Comment on "Quantum theory based on real numbers cannot be experimentally falsified": On the compatibility of physical principles with information theory for fermions

This paper argues that a recently proposed physical postulate intended to rule out real-number-based quantum theories fails when applied to Fermionic Information Theory, thereby demonstrating that the postulate is not a general physical principle and highlighting the necessity of testing foundational principles against fermionic frameworks.

The Gist

The Big Picture: The "Real Number" Debate

Imagine physicists are trying to build a universal rulebook for how the universe works. For a long time, they've used a complex mathematical language involving imaginary numbers (like $i = \sqrt{-1}$) to describe quantum mechanics. It works perfectly.

But recently, some scientists asked: "Do we really need these imaginary numbers? Can we describe the whole universe using only real numbers (like 1, 2, 3.5)?"

Two main theories emerged from this question:

1. Theory A (The Strict One): If you try to build a quantum world with only real numbers, you get a theory that makes different predictions than our current one. Scientists could test this and prove it wrong (falsify it).
2. Theory B (The Flexible One): This is a tweaked version of Theory A. It adds a few extra rules so that it predicts exactly the same things as our current "imaginary number" theory. Because it matches reality perfectly, you can't test it to prove it wrong.

A recent paper (let's call it the "Hoffreumon Paper") argued that Theory B is the only logical choice. They proposed a "Golden Rule" (Postulate 1) to prove this. Their rule was:

"If two people prepare a system independently, and their measurements never show any connection, then they must have prepared it independently."

Basically, they said: "No hidden connections means no hidden preparation."

The Counter-Attack: The "Fermion" Problem

The authors of this paper (Moradi Kalarde, Xu, and Renou) are saying: "Wait a minute. That Golden Rule doesn't work for everything."

They focus on a specific type of particle called a Fermion (think electrons, protons, neutrons—the stuff that makes up matter). These particles have a weird rule called the Parity Superselection Rule.

The Analogy: The "Invisible Wall" of Fermions
Imagine you are in a room with a strict security guard (the Parity Rule).

• The Rule: You can only hold an even number of red balls or an odd number of red balls. You are strictly forbidden from holding a "superposition" (a magical mix) of even and odd numbers.
• The Consequence: Because of this rule, your ability to "look" at the balls (measure them) is limited. You can't see the difference between a specific mix of balls and a completely random pile of balls, because the security guard blocks your view of the "mix."

The "Magic Coin" Experiment

The authors set up a thought experiment to break the "Golden Rule" using these fermions.

1. The Setup: Two scientists, Alice and Bob, are far apart. They have a special "magic coin" system made of fermions.
2. The State: They create a state that looks like a perfect mix of two possibilities.
   • Crucially: This state cannot be made by Alice and Bob working separately. It requires them to coordinate perfectly (a "non-local" preparation).
3. The Test: Alice and Bob start measuring their coins. Because of the "security guard" (the Parity Rule), they can only perform specific types of measurements.
4. The Result: When they compare notes, their results look completely random and unconnected. It looks exactly as if they prepared their coins independently!
   • The Catch: They didn't. They were secretly coordinated. But the "security guard" hid the evidence.

The Conclusion: The Rule is Broken

The "Golden Rule" (Postulate 1) said: "If measurements look independent, the preparation was independent."

But in the Fermion world:

• Preparation: Dependent (They coordinated).
• Measurement: Looks Independent (The security guard hid the link).

The Verdict: The "Golden Rule" fails for fermions. Therefore, it cannot be a universal law of physics.

Why This Matters

The authors are making a broader point about how science works:

1. Don't Trust "Obvious" Intuition: Just because a rule feels right for standard quantum computers (which use distinguishable particles like qubits), it doesn't mean it works for all matter in the universe.
2. The Fermion Test: Any new theory about the foundations of physics must pass the "Fermion Test." If a theory breaks down when you introduce electrons or protons, it's not a general theory of nature.
3. The Lesson: The Hoffreumon paper tried to use a simple principle to choose between two theories. But because that principle ignores the weird rules of fermions, their conclusion is shaky.

Summary Metaphor

Imagine a detective (the Hoffreumon paper) trying to solve a crime. They propose a rule: "If a suspect has no fingerprints on the weapon, they didn't touch it."

The authors of this paper say: "That's a bad rule. What if the suspect wore gloves? (The Fermion Parity Rule). If they wore gloves, they could have touched the weapon, but there would be no fingerprints. Your rule fails for suspects who wear gloves."

Therefore, we cannot use that rule to solve the case. We need a new rule that accounts for the gloves.

In short: You can't build a theory of the whole universe by ignoring the specific, weird rules that govern the particles that make up our bodies.

Technical Summary

1. Problem Statement

The paper addresses a foundational debate in quantum theory: whether quantum mechanics fundamentally requires complex Hilbert spaces or if it can be formulated over real Hilbert spaces.

• Context: Recent works (specifically Ref. [1] in the paper, arXiv:2603.19208) proposed a physical postulate to select the correct formulation of real-number quantum theory. They argued that the standard "Real Quantum Theory" ($T^R_1$) is insufficient and must be modified into an alternative theory ($T^R_2$) to satisfy a specific principle regarding independent preparation.
• The Conflict: The authors of Ref. [1] claim that $T^R_2$ reproduces standard Quantum Information Theory (QIT) predictions and is thus not experimentally falsifiable, whereas $T^R_1$ is.
• The Gap: The current paper argues that the postulate used to select $T^R_2$ (Postulate 1) has not been tested against Fermionic Information Theory (FIT). FIT is the standard framework for describing information encoded in identical fermions, which obeys the parity superselection rule. The authors claim that any "general physical postulate" must hold true within FIT.

2. Methodology

The authors employ a counterexample-based approach using the formalism of Fermionic Information Theory (FIT) to test the universality of Postulate 1.

• Definition of Postulate 1: The postulate states that in a real formulation of quantum theory, independent preparation (states prepared locally by distant parties without communication) must coincide with operational independence (states where all local measurements yield product probability distributions).
• The Counterexample Construction:
  1. The authors consider two fermionic modes, $A$ and $B$.
  2. They define a mixed state $\rho_{AB}$ constructed from an equal mixture of two Bell states with even and odd parity:
     $$\rho_{AB} = \frac{1}{2} (|\phi^+\rangle\langle\phi^+| + |\psi^+\rangle\langle\psi^+|)$$
     where $|\phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and $|\psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$.
  3. They analyze this state under the constraints of FIT, specifically the parity superselection rule, which forbids coherent superpositions between states of even and odd fermion numbers.
• Verification Steps:
  • Step A (Independence Check): They demonstrate that $\rho_{AB}$ cannot be prepared as a product of local states ($\rho_A \otimes \rho_B$) that satisfy the parity superselection rule. Any separable decomposition of $\rho_{AB}$ requires local states that violate the rule. Thus, it is not independently prepared.
  • Step B (Operational Independence Check): They show that for any allowed local measurements (which must commute with the local parity operator), the statistics of $\rho_{AB}$ factorize. By averaging over parity sectors (twirling), the state behaves effectively as the maximally mixed state $I_{AB}/4$ regarding local observables. Consequently, local measurement outcomes are uncorrelated: $p(ab) = p(a)p(b)$. Thus, it is operationally independent.

3. Key Contributions

• Refutation of Universality: The paper proves that Postulate 1 is not a general physical principle. It holds in standard QIT and the proposed $T^R_2$, but it fails in FIT.
• Identification of the Mechanism: The failure arises due to superselection rules. In systems with superselection rules (like fermions with parity), the set of physically allowed local operations is restricted. This restriction makes certain non-locally prepared states (like $\rho_{AB}$) operationally indistinguishable from product states.
• Critique of Recent Works: The authors extend their critique to other recent proposals (Ref. [3] and [4]) that attempt to reconstruct real quantum theory using alternative postulates (Postulate 2). They argue that these postulates, which rely on tensor-product structures and distinguishable subsystems, are not immediately applicable to indistinguishable particles and require non-trivial justification within the second-quantized (mode-based) framework of FIT.
• Theoretical Link: In Appendix C, the authors establish a rigorous mathematical link within Generalized Probabilistic Theories (GPTs): Operational independence coincides with independent preparation if and only if the theory is locally tomographic. Since FIT (and $T^R_1$) is not locally tomographic, Postulate 1 is mathematically guaranteed to fail in these frameworks.

4. Results

• The Counterexample Holds: The state $\rho_{AB}$ is operationally independent but not independently prepared. This directly contradicts Postulate 1.
• Implication for Real Quantum Theory: The selection of $T^R_2$ over $T^R_1$ based solely on Postulate 1 is unjustified because the postulate itself is not a universal law of physics; it is an artifact of theories that are locally tomographic and lack superselection rules.
• Fermionic Specificity: The results highlight that the structural features of quantum theory (specifically regarding composite systems and independence) are deeply sensitive to the nature of the physical carriers (distinguishable vs. indistinguishable particles).

5. Significance

• Foundational Rigor: The paper emphasizes that foundational principles derived from standard QIT (which assumes distinguishable qubits) cannot be blindly applied to all physical systems. It serves as a "reality check" for axiomatic reconstructions of quantum theory.
• Role of Indistinguishability: It underscores the unique information-theoretic challenges posed by indistinguishable particles (fermions), where concepts like "locality" and "independent preparation" differ fundamentally from the qubit-based paradigm.
• Future Directions: The work calls for future reconstructions of real-number quantum theory to explicitly account for superselection rules and the non-local tomographic nature of fermionic systems. It suggests that the "real vs. complex" debate cannot be settled without considering the full spectrum of physical theories, including those involving identical particles.

In summary, the authors demonstrate that the proposed physical postulate used to rule out real-number quantum theory is flawed because it fails to account for the constraints imposed by fermionic superselection rules, thereby invalidating the claim that $T^R_2$ is the unique, physically motivated real-number formulation.