Bounding quantum uncommon information with quantum neural estimators

TL;DR

This work addresses the challenge of quantifying quantum uncommon information $\Upsilon(A:B)$, for which a closed-form expression is unknown, by leveraging the quantum Donsker--Varadhan representation to estimate von Neumann entropies via a variational, neural-augmented method. The authors formulate both loose and tight bounds on $\Upsilon(A:B)$, introducing common subspace and decomposed-state constructions to tighten upper and lower bounds, respectively. They implement a neural estimator for $S(\rho)$ using a parameterized unitary and a DV loss function, and demonstrate numerical experiments on 4–8 qubit systems to validate convergence and rank-based scaling. The results establish a practical, scalable framework for approximating quantum communication costs in state-exchange scenarios, with implications for quantum networks and distributed quantum information tasks, and they discuss avenues for noise mitigation and generalization.

Abstract

In classical information theory, uncommon information refers to the amount of information that is not shared between two messages, and it admits an operational interpretation as the minimum communication cost required to exchange the messages. Extending this notion to the quantum setting, quantum uncommon information is defined as the amount of quantum information necessary to exchange two quantum states. While the value of uncommon information can be computed exactly in the classical case, no direct method is currently known for calculating its quantum analogue. Prior work has primarily focused on deriving upper and lower bounds for quantum uncommon information. In this work, we propose a new approach for estimating these bounds by utilizing the quantum Donsker-Varadhan representation and implementing a gradient-based optimization method. Our results suggest a pathway toward efficient approximation of quantum uncommon information using variational techniques grounded in quantum neural architectures.

Figures (9)

• Figure 1: Classical information quantities associated with messages $X$ and $Y$. Each circle represents the entropy of a variable, with the overlapping region indicating the mutual information, and the non-overlapping regions corresponding to the conditional entropies.
• Figure 2: Quantum information quantities associated with systems $A$ and $B$. In the quantum setting, the total information content of the joint system $AB$ is given by the sum $S(A) + S(B)$ minus the mutual information. Unlike the classical case, the joint entropy $S(AB)$ can be smaller than either $S(A)$ or $S(B)$ due to quantum entanglement.
• Figure 3: Quantum state exchange protocol. The diagram illustrates the quantum state exchange process. Alice (system $A$) holds the reduced state $\rho_A$, and Bob (system $B$) holds $\rho_B$. Their systems are exchanged via a quantum channel $\mathcal{E}$. To facilitate this process, they consume a shared maximally entangled state $S$, which is transformed into another entangled state $S'$ after the protocol. The reduction in entanglement quantifies the quantum uncommon information.
• Figure 4: Ansatz structure in numerical simulations. Shown is the layered ansatz architecture utilized to construct the parameterized unitary $U(\bm{\theta})$ in the numerical experiments. The same structure is uniformly repeated over $L$ layers.
• Figure 5: Convergence of the loose upper bound $S(AB)$. Shown are the convergence profiles of the upper bound $S(AB)$ for quantum states consisting of 4, 6, and 8 qubits.

Theorems & Definitions (11)

• Definition 2.1: Quantum state exchange protocol
• Definition 2.2: Quantum uncommon information
• Definition 2.3: Common subspace
• Definition 2.4: Decomposed states
• Proposition 2.1: Bounds on quantum uncommon information
• Remark 2.1
• Proposition 3.1: Quantum Donsker--Varadhan representation
• Proposition 4.1: Partial spectral alignment and unitary diagonalization
• Theorem 4.1: Unitary mapping of a subspace to a fixed basis segment
• Proposition 4.2: Characterization of common subspaces via a nonzero-structure relation