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arXiv:0912.4740 · Cambridge University Press eBooks

Foliable Operational Structures for General Probabilistic Theories

Lucién Hardy

Published Apr 18, 2011• 64 citations
arXivHTMLPDF

Abstract

Preamble In this chapter, a general mathematical framework for probabilistic theories of operationally understood circuits is laid out. Circuits consist of operations and wires. An operation is one use of an apparatus, and a wire is a diagrammatic device for showing how apertures on the apparatuses are placed next to each other. Mathematical objects are defined in terms of the circuit understood graphically. In particular, we do not think of the circuit as sitting in a background time. Circuits can be foliated by hypersurfaces composed of sets of wires. Systems are defined to be associated with wires. A closable set of operations is defined to be one for which the probability associated with any circuit built from this set is independent both of choices on other circuits and of extra circuitry that may be added to outputs from this circuit. States can be associated with circuit fragments corresponding to preparations. These states evolve on passing through circuit fragments corresponding to transformations. The composition of transformations is treated. A number of theorems are proven, including one that rules out quaternionic quantum theory. The case of locally tomographic theories (where local measurements on a systems components suffice to determine the global state) is considered. For such theories, the probability can be calculated for a circuit from matrices pertaining to the operations that constitute that circuit. Classical probability theory and quantum theory are exhibited as examples in this framework.

DOI
10.1017/cbo9780511976971.013

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Preamble In this chapter, a general mathematical framework for probabilistic theories of operationally understood circuits is laid out. Circuits consist of operations and wires. An operation is one use of an apparatus, and a wire is a diagrammatic device for showing how apertures on the apparatuses are placed next to each other. Mathematical objects are defined in terms of the circuit understood graphically. In particular, we do not think of the circuit as sitting in a background time. Circuits can be foliated by hypersurfaces composed of sets of wires. Systems are defined to be associated with wires. A closable set of operations is defined to be one for which the probability associated with any circuit built from this set is independent both of choices on other circuits and of extra circuitry that may be added to outputs from this circuit. States can be associated with circuit fragments corresponding to preparations. These states evolve on passing through circuit fragments corresponding to transformations. The composition of transformations is treated. A number of theorems are proven, including one that rules out quaternionic quantum theory. The case of locally tomographic theories (where local measurements on a systems components suffice to determine the global state) is considered. For such theories, the probability can be calculated for a circuit from matrices pertaining to the operations that constitute that circuit. Classical probability theory and quantum theory are exhibited as examples in this framework.
In this paper we consider general probabilistic theories pertaining to circuits that satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of space–time employ only mathematical objects associated with that region. We call this formalism locality. It incorporates the idea that space and time should be treated on an equal footing. Formulations that use a foliation of space--time to evolve a state do not have this property, nor do histories-based approaches. An operation has inputs and outputs (through which systems travel), for example, A circuit is built by wiring together operations such that we have no open inputs or outputs left over. A fragment is a part of a circuit and may have open inputs and outputs, for example, We show how each operation is associated with a certain mathematical object, which we call a duotensor (this is like a tensor but with a bit more structure). The following diagram shows how a duotensor is represented graphically: We can link duotensors together such that black and white dots match up to get the duotensor corresponding to any fragment. The following diagram shows the duotensor for the above fragment: Links represent summation over the corresponding indices. We can use such duotensors to make probabilistic statements pertaining to fragments. Since fragments are the circuit equivalent of arbitrary space–time regions, we have formalism locality. The probability for a circuit is given by the corresponding duotensorial calculation (which is a scalar since there are no indices left over). We show how to put classical probability theory and quantum theory into this framework.
A typical quantum experiment has a bunch of apparatuses placed so that quantum systems can pass between them. We regard each use of an apparatus, along with some given outcome on the apparatus (a certain detector click or a certain meter reading for example), as an operation. An operation can have zero or more quantum systems inputted into it and zero or more quantum systems outputted from it. We can wire together operations to form circuits. In the standard framework of quantum theory we must foliate the circuit then calculate the probability by evolving a state through it. This approach has three problems. First, we must introduce an arbitrary foliation of the circuit (such foliations are not unique). Second, we have to pad our expressions with identities every time two or more foliation hypersurfaces intersect a given wire. And third, we treat operations corresponding to preparations, transformations, and results in different ways. In this paper we present the operator tensor formulation of quantum theory which solves all these problems. Corresponding to every operation is an operator tensor. The probability for a circuit is given by simply replacing the operations in the circuit with the corresponding operator tensors. Wires between operator tensors correspond to multiplying the tensors in the associated subspace and then taking the partial trace over that subspace. Operator tensors must be physical (namely, they must have positive input transpose and satisfy a certain normalization condition).
We provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of physical operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Sharpness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Sturdiness. Filters are non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilitieso follows. A more detailed abstract is provided in the paper.
This is a book about operational probabilistic theories. The standard approach in such theories is from a time forward perspective. In this book we mostly take a time symmetric perspective. This presents a branding problem. Is this a niche book merely about time symmetry? No. This is a comprehensive book about operational probabilistic theories, but mostly from a time symmetric perspective. In fact, this book consists of (1) a simple book about simple operations having simple causal structure (where all the inputs are before all the outputs), and (2) a complex book about complex operations that can have complicated causal structure (a complex operation is equipped with a causal diagram). For the simple case we are able to show that the time symmetric perspective is equivalent to the time forward perspective. In each book we set up (A) operational probabilistic theories (OPTs) in terms of operations, (B) Operational Quantum Theory (OQT) in terms of operator tensors which correspond to operations, and (C) the theory of Hilbert objects which can be doubled up to give operator tensors. Operations are required to be physical which guarantees that circuits built out of operations have probabilities between 0 and 1 and that certain causality conditions are met. We prove that when we wire together operations the resulting networks are also physical. We model Sorkin's impossible measurements with complex operations and show that physicality prevents anomalous signalling. We develop diagrammatic notation for Hilbert objects. This includes mirrors for doubling up and mirror theorems. We use this framework to prove time symmetric causal dilation theorems for various causal diagrams.
We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.
In this paper, we present a formulation of quantum theory in terms of bold operator tensors. A circuit is built up of operations where an operation corresponds to a use of an apparatus. We associate collections of operator tensors (which together comprise a bold operator) with these apparatus uses. We give rules for combining bold operator tensors such that, for a circuit, they give a probability distribution over the possible outcomes. If we impose certain physicality constraints on the bold operator tensors, then we get exactly the quantum formalism. We provide both symbolic and diagrammatic ways to represent these calculations. This approach is manifestly covariant in that it does not require us to foliate the circuit into time steps and then evolve a state. Thus, the approach forms a natural starting point for an operational approach to quantum field theory.