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The operator tensor formulation of quantum theory

Lucién Hardy

Published Aug 17, 2016• 42 citations
OpenAlex

Abstract

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).

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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.

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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.

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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).
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.
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.
The conventional phase space of classical physics treats space and time differently, and this difference carries over to field theories and quantum mechanics (QM). In this paper, the phase space is enhanced through two main extensions. First, we promote the time choice of the Legendre transform to a dynamical variable. Second, we extend the Poisson brackets of matter fields to a spacetime symmetric form. The ensuing ``spacetime phase space'' is employed to obtain an explicitly covariant version of Hamilton equations for relativistic field theories. A canonical-like quantization of the formalism is then presented in which the fields satisfy spacetime commutation relations and the foliation is quantum. In this approach, the classical action is also promoted to an operator and retains explicit covariance through its nonseparability in the matter-foliation partition. The problem of establishing a correspondence between the new noncausal framework (where fields at different times are independent) and conventional QM is solved through a generalization of spacelike correlators to spacetime. In this generalization, the Hamiltonian is replaced by the action, and conventional particles by off-shell particles. When the foliation is quantized, the previous map is recovered by conditioning on foliation eigenstates, in analogy with the Page and Wootters mechanism. We also provide an interpretation of the correspondence in which the causal structure of a given theory emerges from the quantum correlations between the system and an environment. This idea holds for general quantum systems and allows one to generalize the density matrix to an operator containing the information of correlators both in space and time.
The study of quantum systems involving two or more qubits necessitates the development of mathematical tools which capture the interactions between the qubits. This leads to the construction of objects called tensors which are elements of a vector space called a tensor product space. This material is essential since tensors are required for defining the means of data representation in quantum computation. The tensor product is the result of combining two vectors in a particular way designed to capture the interaction between the various parts of the individual vectors. The results of finding tensor products reside in a vector space called the tensor product space, the elements of which are called tensors. The formal construction of a tensor product space is somewhat abstract and can be difficult for the student to understand.
Quantum field theory (QFT) is a conceptual framework for understanding the behaviour of subatomic particles-the most successful, mathematically rigorous formulation of QFT is in the language of operator algebras.In this thesis, we describe the construction of specific kinds of QFTs, called gauge theories, using operator-algebraic methods.Once we have described their construction in detail, we use tensor network methods (which are at the centre of modern quantum physics) to build approximations of these QFTs.We finish with a discussion on the relationship between our tensor networks and those used in toy models of the AdS/CFT correspondence. 5.1.2The limit state and the GNS completion . . . . . . . . . . . .5.2 An abelian group with a single measure . . . . . . . . . . . . . . . . .
Quantum algorithms offer the potential for exponential speedups over classical algorithms for specific computational tasks. However, the inherent complexity and probabilistic nature of quantum mechanics pose significant challenges in verifying and understanding these algorithms. Traditional approaches, relying on simulations and theoretical arguments, often struggle to provide definitive proof of correctness and to guide the optimization of quantum circuits. This paper proposes a novel approach to formally specifying quantum algorithms by representing them using tensor networks. Tensor networks provide a powerful mathematical framework for describing multi-dimensional quantum states and operations, allowing for a precise and rigorous description of the algorithm's evolution. We demonstrate how tensor network techniques can be used to capture the unitary transformations that constitute a quantum algorithm, providing a foundation for automated verification and optimization strategies. The key contribution of this work is the application of tensor networks to the formal specification of quantum algorithms, addressing a critical gap in the field and paving the way for more robust and reliable quantum computation. The formal specification allows for the definition of the algorithm's state space and the evolution operator, enabling the application of mathematical techniques to analyze and improve the algorithm's performance.