⚛️ quantum physics

Comment on "Association between quantum paradoxes based on weak values and a realistic interpretation of quantum measurements"

This Comment refutes Aredes and Saldanha's claim that realistic interpretations of weak values inherently lead to inconsistencies by demonstrating that their general argument is formally incorrect and providing Bohmian mechanics as a counterexample where position-based weak values can be consistently interpreted as intrinsic properties of quantum systems.

Original authors: Juan José Seoane, Xabier Oianguren-Asua, Albert Solé, Xavier Oriols

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan José Seoane, Xabier Oianguren-Asua, Albert Solé, Xavier Oriols

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

The Big Picture: A Dispute Over "Ghostly" Numbers

Imagine you are trying to understand a mysterious, invisible world (Quantum Mechanics). Scientists have developed a special tool called a "Weak Value." Think of a Weak Value like a ghostly whisper about a particle's property (like its speed or position). You can't hear it clearly with one ear, but if you listen to thousands of whispers from identical particles and average them out, you get a number that tells you something about the particle's "reality."

Recently, two scientists (Aredes and Saldanha) wrote a paper saying: "These ghostly whispers are dangerous. If you try to treat them as real, physical facts about the particle, you run into logical contradictions and paradoxes. Therefore, weak values cannot be real."

The authors of this new paper (Seoane, Oianguren-Asua, Solé, and Oriols) are saying: "Hold on. You found some contradictions, but you blamed the wrong thing. Your logic is flawed, and we can prove that weak values can be real if you look at them through the right lens."

Part 1: The Flawed Logic (The "Two Roads" Mistake)

The original paper tried to prove that Weak Values are impossible by using a logic trick. Here is how they did it, and why the new authors say it's wrong:

The Setup: They said, "If you believe Weak Values are real (Hypothesis A), you get a specific result. If you believe standard Quantum Measurements are real (Hypothesis B), you get the same result."
The Mistake: They concluded, "Since both hypotheses lead to the same result, Hypothesis A and Hypothesis B must be the same thing."
The Rebuttal: The new authors say this is like saying, "If I drive a Ferrari, I arrive at the airport in 20 minutes. If I take a helicopter, I also arrive in 20 minutes. Therefore, a Ferrari is the same thing as a helicopter."

The Analogy: Just because two different theories predict the same outcome in a specific scenario doesn't mean the theories are identical. The original paper made a logical error by assuming that sharing a result means sharing the same nature.

Part 2: The Counter-Example (The "Bohmian Detective")

To prove their point, the new authors use a specific version of quantum mechanics called Bohmian Mechanics (also known as Pilot Wave theory).

The Analogy of the Detective:
Imagine a detective trying to solve a crime.

Standard Quantum Mechanics (The Orthodox View): The detective says, "The suspect didn't have a location until we looked at the security camera. The act of looking created the location."
Bohmian Mechanics (The Realist View): The detective says, "The suspect was always at a specific location, moving at a specific speed. We just couldn't see them until we looked. The act of looking might have startled them and changed their path, but they were somewhere before we looked."

The new authors show that in the "Bohmian Detective" world:

Weak Values are Real: The "ghostly whisper" (Weak Value) of a particle's position or speed is actually just the particle's real, physical position and speed at that exact moment.
No Contradictions: When you calculate these values, they make perfect sense. A particle has a real speed and a real position. There are no paradoxes.
The Twist: Even though the values are "real," the theory is still "contextual." This means that if you measure the particle, you might disturb it (like the detective startling the suspect). But the value existed before the disturbance.

The Takeaway: The original paper claimed that "Realism leads to contradictions." The new paper says, "Not if you use Bohmian mechanics. In this framework, Weak Values are just the real properties of the particle, and everything works perfectly."

Part 3: How Do We Measure the "Ghost"? (The Ensemble Trick)

You might ask: "If these values are real, why do we need to measure them on thousands of particles? Can't we just look at one?"

The Analogy of the Wind:
Imagine you want to know the speed of the wind at a specific spot.

If you put a single, tiny feather there, the wind might push it wildly, and you won't get an accurate reading of the "true" wind speed because the feather is too light.
However, if you have a million feathers and you gently nudge them all, then look at where they would have gone if they hadn't been nudged, you can calculate the true wind speed.

The Paper's Explanation:

Weak Measurement: This is the "gentle nudge." It disturbs the particle just a tiny bit.
Post-Selection: This is picking out only the particles that ended up in a specific spot.
The Result: By averaging the results of thousands of these "gentle nudges," we can reconstruct the real, pre-existing property (like the Bohmian velocity) that the particle had before we touched it.

The authors argue that this process doesn't create a paradox. It simply reveals a property that was already there, hidden from our direct view, but mathematically accessible through this "ensemble" method.

Summary: What Changed?

The Original Claim: "Weak values cannot be real because they cause logical paradoxes."
The New Critique: "The logic used to prove this is broken. Furthermore, in Bohmian Mechanics, Weak Values are perfectly real and cause no paradoxes."
The Conclusion: You don't have to throw away the idea that particles have real properties. You just need to accept that Weak Values are the real properties of particles, and the "paradoxes" only appear if you try to force them into a framework that doesn't allow for real, pre-existing values.

In one sentence: The paper argues that "ghostly" quantum numbers are actually real physical traits of particles, and the confusion comes from bad logic, not from the nature of reality itself.

1. Problem Statement

The paper is a comment on a recent study by Aredes and Saldanha (Phys. Rev. A 109, 022238, 2024). Aredes and Saldanha argued that "realistic interpretations" of weak values (RIWV)—the idea that weak values represent objective, pre-existing properties of a quantum system—lead to logical inconsistencies and paradoxes.

Their argument relied on a "General Argument" which posits that:

A realistic interpretation of weak values (RIWV) is logically equivalent to a "Realistic Interpretation of Quantum Measurements" (RIQM), defined as the belief that a measurement reveals an underlying ontological value that persists unchanged after the measurement.
Since RIQM contradicts quantum contextuality (specifically the Kochen-Specker theorem), RIWV must also be rejected to avoid paradoxes.

The authors of this comment (Seoane et al.) challenge two main conclusions of Aredes and Saldanha:

Conclusion 1: That RIWV and RIQM are logically equivalent.
Conclusion 2: That realistic interpretations of weak values inevitably lead to inconsistencies.

2. Methodology

The authors employ a counterexample-based approach using Bohmian Mechanics (de Broglie-Bohm pilot-wave theory) to test the validity of Aredes and Saldanha's claims.

Logical Reconstruction: They first reconstruct the "General Argument" from the original paper to demonstrate a formal logical fallacy (confusing shared consequences with logical equivalence).
Theoretical Framework: They utilize Bohmian mechanics, a deterministic hidden-variable theory where particles possess well-defined positions and trajectories at all times, independent of observation.
Specific Analysis:
- They analyze weak values post-selected in position ( $\langle O \rangle_w$ ).
- They construct specific operators (position $\hat{X}$ , velocity $\hat{V}$ , and their sum $\hat{O} = \hat{X} + \hat{V}$ ) that satisfy the mathematical pre-requisites of the "General Argument."
- They compare the "ontological values" derived from the weak value formalism against the actual particle properties defined in Bohmian mechanics.
Contextuality Distinction: They distinguish between the "Faithful Measurement Principle" (which Bohmian mechanics violates, as interactions change particle properties) and the specific definition of weak values as properties of the system before the measurement interaction fully perturbs it.

3. Key Contributions

A. Refutation of Logical Equivalence (Conclusion 1)

The authors demonstrate that Aredes and Saldanha's claim that RIWV $\iff$ RIQM is a logical fallacy.

The Fallacy: Two distinct assumptions can lead to the same specific consequence (e.g., a specific numerical value for an observable) without being logically equivalent to one another.
The Counterexample: In Bohmian mechanics, weak values post-selected in position can be interpreted as real properties of the particle (RIWV holds), yet the theory does not satisfy the "Faithful Measurement Principle" or RIQM (because the measurement interaction alters the particle's trajectory).
Result: It is possible to have a realistic interpretation of weak values without accepting the controversial RIQM, thereby breaking the alleged equivalence.

B. Demonstration of Consistency (Conclusion 2)

The authors prove that realistic interpretations of weak values do not inherently lead to inconsistencies.

Bohmian Consistency: Within Bohmian mechanics, weak values post-selected in position correspond exactly to the local expectation values of the wave function.
- $\langle X \rangle_w = x_0$ (the actual particle position).
- $\langle V \rangle_w = v_\psi(x_0, t)$ (the actual Bohmian velocity).
- $\langle H \rangle_w$ corresponds to the Bohmian energy (kinetic + quantum potential).
Paradox Resolution: They show that famous paradoxes (like the Three-Box Paradox) vanish when weak values are interpreted through the lens of Bohmian mechanics. For instance, a particle cannot be in two boxes simultaneously; the weak value simply reflects the probability current and position distribution of the particle at that specific time.

C. Clarification of "Measurement" vs. "Property"

The paper clarifies the semantic confusion regarding "measurement" in the context of weak values:

Standard Measurement: Usually implies a single-copy interaction that collapses the state.
Weak Value Protocol: Involves an ensemble of identically prepared systems.
The Insight: While the measurement interaction perturbs the individual system (violating RIQM), the weak value obtained from the ensemble average accurately reflects the property the system possessed immediately before the interaction began. Thus, weak values describe pre-existing properties (in Bohmian terms) without requiring the measurement to be non-perturbative in the strong sense.

4. Results

Formal Error Identified: The "General Argument" in the original paper fails because it incorrectly assumes that sharing a consequence implies logical equivalence between RIWV and RIQM.
Bohmian Validity: Weak values post-selected in position are mathematically identical to the real, ontological properties (position, velocity, energy) of particles in Bohmian mechanics.
Contextuality Compatibility: Bohmian mechanics is a contextual theory (satisfying Kochen-Specker constraints) yet allows for a realistic interpretation of weak values. This proves that contextuality does not preclude realism for weak values.
Paradox Elimination: The inconsistencies cited by Aredes and Saldanha arise from specific, non-physical post-selections or misinterpretations of the measurement process. When analyzed within a consistent ontological framework (Bohmian mechanics), these paradoxes disappear.

5. Significance

Theoretical Impact: The paper restores the viability of "realistic interpretations" of weak values, arguing that they are not inherently paradoxical but rather depend on the underlying quantum theory used to interpret them.
Clarification of Weak Values: It provides a robust physical interpretation for weak values, identifying them not as "strange" quantum artifacts, but as measurable properties of a system's state prior to measurement disturbance.
Experimental Relevance: It validates experimental protocols that estimate weak values as legitimate methods for probing the "hidden" dynamical properties (like velocity or energy) of quantum systems, provided one accepts a contextual hidden-variable framework.
Philosophical Shift: It challenges the notion that quantum contextuality necessitates the abandonment of realism for all quantum quantities, suggesting instead that realism is preserved if one accepts the specific nature of quantum contextuality (as in Bohmian mechanics).

In summary, Seoane et al. argue that the "inconsistencies" found by Aredes and Saldanha are artifacts of a flawed logical argument and a restrictive definition of realism, rather than fundamental flaws in the concept of weak values themselves.