Information upper bounds in composite quantum systems

TL;DR

This paper investigates the intrinsic information bounds of quantum systems, revealing that structural constraints limit the information capacity to at most n bits for an n-qubit system, resolving a paradox between different information measures.

Contribution

It analytically derives the upper bound of intrinsic information in quantum systems using MAP estimation and structural constraints, providing a resolution to the information scale contrast paradox.

Findings

The posterior information content of a 2-qubit system is exactly 2.

The intrinsic information of an n-qubit system is bounded by n bits.

Structural constraints cause high interdependence among parameters, limiting information encoding.

Abstract

The intrinsic information of quantum systems refers to the information required to define a quantum state, and may reveal how the nature stores and processes microscopic information. However, there is an evident paradox due to the "\textit{information scale contrast}": Existing analytical results on the information bounds of quantum systems show that the information-carrying capacity of an $n$-qubit system is only $O(n)$. Nonetheless, the intrinsic information content (estimated by, e.g., the number of parameters or the complexity of ontic embedding) indicates that defining an $n$-qubit system often requires information of the order $O(2^n)$. In this paper, we aim to clarify the upper bound of intrinsic information in quantum systems, as well as explain and resolve the aforementioned paradox. Starting with an analysis of the dependence between the Bloch parameters, we take the structural constraints of the quantum state space as a prior and derive the posterior information content of a quantum state through the process of Maximum A Posteriori (MAP) estimation. We analytically prove that the upper bound of the posterior information content of a 2-qubit system is exactly equal to 2. Furthermore, we conjecturally generalized this result to $n$-qubit systems via a convex optimization process and numerical experiments. In summary, our main theoretical observation is that the tight structural constraints among the parameters of a quantum state make them highly interdependent, and thus unable to freely encode information. It turns out that the intrinsic information of an $n$-qubit system defined by posterior information is bounded by $n$ classical bits.