Operationalism, Causality, and Quantum Theory: a mostly time symmetric perspective

Abstract: This is a book about operational probabilistic theories. The standard approach in such theories is from a time forward perspective. In this book we mostly take a time symmetric perspective. This presents a branding problem. Is this a niche book merely about time symmetry? No. This is a comprehensive book about operational probabilistic theories, but mostly from a time symmetric perspective. In fact, this book consists of (1) a simple book about simple operations having simple causal structure (where all the inputs are before all the outputs), and (2) a complex book about complex operations that can have complicated causal structure (a complex operation is equipped with a causal diagram). For the simple case we are able to show that the time symmetric perspective is equivalent to the time forward perspective. In each book we set up (A) operational probabilistic theories (OPTs) in terms of operations, (B) Operational Quantum Theory (OQT) in terms of operator tensors which correspond to operations, and (C) the theory of Hilbert objects which can be doubled up to give operator tensors. Operations are required to be physical which guarantees that circuits built out of operations have probabilities between 0 and 1 and that certain causality conditions are met. We prove that when we wire together operations the resulting networks are also physical. We model Sorkin's impossible measurements with complex operations and show that physicality prevents anomalous signalling. We develop diagrammatic notation for Hilbert objects. This includes mirrors for doubling up and mirror theorems. We use this framework to prove time symmetric causal dilation theorems for various causal diagrams.

• The paper introduces a time-symmetric operational framework unifying time-forward, time-backward, and time-symmetric quantum formulations.
• It develops novel diagrammatic calculus and operator tensor methods to map operational procedures onto Hilbert space representations.
• The study explores extended causality and resource-theoretic pre-/post-selection, with implications for indefinite causal order and quantum gravity.

Time-Symmetric Operationalism and Causality in Quantum Theory

Overview and Key Objectives

This monograph presents a comprehensive treatment of operational probabilistic theories (OPTs) through a predominantly time-symmetric lens, departing from the standard time-forward operational approach that has underpinned much of quantum foundations and quantum information theory. The text is divided into two major conceptual parts: frameworks with simple causal structure and those with complex causal structures, each further dissected into operational, quantum, and Hilbert space formulations. The central aim is to reformulate core probabilistic and quantum-theoretic concepts so that temporal symmetry, rather than a presupposed temporal arrow, is fundamental.

The work introduces new diagrammatic notations and a systematic method for treating both operations and their associated "operator tensors" in Hilbert space, emphasizing the equivalence (and at times potential non-equivalence) between time-forward, time-backward, and time-symmetric perspectives. These foundational efforts are motivated not only by quantum foundations but also by anticipated applications to quantum gravity, where time-symmetric or indefinite causal structures are expected to play a critical role.

Foundations: Operationalism, Physicality, and Diagrammatics

Hardy outlines operationalism as both a methodology and, to some extent, a philosophical stance: physical theories are grounded in descriptions of lab operations—arrangements and readings of apparatuses, with clearly identified "inputs" and "outputs" (for systems) and "incomes" and "outcomes" (for pointer/readout variables). Operational constructs are eventually mapped to mathematical objects (such as operator tensors in quantum theory) via the linear "p" function, which extends probabilities to weighted sums over circuits, allowing for a flexible equivalence structure between operational expressions.

To enforce empirical consistency, the notion of "physicality conditions" is introduced—comprising positivity (probabilities $[0, 1]$) and causality. Notably, the latter is extended to "double causality" in the time-symmetric setting, mandating that no signaling occurs from either future to past or past to future, unless explicitly conditioned.

A significant methodological innovation is the systematic deployment of diagrammatic calculus, extending the traditions of Penrose, Abramsky-Coecke, Selinger, and the process-theoretic community, but adapted to operationalist semantics (rectangles for simple operations, circles for complex). This approach is not only foundationally clarifying but also technically enabling, making complex composition and equivalence proofs tractable.

Causal Structure: Simple Versus Complex Operations

The theory treats two classes of operations:

• Simple operations: All outputs succeed all inputs; circuits are DAGs.
• Complex operations: Allow the outputs of some operations to precede (in causal terms) inputs of others, reflecting more general causal graphs, potentially encompassing indefinite or non-classical causal order.

In the simple case, Hardy rigorously proves that time-symmetric, time-forward, and time-backward operational probabilistic theories are empirically equivalent: the perspectives differ only in how the probabilities are packaged (joint vs. conditional). In contrast, for operations with complex causal structure, it is presently an open question whether this equivalence persists; the additional constraints of double causality may further constrict the admissible theories.

Time Symmetry and Double Causality

A core conceptual development is the generalization of causality conditions to a time-symmetric regime. In the traditional (time-forward) OPT or quantum framework, causality is imposed by demanding that future choices cannot influence past outcomes. The time-symmetric formalism extends this by:

• Introducing both "incomes" and "outcomes" for every operation, corresponding to classical information accessible only before or after the operation, respectively.
• Imposing "double causality": both future-to-past and past-to-future signaling are forbidden, unless conditioning occurs (e.g., via pre- or post-selection).

The time-symmetric formalism thus entails both operational and mathematical modifications:

• In the convex geometries of state/effect spaces, the double-cone structure replaces the usual forward-cone; this symmetry ensures pure states and effects are treated analogously.
• Physicality conditions involve both forward and backward "deterministic preparations/results," with uniqueness (up to equivalence) forced by causality stipulations, generalizing the Chiribella-D'Ariano-Perinotti causality principle.

Quantum Theory: Operator Tensors, Hilbert Objects, and Correspondence

Hardy constructs the time-symmetric version of operational quantum theory (OQT) via a correspondence with operator-tensor objects (generalizations of Choi matrices), facilitating both state-process and higher-order transformations in a unified, type-respecting way.

Major technical innovations include:

• Duotensor and natural basis expansion: Allowing explicit equivalence between operational descriptions and operator tensors in Hilbert space, given suitable choices for fiducial (basis) sets.
• Mirror theorems and adjoint structure: Developing a systematic treatment of the symmetry operations (conjupositions) on Hilbert objects (e.g., left/right, vertical/horizontal adjoints), leading to the diagrammatic "mirror" calculus.
• Dilation theorems: Generalizing Stinespring's theorem to the time-symmetric, and more generally, to complex causal structure, showing when operational processes have unitary or maxometric dilations—although the general existence of unitary dilations in all cases is left as an open question.

Physicality in the operator formulation is enforced via "tester positivity" (positivity under specific kinds of circuit tests composed of pure states/effects) and the appropriate translation of double causality.

Causality and Indefinite Causal Structure

The theory accommodates and clarifies a variety of previously competing frameworks in quantum causality, including quantum combs, general boundary formulations, and higher-order quantum causal models. Hardy reviews the connections with recent quantum causal modeling, indefinite causal order, and process matrix approaches, positioning the time-symmetric OPT as a unifying and foundationally natural generalization.

In complex operational frameworks, Hardy distinguishes causally definite operations from those with indefinite or cyclic causal structure and analyzes the consequences for physicality constraints and informational signaling, including resolution of "impossible measurements" (e.g., Sorkin-type scenarios).

Time Symmetry, Pre-/Post-Selection, and the Resource Theory Perspective

A deep discussion is offered regarding the operational and interpretational status of preselection and postselection:

• Strong preselection (the ability to physically select an income and propagate its value forward, e.g., as memory) is not a feature of the time-symmetric formalism without additional resources.
• Preselection and postselection, when regarded as resources, can be incorporated into a broader "resource-theoretic" operational framework, where their non-amplifiability property is formally proven in time-symmetric theories.

This perspective is tied to conceptual puzzles about temporal asymmetry, the emergence of the thermodynamic arrow, and the role (or absence) of strong preselection in the emergence of classicality, records, and memory. Hardy speculates on possible mechanisms (e.g., coalescence of apparatus and observers from a maximally mixed prior, cosmological selection effects, or mind/agency as a locus of symmetry breaking).

Implications and Prospects for Quantum Gravity

A leading motivation is the utility of time-symmetric operationalism as a framework for quantum gravity, where both time symmetry and indefinite causal structure are anticipated. The framework grants an avenue for generalizing quantum and gravitational theories to situations where neither a canonical time direction nor definite causal order is available, an anticipated characteristic of quantum gravity (superposed spacetimes).

Further, the author raises the possibility of extending the temporal symmetry to general "temporal reference frames," analogous to general covariance in relativity, suggesting a program for future research that would situate quantum theory within a generally temporally-covariant operational framework.

Numerical Results, Structural Theorems, and Open Problems

The text is focused on structural rigor and theorem-proving rather than numerical benchmarks, but several strong and nontrivial results are highlighted:

• The composition theorems guarantee that physicality is stable under network composition, even in the presence of arbitrary causal complexity.
• The equivalence theorems rigorously establish that, for simple cases, time-forward, time-backward, and time-symmetric operational theories are empirically indistinguishable.
• In the complex case, it is asserted but not (yet) shown that time-symmetry may be strictly more constraining than time-forward approaches, opening the door to potential experimental tests of time symmetry at the network/process level.

Open problems are identified, notably:

• The general existence of unitary (rather than just maxometric) dilations for arbitrary time-symmetric processes with complex causal structure.
• The precise extent to which time-symmetry and/or indefinite causal constraints further restrict operational theories beyond standard quantum theory for complex cases.
• The full operationalization of general temporal frames, and the extension to fully temporally-covariant operational theories, possibly informing quantum gravity.

Concluding Remarks

This treatise systematizes and extends operational probabilistic quantum theory into a fully time-symmetric form, both foundationally and technically, introducing a host of new tools (diagrammatic calculus, mirror theorems, complex Hilbert objects, causal dilations), which are poised to be important in quantum foundations, quantum information, and emergent quantum gravity. While for simple causal structure the time-symmetric, forward, and backward operational theories coincide, the greater constraining power of double causality and the flexibility afforded by the operational/diagrammatic machinery set the stage for future penetration of quantum causal structure, indefinite causal order, and the operational underpinnings of quantum spacetime.