maps in quantum information (see [3] for a particular paper that deals with these norms). As with the Ky Fan norms, the Schatten norms are unitarily-invariant and can be equivalently defined via an expression involving the trace:

One of the other nice properties of the Schatten p-norms is a modified submultiplicativity result, which states that if X,Y ∈ M_n then

Everything In Between

We have now seen two families of norms based on the singular values of a matrix, both of which are very important in matrix analysis as well as quantum information theory. The Ky Fan norms are given by summing the first k singular values, while the Schatten norms are given by computing the standard vector p-norm of the vector of singular values. So why have I never seen the natural generalization of these two families of norms – the vector p-norm of the first k singular values – defined? (Update [May 14, 2012]: See the comments for a few references that study these norms.)

Definition. Let X ∈ M_n, p ≥ 1 and 1 ≤ k ≤ n, with k a natural number. Then I define the (p,k)-singular norm of X to be

Notice that these norms are also unitarily-invariant, and as with the previously-defined norms, they are given by a relatively simple trace expression:

One particular case of these norms – the p = 2 case – actually appeared implicitly in [1], though they were referred to as Ky Fan norms. I have also found a need for the p = 2 case of these norms in a recent project of mine that will hopefully be wrapped up in the next month or so.

I will finish by pointing out some special cases of this norm:

• If we allow p = ∞ by taking the limit as p → ∞ in the above definition, then the (∞,k)-singular norm coincides with the standard operator norm, regardless of k.
• When p = 1, the (1,k)-singular norm is exactly the Ky Fan k-norm.
• When k = n, the (p,n)-singular norm is exactly the Schatten p-norm.
• When p = 1, k = n (i.e., the Schatten 1-norm, which equals the Ky Fan n-norm), we recover exactly the trace norm.
• When p = 2, k = n (i.e., the Schatten 2-norm), we recover exactly the Frobenius norm.
• When p = 1, k = 1 (i.e., the Ky Fan 1-norm), we again obtain the operator norm.

References

1. D. Chruscinski, A. Kossakowski, Spectral Conditions for Positive Maps. Commun. Math. Phys. 290, 10511064 (2009). arXiv:0809.4909 [quant-ph]
2. R. Bhatia, Matrix analysis. Volume 169 of Graduate texts in mathematics (1997).
3. I. Devetak, M. Junge, C. King, M. B. Ruskai, Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266, 37-63 (2006). arXiv:quant-ph/0506196

Recall for matrices that a similarity transformation is one that takes a (square) matrix A to SAS^-1, where S is some invertible matrix of the same size as A. Standard linear algebra courses like to beat into students just how important similarity transformations are, which makes sense from their point of view because linear algebra is really more about studying operators than matrices themselves, and similarity transformations correspond to just a change of basis. The downside is that many people end up thinking of similarity as the equivalence relation among matrices and hence the Jordan canonical form as the canonical form. This month’s Lemma of the Month is about the canonical form under a different equivalence relation: *-congruence.

What is *-Congruence?

Two square matrices A and B are said to be *-congruent if there is an invertible matrix S such that SAS^* = B. Even though no inverses are present in that equation, invertibility really is required for any nice results to be obtained (for example, *-congruence would not even define an equivalence relation if S were allowed to be singular). For people who like knowing applications of mathematical tools before knowing the tools themselves, *-congruence has an important role when dealing with quadratic forms. Also, the inertia of a matrix, which will be defined in the next paragraph, comes into play when looking at sign pattern matrices (apologies for the lack of a wiki link; it seems as though sign pattern matrices are at that ugly stage after there are lots of papers written about them but before there is a Wikipedia page written about them).

Sylvester’s Law of Inertia

By far the most well-known result about *-congruence is Sylvester’s Law of Inertia, which gets its name from the seldom-used inertia of a matrix. The inertia of a Hermitian matrix A is defined to be the tuple

where n₊ is the number of positive eigenvalues of A, n₀ is the number of zero eigenvalues of A, and n_– is the number of negative eigenvalues of A (recall that Hermitian matrices have real eigenvalues, so this definition makes sense). Sylvester’s Law of Inertia is as follows:

Theorem [1,2]. Let A, B ∈ M_n be Hermitian. Then A and B are *-congruent if and only if they have the same inertia.

One obvious corollary of this theorem is that every Hermitian matrix A is *-congruent to a diagonal matrix with n₊ ones, n₀ zeroes, and n_– negative ones on the diagonal; this is the canonical form for Hermitian matrices under *-congruence.

Generalizations of Sylvester’s Law

If you’re a linear algebra nerd like me, then you might be thinking to yourself that Sylvester’s Law of Inertia seems trivial in light of the Spectral Decomposition Theorem; for any Hermitian A we could find a unitary so that UAU^* is real and diagonal, and then we could find a diagonal D such that DUAU^*D^* is scaled appropriately to satisfy the theorem. Setting S = DU then shows us that we can always arrive at the canonical form.

Notice, however, that the Spectral Decomposition Theorem applies not only to Hermitian matrices, but to normal matrices as well. Thus, using the same logic outlined in the previous paragraph, it stands to reason that the canonical form under *-congruence for normal matrices should be a diagonal matrix in which each diagonal entry either has modulus one or zero. Indeed, this is essentially the content of the following theorem of Ikramov, which was proved in 2001.

Theorem [3]. Let A, B ∈ M_n be normal. Then A and B are *-congruent if and only if they have the same number of eigenvalues on each open ray from the origin in the complex plane.

This theorem truly generalizes Sylvester’s Law of Inertia; in the Hermitian case, the only two open rays that eigenvalues can lie on are the positive real line and the negative real line. What I would like to know, though, is why on Earth this theorem was first proved/published so recently — it seems like such a natural generalization of Sylvester’s Law of Intertia that it seems like it should be a very old result! Is it just a hole in the literature that wasn’t noticed until recently?

Anyway, for the really interested reader, there is a generalization of these results to the case when A and B need not even be normal, but it’s quite technical so I won’t present it here. See [4] if you’re curious.

Thanks to Roger Horn for his great series of lectures at the Summer School and Advanced Workshop on Trends and Developments in Linear Algebra, which introduced me to the generalizations of Sylvester’s Law.

References:

1. Sylvester, J. J., A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. Philosophical Magazine IV: 138–142 (1852).
2. Horn, R. and Johnson, C., Matrix analysis. Cambridge University Press (1990).
3. Ikramov, Kh. D., On the inertia law for normal matrices. Doklady Math. 64 (2001) 141-142.
4. Horn, R. and Sergeichuk, V., Canonical forms for complex matrix congruence and *congruence. Linear Algebra Appl. 416 (2006) 1010-1032. arXiv:0709.2473v1 [math.RT]

This month’s Lemma of the Month comes all the way from the Summer School and Advanced Workshop on Trends and Developments in Linear Algebra in Trieste, Italy, where Professor Rajendra Bhatia presented a lecture that introduced several simple yet endlessly interesting matrix inequalities. I will briefly present the various results here without proof, though the proof of the first “stepping stone” lemma is provided in the PDF attached to the bottom of this post. The truly interested reader can find full proofs in Professor Bhatia’s notes (follow the link above) or in [1].

Recall that one of the defining properties of a matrix norm is that it satisfies the triangle inequality:

So what can we say about generalizing the backward triangle inequality to matrices? We can of course replace A by A – B in the above equation to find the following backward triangle inequality:

However, what happens if we swap the roles of the absolute value and the matrix norm on the left-hand side? That is, if we recall that |A| is the positive semidefinite part of A (i.e., the square root of A^*A), then can we say something like

It turns out that the answer to this question is heavily norm-dependent, so we will focus on the norm that gives the simplest answer: the Frobenius norm, which I will denote by ||•||₂.

Theorem [Araki-Yamagami]. Let A, B ∈ M_n. Then

Building Up to the Result

In order to prove the result, one can proceed by proving a series of simple commutant-type matrix norm inequalities, which are interesting in their own right.

Lemma. Let A, B ∈ M_n be positive semi-definite and let X ∈ M_n be arbitrary. Then

Lemma. Let A ∈ M_n be Hermitian. Then

Lemma. Let A, B ∈ M_n be Hermitian. Then

Finally, it just wouldn’t seem right to post from Italy without sharing a bit of it. Thus, I leave you with a taste of the highlight of the trip (excepting the mathematics, of course).

The colosseum as of eight hours ago

Related Links

• Lemma of the Month #2: A Backward Triangle Inequality for Matrices [pdf]

References

1. H. Araki and S. Yamagami, An inequality for the Hilbert-Schmidt norm, Commun. Math. Phys., 81 (1981) 89-98.