Quantum resources exist in a hierarchy of multiple levels. At order zero, quantum states are transformed by linear maps (channels, or gates) in order to perform computations or simulate other states. At order one, gates and channels are transformed by linear maps (superchannels) in order to simulate other gates. To develop a full hierarchy of quantum resources, beyond those first two orders, and to account for the fact that quantum protocols can interconvert resources of different orders, we need a theoretical framework that addresses all orders in a uniform manner. We introduce a framework based on a system of types, which label the different kinds of objects that are present at different orders. We equip the framework with a parallel product operation that modifies and generalizes the tensor product so as to be operationally meaningful for maps of distinct and arbitrary orders. Finally, we introduce a family of convex cones that generalize the notion of complete positivity to all orders, with the aim of characterizing the objects that are physically admissible, facilitating an operational treatment of quantum objects at any order.
Quantum resources exist in a hierarchy of multiple levels. At order zero, quantum states are transformed by linear maps (channels, or gates) in order to perform computations or simulate other states. At order one, gates and channels are transformed by linear maps (superchannels) in order to simulate other gates. To develop a full hierarchy of quantum resources, beyond those first two orders, and to account for the fact that quantum protocols can interconvert resources of different orders, we need a theoretical framework that addresses all orders in a uniform manner. We introduce a framework based on a system of types, which label the different kinds of objects that are present at different orders. We equip the framework with a parallel product operation that modifies and generalizes the tensor product so as to be operationally meaningful for maps of distinct and arbitrary orders. Finally, we introduce a family of convex cones that generalize the notion of complete positivity to all orders, with the aim of characterizing the objects that are physically admissible, facilitating an operational treatment of quantum objects at any order.
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.
We develop a general framework for higher-order quantum resources that treats the resourcefulness of quantum states, channels, and supermaps (also called processes) in a unified manner. By generalizing well-established concepts from resource theories of states and channels, we introduce and study supermap resource monotones and supermap distillation and dilution. As free supermaps, we consider free networks (supermaps that can be implemented with free channels), compatible supermaps (sets of supermaps closed under arbitrary composition), and maximally free superchannels. Specializing to approximate channel-to-channel conversions, we show that under mild assumptions, the conversion error with respect to the diamond distance and the worst-case infidelity coincide. We then apply the general framework to resource theories of dynamical coherence, where we establish the existence of largest sets of compatible supermaps. By characterizing these sets with semidefinite constraints, we show that they are strictly larger than the corresponding free networks but do not contain all maximally free superchannels. We finish by investigating the operational differences of these sets in one-shot channel distillation and dilution, providing solutions for both tasks in the parallel setting and for dilution in the adaptive setting.
Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.
We propose a new 'quantum domain theory' in which Scott-continuous functions are replaced by Scott-continuous natural transformations. Completely positive maps are widely accepted as a model of first-order quantum computation. We begin by establishing a categorical characterization of completely positive maps as natural families of positive maps. We explore this categorical characterization by building various representations of quantum computation based on different structures: affine maps between cones of positive elements, morphisms of algebras of effects, and affine maps of convex sets of states. By focusing on convex dcpos, we develop a quantum domain theory and show that it supports some important constructions such as tensor products by quantum data, and lifting.
We initiate the systematic study of resource theories of quantum channels, i.e. of the dynamics that quantum systems undergo by completely positive maps, in abstracto: Resources are in principle all maps from one quantum system to another, but some maps are deemed free. The free maps are supposed to satisfy certain axioms, among them closure under tensor products, under composition and freeness of the identity map (the latter two say that the free maps form a monoid). The free maps act on the resources simply by tensor product and composition. This generalizes the much-studied resource theories of quantum states, and abolishes the distinction between resources (states) and the free maps, which act on the former, leaving only maps, divided into resource-full and resource-free ones. We discuss the axiomatic framework of quantifying channel resources, and show two general methods of constructing resource monotones of channels. Furthermore, we show that under mild regularity conditions, each resource theory of quantum channels has a distinguished monotone, the robustness (and its smoothed version), generalizing the analogous concept in resource theories of states. We give an operational interpretation of the log-robustness as the amount of heat dissipation (randomness) required for resource erasure by random reversible free maps, valid in broad classes of resource theories of quantum channels. Technically, this is based on an abstract version of the recent convex-split lemma, extended from its original domain of quantum states to ordered vector spaces with sufficiently well-behaved base norms (which includes the case of quantum channels with diamond norm or variants thereof). Finally, we remark on several key issues concerning the asymptotic theory.
Processes with indefinite causal orders (ICOs), such as the quantum switch, are higher-order quantum processes that superpose the order in which quantum operations are performed. Such coherent control yields computational advantages but is not faithfully captured by existing quantum programming languages: either they are restricted to the unitary case, and thus cannot combine ICOs with measurement, or they treat coherent control nonlinearly. In both cases, they do not realize the full computational power of ICOs. We introduce a higher-order quantum functional language that supports general quantum computation, not merely the permutation of channels, and whose linear type system allows quantum control to be well-defined beyond the unitary case, on arbitrary quantum channels. We equip this language with a small-step operational semantics that synchronizes measurement outcomes across superposed branches, using device references and a memory function. We also give a denotational semantics by means of completely positive maps. With linearity as the only constraint, some well-typed terms would denote unphysical maps. We therefore impose a typing discipline that goes beyond linearity, and interpret programs in the causal category Caus[CPM], under which every well-typed program is physically meaningful, a property that can be checked statically and efficiently. We prove soundness, and study the language's expressive power: it can express every quantum channel at first order, and at second order a large subclass of the so-called quantum circuits with quantum control (QC-QCs), containing the quantum switch. Last but not least, we show that this language is well-designed enough to be extended to the nonlinear setting with recursion.