Quantum Operations with Indefinite Time Direction

The paper addresses whether and how one can operationally probe quantum processes in the backward time direction, or in coherent combinations of forward and backward directions, given that microscopic laws are time-symmetric while laboratory operations are typically time-asymmetric. The authors propose a framework to formalize operations unconstrained by a fixed time direction, define the class of bidirectional devices that can be used both forward and backward, and construct higher-order transformations (supermaps) that act on such devices. They introduce the quantum time flip, an operation that coherently superposes using a process in the forward and backward time directions, and investigate its information-theoretic power and realizations. The work aims to clarify the operational meaning of time-direction constraints and to explore extensions of quantum theory that allow indefinite time direction and causal order.

The study builds on time-symmetry in fundamental physics (e.g., CPT theorem) and on frameworks treating pre- and postselected states on equal footing, including the two-state vector formalism and generalized multi-time formalisms. It connects to research on indefinite causal order and higher-order quantum transformations (quantum supermaps, process matrices, quantum SWITCH). Prior works on time reversal in quantum theory and thermodynamics are referenced, as well as resource-theoretic thermodynamics and recent explorations of thermodynamic processes with superposed time's arrows. The paper situates its contribution as extending these ideas by defining input-output inversion beyond simple time-reversal (potentially including charge and parity operations) and by characterizing which channels admit such inversion.

- Define bidirectional devices as processes that can be used both forward and backward by two agents. For a process occurring between times t1 and t2, the forward-facing agent observes a quantum channel C: S1→S2 (CPTP). A backward-facing agent observes a channel Θ(C): S2→S1. A map Θ (input-output inversion) is specified to relate the two descriptions. - Impose four requirements on Θ: (1) order-reversing (Θ(DC) = Θ(C)Θ(D)), (2) identity-preserving (Θ(IS)=IS), (3) distinctness-preserving (injective on processes), and (4) compatibility with random mixtures (affine). - Assume S1 and S2 have equal dimension and that all unitary dynamics are bidirectional. Show that for unitary channels the only possible input-output inversions (up to unitary equivalence) are adjoint (U→U†) or transpose (U→UT). - Characterize bidirectional processes among general channels using Kraus representations. Prove that the set of bidirectional channels coincides with the set of bistochastic (unital and trace-preserving) channels C(ρ)=Σi Ci ρ Ci† with Σi Ci Ci† = I and Σi Ci† Ci = I. Show that for such channels the only possible Θ (up to unitary equivalence) are the adjoint C† or the transpose CT. - Highlight a key difference: for dimensions >2, the adjoint is not completely positive on channels when applied locally to parts of bipartite processes, whereas the transpose is. Thus, local application of input-output inversion is valid iff inversion is described by transpose for d>2. - Define quantum operations (supermaps) on bidirectional devices: linear maps S that send bistochastic channels to CPTP channels and preserve complete positivity when acting locally on parts of composite processes. Use the Choi representation to express constraints: positivity of the supermap’s Choi operator and normalization conditions derived by decomposing the Choi operator into orthogonal components subject to trace constraints. - Construct the quantum time flip supermap F: given a bistochastic channel C with Kraus {Ci}, define a controlled channel FC with Kraus Fi := Ci⊗|0⟩⟨0| + Θ(Ci)⊗|1⟩⟨1|. Show: (i) FC is CPTP iff C is bistochastic; (ii) FC is representation-independent iff Θ is unitarily equivalent to the transpose; under these conditions, F is a valid supermap. Prove that F cannot be written as any convex mixture of a forward-only and a backward-only supermap. - Provide a probabilistic realization in a definite-time-direction circuit via teleportation: apply C to one half of a maximally entangled state, use a controlled SWAP to implement transpose vs identity coherently, perform a Bell measurement and post-select on a heralding outcome to realize the time-flipped controlled channel with non-unit probability. - Design an information-theoretic game: given black-box unitary gates U and V promised to satisfy either UVT = UT V or UVT = −UT V, show that applying time flips to generate SU = U|0⟩⟨0| + UT|1⟩⟨1| and SV analogously, and then composing SV followed by SU on |ψ⟩⊗|+⟩ yields orthogonal control outcomes enabling perfect discrimination. - Extend to multipartite supermaps on no-signalling bistochastic channels, including constructions that exhibit both indefinite time direction and indefinite causal order (e.g., combining the time flip with the quantum SWITCH). Provide Choi-based conditions for validity and normalization of such supermaps. - Present a photonic realization concept: interferometric setup where a photon traverses an unknown waveplate from opposite directions, implementing a coherent superposition of U and G U† G† (with G a fixed basis-change), enabling control over U vs U† after undoing G.

- Bidirectional channels = bistochastic channels: A quantum process admits a valid input-output inversion Θ if and only if it is bistochastic (unital and trace-preserving). This is established via linearity arguments and characterization theorems, showing the linear spans of unitary and bistochastic channels coincide. - Only two inversions (up to unitary equivalence): For unitary and bistochastic channels, Θ must be either the adjoint (C→C†) or the transpose (C→CT). For qubits (d=2) these are unitarily equivalent; for higher dimensions they differ. - Local CP property: When applied locally to part of a bipartite process (for d>2), the transpose-based inversion remains completely positive on channels while the adjoint generally does not. Hence, only the transpose defines a universally valid local inversion on subsystems. - Quantum time flip supermap: Defined a controlled channel F(C) that coherently applies C conditioned on |0⟩ and Θ(C) conditioned on |1⟩. Validity requires C to be bistochastic and Θ to be (unitarily equivalent to) transpose; under these conditions F is representation-independent and CPTP. - Indefinite time direction: The quantum time flip cannot be decomposed as any convex mixture of a forward-only and a backward-only supermap. A stronger no-go holds even with two uses of C and even if combined with indefinite causal order: using only a definite time direction for each use cannot reproduce F. - Teleportation-based realization: Provided a probabilistic realization using a maximally entangled resource, controlled SWAP, and Bell measurement. Heralded success reproduces the time-flipped channel. Deterministic realization would require agents with deterministic pre- and postselection capabilities not available in standard circuits. - Information-theoretic advantage: In a discrimination game between promises UVT = UT V vs UVT = −UT V, the time flip enables perfect success via controlled gates SU and SV and measurement in the {|+⟩,|−⟩} basis. Any strategy restricted to definite time direction (including those with indefinite causal order between uses) has a nonzero failure rate, at least about 11%. - Physical illustration: An interferometric photonic setup can implement a coherent superposition of a unitary U and its input-output inverse U† (up to a known basis change G), realizing a controlled-U/U† device without constructing it from an uncontrolled U in a circuit. - Thermodynamic link: Since bistochastic channels are exactly the entropy non-decreasing processes (mapping the maximally mixed state to itself), the processes admitting input-output inversion are precisely those compatible with non-decrease of entropy in both time directions.

The work directly addresses how to formalize operations that do not assume a fixed arrow of time in their usage of quantum devices. By characterizing bidirectional processes as bistochastic channels and limiting valid inversions to adjoint or transpose, it clarifies which physical dynamics can, in principle, be probed in both time directions. The quantum time flip showcases an operation with genuinely indefinite time direction, providing a concrete example with provable advantage in a discrimination task that cannot be matched by any definite-time-direction strategy, even if indefinite causal order is allowed. This separates the notion of indefinite time direction from indefinite causal order and demonstrates new operational capabilities. The framework, expressed via supermaps and their Choi representations, allows systematic composition of processes into higher-order structures that can combine indefinite time direction with indefinite causal order. The observed equivalence between bidirectionality and entropy non-decrease in both directions suggests deeper connections to thermodynamics. The results provide groundwork for exploring whether such operations are physically realizable in regimes like quantum gravity, or if physical principles forbid them.

The paper introduces a rigorous higher-order framework for quantum operations with indefinite time direction, characterizes bidirectional devices as bistochastic channels, and identifies the only consistent input-output inversions (adjoint or transpose, with transpose being locally CP for d>2). It defines and analyzes the quantum time flip, proves it is not realizable by any mixture of forward or backward operations (nor via multiple definite-time uses), gives a probabilistic teleportation-based implementation, and demonstrates a perfect-information advantage in a tailored game unattainable by indefinite causal order alone. It also presents a photonic interferometric realization of a coherent superposition of a process and its input-output inverse. Future research directions include: exploring physical realizability in quantum gravity and constraints that may forbid such operations; developing axiomatic links to thermodynamics and resource theories; generalizing quantum thermodynamics when agents can operate without a definite time direction; and further studying multipartite supermaps that blend indefinite time direction with indefinite causal order.

- Physical implementation constraints: The quantum time flip cannot be perfectly implemented by circuits operating in a definite time direction; only probabilistic, heralded implementations via teleportation are described within standard quantum circuits. - Assumptions on systems: Many results assume equal dimensions for input and output systems and focus on bistochastic channels; non-bistochastic processes fall outside the bidirectional set. - Local inversion: For dimensions greater than two, only the transpose-based inversion is valid when applied locally; the adjoint generally fails complete positivity on channels when localized, limiting applicability. - Hypothetical agents: Deterministic realizations rely on idealized agents capable of deterministic pre- and postselection, which may not be physically accessible. - Scope: Experimental photonic realization describes superposition of U and U† for unitary gates with specific optical implementations; general black-box device implementations remain nontrivial.