Our everyday experience strongly suggests a unidirectional flow of time, from past to future. However, at the microscopic level, the fundamental laws of physics, both classical and quantum, appear time-symmetric. The equations of motion remain valid even when the sign of the time coordinate is reversed (potentially along with other parameters). The CPT theorem in quantum field theory exemplifies this symmetry, stating that backward-in-time evolution is indistinguishable from forward-in-time evolution with charge and parity inversion. While thermodynamics introduces time asymmetry through the second law (entropy increase), even this asymmetry can be reduced to time-symmetric microscopic laws by assuming a low-entropy initial state. Despite this microscopic time symmetry, our interactions with the physical world are inherently unidirectional. Experiments typically involve preparing a system, letting it evolve forward in time, and then making measurements. This asymmetry in experimental design, however, is not a fundamental aspect of the dynamical laws themselves. This observation suggests that time asymmetry might be an emergent property arising from the way agents interact with physical systems, rather than a fundamental feature of the laws of physics. This paper explores the intriguing possibility of agents interacting with quantum systems in a reverse time direction – initializing systems in the future and observing their backward-in-time evolution. This concept is implicit in frameworks treating pre- and post-selected quantum states equally. The authors build on these frameworks to explore agents capable of observing physical processes in arbitrary combinations of forward and backward time directions. While such agents may not exist in reality, their hypothetical existence helps clarify the operational implications of fixed time direction by contrasting the information-theoretic capabilities associated with different temporal operational approaches.

The authors draw upon existing literature on pre- and post-selected quantum states and the associated two-state vector formalism (Aharonov et al., 1964; Aharonov & Vaidman, 2002). This formalism provides a framework for understanding quantum processes that are not constrained to a definite time direction. Previous works have explored related concepts, including quantum time-translation machines (Aharonov et al., 1990) and frameworks for probabilistic theories with non-fixed causal structure (Hardy, 2007). The notion of input-output inversion is linked to time-reversal in quantum mechanics (Wigner, 1959; Messiah, 1965) and quantum thermodynamics (Campisi et al., 2011), but the proposed input-output inversion is more general, encompassing other symmetries such as charge conjugation and parity inversion. The paper also relates to work on quantum computations without definite causal structure (Chiribella et al., 2013) and operations with indefinite causal order (Oreshkov et al., 2012; Chiribella et al., 2013).

The paper introduces a novel mathematical framework to analyze quantum operations with indefinite time direction. It begins by characterizing bidirectional quantum devices, which are quantum processes that can be accessed in both forward and backward time directions. The authors identify the set of quantum channels that are compatible with such bidirectional operation, finding it to coincide with the set of bistochastic channels. A bistochastic channel is a quantum channel with a Kraus representation such that both the sum of the products of Kraus operators and their adjoints is the identity, implying the channel leaves the maximally mixed state invariant. This characterization is achieved by defining an input-output inversion map (Θ), which transforms the quantum channel observed by a forward-facing agent into the channel observed by a hypothetical backward-facing agent. Four natural requirements are imposed on this map: order-reversing, identity-preserving, distinctness-preserving, and compatible with random mixtures. These requirements are motivated by the physical interpretation of the map and its relation to time reversal and other symmetries. The input-output inversion map is found to be either unitarily equivalent to the adjoint or transpose of the original channel. Next, the authors define a broader class of quantum operations using the concept of quantum supermaps (Chiribella et al., 2008; Chiribella et al., 2009; Chiribella et al., 2013). A quantum supermap describes transformations of quantum channels and preserves certain properties under local actions on composite systems. The focus is on supermaps transforming bistochastic channels into standard completely positive trace-preserving (CPTP) maps. This allows for the analysis of operations that employ quantum devices in arbitrary combinations of forward and backward time directions, leading to the concept of quantum operations with indefinite time direction. The authors then introduce a specific example of an operation with indefinite time direction: the quantum time flip. This operation takes a bidirectional device as input and produces a controlled channel that acts as the input channel when a control qubit is in |0⟩ and as its input-output inverse when the control qubit is in |1⟩. The construction of the quantum time flip utilizes the Choi representation of quantum channels and supermaps. The authors prove that the quantum time flip cannot be realized by any quantum circuit with a definite time direction, but a probabilistic realization is demonstrated using quantum teleportation. The paper further extends the framework to multipartite operations, characterized by quantum supermaps that transform lists of bistochastic channels into single CPTP maps. These multipartite operations can exhibit both indefinite time direction and indefinite causal order, providing a rich framework for potential extensions of quantum theory. The Choi representation is employed to characterize these multipartite operations as well.

The key findings of the paper are: 1. **Characterization of Bidirectional Quantum Channels:** The authors rigorously demonstrate that the set of quantum processes compatible with both forward and backward time directions (bidirectional processes) coincides with the set of bistochastic quantum channels. This unexpected connection arises from the properties imposed on the input-output inversion map (Θ). The result implies that only channels that are entropy-nondecreasing in both time directions are bidirectional. 2. **Quantum Operations with Indefinite Time Direction:** The paper introduces a framework for quantum operations that utilize bidirectional channels in superpositions of forward and backward time directions. This framework employs the concept of quantum supermaps, which are maps transforming quantum channels into other quantum channels while respecting certain locality conditions. The authors show that these supermaps can exhibit indefinite time direction, analogous to operations with indefinite causal order. 3. **The Quantum Time Flip:** A novel operation, termed the 'quantum time flip,' is introduced. This operation takes a bistochastic channel as input and generates a controlled channel that acts differently depending on the state of a control qubit, effectively applying the channel or its input-output inverse. Crucially, the quantum time flip cannot be perfectly realized by any quantum circuit with a definite time direction; a probabilistic implementation using quantum teleportation is demonstrated. This result implies that operations with indefinite time direction cannot be simulated by classical or quantum circuits using the components in well-defined time-order, even those with access to multiple copies of the initial resource and indefinite causal order. 4. **Information-Theoretic Advantage:** A game is presented where a player attempts to determine a property of two unknown unitary gates. The authors show that a player equipped with the quantum time flip can win with certainty, whereas players restricted to operations with definite time direction (even those allowing indefinite causal ordering) are limited in their winning probability. This difference provides strong evidence for the unique potential of quantum operations with indefinite time direction. 5. **Photonic Realization:** The authors demonstrate a photonic realization of a superposition of a unitary process and its input-output inverse using polarisation qubits and an interferometric setup. This realization provides a concrete physical example supporting the theoretical framework. The experimental setup exhibits a superposition of forward and backward processes.

The findings of this paper significantly advance our understanding of the operational significance of time direction in quantum theory. The characterization of bidirectional quantum channels reveals a profound connection between the reversibility of quantum processes and the second law of thermodynamics, suggesting new avenues for exploring the foundations of quantum thermodynamics. The introduction of quantum operations with indefinite time direction broadens the operational framework of quantum theory and paves the way for considering more general types of quantum computation and quantum networks. The quantum time flip, as a concrete example of an operation with indefinite time direction, showcases the potential for enhanced information-processing capabilities beyond those achievable with operations restricted to a definite time direction. The experimental proposal for a photonic implementation further strengthens the claim that indefinite time direction operations, while seemingly counterintuitive, are theoretically possible and may be experimentally realizable. This research opens up new avenues for investigating the interface between quantum mechanics and spacetime structure, potentially with implications for quantum gravity.

This paper presents a comprehensive framework for quantum operations with indefinite time direction. It establishes a link between bidirectional quantum channels and bistochastic channels, revealing a fundamental connection to thermodynamics. The quantum time flip serves as a concrete example highlighting the potential advantages of indefinite time-direction operations. The proposed photonic setup lays the groundwork for future experimental investigations. Further research could explore the implications of this framework in quantum gravity and quantum thermodynamics, potentially leading to new insights into the nature of time itself.

The current work focuses primarily on theoretical aspects. While a photonic realization of a superposition of a process and its input-output inverse is proposed, the complete experimental implementation of the quantum time flip remains a challenge for future research. The analysis is largely restricted to unitary operations and bistochastic channels. Generalization to more complex quantum operations and broader classes of quantum channels requires further investigation. The proposed information-theoretic game, while showcasing the potential advantages of the quantum time flip, is a specific example and might not fully capture the broader implications of indefinite time-direction operations in all quantum tasks.