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

The gist

This manuscript proposes that a general physical postulate should be validated within Fermionic Information Theory (FIT), which demonstrates that operational independence does not imply independent preparation in FIT, thereby showing that Postulate 1 cannot be regarded as a fully general physical principle because it fails in FIT<ref:2604.07425#pg2>

The Problem of Real Hilbert Spaces

The question of whether quantum theory fundamentally requires complex Hilbert spaces or can instead be formulated over real Hilbert spaces has attracted renewed interest in recent years<ref:2604.07425#pg2>. When constructing a quantum theory over real Hilbert spaces, one encounters a fundamental ambiguity regarding how to represent composite systems and independent preparations<ref:2604.07425#pg2>

Several approaches have been proposed, leading to two variants: Theory T R1, which is obtained from the standard Quantum Information Theory (QIT) formalism based on complex Hilbert spaces but restricted to R-Hilbert spaces<ref:2604.07425#pg2>. A natural goal in this context is to look for genuine physical principles that enable one to derive or select the appropriate quantum theory over real Hilbert spaces, between T R1 or T R2<ref:2604.07425#pg2>.

Postulate 1 and its Failure in FIT

The authors adopt Postulate 1, which states that the only experimentally motivated assumption for defining independent preparation is operational independence<ref:2604.07425#pg2>. This postulate suggests that the set of independently prepared states should coincide with the set of states for which all local measurements produce product probability distributions<ref:2604.07425#pg2>. The authors show that this Postulate holds in T R2 but not in T R1, thus selecting T R2 as the appropriate quantum theory over real Hilbert spaces<ref:2604.07425#pg2>. However, the claim that any general physics postulate should be shown valid within Fermionic Information Theory (FIT) fails because FIT does not satisfy Postulate 1<ref:2604.07425#pg2>.

Fermionic Counterexample

A counterexample is provided using two fermionic modes A and B and a mixed state ρAB defined as ρAB = 1/2(ϕ+ϕ+ + ψ+ψ+) <ref:2604.07425#pg2>. This state is compatible with the parity superselection rule, meaning it avoids coherent superpositions between states of even and odd fermionic number<ref:2604.07425#pg2>. Although ρAB is operationally independent, it is not independently prepared in FIT because any separable decomposition gives rise to local states that violate the parity superselection rule<ref:2604.07425#pg2>. The state ρAB is operationally independent because the probability p(ab) = Tr(Ma ⊗ Mb)ρAB equals p(a)p(b)<ref:2604.07425#pg2>. This contradiction demonstrates that operational independence does not imply independent preparation in FIT<ref:2604.07425#pg2>.

Information-Theoretic Frameworks

The authors distinguish between Quantum Information Theory (QIT) and Fermionic Information Theory (FIT)<ref:2604.07425#pg2>. QIT describes information carried by distinguishable subsystems, whereas FIT encodes information in the presence or absence of identical fermions, where local operations are described directly in terms of fermionic canonical anticommutation relations<ref:2604.07425#pg2>. A key distinguishing feature is that FIT is not locally tomographic, unlike standard QIT and T R2<ref:2604.07425#pg2>.

Conclusion on Physical Principles

The conclusion drawn is that the intuition behind proposed postulates is rooted in properties of standard QIT, such as the commutation of distant operators, the absence of superselection rules on local operations, and local tomography<ref:2604.07425#pg2>. These properties are not intuitive in the context of indistinguishable particles and FIT specifically violates Postulate 1 because it is not locally tomographic<ref:2604.07425#pg2>. Therefore, operational independence cannot be regarded as a universal characterization of independent preparation<ref:2604.07425#pg2>. The paper concludes that Postulate 1 cannot be regarded as a fully general physical principle because it is not satisfied by FIT<ref:2604.07425#pg2>. The paper also notes that Postulate 2 requires further justification when extended within the second-quantized fermionic framework<ref:2604.07425#pg2>.

Equivalence of Concepts

The paper establishes a formal equivalence between operational independence and independent preparation if and only if the theory is locally tomographic<ref:2604.07425#pg2>. This equivalence is proven by showing that in a locally tomographic GPT, any bipartite state sAB can be decomposed as sA ⊠ sB if and only if (e A ⊠ e B)(sAB) = e A(sA) e B(sB)<ref:2604.07425#pg2>. This equivalence relies on the fact that the holistic subspace HS is invisible to local measurements, meaning any h in HS satisfies (e A ⊠ e B)(h) = 0 for all local effects<ref:2604.07425#pg2>. This property ensures that operational independence implies independent preparation when locality and tomography are present<ref:2604.07425#pg2>.

The Role of Locality in Indistinguishable Particles

For indistinguishable particles, the notion of physical subsystems is more delicate informationtheoretically because particle labels do not define physical subsystems in the same way as for distinguishable systems<ref:2604.07425#pg2>. In FIT, locality is associated with subsets of modes rather than labelled particles<ref:2604.07425#pg2>. Consequently, the applicability of Postulate 2 is not automatic and must be formulated directly in the relevant indistinguishable particle framework, such as second quantization<ref:2604.07425#pg2>. The paper shows that extending Postulate 2 to FIT is nontrivial because physical observables must be invariant under particle exchange<ref:2604.07425#pg2>.

Local Tomography and GPTs

The paper defines local tomography as the property where global states are fully characterized by local measurements, specifically (e A ⊠ e B)(sAB) = (e A ⊠ e B)(tAB) for all sAB, tAB<ref:2604.07425#pg2>. Proposition 1 proves that if a theory is locally tomographic, then operational independence coincides with independent preparation<ref:2604.07425#pg2>. Conversely, Proposition 2 shows that the equivalence of operational independence and independent preparation implies local tomography<ref:2604.07425#pg2>. This framework provides a rigorous context for evaluating proposed physical postulates against established information-theoretic properties<ref:2604.07425#pg2>.

The Importance of Context

The paper emphasizes that the intuition behind proposed postulates is rooted in standard QIT properties, but these are not intuitive in the context of indistinguishable particles and FIT<ref:2604.07425#pg2>. This provides insight into why FIT violates Postulate 1 and why Postulate 2 is not immediate in this setting<ref:2604.07425#pg2>. The paper concludes that accepting the claim that any general physics postulate must be shown valid within FIT cannot be justified without further consideration of these structural differences<ref:2604.07425#pg2>. The discussion highlights the importance of confronting proposed general physical principles with frameworks like Fermionic Information Theory<ref:2604.07425#pg2>.

Final Summary Point

The paper demonstrates that operational independence cannot be regarded as a universal characterization of independent preparation in FIT because FIT violates local tomography<ref:2604.07425#pg2>. This failure invalidates Postulate 1 as a general physical principle for fermionic systems<ref:2604.07425#pg2>. The findings underscore that the choice of information carrier significantly impacts the validity and interpretation of foundational physical principles<ref:2604.07425#pg2>. The paper suggests that future work should focus on formulating postulates directly within the second-quantized fermionic framework<ref:2604.07425#pg2>.

How it works

The core mechanism involves contrasting operational independence with independent preparation within the framework of Fermionic Information Theory (FIT)<ref:2604.07425#pg2>. The authors construct a specific mixed state ρAB that is compatible with the parity superselection rule but is not a product fermionic state, showing it is operationally independent yet not independently prepared<ref:2604.07425#pg2>. This contradiction arises because the restricted set of physically allowed local operations may fail to distinguish between locally prepared and non-locally prepared states due to the presence of a superselection rule<ref:2604.07425#pg2>.

1. Fermionic Information Theory (FIT) integration for quantum computation: An AI system could implement algorithms leveraging indistinguishable particles information by encoding data not in internal degrees of freedom but in the presence or absence of identical fermions, allowing for novel error correction or state preparation methods dictated by anticommutation relations.

2. Selection of appropriate theory based on operational principles: The AI could dynamically select between theories, as the paper demonstrates, by testing whether a proposed postulate (like Postulate 1) is compatible with Fermionic Information Theory (FIT), ensuring the chosen mathematical framework aligns with known physical constraints regarding parity superselection rules.

3. Enhanced subsystem composition modeling: The system can model composite systems using the mode-based, second-quantized one from FIT, where locality is associated with subsets of modes rather than labelled particles, enabling more accurate description of entanglement in fermionic systems compared to standard QIT models that rely on distinguishable subsystems.

4. Local tomography verification: The AI can verify if a given information theory framework is locally tomographic by checking if the equivalence between operational independence and independent preparation holds, which, according to Proposition 2, proves the theory is locally tomographic.