On the Hardness of Detecting Macroscopic Superpositions

Abstract: When is decoherence "effectively irreversible"? Here we examine this central question of quantum foundations using the tools of quantum computational complexity. We prove that, if one had a quantum circuit to determine if a system was in an equal superposition of two orthogonal states (for example, the $|$Alive$\rangle$ and $|$Dead$\rangle$ states of Schr\"{o}dinger's cat), then with only a slightly larger circuit, one could also $\mathit{swap}$ the two states (e.g., bring a dead cat back to life). In other words, observing interference between the $|$Alive$\rangle$and $|$Dead$\rangle$ states is a "necromancy-hard" problem, technologically infeasible in any world where death is permanent. As for the converse statement (i.e., ability to swap implies ability to detect interference), we show that it holds modulo a single exception, involving unitaries that (for example) map $|$Alive$\rangle$ to $|$Dead$\rangle$ but $|$Dead$\rangle$ to -$|$Alive$\rangle$. We also show that these statements are robust---i.e., even a $\mathit{partial}$ ability to observe interference implies partial swapping ability, and vice versa. Finally, without relying on any unproved complexity conjectures, we show that all of these results are quantitatively tight. Our results have possible implications for the state dependence of observables in quantum gravity, the subject that originally motivated this study.

• The paper establishes a complexity-theoretic equivalence between detecting interference in macroscopic superpositions and performing state swaps.
• It leverages quantum circuit models to derive conditions under which observing decoherence reversal becomes computationally infeasible.
• The results highlight intrinsic quantum computational limits that inform our understanding of quantum measurements and foundations in quantum gravity.

An Analysis of the Complexity of Detecting Macroscopic Quantum Superpositions

In the exploration of quantum foundations, particularly the phenomenon of decoherence, a pertinent question arises: when does decoherence become "effectively irreversible"? The paper "On the Hardness of Detecting Macroscopic Superpositions" by Aaronson, Atia, and Susskind addresses this question through the lens of quantum computational complexity. Their investigation specifically focuses on the feasibility of detecting interference between macroscopic superpositions, such as the canonical Schrödinger's cat scenario, where a system exists in a superposition of macroscopically distinct states, like 'alive' and 'dead.'

Core Contributions and Results

The authors establish a significant complexity-theoretic framework to characterize the tasks of detecting quantum superpositions and swapping states within these superpositions. By utilizing quantum circuit models, they derive conditions under which observing interference between macroscopically distinct states becomes infeasible. Their primary result shows an intriguing equivalence: if a quantum circuit can discern whether a system is in an equal superposition of two orthogonal states, then a slightly larger circuit could feasibly swap these states. The implication here is stark—observing such interference amounts to a "necromancy-hard" problem, an infeasible task when death is considered irreversible.

Critically, the paper presents these findings under robust conditions, allowing even partial detection abilities to imply partial swapping capabilities and vice versa. The results are also shown to be quantitatively tight without relying on unverified complexity theory conjectures.

Implications for Quantum Foundations and Computational Complexity

The results have significant implications regarding the state dependence of observables in theories of quantum gravity, providing a complexity-theoretic perspective on the idea of observing coherence in macroscopic superpositions. This framework challenges existing assumptions about measurement and interference in quantum mechanics. In particular, it relates the ability to detect quantum coherence with practical intractabilities, reflecting on the broader issue of state-dependent observables in theories like AdS/CFT correspondence.

The findings also underscore the limits of quantum computational power concerning fundamental physical processes. The equivalence between swap complexity and the observation of coherence paves the way for further exploration into how computational resources impact our ability to perform certain physical operations, making a pronounced case for the complexity inherent in quantum measurements.

Potential for Future Research

This work lays a foundation for further research into both the theoretical underpinnings of quantum mechanics and practical quantum computing. Future inquiries may explore more nuanced aspects of swap versus state complexity or extend this complexity-theoretic perspective to other quantum phenomena. Additionally, the results encourage exploration into error-tolerant or approximate techniques for tackling quantum state operations that are classically intractable.

There is also a pathway for exploring how these complexity insights apply to other interpretations of quantum mechanics or alternative models that contend with macroscopic superpositions. The interconnectivity between physical processes and computational limitations invites broader discussions and integrations across physics and computer science.

In conclusion, this paper provides a pivotal examination of the computational barriers to observing quantum superpositions, proposing an intricate link between quantum circuit complexity and our comprehension of fundamental physical processes. This work advances the dialogue on decoupling theoretical feasibility from practical intractability within the domain of quantum mechanics.