Counterfactual quantum measurements

1. Problem Statement

Counterfactual reasoning—asking "what would have happened if conditions were different?"—is fundamental to science, philosophy, and causal inference. In classical physics, David Lewis's framework evaluates counterfactuals by comparing "possible worlds" based on similarity to the actual world, prioritizing the preservation of physical laws and specific facts.

However, applying this framework to Orthodox Quantum Theory (OQT) presents significant challenges:

• Indeterminism: OQT is inherently probabilistic, whereas Lewis's framework assumes deterministic laws where counterfactual propositions are strictly true or false.
• Ontology: Classical frameworks assume a definite physical state at all times, which contradicts the quantum view where properties often lack definite values until measured.
• Causality: Quantum mechanics violates classical causality (e.g., Bell non-locality), making it unclear how to define "similarity" between worlds without violating quantum laws.

Previous attempts to generalize counterfactuals to quantum mechanics (e.g., by Suresh et al. using Pearl's causal models) often rely on "joint exogeneity" assumptions or treat outcomes as fixed antecedents, leading to trivial results where actual-world evidence becomes irrelevant in the counterfactual scenario.

The Core Question: How can one rigorously calculate the probability (or "supposability") of a specific quantum measurement outcome in a counterfactual scenario (e.g., "If I had measured field quadrature $Y$ instead of photon counts $N$, what would the result be?"), given the actual measurement record and the laws of quantum mechanics?

2. Methodology: The Counterfactual Quantum Calculus

The authors propose a formalism that adapts Lewis's hierarchy of desiderata to the probabilistic nature of OQT. The methodology consists of four key steps:

1. Restriction to Classical Elements: The description is limited to classical variables: measurement settings, measurement outcomes, and uncontrolled classical noise.
2. Generalization to Probabilities (Supposability): Instead of asking if a consequent is true, the framework asks for the probability of the consequent. The authors introduce the term Supposability ($Su$) to distinguish this counterfactual probability from standard empirical probability.
3. Antecedent as Measurement Settings: To avoid violating physical laws (Lewis's first desideratum), the counterfactual antecedent is restricted to measurement settings. In OQT, settings are free choices external to the system; changing a setting does not violate dynamical laws.
4. Fixing "Fixtures" (The Core Innovation): To satisfy Lewis's second desideratum (maximizing the match of particular facts), the framework identifies fixtures: classical variables uninfluenced by the counterfactual antecedent.
   • Definition: A variable is a fixture if it lies outside the future light cone of the counterfactual setting change.
   • Calculation: The supposability is calculated by:
     1. Determining the probability distribution of the unknown fixtures in the actual world, given the evidence.
     2. Calculating the probability of the consequent in the counterfactual world, conditioned on these fixtures.
     3. Averaging the counterfactual probabilities over the distribution of the fixtures.

Mathematical Formulation:
For evidence $E=e$, antecedent $A'=a'$, and consequent $C'=c'$, with fixtures $F=f$:
$$Su(C'=c' | A'=a' \parallel E=e) := \sum_f Pr(C'=c' | A'=a', F=f) \cdot Pr(F=f | E=e)$$
Note the double bar ($\parallel$) indicating conditioning on actual-world evidence, distinct from the single bar ($|$) for counterfactual conditioning.

3. Key Contributions

• Formalizing Quantum Counterfactuals: The paper provides the first rigorous calculus for quantum counterfactuals that respects OQT's indeterminacy without assuming hidden variables or joint exogeneity.
• Distinction from Pearl/Suresh Models: Unlike Pearl-based approaches that "sever" causal arrows and fix only exogenous noise (often rendering actual evidence irrelevant), this approach fixes outcomes (fixtures) that are spacelike separated from the setting change. This allows actual-world evidence to meaningfully constrain the counterfactual prediction.
• Introduction of "Suspectation": For continuous monitoring scenarios, the authors define suspectation ($S$) as the counterfactual counterpart to expectation ($E$), analogous to how supposability relates to probability.

4. Results and Applications

A. The Bell-CHSH Scenario (Discrete Case)

• Setup: Alice and Bob share a singlet state. Alice knows her setting ($X$) and outcome ($A$), and Bob's setting ($Y$). She asks: "If I had chosen setting $X'$ instead of $X$, what would my outcome $A'$ be?"
• Fixture: Bob's outcome ($B$) is spacelike separated from Alice's setting change, making it a fixture.
• Calculation:
  • Alice calculates the probability of Bob's outcome $B$ given her actual evidence ($A, X, Y$).
  • She then calculates the probability of $A'$ given $X'$ and the specific value of $B$.
  • She averages these over the distribution of $B$.
• Result: For specific CHSH parameters, the supposability $Su(A'=\rightarrow | X'=\leftrightarrow \parallel A=\uparrow, X=\uparrow, Y=\uparrow)$ is calculated to be 3/4.
• Significance: This differs from naive answers (e.g., 1/2) derived from ignoring the actual evidence or assuming the state collapses differently. It demonstrates that actual-world evidence does influence counterfactual probabilities in a non-trivial way.

B. Continuous Monitoring of a Driven Atom (Continuous Case)

• Setup: A resonantly driven atom emits fluorescence detected by Alice (photon counting) and Bob (photon counting).
• Counterfactual: Alice asks: "Given my actual photon count record $\vec{N}$, what would the homodyne current record $\vec{Y}$ have been if I had used a homodyne detector instead?"
• Method:
  • The "fixture" is Bob's photon count record $\vec{M}$.
  • The authors use Quantum State Smoothing and Generalized Weak Values to compute the "suspectation" $S[Y_t]$.
  • They generate an ensemble of Bob's possible records weighted by their probability given Alice's actual record.
• Result: The suspectation of the homodyne current $Y_t$ shows a distinct peak around the time of Alice's actual photon detection ($t_A$).
  • Physical Interpretation: In the actual world, Alice's detection suppresses Bob's detection rate nearby (anti-bunching). In the counterfactual world, for Bob's rate to remain suppressed (to match the fixture), the atom's state must be driven closer to the ground state. This requires the homodyne current $Y_t$ to have specific values (peaks) that steer the state accordingly.
  • The "echoes" of this peak appear in the oscillation of the current before and after $t_A$.

5. Significance and Implications

• Bridging Theory and Intuition: The formalism bridges the gap between the abstract power of quantum theory and the intuitive counterfactual reasoning used in scientific explanation. It validates the idea that "what we know now" constrains "what would have happened."
• Quantum Causal Modeling: The approach offers a new tool for quantum causal modeling, moving beyond standard hidden-variable assumptions.
• Experimental Verifiability: The authors note that supposabilities are empirically accessible. Since they are constructed from standard quantum probabilities (relative frequencies), they can be measured. Furthermore, the "suspectation" value corresponds to the expectation value of a smoothed quantum state, which has already been verified experimentally in optics and optomechanics.
• Future Directions: The framework opens avenues for quantum machine learning, improved quantum control, and generalizations to theories of fundamental phenomena like Integrated Information Theory (IIT).

In summary, this paper establishes a robust, non-trivial framework for answering "what if" questions in quantum mechanics, demonstrating that counterfactual outcomes are not independent of actual experimental records but are deeply constrained by them through the preservation of spacelike-separated fixtures.