Archive for July 7th, 2019

7 Jul

Convex Geometry for Quantum Theory

Posted by hishamuddinz in Mathematical Physics, Physics, Quantum Foundations.

Restarting this (technical) blog by reporting what I have been reading few weeks ago. For now, I have returned to the book by Bengtsson & Zyczkowski on “Geometry of Quantum States“. I must thank Prof. Zyczkowski for giving me a complimentary copy of this book, for which I have recommended to the graduate students. The first chapter is on a topic that some (especially students) might not expect, namely on “Convexity, Colours and Statistics” and let me just focus on why convex geometry. Those who are working on quantum foundations will probably know why as I hope to discuss below.

First, convex geometry is simply the study of geometry of convex sets. A convex set is a subspace of Euclidean space $latex  \mathbb{E}^n$ whose points \mathbf{x}_1, \mathbf{x}_2 can be combined as

\mathbf{x}=\lambda_1\mathbf{x}_1+\lambda_2\mathbf{x}_2\ ,\quad \lambda_1+\lambda_2=1

and that \mathbf{x} will still belong to the set. Should be stated here that the fact that we have the linear addition of points necessitates the space it belongs to is a linear space (hence the use of \mathbb{E}^n). One could in fact remove the requirement of a special point (say, the origin) in vector spaces and consider the embedding space being simply an affine space with linear addition operation still in place.

Before we move on to how is this related to quantum theory. Let us mention down some applications where convex spaces or convex geometry are used (see also here). Perhaps the first instance, one hears the word convex is in convex functions (see also here). The use here is very much related to the idea of minimization; minimum of a convex function is a global minimum. Relationship between convex function and convex set is that the epigraph (area above the function’s curve) of a convex function is a convex set. Much more complex problems on convex functions can come in the subject of convex analysis but we will not go into details. Not totally unrelated to optimization is the underlying geometrical ideas that led to the subject of convex geometry which finds uses in tiling problems, robotic motions and image analysis (see also here). One interesting application is the idea of convex relaxation for which hard problems are approximated by approximating them to relevant convex optimization problems. See this lecture by Sam Wong of Microsoft Research at https://www.youtube.com/watch?v=_2pMKksyeD0. An excellent introduction to what is convexity and the various interrelations can be found in the paper of Berger, “Convexity“, Amer. Math. Monthly Vol. 97, No. 8 (1990) 650-678. Note: Berger is the same person who wrote the famous two-volume book on Geometry (see here and here). A worthy new book by Berger is “Geometry Revealed: A Jacob’s Ladder to Modern Higher Geometry“.

Digression: Normally one discussed convex spaces whose embedding space is \mathbb{R}^n and that the convex combination involves a real parameter \mu. One question comes to mind is how this generalizes to complex linear spaces (as in quantum theory). Found a paper, “Convex Sets and Convex Combinations on Complex Linear Spaces” by Matsuzaki, Endou and Shidama, though the contents are quite opaque to me. In any case, they mention the use of a complex \lambda and embed the unit 1_\mathbb{C} to form 1_\mathbb{C} - \lambda in the convex combination. This will imply the loss of ordering. However they also mention the use of inner product of elements of the embedding space to form the necessary reals if required.

Let’s move closer to quantum theory. One field that is close to convex geometry is probability theory. One begins by introducing the idea of probability measures which are maps/functions on sets, normally identified as event spaces. For a simple introduction on probabilities, one can refer to notes by Cozman, “A Few Notes on Sets of Probability Distributions“. Probability measure is defined as a function \mu  on the event space, that takes in the interval [0,1] and it obeys additivity property

\mu\left(\bigcup_{i}\ E_i\right) = \sum_i\mu(E_i) ,

for disjoint events E_i. One can extend this to continuous labelled events and the sum extends to an integral. Generalizing even further one can let the measure be defined for (Minkowski) sum of convex sets (say, A,B) in such a way that they obey Brunn-Minkowski inequality:

(\mu(A+B))^{1/n}\ge(\mu(A))^{1/n}+(\mu(B))^{1/n} .

Putting a convex sum instead, one has

(\mu(\lambda A+(1-\lambda)B))^{1/n}\ge(\lambda\mu(A))^{1/n}+((1-\lambda)\mu(B))^{1/n} .

This is called \mathbf{n}–convex measure. The special case of n=0 gives

(\mu(\lambda A+(1-\lambda)B))\ge\mu(A)^\lambda+\mu(B)^{(1-\lambda)}

or

\log(\mu(\lambda A+(1-\lambda)B))\ge\lambda\log(\mu(A))+(1-\lambda)\log(\mu(B)) .

This is sometimes called log concave measure.

All these are pretty technical but points to construction variability of the subject and certainly requires deeper study. Let me end this by pointing to the article of Milman, “Geometrization of Probability” in the book “Geometry and Dynamics of Groups and Spaces“. Another connection that is worth mentioning is the relation between geometric probabilities with hitting probabilities for connvex bodies – see Schneider’s “Convexity and Geometric Probabilities“.

Continuing the connection between convex geometry and quantum theory; first, we note that quantum states are typically represented by state vectors modulo complex scalars (i.e. rays) that belong to a Hilbert space. However they can also be represented by projection or density operators \rho (typically density matrices if the Hilbert space is finite-dimensional). Generally density operators \rho have the following properties:

  • Hermiticity: \rho^\dagger=\rho;
  • Unit trace: \textrm{Tr}(\rho)=1;
  • Positive-definite: \langle\phi\vert\rho\vert\phi\rangle\ge 0.

Projectors corresponding to rays/states have the further property of being

  • \rho^2=\rho (twice projection has the same effect as the first single one).

Such states are called pure states. Are there any other states? Indeed, there are, which are the convex combination of pure states called mixed states. These typically correspond to states of partial knowledge (probabilistic mixture). In this particular case, \textrm{Tr}(\rho^2)\le 1. Hence the relation of convex geometry to quantum theory. While mixed states are the more general form (which includes pure ones), it is normally the case that one needs to define what the pure states are first (through some procedure, say, quantization). The other interesting point is that the expression of the mixed state in terms of pure state is not unique; physically translating as two statistical mixtures with same statistical averages cannot be distinguished. Mielnik likened this to be similar to the use of local coordinates in Riemannian geometry and promoted convex geometry as the geometry for (even a generalised) quantum mechanics – see Mielnik, “Generalized Quantum Mechanics“, Comm. Math. Phys. 37 (1974) 221-256. His views are summarized in the relatively preprint “Convex Geometry: A Travel to the Limits of Our Knowledge” (arXiv:1202.2164) – published version appeared in “Geometric Methods in Physics” (eds.) P. Kielanowski, S.T. Ali, A. Odzijewicz, M. Schlichenmaier & T. Voronov (Springer, 2013).

One has actually more stories to convex geometry. As stated above, the pure states are special points in the convex geometry of states (extremal points). The bulk of (mixed) states live elsewhere in the convex hull, which is the smallest convex set containing the pure states. If these are of finite number, the body is called a convex polytope. One can start building a hierarchy of simplexes: A p-simplex consists of points taking up the convex combination:

\mathbf{x}=\lambda_0\mathbf{x}_0+\lambda_1\mathbf{x}_1+\cdots +\lambda_p\mathbf{x}_p

where \lambda_i\ge 0 and \lambda_0+\lambda_1+\cdots +\lambda_p=1. A subset F of a convex set, stable under mixing and purification is called a face of the convex set i.e. if

\mathbf{x}=\lambda_1\mathbf{x}_1+(1-\lambda)\mathbf{x}_2\ ;\quad 0\le\lambda\le 1 ,

then \mathbf{x} lies in F if and only if \mathbf{x}_1,\mathbf{x}_2 lie in F. A face of dimension k is a k-face. Special case: 0-face is an extremal point and (n-1)-face is a facet. Given the set of all feces of a convex body, one can form a partial ordering i.e. F_1\le F_2 if face F_1 is contained in F_2. This leads to another idea: partial ordering is the essential ingredient to form a logic, allowing what statements can be made. Thus convex geometry is related to the idea of quantum logic proposed much earlier and this is discussed in Mielnik, “Geometry of Quantum States“, Comm. Math. Phys. 9 (1968) 55-80. More recent developments and discussions can be found in Coecke, “An Alternative Gospel of Structure: Order, Composition, Processes” (arXiv: 1307.4038) and Coecke & Martin, “A Partial Order on Classical and Quantum States” (downloadable here).

Thus, with all these fundamental interconnections, one can understand convex geometry is a convenient place to start discussing quantum states in the book of Bengtsson and Zyczkowski.