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

October 23rd, 2009

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 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

\sigma=\sum_ip_i\sigma_i\otimes\tau_i,

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

\Phi(X)=\sum_iA_iXA_i^*,

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

\Phi(X)=\sum_iA_iXA_i^*,

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

{\rm Tr}_1(C_\Phi)=I_m,

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

{\rm Tr}_2(C_\Phi)=I_n,

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

(id_n\otimes T)(C_\Phi)\geq 0.

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

C_\Phi:=\sum_{i,j=1}^n|e_i\rangle\langle e_j|\otimes\Phi(|e_i\rangle\langle e_j|),\text{ where }\big\{|e_i\rangle\big\}\text{ is an orthonormal basis of $\mathbb{C}^n$}.

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

\Phi(X)=\sum_iA_iXB_i.

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

\Phi(X)=\sum_i\lambda_iA_iXA_i^*.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

(\langle a|\otimes\langle b|)C_\Phi(|a\rangle\otimes|b\rangle)\geq 0\quad\forall\,|a\rangle,|b\rangle.

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

\Phi(X)=\sum_iA_iXA_i^*.

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

\langle x|C_\Phi|x\rangle\geq 0\quad\forall\,|x\rangle\text{ with }SR(|x\rangle)\leq k.

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

{\bf v}=\sum_{j=1}^k\alpha_j{\bf e_j}\otimes{\bf f_j}

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

\big\|X\big\|_{S(k)}:=\sup_{{\bf v},{\bf w}}\big\{|{\bf w}^*X{\bf v}| : \|{\bf v}\|,\|{\bf w}\|\leq 1,SR({\bf v}),SR({\bf w})\leq k\big\}

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

X=\sum_jA_j\otimes B_j,\ \ {\bf v}={\bf v_1}\otimes{\bf v_2},\text{ and } \ {\bf w}={\bf w_1}\otimes{\bf w_2}.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:

{\bf w_2}^*\Big(\sum_jc_jB_j\Big){\bf v_2}=0 \ \text{ where }c_j={\bf w_1}^*A_j{\bf v_1} \ \ \forall \, j.

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

\sum_jc_jB_j=0.

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]

Quantum Semidefinite Programs

September 25th, 2009

In quantum information theory, semidefinite programs are often useful, as one is often interested in the behaviour of linear maps over convex sets. For example, they have very recently been used to compute the completely bounded norm of a linear map [1], prove that QIP = PSPACE [2], and bound a new family of norms of operators [3]. However, if you were to look at the standard form of a semidefinite program provided on the Wikipedia page linked above, you would likely only see some very superficial connections with the standard form of quantum semidefinite programs in references [1-3] — this post aims to bridge that gap and show that the two forms are indeed equivalent (or at the very least outline the key steps in proving they are equivalent).

The “Quantum” Form

Let Mn denote the space of n×n complex matrices. Assume that we are given Hermitian matrices A = A* ∈ Mn and B = B* ∈ Mm, as well as a Hermicity-preserving linear map Φ : Mn → Mm (i.e., a map such that Φ(X) is Hermitian whenever X is Hermitian). Then we can define a “quantum” semidefinite program to be the following pair of optimization problems:

Quantum Semidefinite Program

In the dual problem, Φ refers to the dual map of Phi — that is, the adjoint map in the sense of the Hilbert-Schmidt inner product. It is not surprising that many problems in quantum information theory can be formulated as an optimization problem of this type — completely positive maps (a special class of Hermicity-preserving maps) model quantum channels, positive semidefinite matrices represent quantum states, and the trace of a product of two positive semidefinite matrices represents an expectation value.

The Standard Form

In the more conventional set up of semidefinite programming, we are given matrices D and {G_i} ∈ Mr and a complex vector c ∈ Cs. The associated semidefinite program is given by the following pair of optimization problems:

Semidefinite Programming Standard Form

The interested reader should read on Wikipedia about how semidefinite programs generalize linear programs and how their theory of duality works. It is also important to note that semidefinite programs can be solved efficiently to any desired accuracy by a variety of different solvers, using a number of different algorithms. Thus, once we show that quantum semidefinite programs can be put into this standard form, we will be able to efficiently solve quantum semidefinite programs.

Converting the Quantum Form to the Standard Form

Define a linear map Ψ : Mn → (Mm ⊕ Mn) by

Psi

Then the requirement that $\Phi(P) \leq B$ and $P \geq 0$ is equivalent to
\[
\Psi(X) \leq \begin{bmatrix}B & 0 \\ 0 & 0 \end{bmatrix}.

Then the requirement that Ψ(P) ≤ B and P ≥ 0 is equivalent to

Psi Inequality

The dual map Ψ is given by

Psi Dual

By putting these last few steps together, we see that our original quantum semidefinite program is of the following form:

Simplified Quantum SDP

The inequality in the dual problem was able to be replaced by equality because of the flexibility that was introduced by the arbitrary positive operator R. Now let {Ea} and {Fa} be families of left and right generalized Choi-Kraus operators for Ψ. Denote the (k,l)-entry of P by pkl and the (i,j)-entry of Ea or Fa by eaij or faij, respectively. Then

Psi Reductionwhere

G_{kl}

Finally, defining x := vec(P) and c := vec(A) (where vec refers to the vectorization of a matrix, which stacks each of its columns on top of each other into a column vector) shows that the quantum primal problem is in the form of the standard primal problem. Some simple linear algebra can be used to show that the quantum dual form reduces to the standard dual form as well.

Downloads:

References:

  1. J. Watrous, Semidefinite programs for completely bounded norms. Preprint (2009). arXiv:0901.4709 [quant-ph]
  2. R. Jain, Z. Ji, S. Upadhyay, J. Watrous, QIP = PSPACE. Preprint (2009). arXiv:0907.4737 [quant-ph]
  3. 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]

A Brief Introduction to the Multiplicative Domain and its Role in Quantum Error Correction

July 24th, 2009

Given a completely positive linear map E: Mn → Mn, its multiplicative domain, denoted MD(E), is an algebra defined as follows:

MD1

Roughly speaking, MD(E) is the largest subalgebra of Mn on which E behaves multiplicatively. It will be useful to make this notion precise:

Definition. Let A be a subalgebra of Mn and let π : A → Mn. Then π is said to be a *-homomorphism if π(ab) = π(a)π(b) and π(a*) = π(a)* for all a,b ∈ A.

Thus, MD(E) is roughly the largest subalgebra of Mn such that, when E is restricted to it, E is a *-homomorphism (I keep saying “roughly speaking” because of the “∀b ∈ Mn” in the definition of MD(E) — the definition of a *-homomorphism only requires that the multiplicativity hold ∀b ∈ A). Probably the most well-known result about the multiplicative domain is the following theorem of Choi [1,2], which shows how the multiplicative domain simplifies when E is such that E(I) = I (i.e., when E is unital):

Theorem [Choi]. Let E: Mn → Mn be a completely positive map such that E(I) = I. Then

MD2

Let $\phi : \cl{L}(\cl{H}) \rightarrow \cl{L}(\cl{H})$ be a completely positive, unital map. Then
\begin{align*}
MD(\phi) = & \big\{ a \in \cl{L}(\cl{H}) : \phi(a)^{*}\phi(a) =
\phi(a^*a)\text{ and } \phi(a)\phi(a)^{*} =
\phi(aa^*)\big\}.
\end{align*}

One thing in particular that this theorem shows is that, when E(I) = I, the multiplicative domain of E only needs to be multiplicative within MD(E) (i.e., we can remove the “roughly speaking” that I spoke of earlier).

MD(E) in Quantum Error Correction

Before moving onto how MD(E) plays a role in quantum error correction, let’s consider some examples to get a better feeling for what the multiplicative domain looks like.

  • If E is the identity map (that is, it is the map that takes a matrix to itself) then MD(E) = Mn, the entire matrix algebra.
  • If E(a) = Diag(a) (i.e., E simply erases all of the off-diagonal entries of the matrix a), then MD(E) = {Diag(a)}, the set of diagonal matrices.

Notice that in the first example, the map E is very well-behaved (as well-behaved as a map ever could be); it preserves all of the information that is put into it. We also see that MD(E) is as large as possible. In the second example, the map E does not preserve information put into it (indeed, one nice way to think about matrices in the quantum information setting is that the diagonal matrices are “classical” and rest of the matrices are “quantum” — thus the map E(a) = Diag(a) is effectively removing all of the “quantumness” of the input data). We also see that MD(E) is tiny in this case (too small to put any quantum data into).

The above examples then hint that if the map E preserves quantum data, then MD(E) should be large enough to store some quantum information safely. This isn’t quite true, but the intuition is right, and we get the following result, which was published as Theorem 11 in this paper:

Theorem. Let E: Mn → Mn be a quantum channel (i.e., a completely positive map such that Tr(E(a)) = Tr(a) for all a ∈ Mn) such that E(I) = I. Then MD(E) = UCC(E), the algebra of unitarily-correctable codes for E.

What this means is that, when E is unital, its multiplicative domain encodes exactly the matrices that we can correct via a unitary operation. This doesn’t tell us anything about correctable codes that are not unitarily-correctable, though (i.e., matrices that can only be corrected by a more complicated correction operation). To capture these codes, we have to generalize a bit.

Generalized Multiplicative Domains

In order to generalize the multiplicative domain, we can require that the map E be multiplicative with another map π that is already a *-homomorphism, rather than require that it be multiplicative with itself. This is the main theme of this paper, which was submitted for publication this week. We define generalized multiplicative domains as follows:

Definition. Let A be a subalgebra of Mn, let E : Mn → Mn be completely positive, and let π : A → Mn be a *-homomorphism. Then the multiplicative domain of E with respect to π, denoted MDπ(E), is the algebra given by

MD3

It turns out that these generalized multiplicative domains are reasonably well-behaved and generalize the standard multiplicative domain in exactly the way that we wanted: they capture all correctable codes for arbitrary quantum channels (see Theorem 11 of the last paper I mentioned). Furthermore, there are even some characterizations of MDπ(E) analogous to the theorem of Choi above (see Theorems 5 and 7, as well as Corollary 12).

References:

  1. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras. Illinois Journal of Mathematics, 18 (1974), 565-574.
  2. V. I. Paulsen, Completely Bounded Maps and Operator AlgebrasCambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2003.

Unital Channel Eigenvalue Majorization

June 6th, 2009

I’ve decided that, starting today, I will once a month post a mathematical result that I find interesting and/or useful, but I feel sadly gets less attention than it deserves (Update: well, that lasted about four months). I will try to present all relevant preliminaries along with the result to provide context, so hopefully the results and proofs will be accessible to someone at the upper undergraduate level.

Since I’m a quantum kinda guy, it seems natural that the first such lemma deals with quantum information science. In particular, it helps quantify the behaviour of unital quantum channels acting on density operators. Before delving into the result, let’s begin with…

Quantum Preliminaries

Given the complex matrix space Mn, a quantum channel E is defined to be a completely positive, trace-preserving map. That is, it is a map of the form

Choi-Kraus representationwhere {Aj} ∈ Mn is a family of matrices. Trace-preservation of E is equivalent to the requirement that

Trace-preservationIn many physical situations we are interested in unital quantum channels; that is, channels that satisfy E(I) = I. Such channels in general exhibit much nicer behaviour than arbitrary quantum channels, and this month’s lemma will show one particular instance of this fact.

The Hardy-Littlewood-Polya Theorem

The proof of the lemma relies on a classical result known as the Hardy-Littlewood-Polya Theorem. The result explains how doubly stochastic matrices act on vectors. Since it seems to be surprisingly difficult to find on Wikipedia and other popular (read: non-research) websites, I will state it here.

Theorem [Hardy-Littlewood-Polya]. Let x,y ∈ Rn be real vectors. Then x majorizes y if and only if y = Dx for some doubly stochastic matrix D ∈ Mn.

It might be worth mentioning that the “if” direction of the proof is borderline trivial; the real meat and potatoes of the theorem is the “only if” direction.

The Lemma Itself

The lemma makes precise something that feels quite natural when thought of physically: a unital channel (that is, a completely positive, trace-preserving map E for which E(I) = I) can only increase the impurity (or “mixedness”) of quantum states. It has several simple consequences that are of great use when dealing with unital channels, and furthermore its proof makes excellent use of classical machinery. It was originally due to Uhlmann [1,2], but has recently appeared in [3]. The proof provided in the PDF attached at the end of this post is from the latter source.

Lemma [Unital Channel Eigenvalue Majorization]. Suppose ρ = E(σ) for a unital channel E. Then the ordered spectrum r of ρ is majorised by the ordered spectrum s of σ.

One particularly useful corollary of this lemma is presented here, and its proof is omitted (and dare I say left as an exercise for the reader?)

Corollary. If E is a unital quantum channel and ρ is a positive operator, then rank(E(ρ)) ≥ rank(ρ).

Related Links

References

  1. A. Uhlmann, Commun. Math. Phys. 54, 21 (1977).
  2. I. Bengtsson, and K. Zyczkowski, Geometry of quantum states, Cambridge University Press (2006).
  3. D. W. Kribs, R. W. Spekkens, Phys. Rev. A 74, 042329 (2006). arXiv:quant-ph/0608045v2