Accessible Quantum Correlations Under Complexity Constraints

TL;DR

This paper introduces a framework to analyze quantum correlations accessible under computational constraints, revealing fundamental limits on observable quantum entanglement for bounded observers.

Contribution

It defines a complexity-constrained min-entropy for bipartite states and demonstrates strong separations from standard entropy measures, highlighting the impact of computational limits.

Findings

Highly entangled states can have exponentially suppressed accessible entanglement under computational constraints.

For mixed states, the accessible min-entropy can be nearly maximal even when the standard entropy is highly negative.

Computational constraints fundamentally limit observable quantum correlations in practice.

Abstract

Quantum systems may contain underlying correlations which are inaccessible to computationally bounded observers. We capture this distinction through a framework that analyses bipartite states only using efficiently implementable quantum channels. This leads to a complexity-constrained max-divergence and a corresponding computational min-entropy. The latter quantity recovers the standard operational meaning of the conditional min-entropy: in the fully quantum case, it quantifies the largest overlap with a maximally entangled state attainable via efficient operations on the conditional subsystem. For classical-quantum states, it further reduces to the optimal guessing probability of a computationally bounded observer with access to side information. Lastly, in the absence of side information, the computational min-entropy simplifies to a computational notion of the operator norm. We then establish strong separations between the information-theoretic and complexity-constrained notions of min-entropy. For pure states, there exist highly entangled families of states with extremal min-entropy whose efficiently accessible entanglement in terms of computational min-entropy is exponentially suppressed. For mixed states, the separation is even sharper: the information-theoretic conditional min-entropy can be highly negative while the complexity-constrained quantity remains nearly maximal. Overall, our results demonstrate that computational constraints can fundamentally limit the quantum correlations that are observable in practice.