Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between general linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.
Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between general linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.
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.
Bidirectional devices are devices for which the roles of the input and output ports can be exchanged. Mathematically, these devices are described by bistochastic quantum channels, namely completely positive linear maps that are both trace-preserving and identity-preserving. Recently, it has been shown that bidirectional quantum devices can, in principle, be used in ways that are incompatible with a definite input-output direction, giving rise to a new phenomenon called input-output indefiniteness. Here we characterize the most general forms of input-output indefiniteness, associated with a hierarchy of higher-order transformations built from transformations of bistochastic quantum channels. Some levels of the hierarchy correspond to transformations that combine bistochastic channels in a definite causal order, while generally using each channel in an indefinite input-output direction. For other levels of the hierarchy, the indefiniteness can involve both the local input-output direction of each process and the global causal order among the processes. On the foundational side, the hierarchy of higher-order transformations characterized here can be regarded as the largest set of physical processes compatible with a time-symmetric variant of quantum theory, where the possible state transformations are restricted to bistochastic channels.
The study of quantum information protocols that process quantum gates has led to an interest in ‘higher-order’ quantum transformations, which are transformations that act on other quantum transformations as inputs and/or return them as outputs. Mathematically, such transformations form an inductive hierarchy of infinitely many levels, or ‘orders’: quantum states are at order zero; channels, which transform quantum states, are at order one; superchannels, which transform channels, are at order two; maps that transform superchannels are at order three, and so on. To proceed toward a general understanding of higher-order quantum theory, we need methods to study the full inductive hierarchy in a uniform manner. In the context of the Hilbert space formalism of quantum theory, this means studying operator spaces (used to represent states), linear operators between operator spaces (used to represent channels), etc.; in other words, the inductive hierarchy of higher-order linear maps. We introduce a methodological framework for higher-order quantum theory, partly modeled on that of [Bisio \& Perinotti, 2019]. The framework is based on a set of types, denoted TypesA. Types may be thought of as labels that we use to identify and differentiate all the different kinds of objects of higher-order quantum theory. The central objects of the framework are typed (higher-order) linear maps, as given by the construction of a set L(x) of linear maps of type x for each x ∈ TypesA. We equip the framework with a parallel product operation for typed linear maps, which modifies and generalizes the tensor product so as to be physically meaningful when applied to maps of distinct and arbitrary orders. With the aim of characterizing the higher-order linear maps that represent valid quantum objects, we give a construction, consisting of a convex cone K(x) for each x ∈ TypesA, which generalizes the notion of complete positivity to all orders. We construct a version of the Choi isomorphism for our framework, and we use it to show that K(x) is self-dual for all x ∈ TypesA.
Quantum thermodynamics is traditionally formulated as a theory of equilibrium states and state transformations. Recent advances in higher-order quantum transformations, which describe physical scenarios beyond states and channels and provide a systematic framework for causal order, raise the question of how equilibrium should be defined and preserved in this more general setting. Starting from the Gibbs state as the unique equilibrium state, we identify equilibrium preservation as its natural higher-order extension. We show that, while this principle can in general give rise to distinct classes of transformations, all such distinctions disappear when equilibrium preservation is required completely, namely under arbitrary ancillary extensions. Remarkably, every transformation satisfying this condition is causally ordered, making causal order an emergent consequence of thermodynamic equilibrium. We establish this result for arbitrary higher-order maps and use the resulting framework to introduce free-energy-like quantities for quantum channels. Our findings reveal a fundamental connection between thermodynamic equilibrium and causal structure, providing a foundation for a fully fledged theory of higher-order quantum thermodynamics.
Quantum superchannels are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps we motivate and define superchannels on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.
In recent years, the investigation of causal relations in quantum theory has attracted a lot of interest. In particular, it has been found that it is possible to conceive of scenarios where some parties perform operations that are compatible with quantum theory locally, but that cannot be embedded into a global background causal structure. Such indefinite causal structures are of interest from a fundamental point of view, but also from the perspective of quantum information processing, since they do not fit into the usual paradigm of quantum circuits, which assumes a definite causal order. The main aim of this thesis is to study indefinite quantum causal relations involving more than two parties. Compared to the bipartite case, there are many new aspects and complications that arise in multipartite situations, which need to be clarified in order to fundamentally understand quantum causal structures, and to shed light on their implications and potential usefulness for quantum information processing.A suitable mathematical framework for the investigation of quantum causal relations is the process matrix formalism. In this framework, the notion of causal nonseparability was introduced in order to qualify scenarios that are incompatible with a definite causal order. In Chapter 2, we study how to generalise this concept to the multipartite case, how to characterise multipartite causally (non)separable quantum processes, and how to certify multipartite causal nonseparability. Another important topic is to determine which quantum scenarios with indefinite causal order are physically implementable, and how they can be realised concretely. In Chapter 3, we introduce two new classes of physically realisable multipartite quantum processes, and characterise them in terms of their process matrix descriptions. In particular, we define the class of quantum circuits with quantum control of causal order. The simplest example of such a circuit is the quantum switch, a protocol in which the order between two operations is controlled by a qubit in a superposition state, and which defines a causally nonseparable process. The class we introduce also contains more general examples of causally nonseparable processes with new features. We then show how the process matrix characterisation of this class allows us to search for new quantum information processing applications of causal nonseparability. In Chapter 4, we investigate a particular quantum communication effect in a scenario involving coherent control between two quantum channels. This leads us to a more general analysis of the notion of a coherently controlled channel, which involves certain subtilities. In Chapter 5, we turn to another problem that arises in multipartite scenarios, namely whether a given phenomenon is genuinely multipartite or not. More particularly, we study genuinely multipartite (non)causal correlations. In Chapter 6, we show that anomalous weak values are possible without post-selection. In Chapter 7, we show that certain tripartite process matrices, namely those that are unitarily extensible, have a realisation on so-called time-delocalised subsystems, i.e., quantum subsystems that are not associated with a definite time. The class of unitarily extensible tripartite process matrices is larger than the class of quantum circuits with quantum control, and in particular contains processes that violate so-called causal inequalities.