Lab #1: Intervals with Braid

The first week’s problem was based off the video game Braid. This problem ended up not working too well due to about 10 people in the class of ~600 having played of the game, and the rest being very confused by the idea of the cloud platform moving along with the main character (it makes sense if you’ve played the game, honest!).

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Lab #2: The Bat Man

The next week I decided to go a bit more mainstream and have the problem based on Batman chasing the Joker. The question doesn’t make a lick of sense if you think about it physically (the cars have negative acceleration for one thing), but this being a math class I decided not to care. I feel like this was the most successful of the weekly problems because the Batman/Joker stuff was completely incidental and the question was still easy enough for the students to understand and tackle.

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Lab #3: Calvin Reaches his Limit

By this point I had learned that even if I am going to sugar-coat the question under a picture of Calvin and Hobbes, it’s a good idea to include a sentence at the end summarizing what the heck it is I’m asking them to compute or prove. I think this is a fun question no matter how advanced of a mathematician you are, and it was probably a bit mean of me to present it to first-year students.

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Lab #4: Continuity with Mario

The last of these problems that I presented was a (very) simple continuity question based on Super Mario World. I originally wanted to use Super Metroid for this question since it would allow for more varied movements from the hero, but I decided that (as I learned in Lab #1) it would be best to stick with a more recognizable game. It was a pain to come up with a semi-nice-looking branch function that resembled Mario’s movement in a believable way and led to simple (i.e., non-fractional) limits at the points of interest.

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Recall from Section 1 of this post that any superoperator Φ can be written as

for some operators {A_i} and {B_i}. The isomorphism that I am going to focus on in this post is the one given by associating Φ with the operator

The main reason that M_Φ can be so useful is that it retains the operator structure of Φ. In particular, if you define vec(X) to be the vectorization of the operator X, then

In other words, if you treat X as a vector, then M_Φ is the operator describing the action of Φ on X. From this it becomes simple to compute some basic quantities describing Φ. For example, the induced Frobenius norm,

is equal to the standard operator norm of M_Φ. If n = m then we can define the eigenvalues {λ} and the eigenmatrices {V} of Φ in the obvious way via

Then the eigenvalues of Φ are exactly the eigenvalues of M_Φ, and the corresponding eigenvectors of M_Φ are the vectorizations of the eigenmatrices of Φ. It is similarly easy to check whether Φ is invertible (by checking whether or not det(M_Φ) = 0), find the inverse if it exists, or find the nullspace (and a pseudoinverse) if it doesn’t.

Finally, here’s a question for the interested reader to think about: why is the transpose required on the B_i operators for this isomorphism to make sense? That is, why can we not define an isomorphism between Φ and the operator

In the above definition, SR(w) refers to the Schmidt rank of the vector w and so these norms are in some ways like a measure of entanglement for pure state vectors. One of the results of [2] shows how to compute these norms efficiently, so with that in mind we can perform all sorts of fun numerical analysis on them. Analytic results are provided in the paper, so I’ll provide more hand-wavey stuff and pictures here. In particular, let’s look at what the distributions of the Schmidt vector norms look like.

Figure 1: The distribution of the Schmidt 1 and 2 vector norms in (3 ⊗ 3)-dimensional space

Figure 1 shows the distributions of the Schmidt 1 and 2 norms of unit vectors distributed according to the Haar measure in C³ ⊗ C³, based on 5×10⁵ vectors generated randomly via MATLAB. Note that the Schmidt 3-norm just equals the standard Euclidean norm so it always equals 1 and is thus not shown. Figures 2 and 3 show similar distributions in C⁴ ⊗ C⁴ and C⁵ ⊗ C⁵.

Figure 2: The distribution of the Schmidt 1, 2, and 3 vector norms in (4 ⊗ 4)-dimensional space

Figure 3: The distribution of the Schmidt 1, 2, 3, and 4 vector norms in (5 ⊗ 5)-dimensional space

The following table shows various basic statistics about the above distributions. I suppose the natural next step is to ask whether or not we can analytically determine the distribution of the Schmidt vector norms. Since these norms are essentially just the singular values of an operator that is associated with the vector, it seems like this might even already be a (partially) solved problem, since many results are known about the distribution of the singular values of random matrices. The difficulty comes in trying to interpret the Haar measure (or any other natural measure on pure states, such as the Hilbert-Schmidt measure) on the associated operators.

References:

1. D. Chruscinski, A. Kossakowski, G. Sarbicki, Spectral conditions for entanglement witnesses vs. bound entanglement, Phys. Rev A 80, 042314 (2009). arXiv:0908.1846v2 [quant-ph]
2. N. Johnston and D.W. Kribs, Schmidt norms for quantum states. Preprint (2009). arXiv:0909.3907 [quant-ph]

The glider

Lightweight spaceship

A natural question to ask is whether or not there are any spaceships that travel faster than c/4 diagonally or c/2 orthogonally. John Conway proved in 1970 (very shortly after inventing the Game of Life) that the answer is no. I present this proof here, since it’s a bit difficult to find online (though Dave Greene was kind enough to post a copy of it on the ConwayLife.com forums).

Theorem 1. The maximum speed that a spaceship can travel in Conway’s Game of Life is c/4 diagonally and c/2 orthogonally.

Proof. We begin by proving the c/4 speed limit for diagonal spaceships. Consider the grid given in Figure 1 (below). If the spaceship is on and to the left of the diagonal line of cells defined by A, B, C, D, and E in generation 0, then suppose that cell X can be alive in generation 2.

Figure 1: The spaceship is to the left of A, B, C, D, and E

Well, if cell X is alive in generation 2, then cells C, U, and V must be alive in generation 1. This means that U and V must have had 3 alive neighbours in generation 0, so each of B, C, D, J, and K must be alive in generation 0. This means that C must have at least four live neighbours in generation 0 though, so there is no way for it to survive to generation 1, which gives a contradiction.

It follows that X can not be alive in generation 2. In other words, if the spaceship is behind the diagonal line A, B, C, D, E in generation 0, then it must be behind the diagonal line defined by U and V in generation 2. It follows that can not travel faster than c/4 diagonally.

To see the corresponding result for orthogonal spaceships, just use two diagonal lines as in Figure 2. If a spaceship is on and below the diagonal lines defined by the solid black cells in generation 0, then we already saw that it must be on and below the diagonal lines defined by the striped cells in generation 2. It follows that it can not travel faster than c/2 orthogonally.

Figure 2: The spaceship is on and below the solid black cells in generation 0

Notice that this result doesn’t only apply to spaceships, but also to other configurations that are (initially) finite and travel across the grid, such as puffers and wickstretchers. Also, this result applies to many Life-like cellular automata — not just Conway’s Game of Life.

In particular, these speed limits apply to any of the 2¹² = 4096 Life-like cellular automata in the range B3/S – B345678/S0123678. That is, these speed limits apply to any rule on the 2D square lattice such that birth occurs for 3 neighbours but not 0, 1, or 2 neighbours, and survival does not occur for 4 or 5 neighbours. But are the spaceship speed limits attained in each of these rules? The regular c/4 glider only works in the 2⁸ = 256 rules from B3/S23 – B3678/S0235678. In the remaining rules, not much is known; some of them have c/3 orthogonal spaceships, some have c/5 orthogonal spaceships, and some have no spaceships at all (such as any of the rules containing S0123, which can not contain spaceships because the trailing edge of the spaceship could never die). Of particular interest are the sidewinder and this spaceship, which play the c/4 diagonal and c/2 orthogonal roles of the glider and lightweight spaceship, respectively, in B3/S13 (as well as several other rules).

So what about the other B3 (but not B0, B1, or B2) rules? If cells survive when they have 4 or 5 cells, then it’s conceivable that spaceships might be able to travel faster than c/4 diagonally or c/2 orthogonally because Theorem 1 does not apply to them. It turns out that they indeed can travel faster diagonally, but somewhat surprisingly they can not travel faster orthogonally.

Theorem 2. In any Life-like cellular automaton in which birth occurs when a cell has 3 live neighbours but not 0, 1, or 2 live neighbours, the maximum speed that a spaceship can travel is c/3 diagonally and c/2 orthogonally.

Proof. The trick here is to consider lines of slope -1/2 as in Figure 3 below. It is possible (though a bit more complicated) to prove the c/3 diagonal speed limit using a diagonal line as in Figure 1 for Theorem 1, but the orthogonal speed limit that results is 2c/3. What is presented here is the only method I know of proving both the diagonal speed limit of c/3 and the orthogonal speed limit of c/2.

Figure 3: The spaceship is below A, B, C, D, E, and F in generation 0

Suppose that a spaceship is on and below the line defined by the cells A, B, C, D, E, and F in Figure 3 in generation 0. It is clear that Y can not be alive in generation 2, since its only neighbour that could possibly be alive in generation 1 is K. Similarly, X can not be alive in generation 2 because its only neighbours that can be alive in generation 1 are B and K. It follows that in generation 2, the spaceship can not be more than 1 cell above the line A, B, C, D, E, F.

More mathematically, this tells us that the maximum speed of a spaceship that travels x cells horizontally for every y cells vertically can not travel faster than max{x,y}c/(x+2y). Taking x = y = 1 (diagonal spaceships) gives a speed limit of c/3. Taking x = 0, y = 1 (orthogonal spaceships) gives a speed limit of c/2.

Finally, it should be noted that even though these spaceship speed upper bounds apply to a wide variety of different rules, many rules don’t even have spaceships (even relatively simple rules containing B3 in their rulestring). For example, no spaceships are currently known in the rule “maze” (B3/S12345), and it seems quite believable that there are no spaceships to be found in that rule. I would love to see a proof that maze contains no spaceships, but it seems that there are too many cases to check by hand. I may end up trying a computer proof sometime in the near future.

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

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 deterministic method of testing whether or not an operator is block positive (and thus we do not have a 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).

Schmidt Decomposition Theorem

The Schmidt decomposition theorem says that any complex vector v ∈ Cⁿ ⊗ Cⁿ can be written as

where k ≤ n, {α_j} ⊆ R is a family of non-negative real scalars, and {e_j}, {f_j} ⊆ 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 ∈ M_n is defined to be

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

Furthermore, we may write X in this way using matrices B_j 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:

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

Because we chose the B_j matrices to be linearly independent, it follows that c_j = 0 for all j. By referring back to the definition of c_j, we see that this then implies A_j = 0 for all j, so X = 0 as desired. QED.

References:

1. N. Johnston and D.W. Kribs, Schmidt norms for quantum states. Preprint (2009). 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]

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]

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.

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).