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## How to compute hard-to-compute matrix norms

January 11th, 2016

There are a wide variety of different norms of matrices and operators that are useful in many different contexts. Some matrix norms, such as the Schatten norms and Ky Fan norms, are easy to compute thanks to the singular value decomposition. However, the computation of many other norms, such as the induced p-norms (when p ≠ 1, 2, ∞), is NP-hard. In this post, we will look at a general method for getting quite good estimates of almost any matrix norm.

The basic idea is that every norm can be written as a maximization of a convex function over a convex set (in particular, every norm can be written as a maximization over the unit ball of the dual norm). However, this maximization is often difficult to deal with or solve analytically, so instead it can help to write the norm as a maximization over two or more simpler sets, each of which can be solved individually. To illustrate how this works, let’s start with the induced matrix norms.

### Induced matrix norms

The induced p → q norm of a matrix B is defined as follows:

$\displaystyle\|B\|_{p\rightarrow q}:=\max\big\{\|B\mathbf{x}\|_q : \|\mathbf{x}\|_p = 1\big\},$

where

$\displaystyle\|\mathbf{x}\|_p := \left(\sum_{i}|x_i|^p\right)^{1/p}$

is the vector p-norm. There are three special cases of these norms that are easy to compute:

1. When p = q = 2, this is the usual operator norm of B (i.e., its largest singular value).
2. When p = q = 1, this is the maximum absolute column sum: $\|B\|_{1\rightarrow 1} = \max_j\sum_i|b_{ij}|$.
3. When p = q = ∞, this is the maximum absolute row sum: $\|B\|_{\infty\rightarrow \infty} = \max_i\sum_j|b_{ij}|$.

However, outside of these three special cases (and some other special cases, such as when B only has real entries that are non-negative [1]), this norm is much messier. In general, its computation is NP-hard [2], so how can we get a good idea of its value? Well, we rewrite the norm as the following double maximization:

$\displaystyle\|B\|_{p\rightarrow q}=\max\big\{|\mathbf{y}^*B\mathbf{x}| : \|\mathbf{x}\|_p = 1, \|\mathbf{y}\|_{q^\prime} = 1\big\},$

where $q^\prime$ is the positive real number such that $1/q + 1/q^\prime = 1$ (and we take $q^\prime = 1$ if $q = \infty$, and vice-versa). The idea is then to maximize over $\mathbf{x}$ and $\mathbf{y}$ one at a time, alternately.

1. Start by setting $j = 1$ and fixing a randomly-chosen vector $\mathbf{x}_0$, scaled so that $\|\mathbf{x}_0\|_{p} = 1$.
2. Compute

$\max\big\{|\mathbf{y}^*B\mathbf{x}_{j-1}| : \|\mathbf{y}\|_{q^\prime} = 1\big\},$

keeping $\mathbf{x}_{j-1}$ fixed, and let $\mathbf{y}_j$ be the vector attaining this maximum. By Hölder’s inequality, we know that this maximum value is exactly equal to $\|B\mathbf{x}_{j-1}\|_{q}$. Furthermore, the equality condition of Hölder’s inequality tells us that the vector $\mathbf{y}_j$ attaining this maximum is the one with complex phases that are the same as those of $B\mathbf{x}_{j-1}$, and whose magnitudes are such that $|\mathbf{y}_j|^{q^\prime}$ is a multiple of $|B\mathbf{x}_{j-1}|^q$ (here the notation $|\cdot|^q$ means we take the absolute value and the q-th power of every entry of the vector).

3. Compute

$\max\big\{|\mathbf{y}_j^*B\mathbf{x}| : \|\mathbf{x}\|_{p} = 1\big\},$

keeping $\mathbf{y}_j$ fixed, and let $\mathbf{x}_j$ be the vector attaining this maximum. By an argument almost identical to that of step 2, this maximum is equal to $\|B^*\mathbf{y}_j\|_{p^\prime}$, where $p^\prime$ is the positive real number such that $1/p + 1/p^\prime = 1$. Furthermore, the vector $\mathbf{x}_j$ attaining this maximum is the one with complex phases that are the same as those of $B^*\mathbf{y}_j$, and whose magnitudes are such that $|\mathbf{x}_j|^p$ is a multiple of $|B^*\mathbf{y}_j|^{p^\prime}$.

4. Increment $j$ by 1 and return to step 2. Repeat until negligible gains are made after each iteration.

This algorithm is extremely quick to run, since Hölder’s inequality tells us exactly how to solve each of the two maximizations separately, so we’re left only performing simple vector calculations at each step. The downside of this algorithm is that, even though it will always converge to some local maximum, it might converge to a value that is smaller than the true induced p → q norm. However, in practice this algorithm is fast enough that it can be run several thousand times with different (randomly-chosen) starting vectors $\mathbf{x}_0$ to get an extremely good idea of the value of $\|B\|_{p\rightarrow q}$.

It is worth noting that this algorithm is essentially the same as the one presented in [3], and reduces to the power method for finding the largest singular value when p = q = 2. This algorithm has been implemented in the QETLAB package for MATLAB as the InducedMatrixNorm function.

### Induced Schatten superoperator norms

There is a natural family of induced norms on superoperators (i.e., linear maps $\Phi : M_n \rightarrow M_n$) as well. First, for a matrix $X \in M_n$, we define its Schatten p-norm to be the p-norm of its vector of singular values:

$\|X\|_p := \left(\sum_{i=1}^n \sigma_i(X)^p\right)^{1/p}.$

Three special cases of the Schatten p-norms include:

• p = 1, which is often called the “trace norm” or “nuclear norm”,
• p = 2, which is often called the “Frobenius norm” or “Hilbert–Schmidt norm”, and
• p = ∞, which is the usual operator norm.

The Schatten norms themselves are easy to compute (since singular values are easy to compute), but their induced counter-parts are not.

Given a superoperator $\Phi : M_n \rightarrow M_n$, its induced Schatten p → q norm is defined as follows:

$\|\Phi\|_{p\rightarrow q} := \max\big\{ \|\Phi(X)\|_q : \|X\|_p = 1 \big\}.$

These induced Schatten norms were studied in some depth in [4], and crop up fairly frequently in quantum information theory (especially when p = q = 1) and operator theory (especially when p = q = ∞). The fact that they are NP-hard to compute in general is not surprising, since they reduce to the induced matrix norms (discussed earlier) in the case when $\Phi$ only acts on the diagonal entries of $X$ and just zeros out the off-diagonal entries. However, it seems likely that this norm’s computation is also difficult even in the special cases p = q = 1 and p = q = ∞ (however, it is straightforward to compute when p = q = 2).

Nevertheless, we can obtain good estimates of this norm’s value numerically using essentially the same method as discussed in the previous section. We start by rewriting the norm as a double maximization, where each maximization individually is easy to deal with:

$\|\Phi\|_{p\rightarrow q} = \max\big\{ |\mathrm{Tr}(Y^*\Phi(X))| : \|X\|_p = 1, \|Y\|_{q^\prime} = 1\big\},$

where $q^\prime$ is again the positive real number (or infinity) satisfying $1/q + 1/q^\prime = 1$. We now maximize over $X$ and $Y$, one at a time, alternately, just as before:

1. Start by setting $j = 1$ and fixing a randomly-chosen matrix $X_0$, scaled so that $\|X_0\|_p = 1$.
2. Compute

$\max\big\{|\mathrm{Tr}(Y^*\Phi(X_{j-1})| : \|Y\|_{q^\prime} = 1\big\},$

keeping $X_{j-1}$ fixed, and let $Y_j$ be the matrix attaining this maximum. By the Hölder inequality for Schatten norms, we know that this maximum value is exactly equal to $\|\Phi(X_{j-1})\|_{q}$. Furthermore, the matrix $Y_j$ attaining this maximum is the one with the same left and right singular vectors as $\Phi(X_{j-1})$, and whose singular values are such that there is a constant $c$ so that $\sigma_i(Y_j)^{q^\prime} = c\sigma_i(\Phi(X_{j-1}))^q$ for all $i$ (i.e., the vector of singular values of $Y_j$, raised to the $q^\prime$ power, is a multiple of the vector of singular values of $\Phi(X_{j-1})$, raised to the $q$ power).

3. Compute

$\max\big\{|\mathrm{Tr}(Y_j^*\Phi(X)| : \|X\|_{p} = 1\big\},$

keeping $Y_j$ fixed, and let $X_j$ be the matrix attaining this maximum. By essentially the same argument as in step 2, we know that this maximum value is exactly equal to $\|\Phi^*(Y_j)\|_{p^\prime}$, where $\Phi^*$ is the map that is dual to $\Phi$ in the Hilbert–Schmidt inner product. Furthermore, the matrix $X_j$ attaining this maximum is the one with the same left and right singular vectors as $\Phi^*(Y_j)$, and whose singular values are such that there is a constant $c$ so that $\sigma_i(X_j)^{p} = c\sigma_i(\Phi^*(Y_j))^{p^\prime}$ for all $i$.

4. Increment $j$ by 1 and return to step 2. Repeat until negligible gains are made after each iteration.

The above algorithm is almost identical to the algorithm presented for induced matrix norms, but with absolute values and complex phases of the vectors $\mathbf{x}$ and $\mathbf{y}$ replaced by the singular values and singular vectors of the matrices $X$ and $Y$, respectively. The entire algorithm is still extremely quick to run, since each step just involves computing one singular value decomposition.

The downside of this algorithm, as with the induced matrix norm algorithm, is that we have no guarantee that this method will actually converge to the induced Schatten p → q norm; only that it will converge to some lower bound of it. However, the algorithm works pretty well in practice, and is fast enough that we can simply run it a few thousand times to get a very good idea of what the norm actually is. If you’re interested in making use of this algorithm, it has been implemented in QETLAB as the InducedSchattenNorm function.

### Entanglement Norms

The central idea used for the previous two families of norms can also be used to get lower bounds on the following norm on $M_m \otimes M_n$ that comes up from time to time when dealing with quantum entanglement:

$\|X\|_{S(1)} := \max\Big\{\big|(\mathbf{v}\otimes\mathbf{w})^*X(\mathbf{x} \otimes \mathbf{y})\big| : \|\mathbf{v}\| = \|\mathbf{w}\| = \|\mathbf{x}\| = \|\mathbf{y}\| = 1\Big\}.$

(As a side note: this norm, and some other ones like it, were the central focus on my thesis.) This norm is already written for us as a double maximization, so the idea presented in the previous two sections is somewhat clearer from the start: we fix randomly-generated vectors $\mathbf{v}$ and $\mathbf{x}$ and then maximize over all vectors $\mathbf{w}$ and $\mathbf{y}$, which can be done simply by computing the left and right singular vectors associated with the maximum singular value of the operator

$(\mathbf{v} \otimes I)^*X(\mathbf{x} \otimes I) \in M_n.$

We then fix $\mathbf{w}$ and $\mathbf{y}$ as those singular vectors and then maximize over all vectors $\mathbf{v}$ and $\mathbf{x}$ (which is again a singular value problem), and we iterate back and forth until we converge to some value.

As with the previously-discussed norms, this algorithm always converges, and it converges to a lower bound of $\|X\|_{S(1)}$, but perhaps not its exact value. If you want to take this algorithm out for a spin, it has been implemented in QETLAB as the sk_iterate function.

It’s also worth mentioning that this algorithm generalizes straightforwardly in several different directions. For example, it can be used to find lower bounds on the norms $\|\cdot\|_{S(k)}$ where we maximize on the left and right by pure states with Schmidt rank not larger than k rather than separable pure states, and it can be used to find lower bounds on the geometric measure of entanglement [5].

References:

1. D. Steinberg. Computation of matrix norms with applications to robust optimization. Research thesis. Technion – Israel University of Technology, 2005.
2. J. M. Hendrickx and A. Olshevsky. Matrix p-norms are NP-hard to approximate if p ≠ 1,2,∞. 2009. E-print: arXiv:0908.1397
3. D. W. Boyd. The power method for ℓp norms. Linear Algebra and Its Applications, 9:95–101, 1974.
4. J. Watrous. Notes on super-operator norms induced by Schatten norms. Quantum Information & Computation, 5(1):58–68, 2005. E-print: arXiv:quant-ph/0411077
5. T.-C. Wei and P. M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Physical Review A, 68:042307, 2003. E-print: arXiv:quant-ph/0212030
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## Introducing QETLAB: A MATLAB toolbox for quantum entanglement

April 14th, 2015

After over two and a half years in various stages of development, I am happy to somewhat “officially” announce a MATLAB package that I have been developing: QETLAB (Quantum Entanglement Theory LABoratory). This announcement is completely arbitrary, since people started finding QETLAB via Google about a year ago, and a handful of papers have made use of it already, but I figured that I should at least acknowledge its existence myself at some point. I’ll no doubt be writing some posts in the near future that highlight some of its more advanced features, but I will give a brief run-down of what it’s about here.

### The Basics

First off, QETLAB has a variety of functions for dealing with “simple” things like tensor products, Schmidt decompositions, random pure and mixed states, applying superoperators to quantum states, computing Choi matrices and Kraus operators, and so on, which are fairly standard daily tasks for quantum information theorists. These sorts of functions are somewhat standard, and are also found in a few other MATLAB packages (such as Toby Cubitt’s nice Quantinf package and Géza Tóth’s QUBIT4MATLAB package), so I won’t really spend any time discussing them here.

### Mixed State Separability

The “motivating problem” for QETLAB is the separability problem, which asks us to (efficiently / operationally / practically) determine whether a given mixed quantum state is separable or entangled. The (by far) most well-known tool for this job is the positive partial transpose (PPT) criterion, which says that every separable state remains positive semidefinite when the partial transpose map is applied to it. However, this is just a quick-and-dirty one-way test, and going beyond it is much more difficult.

The QETLAB function that tries to solve this problem is the IsSeparable function, which goes through several separability criteria in an attempt to prove the given state separable or entangled, and provides a journal reference to the paper that contains the separability criteria that works (if one was found).

As an example, consider the “tiles” state, introduced in [1], which is an example of a quantum state that is entangled, but is not detected by the simple PPT test for entanglement. We can construct this state using QETLAB’s UPB function, which lets the user easily construct a wide variety of unextendible product bases, and then verify its entanglement as follows:

>> u = UPB('Tiles'); % generates the "Tiles" UPB
>> rho = eye(9) - u*u'; % rho is the projection onto the orthogonal complement of the UPB
>> rho = rho/trace(rho); % we are now done constructing the bound entangled state

>> IsSeparable(rho)
Determined to be entangled via the realignment criterion. Reference:
K. Chen and L.-A. Wu. A matrix realignment method for recognizing entanglement.
Quantum Inf. Comput., 3:193-202, 2003.

ans =

0

And of course more advanced tests for entanglement, such as those based on symmetric extensions, are also checked. Generally, quick and easy tests are done first, and slow but powerful tests are only performed if the script has difficulty finding an answer.

Alternatively, if you want to check individual tests for entanglement yourself, you can do that too, as there are stand-alone functions for the partial transpose, the realignment criterion, the Choi map (a specific positive map in 3-dimensional systems), symmetric extensions, and so on.

### Symmetry of Subsystems

One problem that I’ve come across repeatedly in my work is the need for robust functions relating to permuting quantum systems that have been tensored together, and dealing with the symmetric and antisymmetric subspaces (and indeed, this type of thing is quite common in quantum information theory). Some very basic functionality of this type has been provided in other MATLAB packages, but it has never been as comprehensive as I would have liked. For example, QUBIT4MATLAB has a function that is capable of computing the symmetric projection on two systems, or on an arbitrary number of 2- or 3-dimensional systems, but not on an arbitrary number of systems of any dimension. QETLAB’s SymmetricProjection function fills this gap.

Similarly, there are functions for computing the antisymmetric projection, for permuting different subsystems, and for constructing the unitary swap operator that implements this permutation.

### Nonlocality and Bell Inequalities

QETLAB also has a set of functions for dealing with quantum non-locality and Bell inequalities. For example, consider the CHSH inequality, which says that if $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are $\{-1,+1\}$-valued measurement settings, then the following inequality holds in classical physics (where $\langle \cdot \rangle$ denotes expectation):

$\langle A_1B_1 \rangle + \langle A_1B_2 \rangle + \langle A_2B_1 \rangle - \langle A_2B_2 \rangle \leq 2.$

However, in quantum-mechanical settings, this inequality can be violated, and the quantity on the left can take on a value as large as $2\sqrt{2}$ (this is Tsirelson’s bound). Finally, in no-signalling theories, the quantity on the left can take on a value as large as $4$.

All three of these quantities can be easily computed in QETLAB via the BellInequalityMax function:

>> coeffs = [1 1;1 -1]; % coefficients of the terms <A_iB_j> in the Bell inequality
>> a_coe = [0 0]; % coefficients of <A_i> in the Bell inequality
>> b_coe = [0 0]; % coefficients of <B_i> in the Bell inequality
>> a_val = [-1 1]; % values of the A_i measurements
>> b_val = [-1 1]; % values of the B_i measurements
>> BellInequalityMax(coeffs, a_coe, b_coe, a_val, b_val, 'classical')

ans =

2

>> BellInequalityMax(coeffs, a_coe, b_coe, a_val, b_val, 'quantum')

ans =

2.8284

>> BellInequalityMax(coeffs, a_coe, b_coe, a_val, b_val, 'nosignal')

ans =

4.0000

The classical value of the Bell inequality is computed simply by brute force, and the no-signalling value is computed via a linear program. However, no reasonably efficient method is known for computing the quantum value of a Bell inequality, so this quantity is estimated using the NPA hierarchy [2]. Advanced users who want more control can specify which level of the NPA hierarchy to use, or even call the NPAHierarchy function directly themselves. There is also a closely-related function for computing the classical, quantum, or no-signalling value of a nonlocal game (in case you’re a computer scientist instead of a physicist).