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arXiv:1509.08339

A diagrammatic approach to map-state and channel-state dualities

Ville Bergholm

Published Sep 28, 2015• 1 citation
arXivHTMLPDF

Abstract

Diagrammatic representation and manipulation of tensor networks has proven to be a useful tool in mathematics, physics, and computer science. Here we present several important and mostly well-known theorems regarding the dualities between linear maps and bipartite pure quantum states, and the dualities between quantum channels and bipartite mixed quantum states, in diagrammatic form. The graphical presentation makes the proofs very compact and in some cases even intuitive.

DOI
10.48550/arxiv.1509.08339

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Diagrammatic representation and manipulation of tensor networks has proven to be a useful tool in mathematics, physics, and computer science. Here we present several important and mostly well-known theorems regarding the dualities between linear maps and bipartite pure quantum states, and the dualities between quantum channels and bipartite mixed quantum states, in diagrammatic form. The graphical presentation makes the proofs very compact and in some cases even intuitive.
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The intuitiveness of the tensor network graphical language is becoming well known through its use in numerical simulations using methods from tensor network algorithms. Recent times have also seen rapid progress in developing equations of motion to predict the time evolution of quantum entanglement [Nature Physics, 4(\textbf{4}):99, 2008]. Here we cast these recent results into a tensor network framework and in doing so, construct a theory which exposes the topological equivalence of the evolution of a family of entanglement monotones in arbitrary dimensions. This unification was accomplished by tailoring a form of channel state duality through the interpretation of graphical tensor network rewrite rules. The introduction of tensor network methods to the theory of entanglement evolution opens the door to apply methods from the rapidly evolving area known as tensor network states.
We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits . We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.
This book serves as an introductory yet thorough guide to tensor networks and their applications in quantum computation and quantum information, designed for advanced undergraduate and graduate-level readers. In Part I, foundational topics are covered, including tensor structures and network representations like Matrix Product States (MPS) and Tree Tensor Networks (TTN). These preliminaries provide readers with the core mathematical tools and concepts necessary for quantum physics and quantum computing applications, bridging the gap between multi-linear algebra and complex quantum systems. Part II explores practical applications of tensor networks in simulating quantum dynamics, with a particular focus on the efficiency they offer for systems of high computational complexity. Key topics include Hamiltonian dynamics, quantum annealing, open system dynamics, and optimization strategies using TN frameworks. A final chapter addresses the emerging role of “quantum magic” in tensor networks. It delves into non-stabilizer states and their contribution to quantum computational power beyond classical simulability, featuring methods such as stabilizer-enhanced MPS and the Clifford-dressed TDVP. In presenting tensor networks as tools for understanding quantum complexity, this book aims to foster a deeper collaboration between the many-body physics and quantum computing communities, inviting a broader audience to engage with these recent developments.
We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.
We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we demonstrate the construction of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of quantum states, review how these representations interrelate, and illustrate how graphical manipulations of the tensor networks may be used to concisely transform between them. To further demonstrate the utility of the presented graphical calculus we include several examples where we provide arguably simpler graphical proofs of several useful quantities in quantum information theory including the composition and contraction of multipartite channels, a condition for whether an arbitrary bipartite state may be used for ancilla assisted process tomography, and the derivation of expressions for the average gate fidelity and entanglement fidelity of a channel in terms of each of the different representations of the channel.