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.
AI Summary
The paper asks how to characterize arbitrary higher-order transformations when elementary devices are bidirectional, represented by bistochastic channels, rather than channels with fixed input–output roles. It extends earlier characterizations of transformations taking several bistochastic channels to a single channel to a recursively defined hierarchy with arbitrary higher-order inputs and outputs.
An operational admissibility condition requires preservation of valid events under ancillary extensions and completion of probabilistic events into deterministic ones. For every type $x$, admissible Choi operators are characterized by $M\geq0$ and $M\leq D$ for some deterministic operator $D$. Deterministic operators have the form $D=\lambda_x\mathds{1}+X_x\geq0$, with explicit recursive rules for the normalization $\lambda_x$ and linear subspace containing $X_x$. The proofs adapt existing higher-order transformation arguments by verifying the additional elementary cases involving partially bistochastic channels.
The framework yields characterizations and sequential realization theorems for networks of higher-order maps. In particular, bi-slot combs decompose into sequences of bistochastic supermaps: laboratories have a definite global causal order even when individual devices have indefinite input–output direction. Bistochastic process matrices additionally permit indefinite global order. Examples include sequential time flips and a construction combining coherent order reversal with direction reversal; previously introduced processes allowing perfect two-way signaling are incorporated into the framework.
The characterization concerns operational consistency, not universal physical implementability. Resource advantages in communication, computation and discrimination, thermodynamic applications, and the relation between composition and signaling remain open directions.
Some signs of AI-generated textIncrementalPossible error
Error found: The explicitly expanded LC R_2 and R_3 operators are not deterministic bistochastic processes: valid local bistochastic channels yield normalization values 2 and 3, respectively, rather than 1.
The identity-on-output expansions for the quantum time flip and flippable SWITCH incorrectly remove branch-dependent input-control projectors, although correcting those expansions preserves the constructions’ claimed admissibility.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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 paper asks how to characterize arbitrary higher-order transformations when elementary devices are bidirectional, represented by bistochastic channels, rather than channels with fixed input–output roles. It extends earlier characterizations of transformations taking several bistochastic channels to a single channel to a recursively defined hierarchy with arbitrary higher-order inputs and outputs.
An operational admissibility condition requires preservation of valid events under ancillary extensions and completion of probabilistic events into deterministic ones. For every type x, admissible Choi operators are characterized by M≥0 and M≤D for some deterministic operator D. Deterministic operators have the form D=λx\mathds1+Xx≥0, with explicit recursive rules for the normalization λx and linear subspace containing Xx. The proofs adapt existing higher-order transformation arguments by verifying the additional elementary cases involving partially bistochastic channels.
The framework yields characterizations and sequential realization theorems for networks of higher-order maps. In particular, bi-slot combs decompose into sequences of bistochastic supermaps: laboratories have a definite global causal order even when individual devices have indefinite input–output direction. Bistochastic process matrices additionally permit indefinite global order. Examples include sequential time flips and a construction combining coherent order reversal with direction reversal; previously introduced processes allowing perfect two-way signaling are incorporated into the framework.
The characterization concerns operational consistency, not universal physical implementability. Resource advantages in communication, computation and discrimination, thermodynamic applications, and the relation between composition and signaling remain open directions.
Abstract Recent developments in quantum information science have revealed that the quantum circuit model, while foundational, does not exhaust the computational possibilities permitted by quantum mechanics. It allows for higher-order models of quantum computing based on an hierarchy of supermaps starting by transformations of quantum operations instead of quantum states. Experimental implementations of quantum SWITCH and quantum time flip, supermaps that allow for indefinite causal order and input-output direction of operations, respectively, and their applications have triggered a high interest on higher-order generalizations of quantum theory. In this paper, we review a framework for higher-order quantum theory with indefinite time direction based on an axiomatic type-theoretic characterization of quantum operations of arbitrary complexity. This framework can be naturally used to generalize process matrices, which introduce indefinite causal order of local agents’ operations into quantum theory, to include indefinite time direction.
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.
Abstract Certain quantum devices, such as half-wave plates and quarter-wave plates in quantum optics, are bidirectional, meaning that the roles of their input and output ports can be exchanged. Bidirectional devices can be used in a forward mode and a backward mode, corresponding to two opposite choices of the input-output direction. They can also be used in a coherent superposition of the forward and backward modes, giving rise to new operations with indefinite input-output direction. In this work we explore the potential of input-output indefiniteness for the transfer of classical and quantum information through noisy channels. We first formulate a model of communication from a sender to a receiver via a noisy channel used in indefinite input-output direction. Then, we show that indefiniteness of the input-output direction yields advantages over standard communication protocols in which the given noisy channel is used in a fixed input-output direction. These advantages range from a general reduction of noise in bidirectional processes, to heralded noiseless transmission of quantum states, and, in some special cases, to a complete noise removal. The noise reduction due to input-output indefiniteness can be experimentally demonstrated with current photonic technologies, providing a way to investigate the operational consequences of exotic scenarios characterised by coherent quantum superpositions of forward-time and backward-time processes.
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.
Abstract At the fundamental level, the dynamics of quantum fields is invariant under the combination of time reversal, charge conjugation, and parity inversion. This symmetry implies that a broad class of effective quantum evolutions are bidirectional, meaning that the exchange of their inputs and outputs gives rise to valid quantum evolutions. Recently, it has been observed that quantum theory is theoretically compatible with a family of operations in which the roles of the inputs and outputs is indefinite. However, such operations have not been demonstrated in the laboratory so far. Here we experimentally demonstrate input-output indefiniteness in a photonic setup, demonstrating an advantage in a quantum game and showing incompatibility with a definite input-output direction by more than 69 standard deviations. Our results establish input-output indefiniteness as a new resource for quantum information protocols, and enable the table-top simulation of hypothetical scenarios where the arrow of time could be in a quantum superposition.
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.