Axiological Time Symmetry in Quantum Foundations

• ATS is a principle asserting invariant joint probabilities when exchanging preparation and measurement roles in quantum experiments.
• It underpins the derivation of CHSH-type causal inequalities by enforcing mediator symmetry across pseudo-event and observed outcomes.
• ATS exposes a foundational tension between classical event absoluteness and quantum predictions, challenging standard temporal causality.

Axiological Time Symmetry (ATS) is a principle introduced in the context of quantum foundations, particularly in analyzing timelike Wigner’s Friend-type scenarios where the usual locality assumption is replaced by a symmetry involving both observed and “pseudo-” events. ATS posits a specific invariance of joint probabilities under the exchange of time-ordered roles of preparation and measurement, thereby constituting a key assumption in deriving causal inequalities analogous to Bell-type bounds in these setups. Experimental incompatibility of quantum mechanics with the conjunction of ATS and related classical constraints foregrounds the foundational tension between classical event absolutism and quantum predictions (Mukherjee et al., 30 Oct 2025).

1. Mathematical Formulation of ATS

Let $c, d \in \{\pm1\}$ denote “pseudo-events” (intermediate outcomes that may be unitarily erased), and $a, b \in \{\pm1\}$ denote truly observed outcomes, with $x, y \in \{0,1\}$ controlling whether the observer undoes or reads the friend’s measurement. In the forward-time ordering “Charlie–Alice → Debbie–Bob,” define the joint distribution $p_{(\rightarrow)}(c,a,d,b\,|\,x,y)$. In the time-reversed ordering, “Debbie–Bob → Charlie–Alice,” define $p_{(\leftarrow)}(d,b,c,a\,|\,y,x)$.

Axiological Time Symmetry requires invariance under this exchange: $$\forall\, x,y,\, a,b,c,d:\quad p_{(\leftarrow)}(d,b,c,a\,|\,y,x) = p_{(\rightarrow)}(c,a,d,b\,|\,x,y)$$ This condition expresses that swapping the composite “preparation” and “measurement” blocks (with appropriate permutations) leaves the joint statistics invariant.

2. Conceptual Distinction from Conventional Time Symmetry

Physical theories traditionally distinguish between:

• Dynamical $T$-invariance: Time-reversal symmetry of fundamental dynamical equations, such as those governing Hamiltonian evolution.
• Operational time symmetry: In variance of observed statistics under reversal of prepare-and-measure protocols.

ATS generalizes operational time symmetry to contexts where certain events are only “pseudo-events” (potentially unphysical, subject to erasure). Here, one cannot simple swap preparation and measurement devices, but must interchange composite systems (e.g., (c,a) with (d,b)), ensuring valuation-preserving (axiological) symmetry of outcome assignments. ATS thus encodes a temporally unbiased principle for analyzing scenarios with both observed and pseudo-observed events, going beyond dynamical or operational notions of time symmetry.

3. Role of ATS in Causal-Friendliness Inequality Derivation

When combined with Absoluteness of Observed Events (AOE), No Retrocausality (NRC), and Screening via Pseudo Events (SPE), ATS enables a full reduction of the joint probability distribution, leading to a CHSH-type causal-inequality: $$| \langle A_0B_0 \rangle + \langle A_0B_1 \rangle + \langle A_1B_0 \rangle - \langle A_1B_1 \rangle | \leq 2$$ The derivation proceeds as follows:

1. Causal Factorization under NRC: NRC imposes the lack of retrocausal influences, dictating a temporal factorization of joint probabilities.
2. Bayesian Relations: Bayes’ theorem relates probabilities in different time orderings, generating equivalent but distinct expressions for the joint distribution.
3. Mediator Independence and Screening Lemmas (from ATS):
   • $p(c,d\,|\,x,y)=p(c,d)$ (pseudo-event independence from choices)
   • $p(a\,|\,c,d,x,y)=p(a\,|\,c,x)$, $a, b \in \{\pm1\}$0 (screening of truly observed outcomes)
4. Insertion of SPE: Ensures pseudo-events fully mediate past-to-future influences.
5. Factorized Joint: The above yield

$a, b \in \{\pm1\}$1

and upon summing over pseudo-events, the correlator factorization necessary for the CHSH bound.

Quantum predictions, achievable by appropriate observable choices (e.g., $a, b \in \{\pm1\}$2, $a, b \in \{\pm1\}$3 at $a, b \in \{\pm1\}$4), violate this bound with $a, b \in \{\pm1\}$5. Hence, quantum mechanics is incompatible with the full set of assumptions including ATS in this framework (Mukherjee et al., 30 Oct 2025).

4. Operational Weakenings of Event Absoluteness

The full AOE assumption (existence of a global four-way joint $a, b \in \{\pm1\}$6) can be replaced by a purely operational—yet sufficient—set of conditions termed Operational Pseudo-Event Mediation (OPEM):

• Existence of Marginals (EOM): Only the empirical marginals $a, b \in \{\pm1\}$7 are required.
• Operational Mediation (OM): $a, b \in \{\pm1\}$8.

With ATS and NRC, this allows construction of a normalized four-way function and recovery of the same causal-inequality: $a, b \in \{\pm1\}$9 Thus, the CHSH constraint persists and quantum predictions remain incompatible. If the absoluteness of pseudo-events is dropped (allowing dependence $x, y \in \{0,1\}$0 on $x, y \in \{0,1\}$1), mediator-independence is lost and the causal bound can reach the Box-world maximum ($x, y \in \{0,1\}$2), showing that pseudo-event absolutness is essential for the CHSH-style constraint.

5. Illustrative Scenario and Structure

The Causal-Friendliness scenario involves a sequential measurement process:

• Charlie measures a qubit, producing $x, y \in \{0,1\}$3.
• Alice either undoes the measurement or records $x, y \in \{0,1\}$4 (depending on $x, y \in \{0,1\}$5), then forwards to Debbie.
• Debbie analogously produces $x, y \in \{0,1\}$6 or not, depending on future choices.
• Bob either reads or undoes, yielding $x, y \in \{0,1\}$7.

Causal influence propagates $x, y \in \{0,1\}$8, with $x, y \in \{0,1\}$9 and $p_{(\rightarrow)}(c,a,d,b\,|\,x,y)$0 controlling interventions at Alice and Bob. Pseudo-events $p_{(\rightarrow)}(c,a,d,b\,|\,x,y)$1 may be unitarily erased, and the operational distinction between observed and pseudo-observed outcomes is central. ATS symmetry is realized by exchanging (c,a) with (d,b) and swapping $p_{(\rightarrow)}(c,a,d,b\,|\,x,y)$2, as made explicit in the scenario diagrams and causal graphs.

6. Foundational Significance and Quantum Incompatibility

The conjunction of AOE, ATS, NRC, and SPE yields a constraint structurally analogous to the Bell-CHSH inequality. The experimental violation of the related causal-inequality by quantum correlations underpins the foundational conclusion: quantum mechanics cannot uphold ATS in conjunction with even weakened classical-style assumptions for event absoluteness and temporal causality.

The only consistent theoretical routes for accommodating quantum predictions are giving up ATS (entailing a fundamental temporal asymmetry in event assignment), abandoning event absolutness (admitting observer-relativity as in Everettian or QBist interpretations), or violating NRC or screening. ATS therefore occupies, for timelike Wigner-Friend-type scenarios, a role structurally parallel to that of Locality in spacelike Bell-type scenarios: it anchors a classical symmetry principle whose tension with quantum theory is exposed via operational-theoretic no-go theorems (Mukherjee et al., 30 Oct 2025).

7. Summary Table of Key Concepts

Principle/Assumption	Description	Role in CHSH Inequality
ATS	Invariance under preparation/measurement block exchange	Enforces mediator symmetry
AOE	Absoluteness of all (pseudo-/real) observed events	Allows joint distributions
NRC	No influence from future choices to past outcomes	Enables causal factorization
SPE	Pseudo-events mediate all past influences on future outcomes	Simplifies response functions