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]

One of the things that has bothered me severely with the status of Conway’s Game of Life on the internet (and the main reason that I started the LifeWiki) is the severe fragmentation of information about the game — there are tidbits of knowledge sprinkled all over the place, but it’s quite a task to find a complete collection of patterns of a specific type unless you already know where to look. Fortunately, this fragmentation problem just got knocked around quite a bit by the release of Golly 2.1.

Golly is an open-source, cross-platform application for exploring Conway’s Game of Life (and it is probably currently the most widely-used such program). Version 2.1 was just released this week, and it’s a particularly exciting update from my point of view because it introduces a feature that has been long-needed in the Game of Life world — access to online pattern collections.

The pattern collections that Golly 2.1 can access by default are as follows:

• The LifeWiki pattern archive
• David Eppstein’s database of gliders in Life-like cellular automata
• Alan Hensel’s pattern collections
• Jason Summers’ pattern collections

Additionally, Golly can directly download rules from the cellular automata Rule Table Repository and scripts from the Golly Scripts Database. So now all the interested Lifer has to do to find out about (for example) period 51 oscillators is open up the LifeWiki pattern archive, select “oscillators”, and either load a relevant pattern or click on the help link beside it to bring up the corresponding page at LifeWiki. Take that, fragmentation of information.

Anyway, other changes have of course been made for the new release of Golly as well — a complete list can be found here. Or just go right ahead and…

Download Golly

About once a year, the internet news fills up for a week or so with talk of how a new largest-known prime has just been found. This largest-known prime has invariably been found by GIMPS, a distributed computing project designed to find large Mersenne primes.  Of course, mainstream media doesn’t like reporting things unless they can give people the illusion of some sort of immediate practical purpose. So what to do when you can’t think of a practical use for some recently-discovered 10-million-digit prime numbers? Make one up, of course! Just say that they have applications in cryptography:

Scientists in the US and Germany have found the two largest prime numbers ever calculated in a discovery which could dramatically increase the effectiveness of cryptographic systems.

– v3.co.uk

The Source of the Myth: RSA Encryption

Like all good myths, the Mersenne prime cryptography myth is so widespread because it is so close to being true. The most widely-used form of encryption used on the internet is RSA encryption, which works by multiplying two huge prime numbers together to form an even larger number with exactly two prime factors. Since factoring numbers is believed to be computationally difficult, reversing this process is currently a very difficult problem, which leads to RSA providing reasonably strong encryption. The thing is, RSA typically uses primes that have a few hundred digits, not a few million digits. Some of the reasons for this are as follows:

1. You don’t need to use million-digit primes. Considering that even cracking RSA that uses 250-digit primes is an extremely difficult problem that hasn’t been completed yet, and the problem gets exponentially more difficult as you add more digits, even the most paranoid of people should be comfortable using primes with a couple thousand digits. You might argue that some big government agencies would want RSA to be as secure as possible for their transactions, so they might want to use million-digit primes, but any agency that is that worried about security shouldn’t be using public key cryptography in the first place.
2. Using primes with millions of digits actually decreases security. As of this writing, there are 26 known primes with more than one million digits, so to break RSA encryption that makes use of primes with millions of digits you can just test each one of the known million-digit primes to see if they are one of the factors. RSA only works because there are lots of primes with hundreds of digits to choose from (as in billions of billions of billions of them, and then some).
3. Manipulating numbers with millions of digits is slow. Internet-based public key cryptography systems need to be fast if they’re to be of any practical use, so it doesn’t make much sense to try to use a cryptography system that relies on multiplying and finding residues with numbers that take several megabytes just to store. Just imagine trying to do some online banking when you have to transmit this number along with every other piece of data that you send back to the server.

Not all media outlets are so bad as to directly say that the primes found by GIMPS are useful for cryptography, but the vast majority of them imply it at some point throughout the story. Consider the following examples, which are taken from stories about newly-discovered GIMPS primes:

Mersenne primes are important for the theory of numbers and they may help in developing unbreakable codes and message encryptions.

– BBC News

Current cryptographic systems rely on the challenge of factoring large primes.

– ScienceNews.org

While those tidbits of information are quite true (well, almost — see the comments), when taken in context they are entirely misleading and cause the reader to think that GIMPS primes have applications in today’s cryptography systems. It’s like running a story about a recent plane crash that includes a sentence about how it’s a good idea to wear a helmet when riding a bicycle.

So Why Do We Search for Huge Primes?

The main reason that we search for huge primes is simply for sport. It gives our idle CPU cycles something to do. Non-mathematicians seem to balk at that idea and call it a huge waste of CPU cycles/time, and they’re probably right, but so what? Have you ever played a video game? This is our version of going for a high score. If that doesn’t seem like a particularly good reason to you, perhaps one of the reasons given by GIMPS itself will satisfy you. One thing that you’ll notice though is that cryptography is not mentioned anywhere on that page.

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).

One of the most interesting patterns that has ever been constructed in Conway’s Game of Life is primer, a gun that fires lightweight spaceships that represent exactly the prime numbers. It was constructed by Dean Hickerson way back in 1991, yet arguably no pattern since then has been constructed that’s as interesting. It seems somewhat counter-intuitive at first that the prime numbers, which seem somehow “random” or “unpredictable”, can be generated by this (relatively simple) pattern in the completely deterministic Game of Life.

Primer, the prime-generating gun

The gun works by firing lightweight spaceships westward, and destroying them via glider guns that emulate the Sieve of Eratosthenes. A lightweight spaceship makes it past the left edge of the gun at generation 120N if and only if N is a prime number (though for technical reasons, 2 and 3 are not outputted).

The first six lightweight spaceships output by primer

It wasn’t too long after making primer that Hickerson realized that he could attach a gun to the bottom-left corner of it to turn it into a twin prime calculator by allowing each lightweight spaceship through only if another lightweight spaceship passed through 240 generations earlier. Similarly, Jason Summers constructed a Fermat prime calculator in 2000 by shooting a glider at the lightweight spaceship stream every generation of the form 120(2^N + 1), which ends up detecting exactly the Fermat primes.

So what other families of primes can we compute in Life by altering the output of the original prime-generating gun?

Mersenne Primes

Mersenne primes can easily be computed using the exact same method as was used in the Fermat prime calculator — use a 7-engine Cordership (in blue below) to bounce a glider back at the stream of lightweight spaceships, with the time required for the glider to reach the stream doubling each time. An inverter (in green below) eliminates all lightweight spaceships that try to get past it unless it just received a glider from the Cordership. By fiddling around with timing a tiny bit, we then have a Mersenne prime calculator:

Mersenne Prime Calculator

• Java link: here
• RLE file: here

Prime Quadruplets

Four prime numbers are said to form a prime quadruplet if they are of the form (p, p+2, p+6, p+8) for some prime number p, which is the closest that four prime numbers can be together (except for the degenerate cases of (2,3,5,7) and (3,5,7,11)). Prime quadruplets are easy to compute because they can be thought of as consecutive pairs of twin primes. Since we already have a twin prime calculator, we can just repeat its reaction.

The twin prime calculator works by attaching a period 240 gun (in green below) to the bottom-left corner of primer. If it is timed correctly, it has the effect of allowing a lightweight spaceship through at generation 240N if and only if a lightweight spaceship tried to pass through at generation 240(N-1). Thus, it will only allow a lightweight spaceship through if it represents a prime number of the form p+2, where p is another prime number. Well, simply attaching a period 720 gun (in blue below) then allows a spaceship through at generation 720N if and only if a lightweight spaceship tried to pass through at generation 720(N-1). This has the effect of allowing a lightweight spaceship to pass through only if it represents a twin prime pair (p,p+2), and there is another twin prime pair of the form (p-6,p-4). That is, the only lightweight spaceships allowed through are those representing the upper members of prime quadruplets.

Prime quadruplet calculator

• Java link: here
• RLE file: here

Prime Pairs of the Form (p, p+2k)

The twin prime calculator mentioned earlier gives a way of computing prime pairs of the form (p,p+2), but what about pairs where the gap is larger than 2? For example, the k=2 case gives what are known as cousin primes, and the k=3 case gives sexy primes (yes, really).

For the case of cousin primes, the thing to notice is that every pair of cousin primes (except for the first pair, (3,7)) must be of the form (6n+1, 6n+5) for some natural number n. Thus, we can use two period 720 guns (in blue below) to allow only the upper prime in a cousin prime pair to pass through. This is achieved by having the top gun fire at the lightweight spaceships representing primes of the form 6n+1 — if a lightweight spaceship is hit, then a block is created in the path of the other gun, which is fired at lightweight spaceships representing primes of the form 6n+5. If a prime was present at 6n+1, then the lightweight spaceship makes it through unharmed at 6n+5. If there was no prime present at 6n+1, then the bottom gun destroys the lightweight spaceship representing 6n+5.

Cousin prime calculator

• Java link: here
• RLE file: here

Extending this idea to prime pairs of the form (p,p+2k) for k ≥ 3 is a bit more challenging, however, because it is possible for pairs to overlap. For example, (37,43) is a sexy prime pair, as is (41,47). Up until now we have only been able to detect single pairs at a time, since the block that acts as our “counter” that keeps track of whether a prime was detected earlier is placed in the stream of incoming lightweight spaceships. Thus, if it’s possible for two pairs to overlap, we will get lightweight spaceships colliding with the block, causing a mess.

To get around this problem, we use a device (known as a fanout, in green below) that duplicates the stream of lightweight spaceships. We then check for certain pairs on one stream, and the rest of the pairs on the other stream (these devices are outlined in blue below). Once we’re done, we merge the resulting streams of lightweight spaceships back together (using the devices in purple below).

To make this process a bit more explicit, I present a gun that computes prime pairs of the form (p,p+8). In particular, a lightweight spaceship will make it past the left edge of this pattern at about generation 1620+120N if and only if both N and N+8 are prime.

(p, p+8) prime calculator

• Java link: here
• RLE file: here

We now have all of the tools needed to build any pattern that computes prime pairs of the form (p, p+2k) as long as k = 1 or 2 (mod 3), though we may need to use the fanout device multiple times if it’s possible for more than one pair to overlap. If k = 0 (mod 3), however, it’s much more difficult to construct the desired pattern, because not only can you have overlapping prime pairs like (5, 11) and (7, 13), but you can have prime pairs in sequence such as (5, 11) and (11, 17). This problem can be remedied using the same tools as used in the (p,p+8) prime calculator, though you may need to use a lot of fanout devices to make things work. For example, computing the sexy primes using these tools would require at least four fanouts, and some clever elimination logic on each of the resulting five lightweight spaceship streams. I don’t feel up to that task myself, but it’s nice to know that we have a method for constructing a sexy prime calculator.

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