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arXiv:1612.05099 · Tutorials, schools, and workshops in the mathematical sciences

Causal Structures and the Classification of Higher Order Quantum Computations

Paolo Perinotti

Published Jan 1, 2017• 39 citations
arXivHTMLPDF
DOI
10.1007/978-3-319-68655-4_7

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Higher-order maps account for the most general transformations in quantum mechanics—they transform input quantum maps to output quantum maps. In this thesis, I focus on the application of such higher-order maps representing two important processes: (1) a particular type of non-Markovian quantum dynamics, and  (2) indefinite causal order. First, I will introduce our work on the non-Markovian evolution of quantum states. Such a situation occurs when the past system-environment interaction leads the environment to memorize the information about the system. The measure of the non-Markovianity is given by the temporal correlation among the operations on the system. One way to estimate such correlation is to employ a full tomography of the process matrix, which is computationally intractable. Instead, we utilize machine learning algorithms without using tomographically complete measurement, to estimate the measure of non-Markovianity. We apply our model to a dataset obtained from an optical experiment and show that we are able to predict the measure of non-Markovianity with 90% accuracy. I then introduce various aspects of indefinite causal order. Ordinary experience points only to a definite causal structure, but quantum mechanics admits a more general causal structure that allows for the superposition of different causal orders. That means that the statements, is in the causal past of event B, and B is in the causal past of event A can both be true. The quantum switch can physically realise such an indefinite causal structure where a control quantum system controls the order of operations acting on a target quantum system. Superposition in the control leads to the superposition of the order. We experimentally realise a quantum switch with the control qubit being the polarisation of light and the target qubit being the transverse spatial mode of light. We first verify our quantum switch by measuring a `causal witness'. We then demonstrate an advantage of the quantum switch for classical communication. Finally, I present our work on classical communication through the most general quantum causal structures, represented by the process matrix. We formulate different classical capacities of a bi-partite process matrix. We also study one-way and bidirectional communication protocols through such a process and establish the relevant communication bounds. These enable further analyses of higher-order maps that take advantage of the full suite of processes that quantum mechanics allows.

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Quantum processes with indefinite causal structure emerge when we wonder which are the most general evolutions, allowed by quantum theory, of a set of local systems which are not assumed to be in any particular causal order. These processes can be described within the framework of higher−order quantum theory which, starting from considering maps from quantum transformations to quantum transformations, recursively constructs a hierarchy of quantum maps of increasingly higher order. In this work, we develop a formalism for quantum computation with indefinite causal structures; namely, we characterize the computational structure of higher order quantum maps. Taking an axiomatic approach, the rules of this computation are identified as the most general compositions of higher order maps which are compatible with the mathematical structure of quantum theory. We provide a mathematical characterization of the admissible composition for arbitrary higher order quantum maps. We prove that these rules, which have a computational and information-theoretic nature, are determined by the more physical notion of the signalling relations between the quantum systems of the higher order quantum maps.
In quantum causality and quantum information, there is a vast landscape of abstract quantum protocols permitting cyclic or non-acyclic causal structures between operations, including frameworks for indefinite causal order and higher-order quantum processes such as process matrices. A longstanding open question is what is the largest class of abstract processes that admit physical realisations without post-selection. In this work, we provide a rigorous answer using a top-down approach grounded in relativistic causality principles. Building on the framework of causal boxes, which characterise the most general quantum information-processing protocols compatible with fixed background spacetimes, we formalise additional constraints (Acting Once + Local Order) capturing the closed-laboratory assumptions of the process matrix framework at a fine-grained spacetime level. We prove that any protocol in a classical acyclic spacetime satisfying these conditions is behaviourally equivalent to a quantum circuit with quantum control of causal order (QC-QC), providing a top-down derivation of QC-QCs from physical principles. Our results show that QC-QCs constitute precisely the class of higher-order quantum processes, including those with indefinite order, that can be physically realised within classical spacetime, ruling out more general non-causal processes under the closed-labs assumption. This clarifies the relationship between abstract higher-order process matrix frameworks and experimentally accessible quantum protocols, as well as the interplay between coarse-grained cyclic and fine-grained acyclic operational causal structures. We also develop characterisation techniques for process box protocols that lead to new causality-based open questions concerning spacetime quantum protocols and relativistic quantum experiments.
Higher-order maps account for the most general transformations in quantum mechanics—they transform input quantum maps to output quantum maps. In this thesis, I focus on the application of such higher-order maps representing two important processes: (1) a particular type of non-Markovian quantum dynamics, and  (2) indefinite causal order. First, I will introduce our work on the non-Markovian evolution of quantum states. Such a situation occurs when the past system-environment interaction leads the environment to memorize the information about the system. The measure of the non-Markovianity is given by the temporal correlation among the operations on the system. One way to estimate such correlation is to employ a full tomography of the process matrix, which is computationally intractable. Instead, we utilize machine learning algorithms without using tomographically complete measurement, to estimate the measure of non-Markovianity. We apply our model to a dataset obtained from an optical experiment and show that we are able to predict the measure of non-Markovianity with 90% accuracy. I then introduce various aspects of indefinite causal order. Ordinary experience points only to a definite causal structure, but quantum mechanics admits a more general causal structure that allows for the superposition of different causal orders. That means that the statements, is in the causal past of event B, and B is in the causal past of event A can both be true. The quantum switch can physically realise such an indefinite causal structure where a control quantum system controls the order of operations acting on a target quantum system. Superposition in the control leads to the superposition of the order. We experimentally realise a quantum switch with the control qubit being the polarisation of light and the target qubit being the transverse spatial mode of light. We first verify our quantum switch by measuring a `causal witness'. We then demonstrate an advantage of the quantum switch for classical communication. Finally, I present our work on classical communication through the most general quantum causal structures, represented by the process matrix. We formulate different classical capacities of a bi-partite process matrix. We also study one-way and bidirectional communication protocols through such a process and establish the relevant communication bounds. These enable further analyses of higher-order maps that take advantage of the full suite of processes that quantum mechanics allows.