General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space

In our previous work, we proposed a mathematical framework for PT-symmetric quantum theory, and in particular constructed a Krein space in which PT-symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral property and PT symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical-quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding PT-symmetric quantum system, which indicates that PT-symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in the framework.Comment: 32 pages, no figures; explanation, discussion and references adde

Completeness in supergravity constructions

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.Comment: 29 page

Lamination exact relations and their stability under homogenization

Relations between components of the effective tensors of composites that hold regardless of composite's microstructure are called exact relations. Relations between components of the effective tensors of all laminates are called lamination exact relations. The question of existence of sets of effective tensors of composites that are stable under lamination, but not homogenization was settled by Milton with an example in 3D elasticity. In this paper we discuss an analogous question for exact relations, where in a wide variety of physical contexts it is known (a posteriori) that all lamination exact relations are stable under homogenization. In this paper we consider 2D polycrystalline multi-field response materials and give an example of an exact relation that is stable under lamination, but not homogenization. We also shed some light on the surprising absence of such examples in most other physical contexts (including 3D polycrystalline multi-field response materials). The methods of our analysis are algebraic and lead to an explicit description (up to orthogonal conjugation equivalence) of all representations of formally real Jordan algebras as symmetric $n\times n$ matrices. For each representation we examine the validity of the 4-chain relation|a 4th degree polynomial identity, playing an important role in the theory of special Jordan algebras

Tasks and premises in quantum state determination

The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (POVMs) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.Comment: minor changes in v

A Diagrammatic Axiomatisation for Qubit Entanglement

Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic classification of multipartite qubit entanglement that would highlight the communicational properties of quantum states, and their potential uses in cryptographic schemes. In this paper, we present a full graphical axiomatisation of the relations between GHZ and W: the ZW calculus. This refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirectedness, a large degree of symmetry, and an algebraic underpinning. We prove that the ZW calculus is complete for the category of free abelian groups on a power of two generators - "qubits with integer coefficients" - and provide an explicit normalisation procedure.Comment: 12 page

Symmetric Informationally Complete Quantum Measurements

We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.Comment: 8 page

Generalised Compositional Theories and Diagrammatic Reasoning

This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular case, namely the study of complementarity and non-locality, two fundamental concepts of quantum theory whose relationship we explore in later part of this chapter. The diagrammatic calculus that we are concerned with here is not merely an illustrative tool, but it has both (i) a conceptual physical backbone, which allows it to act as a foundation for diverse physical theories, and (ii) a genuine mathematical underpinning, permitting one to relate it to standard mathematical structures.Comment: To appear as a Springer book chapter chapter, edited by G. Chirabella, R. Spekken

Coherent states in fermionic Fock-Krein spaces and their amplitudes

We generalize the fermionic coherent states to the case of Fock-Krein spaces, i.e., Fock spaces with an idefinite inner product of Krein type. This allows for their application in topological or functorial quantum field theory and more specifically in general boundary quantum field theory. In this context we derive a universal formula for the amplitude of a coherent state in linear field theory on an arbitrary manifold with boundary.Comment: 20 pages, LaTeX + AMS + svmult (included), contribution to the proceedings of the conference "Coherent States and their Applications: A Contemporary Panorama" (Marseille, 2016); v2: minor corrections and added axioms from arXiv:1208.503