⚛️ quantum physics

Time symmetry in quantum theories and beyond

This paper introduces the process theory QPhys to address the tension between time-symmetric and time-asymmetric quantum formulations by proposing three distinct methods to incorporate time symmetry: enforcing retrocausality, extending causality concepts to distinguish particles from anti-particles, or developing a supertheory that relaxes both causal constraints.

Original authors: John H. Selby, Maria E. Stasinou, Stefano Gogioso, Bob Coecke

Published 2026-09-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: John H. Selby, Maria E. Stasinou, Stefano Gogioso, Bob Coecke

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Time is usually thought of as a one-way street. In our daily lives, we see eggs break but never un-break, and coffee cool down but never spontaneously heat up. This directionality is so deeply woven into our experience that we rarely question it. Yet, in the fundamental laws of physics, particularly in the strange realm of quantum mechanics, the picture is more complicated. The equations that describe how particles move and interact often work just as well if you run them backward in time as they do forward. This creates a tension: the math suggests a universe that is perfectly balanced between past and future, but our reality is stubbornly asymmetric, with a clear distinction between cause and effect. Scientists have long tried to reconcile these two views, asking whether the arrow of time is a fundamental feature of nature or merely a result of how we choose to look at the world.

A new study by a team of researchers offers a fresh perspective on this ancient puzzle by treating physical theories not as collections of equations, but as maps of processes. Imagine a physical theory as a set of instructions for how things change and interact. In this view, a "process" is simply something that takes an input and produces an output. Standard quantum theory, which describes the behavior of atoms and light, is built on a specific kind of process map that includes a rule called causality. This rule ensures that information flows only from the past to the future, preventing signals from traveling backward in time. While this makes the theory match our everyday experience, it breaks the perfect time symmetry found in the underlying math. The researchers set out to explore what happens if we try to build quantum theory without this one-way restriction, or even without any distinction between past and future at all.

The team began by examining the standard quantum framework, which they describe using diagrams where wires represent systems and boxes represent changes. In this standard version, there is a fundamental imbalance: you can prepare many different states of a system, but there is only one way to "discard" or ignore a system, which corresponds to the mathematical operation of taking a trace. This single discard operation is what enforces the arrow of time, ensuring that the theory respects the rule that you cannot send signals to the past. The researchers showed that if you simply reverse the direction of time in this standard theory, you get a completely different, and rather strange, theory where there is only one possible state for every system, effectively describing a world of eternal noise.

To resolve this tension, the researchers explored three distinct ways to reconstruct quantum theory so that it treats time more fairly. The first approach involves creating a theory that is both causal and "retrocausal." In this scenario, every system has a unique state and a unique effect, forcing the theory to be perfectly symmetric. While mathematically consistent, this version of reality is very restrictive; it allows for only one state per system, which makes it difficult to explain the rich variety of states we observe in the real world. The researchers suggest this might not describe our universe directly, but it could be a useful tool for understanding how the appearance of time's arrow might emerge from a deeper, symmetric reality.

The second approach takes a different path, inspired by the idea that particles and their antimatter counterparts might be related by time reversal. In this model, the researchers construct a theory where some systems flow forward in time while others flow backward. To prevent this from creating impossible time-travel paradoxes, they impose a strict rule: no information can be sent between the forward-flowing and backward-flowing systems. This creates a time-symmetric theory that still allows for the rich interactions we see in particle physics, effectively treating antiparticles as particles moving backward in time, but with a built-in safety mechanism that keeps the universe consistent.

The third and most radical approach aims for "time neutrality," a state where the theory makes no distinction between inputs and outputs at all. In this framework, the usual rules of wiring diagrams are relaxed, allowing connections that would normally be forbidden, such as linking an output back to an input. The researchers found that the standard way of handling probabilities in such a theory leads to nonsensical results, like probabilities greater than one. To fix this, they developed a new method of constructing the theory by grouping processes that are essentially the same up to a scaling factor. This leads to a deterministic, time-neutral theory that contains all the mathematical structures needed to describe complex scenarios, including those where the order of cause and effect is not fixed.

The researchers demonstrated that these different approaches, which have appeared in various forms in previous literature, are actually different ways of describing the same underlying mathematical structure. They showed that by carefully choosing how to group and interpret the processes, one can move seamlessly between a theory that is time-symmetric and one that is time-neutral. Crucially, they constructed a specific version of this time-neutral theory that is fully deterministic, meaning it has a single, unique way to describe the outcome of any process, just as our standard theories do. This new formulation includes all the known "process matrices," which are mathematical tools used to study scenarios where the causal order of events is indefinite or unknown.

This work does not claim to have solved the mystery of time or to have proven that our universe is time-neutral. Instead, it provides a powerful new toolkit for thinking about these questions. By showing that time symmetry, time neutrality, and the standard causal view are all connected through a common mathematical language, the researchers have clarified the landscape of possibilities. They have shown that the tension between the symmetric laws of physics and the asymmetric world we experience is not a contradiction to be feared, but a feature that can be understood, manipulated, and potentially resolved through careful theoretical construction. The study suggests that the arrow of time might not be a fundamental brick in the foundation of the universe, but rather a consequence of how we choose to interact with it, opening the door to new ways of thinking about the nature of reality itself.

Technical Summary: Time Symmetry in Quantum Theories and Beyond

1. Problem Statement

The paper addresses a fundamental tension in the formulation of quantum theory: while the unitary dynamics of the theory are time-symmetric, the theory as a whole is radically time-asymmetric due to the principle of causality (or terminality). In standard quantum theory (QPhys), there are many pure states but only a single pure effect (the discarding map/partial trace) that can be realized without post-selection. This asymmetry prevents the theory from being time-symmetric or time-neutral.

The authors identify three distinct levels of temporal structure:

Time Asymmetry: Theories with causality (unique discarding effect) but no unique state (e.g., standard QPhys).
Time Symmetry: Theories where time-reversal is an internal symmetry (possessing a dagger structure), implying the theory is isomorphic to its time-reverse.
Time Neutrality: Theories where there is no essential distinction between inputs and outputs (possessing cups and caps), allowing for cyclic wiring and indefinite causal order.

The core problem is that standard quantum theory is causal but not time-symmetric. Attempts to make it time-symmetric often result in theories that are either trivial (e.g., "eternal noise") or lack sensible operational predictions (e.g., allowing probabilities $>1$ ). The paper seeks to resolve this tension by categorizing existing approaches and proposing new process-theoretic frameworks that achieve time symmetry or neutrality while maintaining operational consistency.

2. Methodology

The authors employ the framework of Process Theories (categorical quantum mechanics), where physical theories are described as symmetric monoidal categories (SMCs) of processes. They utilize diagrammatic reasoning to characterize structural properties:

Causality: Defined by the existence of a unique discarding map for every system.
Time Reversal: Defined by reading diagrams top-to-bottom or flipping them, leading to the opposite category ( $Proc^R$ ).
Time Symmetry: Characterized by the existence of a dagger ( $\dagger$ ), an involutive contravariant functor that reflects diagrams.
Time Neutrality: Characterized by the existence of cups and caps (bent wires), which allow inputs to be connected to inputs and outputs to outputs, effectively removing the input/output distinction.

The methodology involves:

Analyzing the structural constraints of standard quantum theory (QPhys) and its supertheory (QCalc, which drops trace-preservation).
Constructing new process theories by imposing specific constraints (e.g., retrocausality, non-signalling) or by modifying composition rules and quotienting spaces.
Proving equivalences between different mathematical formulations of time-symmetric and time-neutral theories.
Investigating the relationship between these theories and higher-order processes (process matrices).

3. Key Contributions and Results

A. Characterization of Temporal Structures

The paper rigorously defines the hierarchy of temporal theories:

Causal Theories: Have a unique effect (discarding) but many states. (e.g., QPhys).
Retrocausal Theories: Have a unique state but many effects. The time-reverse of a causal theory is retrocausal.
Time-Symmetric Theories: Possess a dagger. Proposition 2.9 states that a time-symmetric theory is causal if and only if it is retrocausal. Since QPhys is causal but not retrocausal (it has many states), it is not time-symmetric.
Time-Neutral Theories: Possess cups and caps. These are necessarily time-symmetric. However, Proposition 2.12 shows that a time-neutral theory that is also causal must be trivial (only one process between any two systems). Thus, QPhys is neither time-symmetric nor time-neutral.

B. Three Approaches to Resolving the Tension

The authors explore three distinct strategies to construct time-symmetric or time-neutral theories:

1. Causal and Retrocausal (Bicausality)

Approach: Restrict the theory to processes that are both causal and retrocausal. This requires selecting a unique state for every system (e.g., the maximally mixed state) and restricting processes to those that preserve this state (unital maps).
Result: The resulting theory, QPhys|unital, is time-symmetric. However, it is highly restrictive, containing only one state per system (the maximally mixed state) and bistochastic classical maps. It lacks the ability to copy classical information.
Significance: This approach aligns with recent work by Lucien Hardy, suggesting that observed asymmetry arises from conditioning on specific variables rather than fundamental physics.

2. Causal or Retrocausal (Dual-Causality)

Approach: Introduce a theory with dual systems where some systems are causal ( $A^\rightarrow$ ) and others are retrocausal ( $A^\leftarrow$ ). This is modeled using the category QRepG (representations of a group $G$ ), where particles are causal and antiparticles are retrocausal.
Constraint: To prevent causal paradoxes (closed timelike curves), a non-signalling condition is imposed between causal and retrocausal systems.
Result: The resulting theory, QPartG, is time-symmetric. It allows for non-trivial interactions (e.g., entanglement) between causal and retrocausal systems without permitting signalling. It serves as a toy model for particle physics where antiparticles are particles travelling backwards in time.
Significance: This provides a process-theoretic realization of Feynman's intuition about antiparticles, ensuring consistency without trivializing the theory.

3. Time Neutrality via Quotienting

Approach: Start with the supertheory QCalc (all completely positive maps, no trace condition), which is time-neutral but operationally flawed (allows scalars $>1$ ).
Method 1 (Modified Composition): Define a new composition rule that renormalizes probabilities (modified Born rule) whenever a state is composed with an effect. This yields QCalc $^\bullet$ .
Method 2 (Quotienting): Define an equivalence relation where processes are equivalent if they differ by a non-zero scalar. The quotient theory QCalc/ $\sim$ naturally captures the renormalization.
Result: Theorem 4.4 proves that QCalc $^\bullet$ and QCalc/ $\sim$ are equivalent. Both are time-symmetric and time-neutral.
Determinism Restoration: To resolve the issue of "zero-processes" (events with probability 0) and ensure determinism, the authors introduce QNeut. This is constructed by starting with noisy processes (QCalc|noise, where $\epsilon > 0$ ) and then quotienting.
Significance: QNeut is a deterministic and time-neutral process theory. It contains all CPTP maps, and the action of process matrices is represented or simulable within the formalism using cups and caps, though strictly speaking, "diagrams with holes" are not part of the process-theory framework.

C. Equivalence of Existing Approaches

The paper demonstrates that various existing formulations of time-symmetric and time-neutral quantum theory (e.g., those by Oreshkov, Cerf, Hardy, and others) are mathematically equivalent. They differ only in convention (e.g., choice of representative elements in an equivalence class) rather than underlying mathematics. Specifically, the modified Born rule approaches are shown to be particular presentations of the quotient theory QCalc/ $\sim$ .

4. Significance and Claims

The paper claims three main contributions:

Mathematical Unification: It establishes that several disparate approaches to time symmetry and neutrality in quantum theory are mathematically equivalent quotients of a single time-neutral theory. This allows tools and results to transfer freely between these frameworks.
New Theoretical Constructions:
- QPhys|unital: A precise process-theoretic realization of a theory that is both causal and retrocausal.
- QPartG: A toy model of particle physics based on Feynman's intuition, where antiparticles are retrocausal particles, constrained by non-signalling to avoid paradoxes.
- QNeut: A fully time-neutral, deterministic formulation of quantum theory that encompasses all CPTP maps and connects the framework to the study of indefinite causal structure by simulating process matrices.
Framework Flexibility: The process-theoretic approach is shown to be applicable beyond standard quantum theory to Generalized Probabilistic Theories (GPTs) and Operational Probabilistic Theories (OPTs). The authors note that symmetrized versions of GPTs are typically not GPTs themselves, highlighting the utility of process theories as a broader framework for exploring physical theories.

The paper concludes modestly, noting that while it provides a broad overview and unification, each approach (e.g., the physical implications of QPartG or the reconstruction of QNeut from first principles) warrants further independent study. It identifies open problems, such as whether QPartG can be distinguished from QPhys experimentally and whether QNeut subsumes all frameworks for higher-order processes. The authors emphasize that while QNeut makes sensible operational predictions in the context of its construction, the underlying supertheory (QCalc) is not a good description of nature, and QNeut serves as a formalism for calculations and exploring indefinite causal structures rather than a definitive physical theory.