The arrow of time in operational formulations of quantum theory

Abstract: The operational formulations of quantum theory are drastically time oriented. However, to the best of our knowledge, microscopic physics is time-symmetric. We address this tension by showing that the asymmetry of the operational formulations does not reflect a fundamental time-orientation of physics. Instead, it stems from built-in assumptions about the $users$ of the theory. In particular, these formalisms are designed for predicting the future based on information about the past, and the main mathematical objects contain implicit assumption about the past, but not about the future. The main asymmetry in quantum theory is the difference between knowns and unknowns.

• The paper shows that quantum indeterminism remains time-reversal invariant, with perceived asymmetry arising from inference directionality.
• It introduces operational definitions for prediction and postdiction that yield symmetric probabilities under specific normalization conditions.
• The study attributes the observed time orientation in quantum formalism to user assumptions and entropy gradients rather than intrinsic quantum laws.

The Arrow of Time in Operational Formulations of Quantum Theory

Introduction

Classical mechanics is inherently time-reversal invariant, where its fundamental laws don't differentiate between past and future. The observed arrow of time is a macroscopic phenomenon dependent on macroscopic variables and the fact that entropy characterized by these variables was lower in the past. In contrast, quantum mechanics presents a dual image. On one hand, the Schrödinger equation and quantum field theories are time reversal invariant, indicating no preferred direction of time in elementary quantum phenomena. On the other hand, quantum theory formalism often appears heavily time-oriented. This paper examines the tension between microscopic time symmetry and the time-oriented nature of quantum formalism, proposing that inherent asymmetries arise from inference directionality and user assumptions rather than fundamental physics.

Prediction and Postdiction

The paper argues that quantum indeterminism is fundamentally time-reversal invariant, challenging the common emphasis on future uncertainty in quantum systems. It asserts that given current interactions, past uncertainty mirrors future uncertainty. Operational definitions for prediction (calculating future probabilities) and postdiction (calculating past probabilities) tasks are proposed. In a closed system, the symmetry in tasks is evident as demonstrated through direct application of the Born rule, highlighting that postdiction probabilities equal prediction probabilities under flat priors.

For open systems, where some components or interactions remain unobserved, prediction and postdiction probabilities differ by normalization factors dictated by inferential asymmetries, not the arrow of time. The normalisation attached to identity operators is shown to stem from the direction of inference rather than temporal direction.

Quantum Channels

The paper extends its examination to quantum operations, including channels, preparations, and tests. In operational quantum theory, distinct time-oriented assumptions arise from users seeking answers about future events with a marked disregard for potential past-directed inference problems. It suggests that inference-asymmetry in channels arises not from temporal asymmetry but from implicit data assumptions about ancillary systems involved. Inferential symmetry in bistochastic channels reveals the connection between time-reversal invariance and inference symmetry.

Time-Reversal Symmetry

Quantum theory is examined for time-reversal symmetry, distinguishing between passive (switching prediction and postdiction tasks) and active (undertaking time-reversed transformations) forms. Active time-reversal is explored through the adjoint of quantum channels, finding that bistochastic channels uniquely support a fully symmetric approach. Conversely, general channels reveal insights into understanding postdiction as scenarios with implied fixed future data.

Implications and Discussion

The discussion challenges the insistence on the uniqueness of deterministic effects and the notion of 'no signalling from the future' as expressions of quantum time-asymmetry. Instead, these are interpretations of inference asymmetry within established operational formalism shaped by macroscopic agents in thermodynamically oriented contexts. Time orientation in quantum formalisms often reflects external assumptions related to entropy gradients and agent capabilities rather than intrinsic quantum properties.

The paper emphasizes the need for separating the inferential process from quantum mechanical descriptions and cautions against extending laboratory-centred operational quantum formalisms indiscriminately to broader phenomena. It reviews time orientation across various interpretations of quantum mechanics, noting that such orientations often result from external assumptions, entropy considerations, or agent-centric perspectives.

Conclusion

The paper concludes by asserting that the quantum formalism's perceived time-asymmetry originates not from quantum mechanics itself but from external assumptions linked to the arrow of inference and user-agent interactions in thermodynamically oriented settings. It emphasizes that quantum theory remains fundamentally time-symmetric, with operational asymmetries grounded in inference directionality, necessitating careful delineation between user assumptions and core quantum mechanics in future research and application.