Atemporal diagrams for quantum circuits

Abstract

A system of diagrams is introduced that allows the representation of various elements of a quantum circuit, including measurements, in a form which makes no reference to time (hence ``atemporal''). It can be used to relate quantum dynamical properties to those of entangled states (map-state duality), and suggests useful analogies, such as the inverse of an entangled ket. Diagrams clarify the role of channel kets, transition operators, dynamical operators (matrices), and Kraus rank for noisy quantum channels. Positive (semidefinite) operators are represented by diagrams with a symmetry that aids in understanding their connection with completely positive maps. The diagrams are used to analyze standard teleportation and dense coding, and for a careful study of unambiguous (conclusive) teleportation. A simple diagrammatic argument shows that a Kraus rank of 3 is impossible for a one-qubit channel modeled using a one-qubit environment in a mixed state.

Minor changes in references. Latex 32 pages, 13 figures in text using PSTricks

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• Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality
• Local distinguishability with preservation of entanglement
• Tensor Network Contractions for #SAT
• Separable Operations on Pure States
• Deterministic and Unambiguous Dense Coding
• Categorical Tensor Network States
• Quantum source-channel coding and non-commutative graph theory
• Categorical Quantum Circuits
• Entanglement requirements for implementing bipartite unitary operations
• Teleportation, Braid Group and Temperley--Lieb Algebra
• Types of quantum information
• Lectures on Quantum Tensor Networks
• Tripartite Entanglement in Qudit Stabilizer States and Application in Quantum Error Correction
• Simulating a single qubit channel using a mixed state environment
• Entanglement transformations using separable operations
• Local cloning of entangled states
• Computing the Tutte Polynomial of Lattice Path Matroids Using Determinantal Circuits
• Restrictions on the Schmidt rank of bipartite unitary operators beyond dimension two
• Complementarity of information sent via different bases
• Graphical Calculus for products and convolutions
• Topological-Like Features in Diagrammatical Quantum Circuits
• Separable Operations, Graph Codes and the Location of Quantum Information
• Local and global statistical distances are equivalent on pure states