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 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]

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

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 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 ∈ Rⁿ be real 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

I spent the beginning of this week at the 4th Workshop on Theory of Quantum Computation, Communication and Cryptography, and I decided that I might as well post some information and thoughts about it since it was probably the best quantum workshop that I’ve attended thus far (not least because it was only three days long and the Institute for Quantum Computing knows how to do food right). There were some fifteen or so talks, and here I’ll describe two of my personal favourites.

Quantum algorithm for online memory checking

Wim van Dam and Qingqing Yuan

In his talk on online memory checking, Wim van Dam described the problem of trying to retrieve data from an unreliable and public server. Because the server is public, an adversary could tamper with its data, rendering the information that you retrieve useless. The online memory checking question of interest is then how to make sure that the requested information has not been tampered with.

One obvious (albeit largely useless) approach to making sure that the public server has not been tampered with is to simply keep a copy of its entire contents on a private server as well. When data is requested from the public server, check to see if it matched the data on the private server, and you’re good to go. This carries the obvious problem that you have to keep two copies of all of the data.

The obvious improvement to this is to use some sort of hashing function on the contents of the public server, store the results on a private server, and then check to see if the hash of your retrieved data equals the hash of the stored data whenever you access the public server. The question of how large the hash (or whatever encoding of the information you use) must be was answered in 1994 by Blum et al. and in 2005 by Naor and Rothblum: s × t ∈ Ω(n), where n is the size of the data on the public server, s is the size of the data stored on the private server, and t is the number of queries made to the public server.

Wim van Dam and Qingqing Yuan show that if the memory checker is allowed to act in a quantum mechanical manner, then the complexity of the problem is reduced to s × t ∈ Ω(polylog(n)). Score 1 for quantum algorithms.

A necessary condition for approximate quantum error correction

Cedric Bény

It’s not particularly surprising that this talk spoke to me pretty directly, as it deals with the same approach to the same subarea of the same field of research that I do. That and my name was mentioned in the talk. Score 10 for Cedric Bény.

The main result of the talk was a generalization of the Knill-Laflamme conditions to the realm of approximate quantum error correction (as opposed to perfect quantum error correction).  The idea was that the Knill-Laflamme conditions can be restated in terms of the complementary channel in a very natural way: a given subspace is correctable for a channel if and only if the complementary channel sends everything supported on that subspace to a single fixed operator (another way of saying this is that the subspace is private for the complementary channel, a connection that was explored by Kretschmann, Kribs and Spekkens in 2007).

It turns out that there is a natural way to generalize this formulation of the Knill-Laflamme conditions. The idea is that if the complementary channel, when restricted to a given subspace, is within a small distance of a channel that maps everything to a fixed operator (where distance is measured via the diamond norm), then that subspace is approximately correctable (and he of course quantifies the “small distance” and “approximately”).

He also provides a generalization of these results to approximate error correction of subsystem codes, which is a nice bonus.  Finally, it’s interesting to note that the set of operators that are approximately correctable by a given recovery operation does not necessarily form an algebra, in contrast to the perfect error correction situation.