Unobservables and Decoherence from Complexity

Abstract: The interface between the quantum and the classical is an intriguing and, at times, hotly contested subject of ongoing research. The quantum regime is characterized by interference, made possible by the superposition principle, while such phenomena are absent in macroscopic, everyday experience. Here, we investigate the link of this absence (or, as we will argue, unobservability) to computational complexity. We show how the assumption that quantum systems cannot solve NP-complete problems efficiently implies that certain formally valid quantum measurements on finite-dimensional systems are unperformable. We study several consequences of this restriction. First, Pauli matrices in an inconveniently transformed basis are a simple example of unobservables. Furthermore, some quantum states are not connected by any physically realizable time evolution. Finally there are quantum states whose coherence cannot be observed, i.e. superpositions of pure quantum states which are indistinguishable from mixtures. We discuss the connection of this phenomenon to the presence of superselection sectors. Our results suggest that the apparent classicality of macroscopic systems may be partly due to limitations on measurements and time evolutions imposed by computational complexity.

• The paper shows computational complexity imposes fundamental limits on quantum measurement observability by linking them to intractable NP-complete problems.
• It employs a sandwich-POVM construction to illustrate that physically realizable operations must conform to computational boundaries, thereby enforcing decoherence.
• The study suggests complexity-induced decoherence offers an alternative mechanism for the quantum-to-classical transition, impacting quantum algorithm design.

Unobservables and Decoherence from Complexity

Overview

"Unobservables and Decoherence from Complexity" (2606.20927) systematically investigates the relationship between computational complexity and the quantum-to-classical transition, specifically focusing on the operational unobservability of certain quantum measurements. The paper establishes that the existence of computationally hard problems, such as 3SAT, imposes strong restrictions on which measurements and time-evolution operations are physically realizable. By constructing a formally valid quantum measurement (sandwich-POVM) that would enable efficient solutions to NP-complete problems, the authors argue that such measurements must be fundamentally unobservable, given the widely accepted assumption that NP-complete problems remain intractable for quantum computers. This premise leads to a rigorous exploration of consequences, including decoherence from complexity and connections to superselection rules.

Quantum Measurements and Complexity Limitations

The authors demonstrate, using the sandwich-POVM $\mathcal{N}(F)$, that there exist mathematically valid quantum measurements whose operational realization would allow efficient solving of NP-complete problems—contradicting the assumption that NP $\not\subseteq$ BQP. They construct a hypothetical algorithm that sequentially measures projectors associated with clauses of a CNF formula, repeatedly undoing undesired measurement outcomes via tailored measurements. Specialized dilution techniques ensure that each step of the algorithm has non-negligible probability of success, circumventing typical exponential decay seen in hard SAT instances.

Key claim: For formulas of sufficient complexity and size, $\mathcal{N}(F)$ constitutes an unobservable measurement, since its physical implementation would violate accepted complexity-theoretic boundaries. Furthermore, any measurement operationally close in diamond norm to $\mathcal{N}(F)$ is similarly unobservable, ensuring robustness of the argument.

Consequences: Unobservable Operators, States, and Decoherence

Unobservable Operators

Applying Naimark's dilation theorem, the authors show that unobservable POVMs correspond to unobservable PVMs in larger Hilbert spaces. Explicit constructions demonstrate that transformed Pauli operators in these spaces become unobservable, even though their canonical forms remain observable.

Restricted Physical Time Evolution

Corresponding unitary operations that map observable operators to unobservable ones cannot be realized as physical time evolutions. Formally, the set $\mathcal{T}$ of physically realized time evolutions omits such unitaries, connecting hard-to-realize measurements to unreachable states in Hilbert space.

Decoherence from Complexity

The paper rigorously proves that certain pure superpositions cannot be operationally distinguished from corresponding mixed states due to the inability to physically realize requisite measurements and evolutions. In technical terms, for states $\ket{\psi_1}$ and $\ket{\psi_2}$ not connected by physical time evolution, their coherent superposition exhibits unobservable coherence, effectively rendering them operationally classical. The authors link this "complexity-induced decoherence" to the formal structure of superselection rules, arguing that observer limitations can induce effective SSRs.

Strong claim: Decoherence may arise from computational complexity limitations, not solely from environmental interactions, constituting an alternative or complementary mechanism for the emergence of classicality.

Theoretical and Practical Implications

The results have far-reaching implications for the foundations of quantum mechanics and quantum computing:

• Physical Limitations of Quantum Formalism: Not all mathematically valid measurements can be physically realized, urging a revision of operational quantum theory to incorporate complexity-theoretic constraints.
• Emergence of Classicality: Complexity-induced superselection rules provide a mechanism for the loss of observable coherence, supporting classical behavior in macroscopic systems even in absence of environmental interaction.
• Quantum Algorithm Design: The argument reinforces limitations on measurement-based quantum algorithms, especially those that could, in principle, solve NP-complete problems efficiently.
• Decoherence as Complexity Phenomenon: The study motivates integrating computational complexity as a central factor in quantum-to-classical transition models, complementing standard approaches such as environmental decoherence and einselection.

Speculation on Future Developments

Future work may quantitatively estimate the prevalence of unobservables in macroscopic systems, potentially accounting for the lack of visible quantum phenomena at scale. Extending complexity-based restrictions to other physical operations and measurement settings could further refine operational quantum mechanics. Moreover, complexity-induced decoherence may inform new models of observer-dependent superselection rules, with broad implications for quantum information theory and measurement foundations.

Conclusion

The paper rigorously establishes that computational complexity imposes a structural barrier on the observability of certain quantum measurements and evolutions. By explicitly linking unobservable measurements to the inability to solve NP-complete problems efficiently in quantum mechanics, the authors provide a foundational explanation for the operational emergence of classicality. Complexity-induced decoherence, manifesting as effective superselection rules, represents a significant theoretical advance, suggesting that macroscopic classicality may fundamentally arise from observer limitations rooted in computational hardness.