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Interacting quantum observables: Categorical algebra and diagrammatics
 In Automata, Languages and Programming, ICALP 2008, number 5126 in Lecture Notes in Computer Science
, 2008
"... Abstract: Within an intuitive diagrammatic calculus and corresponding highlevel categorytheoretic algebraic description we axiomatise complementary observables for quantum systems described in finite dimensional Hilbert spaces, and study their interaction. We also axiomatise the phase shifts rela ..."
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Abstract: Within an intuitive diagrammatic calculus and corresponding highlevel categorytheoretic algebraic description we axiomatise complementary observables for quantum systems described in finite dimensional Hilbert spaces, and study their interaction. We also axiomatise the phase shifts relative to an observable. The resulting graphical language is expressive enough to denote any quantum physical state of an arbitrary number of qubits, and any processes thereof. The rules for manipulating these result in very concise and straightforward computations with elementary quantum gates, translations between distinct quantum computational models, and simulations of quantum algorithms such as the quantum Fourier transform. They enable the description of the interaction between classical and quantum data in quantum informatic protocols. More specifically, we rely on the previously established fact that in the symmetric monoidal category of Hilbert spaces and linear maps nondegenerate observables correspond to special commutative †Frobenius algebras. This leads to a generalisation of the notion of observable that extends to arbitrary †symmetric monoidal categories (†SMC). We show that any observable in a †SMC comes with an abelian group of phases. We define complementarity of observables in arbitrary †SMCs and prove an elegant diagrammatic characterisation thereof. We show that an important class of complementary observables give rise to a Hopfalgebraic structure, and provide equivalent characterisations thereof. Contents 1. Introduction................................
The group theoretic origin of nonlocality for qubits
, 2009
"... We describe a general framework in which we can precisely compare the structures of quantumlike theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operation ..."
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We describe a general framework in which we can precisely compare the structures of quantumlike theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and a toy theory proposed by Spekkens. We discover that viewed within our framework these theories are very similar, but differ in one key aspect a four element group we term the phase group which emerges naturally within our framework. In the case of the stabiliser theory this group is Z4 while for Spekkens’s theory the group is Z2 × Z2. We further show that the structure of this group is intimately involved in a key physical difference between the theories: whether or not they can be modelled by a local hidden variable theory. This is done by establishing a connection between the phase group, and an abstract notion of GHZ state correlations. We go on to formulate precisely how the stabiliser theory and toy theory are ‘similar ’ by defining a notion of ‘mutually unbiased qubit theory’, noting that all such theories have four element phase groups. Since Z4 and Z2 × Z2 are the only such groups we conclude that the GHZ correlations in this type of theory can only take two forms, exactly those appearing in the stabiliser theory and those appearing in Spekkens’s theory. The results point at a classification of local/nonlocal behaviours by finite Abelian groups, extending beyond qubits to any finitary theory whose observables are all mutually unbiased. 1
The Expectation Monad in Quantum Foundations
"... The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two pr ..."
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The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to socalled Banach effect algebras. These structures capture states and effects in quantum foundations, and the duality between them. Moreover, the approach leads to a new reformulation of Gleason’s theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.
Division Algebras and Quantum Theory
, 2011
"... Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with rea ..."
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Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘threefold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex ’ representations), those that are selfdual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are selfdual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This threefold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure. 1
THE ROAD TO A NEW QUANTUM FORMALISM: CATEGORIES AS A CANVAS FOR QUANTUM FOUNDATIONS
"... 3 Highlevel structures governing physics: a categorical approach 4 3.1 Structure as phenomenology............................................. 5 ..."
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3 Highlevel structures governing physics: a categorical approach 4 3.1 Structure as phenomenology............................................. 5
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"... Dedicated to the many bright young theoretical physicists that failed to escape the fate of having to work in institutions like banks. Preface New? In what sense? Surely I am not the only person who, after extensively justifying why certain mathematical structures naturally arise in physics, gets qu ..."
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Dedicated to the many bright young theoretical physicists that failed to escape the fate of having to work in institutions like banks. Preface New? In what sense? Surely I am not the only person who, after extensively justifying why certain mathematical structures naturally arise in physics, gets questions like: “this is all nice maths but what’s the physics? ” Meanwhile I figured out what this truly means: “I don’t see any differential equations! ” Okay, this is indeed a bit overstated. Nowadays any mathematical argument involving groups, when these are moreover referred to as ‘symmetry groups’, stands a serious chance of being eligible for carrying the label ‘physics’. But it hasn’t always been like this. John Slater (cf. the Slater determinant in quantum chemistry) referred to the use of group theory in quantum physics by Weyl, Wigner et al. as der Gruppenpest, what translates as the ‘plague of groups’. Even in 1975 he wrote [14]: “As soon as [my] paper became known, it was obvious that a great many other physicists were as ‘disgusted ’ as I had been with the grouptheoretical approach to the problem. As I heard later, there were remarks made such as ‘Slater has slain the Gruppenpest’. I believe that no other