Mathematics Home / 2011 Clifford Lectures

**Dr. Christopher Fuchs, Perimeter Institute,Waterloo, Canada**

**1. Charting the Shape of Quantum State Space**

**Abstract**

Physicists have become accustomed to the idea that a theory's content is always most transparent when written in coordinate-free language. Sometimes though the choice of a good coordinate system can be quite useful for settling deep conceptual issues. This is particularly so for an information-oriented or Bayesian approach to quantum foundations: One good coordinate system may be worth more than a hundred blue-in-the-face arguments. This talk will motivate and chronicle the search for one such class of coordinate systems for finite dimensional operator spaces, the so-called Symmetric Informationally Complete (SIC, pronounced "seek") measurements. The desired class will take little more than five minutes to define, but the quest to construct these objects will carry us down a 35 year journey, with the most frenzied activity only recently. Beyond this, we will turn the tables and discuss how one might hope to get the formal content of quantum mechanics out of the very existence of such a coordinate system. It has to do with seeing the Born Rule as a "tiny" modification to the Bayesian Law of Total Probability.

**2. The Platform of Quantum Bayesianism **

**Abstract**

Quantum Bayesianism is a point of view on quantum mechanics that says that there is no such thing as a “measurement problem” in it because there is no such THING as a quantum state: Quantum states are not things, but rather tables of (personalist) Bayesian degrees of belief. The view doesn’t stop with the writing of a philosophical manifesto, however; it starts there! Taking the idea seriously over the last 15 years has been the direct motivation for a number of theorems and objects in quantum information theory: from the quantum no-broadcasting theorem, to the quantum de Finetti theorem, and even some quantum cryptographic alphabets. I will review the reasons for developing this framework as well as some of the theorems just mentioned to lay the groundwork for the holy grail of the present meeting: Finding an efficient, elegant representation of quantum states in terms of a single probability function---the one for the outcomes of a fiducial Symmetric Informationally Complete (SIC) quantum measurement.

**3. Seeking SIC's**

**Abstract**

But what is so damned consternating about the SICs is that we don’t know whether they exist for all finite dimensional Hilbert spaces. Existence proofs (constructions) have so far been given only in dimensions 2-15, 19, 24, 35, and 48, though numerical evidence indicates that they exist in all dimensions 2-67 (where computer time becomes too demanding to search further). In this lecture, I will give several ways to formulate the existence problem: From a question about algebraic varieties to a question about convex optimization to a question about structure coefficients for the Lie algebra gl(d,C). Along the way, I will describe the information that has been gleaned about SIC structure from these various formulations.

**4. Properties of Quantum-Bayesian State Spaces**

**Abstract**

Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum-state space may thus be thought of as a restricted subset of all potentially available probabilities. The previous lectures have advocated such a representation via the SIC measurements. Building upon those talks we study how this subset might be characterized. Our leading characteristic is that the inner products of the probabilities are bounded above and below, a simple condition with quite nontrivial consequences. To get quantum-state space precisely, something more detailed about the extreme points is needed. No definitive characterization is reached, but we will see several interesting features, all in conformity with actual quantum theory.

**A Unified Sheaf-Theoretic Account of Contextuality and Non-Locality: I and II**

A number of landmark results in the foundations of quantum mechanics show that quantum systems exhibit behaviour that defies explanation in classical terms, and which cannot be accounted for in such terms even by postulating "hidden variables" as additional unobserved factors. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the inter-relationships among the various concepts and results.

We present an approach which exposes some of the key mathematical structures involved, and allows definitions to be formulated and results to be proved at a high level of generality. We use the mathematical language of sheaves and monads. This allows for a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed.

-The notion of context is parameterized by the poset over which the sheaves and presheaves are defined. This allows locality, contextuality, and combinations of the two to be covered in a uniform fashion. Such partial orders arise naturally from the compatibility relations which hold between measurements; and *incompatibility* has a natural order-theoretic interpretation.

-The notion of behaviour or event is parameterized as a sheaf over the poset of contexts.

-The notion of weight assigned by models to events is parameterized by a commutative monad, which can be regarded as expressing a general notion of effect. In particular, any commutative semiring gives rise to a distribution monad. Thus both probabilistic and relational models are covered, as very special cases of the general theory.

In this setting, we can formalize the notion of an *empirical model*, which assigns weights (e.g., probabilities) to the outcomes for each choice of a compatible family of measurements. A basic requirement on such models is *naturality*, which translates into a generalized form of No-Signalling and which holds within compatible families of measurements on the same system, as well as among spatially distributed local measurements.

We can also formalize the notion of a *hidden-variable model*, and the realization of an empirical model by a hidden-variable model, at this very general level. In the probabilistic case, this specializes to the familiar notion of averaging over the values of the hidden variable.

We formulate a notion of *factorizability* in hidden-variable models, which subsumes both Bell locality and a form of non-contextuality. We define deterministic hidden-variable models, and show that these always satisfy factorizability. The key notion of lambda-independence is expressed as constancy of the presheaf of hidden variables. Without this constraint, the possibility of introducing deterministic hidden variables holds trivially. We say that an empirical model is *contextual* if it cannot be realized by a factorizable hidden variable model.

A central result is the equivalence of contextuality and *joint measurability*, i.e. the existence of an extension of the empirical model to one defined on *all* subsets of measurements, regardless of compatibility. This has many consequences, and leads to a striking simplification of many results. For example, it becomes very easy to show that if a model has a realization by a factorizable hidden variable model, then it has a realization by a deterministic hidden variable model.

We use our approach to derive general forms of two types of No-Go theorems:

-We prove model-specific results. For illustration, we use a Hardy-style construction. As an alternative to the standard type of argument for the impossibility of a deterministic hidden variable model, we give a direct argument showing the non-existence of a joint measure.

-We formulate a general version of a Kochen-Specker result, by extracting a necessary conditionfor any non-contextual model, namely the existence of a global section in itssupport. Thus showing the non-existence of global sections for a sub-presheaf of events immediately implies the contextuality of any model with support contained in that presheaf. A simply defined presheaf, asserting that *exactly one outcome can occur* on each maximal context, yields the standard applications to quantum mechanics. There are some interesting connections to combinatorics and computational complexity, including a purely graph-theoretic formulation of "Kochen-Specker graphs."

**SIC-POVMs, Theta Functions and Squeezed States**

**Abstract**

This talk is motivated by the speculation that the reason the SIC-POVM existence problem has so far proved intractable is that we are looking at it in the wrong way. Posed in the usual way, as the problem of proving the existence of d^2 vectors having constant overlap, the fact that SICs exist in every dimension up to 67 seems like a miracle, depending as it does on a highly over-determined set of equations having a solution. In spite of strenuous attempts by numerous investigators over a period of more than 10 years we still have essentially zero insight into the structural features of the equations which causes them to be soluble. Yet one feels that there must surely be such a structural feature. This suggests the thought that the reason existence has been so hard to prove is that the natural structure of the problem is obscured by the usual way of formulating it. Perhaps if the problem was re-formulated in a different mathematical context the solution would become obvious. This was the motivation for Appleby, Flammia and Fuchs arXiv:1001.0004, in which we showed that SIC existence in dimension d is equivalent to the existence of a certain structure in the adjoint representation of the Lie Algebra gl(d,C). In this talk we will describe a different approach, in which properties of the Jacobi theta functions are used to embed Weyl-Heisenberg SICs in the space of complex analytic functions. This embedding is interesting for several reasons. In the first place it allows us to embed every Weyl-Heisenberg SIC, of every dimension, in a single infinite dimensional structure. It therefore offers the prospect of getting a handle on what the different SICs all have in common (one of the frustrating features of the problem as it is currently formulated is that the properties of an individual SIC seem to be highly sensitive to the dimension). In the second place it offers the prospect of bringing the full apparatus of complex function theory to bear on the problem. In the third place it suggests some intriguing connections with the theory of elliptic curves and modular forms. Finally it may be of some experimental significance. As we will show, theta functions are intimately connected with squeezed states. There is therefore some hope that the results described in this talk will enable us to realize a SIC in a quantum optical system.

**Multipartite Quantum Entanglement as Rational Arithmetic**

**Abstract**

The GHZ-state and W-state arise as the incomparable maximally entangled three qubit states. We show that on-the-nose they respectively correspond to arithmetic multiplication and addition on a (dense) encoding of the rational numbers on the real part of the Bloch sphere. From this it also follows that arbitrary multipartite states can be represented by polynomials.

**Simplified Representations of the Clifford Group**

**Abstract**

The Weyl-Heisenberg group, also called the Pauli group, is related to many areas and has applications within quantum information theory. It is the group of Pauli-spin operators when generalized to any dimension. Naturally also its normalizer---the Clifford group---is of interest. We will see how all Clifford operators can be represented as monomial matrices when the dimension is a square number, and how a related simplification is possible when the dimension is a multiple of a square.

An application---and this is the motivation we had for studying the Clifford group---is to the search for Symmetric Informationally Complete quantum measurements (SICs). What the relation to this problem is and how the new bases are of help will be reviewed.

**The Bayesian Order on Probability Distributions**

Abstract: Partial orders on probability distributions have been of interest for a very long time, but research in the area has seen an explosion in the past 30 years. Majorization and Schur convexity has found a wide range of applications in many different areas of mathematics, particularly since the 1979 publication of Marshall and Olkin's Inequalities: Theory of Majorization and Its Applications. In 2002, Keye Martin and Bob Coecke discovered the Bayesian order on finite probability distributions, and extended this to the spectral order on quantum states. The Bayesian and spectral orders opened the door for us to use techniques from Domain Theory to analyze classical probability distributions and quantum states under a unified framework. The Bayesian order is particularly special, as it is the only order currently known to extend to an order on quantum states.

In this talk, we will provide an introduction to the Bayesian order, along with several interpretations of what it means - from comparisons between states of knowledge to statements about relative wealth distribution, and then review its extension to the spectral order on quantum states. We will then discuss recent work on extending the Bayesian order in other directions, including a new relation that extends it from finite probability distributions to probability measures. On a large class of measures, which includes several different embeddings of the Bayesian order on classical distributions, this relation is known to be a partial order. Attempts to characterize this relation, and to prove that it is a partial order on all probability measures, have led to some interesting questions and constructions, which we will explore.

**Geometry, Probability, and Quantum Theory**

**Abstract**

It is well known now that many aspects of quantum theory admit a natural interpretation in the classical geometry of complex projective space. It is curious that this particular approach to the theory seems to have been overlooked by the founders of the subject. Dirac, to be sure, was well aware of the advantages of projective geometry, and evidently carried out many of his private calculations in that way; but it seems that he was nevertheless unaware of the natural metrical and symplectic geometries associated with the space of pure states in quantum mechanics. We know now that the pure state space, when it is regarded as complex projective space endowed with the unitary-invariant Fubini-Study metric, has a Riemannian structure, and that much of the standard theory can be developed in the language of this geometry. For example, the transition probability between two states can be expressed as a function of the geodesic distance between the two states in this geometry. For a long time the geometric approach was to some extent neglected, perhaps because its development in the infinite dimensional situation of a general Hilbert space involves technical issues. But over the last decade or two, with the advent of quantum information and various allied lines of investigation that have tended to emphasize the role of finite-dimensional state spaces, the situation has changed, and the geometric method in this context provides a powerful tool, allowing one to bring into play a variety of different constructions from classical algebraic geometry as an aid to understanding quantum theory and gaining new insights. In this talk I shall present an overview of such methods, focusing on probabilistic aspects of the theory, including especially the role of general measurements (POVMs) and various explicit geometrical constructions associated with such operations.

This work builds on material developed largely with D.C. Brody and other collaborators --- for background reading see, for example, D.C. Brody & L.P. Hughston (2000) Geometric Quantum Mechanics, J. Geometry and Physics 38, 19-53, and references cited therein.

**A Different Way of Organizing Quantum States: The Quantum Loop**

**Abstract**

We view the density matrices (positive semidefinite linear operators) on a finite dimensional complex Hilbert space as the states of a quantum system. We observe that via a straightforward construction the set of invertible density matrices may be endowed with both a loop (i.e., non-associative group) operation and a scalar multiplication so that the resulting structure bears resemblances to a vector space. The loop structure arises from a natural one on all positive definite matrices given by $A\oplus B=A^{1/2}BA^{1/2}$ by dividing out the subgroup of positive scalar matrices and identifying each of these cosets with a unique density matrix.

We consider an appropriate axiom scheme for this type of structure and some of the associated theory that has been built up. We consider also the extent to which the loop operation may be extended to the positive semidefinite density matrices. We also point out some connections with and potential applications to quantum theory. Finally we allude to other settings where this type of loop structure arises, namely the loop of Einstein velocity addition in the theory of special relativity and the Mobius loop of the complex unit ball. The former actually turns out to be isomorphic to the special case of the qubit loop, defined for the two-dimensional Hilbert space.

**Retractive Groups**

**Abstract**

The goal of our current research is to define a new area called "algebraic information theory." It began with the realization that many important classes of channels, both quantum and classical, possess the structure of a compact affine monoid. The idea is then to use this structure as the basis for new techniques in information theory. In a large number of cases, these classes arise as the convex closure of a certain underlying group, and among these, certain groups distinguish themselves in that they generate very restricted classes, which are remarkable in that they always contain the solution to optimization problems posed over the set of all channels. These groups, which we call "retractive," can also be used to derive useful inequalities, devise methods for tomography of quantum channels and appear to always generate channels whose capacities can be expressed in closed form.

**A Topological Approach to Channel Capacity**

**Abstract**

The classical approach to Shannon Information relies heavily on inequalities. Recent work by Martin, Allwein and Moskowitz [1] took a new approach, in which domain theory and compact monoids were major tools for analyzing the capacity of binary channels. Inspired by this work, the author has explored possible extensions of their results to higher dimensions. In this talk, we extend their approach to present a new understanding of channel capacity that also utilizes topology as a primary tool.

1. Martin, K., G. Allwein and I. Moskowitz, Algebraic information theory for binary channels, Proceedings of MFPS 22, Electronic Notes in Theoretical Computer Science (ENTCS) Volume 158, May, 2006

**The Group Theoretical Magic of the Harmonic Oscillator and Its Deformed Cousins**

**Abstract**

In the 1960s Julian Schwinger showed how to work out the representation theory of the rotation group using the formalism of raising and lowering operators from the harmonic oscillator. This is sometimes called the boson calculus. This was later generalized to SU(n) and other semi-simple Lie groups. It turns out that one can define a so-called q-deformation of the harmonic oscillator and analyze the representations of quantum groups, a topic of importance for topological quantum computing. In this purely expository talk, I will describe the Schwinger oscillator representation and the q-deformed variants. I will give a brief account of how we used the Schwinger representation to prove covariance of the Unruh channel and will close with how I hope to use the q-deformed version to study questions in topological quantum computing.