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 M_n denote the space of n×n complex matrices. Assume that we are given Hermitian matrices A = A^* ∈ M_n and B = B^* ∈ M_m, as well as a Hermicity-preserving linear map Φ : M_n → M_m (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:

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} ∈ M_r and a complex vector c ∈ C^s. The associated semidefinite program is given by the following pair of optimization problems:

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 : M_n → (M_m ⊕ M_n) by

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

The dual map Ψ^† is given by

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

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 {E_a} and {F_a} be families of left and right generalized Choi-Kraus operators for Ψ. Denote the (k,l)-entry of P by p_kl and the (i,j)-entry of E_a or F_a by e_aij or f_aij, respectively. Then

where

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:

• QuantumSDP.pdf -- PDF version of this blog post

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, D.W. Kribs, Schmidt norms for quantum states. Preprint (2009). arXiv:0909.3907 [quant-ph]

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

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

Definition. Let A be a subalgebra of M_n and let π : A → M_n. 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 M_n such that, when E is restricted to it, E is a *-homomorphism (I keep saying “roughly speaking” because of the “∀b ∈ M_n” 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: M_n → M_n be a completely positive map such that E(I) = I. Then

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) = M_n, 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: M_n → M_n be a quantum channel (i.e., a completely positive map such that Tr(E(a)) = Tr(a) for all a ∈ M_n) 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 M_n, let E : M_n → M_n be completely positive, and let π : A → M_n be a *-homomorphism. Then the multiplicative domain of E with respect to π, denoted MD_π(E), is the algebra given by

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 MD_π(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 Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2003.

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. 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 M_n, a quantum channel E is defined to be a completely positive, trace-preserving map. That is, it is a map of the form

where {A_j} ∈ M_n is a family of matrices. Trace-preservation of E is equivalent to the requirement that

In 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 ∈ Cⁿ be complex vectors. Then x majorizes y if and only if y = Dx for some doubly stochastic matrix D ∈ M_n.

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

• Lemma of the Month #1: Unital Channel Eigenvalue Majorization [pdf]

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

This page contains information about all of my publications, both published and pending. Note that some entries below contain links to additional discussion, code, or slides that are not included with the publications themselves.

Submitted

1. N. Johnston and D. W. Kribs, A Family of Norms With Applications in Quantum Information Theory II. Preprint (2010).

Published/Accepted

1. N. Johnston and D. W. Kribs, A Family of Norms With Applications in Quantum Information Theory. To appear in Journal of Mathematical Physics (2010).
2. N. Johnston and D. W. Kribs, Generalized Multiplicative Domains and Quantum Error Correction. To appear in Proceedings of the American Mathematical Society (2010).
3. N. Johnston, The B36/S125 “2×2” Life-Like Cellular Automaton. Book chapter to appear in Game of Life Cellular Automaton (2010).
4. N. Johnston and D. W. Kribs, Schmidt Operator Norms and Entanglement Theory. Fourth International Conference on Quantum, Nano and Micro Technologies, 92-95 (2010).
5. M.-D. Choi, N. Johnston and D. W. Kribs, The Multiplicative Domain in Quantum Error Correction. Journal of Physics A: Mathematical and Theoretical 42 245303 (2009).
6. N. Johnston, D. W. Kribs and V. Paulsen, Computing Stabilized Norms for Quantum Operations. Quantum Information & Computation 9 1 & 2, 16-35 (2009).
7. N. Johnston, D. W. Kribs and C.-W. Teng, An Operator Algebraic Formulation of the Stabilizer Formalism for Quantum Error Correction. Acta Applicandae, 0167-8019 (2009).

Unpublished Notes

1. N. Johnston, Partially Entanglement Breaking Maps and Right CP-Invariant Cones. Unpublished preprint (2008).

My main research interests lie with solving mathematical and computational problems motivated by questions in quantum information theory. In particular, I make use of operator algebras and matrix analysis to tackle the field of quantum error correction. My recent work includes implementing an algorithm to estimate the completely bounded norm for arbitrary linear maps between complex square matrix spaces, and developing a link between the multiplicative domain of a quantum channel and its correctable subsystems.

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• Standard CV [pdf]
• Detailed CV [pdf]

Related Links:

• LaTeX CV Template – A description of the template that my CV uses.