One-Shot Distributed Source Simulation: As Quantum as it Can Get

TL;DR

The paper develops a near-tight, one-shot framework for distributed source simulation using smooth entropy tools, covering both classical seeds and quantum (including entangled) targets. It introduces max Wyner common information quantities and their smooth variants to characterize the minimal shared randomness required, and proves one-shot achievability and converse bounds that recover Wyner’s common information in the asymptotic limit. A key technical advance is a generalized cardinality bound (generalized support lemma) and a refined one-shot analysis that accommodates a Markov-chain structure and subnormalized states. The work further extends the framework to multi-receiver and fully quantum settings via embezzling states, catalysis, and entanglement-assisted variants, and it outlines weak-asymptotic equipartition properties and the challenges toward strong second-order results. The results provide a robust, quantum-network-oriented toolbox for correlation conversion without communication, with implications for one-shot quantum information theory and network coding problems.

Abstract

Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner's seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner's common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.

Figures (5)

• Figure 1: Diagram of distributed source simulation of a quantum state from a classical 'seed'. After the copying procedure (at the light blue line), the two parties share a perfectly correlated state $\chi^{|p}_{XX'}$. After their local processing (at the dark blue line), the parties share a state $\widetilde{\rho}_{AC}$ which should be approximately the target state $\rho_{AC}$. We distinguish between the scenarios of uniform and non-uniform random seed $p_X$.
• Figure 2: Primary versions of entangled state source simulation considered. (a) Target state $\rho_{AC}$ is prepared using local operations on an embezzling state $\ket{\mu(n)}$ and arbitrary classical correlation. (b) Target state $\rho_{AC}$ is constructed using local operations with $\sigma_{A'C'}$ as a catalysis. This means the $A'$ and $C'$ registers are relevant to the error criterion rather than implicitly ignored as in (a).
• Figure 3: Diagram of distributed source simulation of a quantum state. After the copying procedure (at the light blue line), the two parties share a perfectly correlated state $\chi^{|p}_{XX'}$. After their local processing (at the dark blue line), the parties share a state $\widetilde{\rho}_{AC}$ which should be approximately the target state $\rho_{AC}$.
• Figure 4: Comparison between distributed source simulation and the entangled state versions. Dashed boxes denote that those registers are not considered in the approximation criterion. (a) Distributed source simulation where the $X$ register is possibly strongly correlated to $A$ and $C$. (b) Entangled source simulation where the input is an arbitrary quantum state and an appropriate marginal of the output must achieve the target state to tolerable error $\varepsilon$. For arbitrary $\sigma_{A'C'}$, this depicts Definition \ref{['def:ent-of-simulation']}, but if one restricts $\sigma_{A'C'}$ to an embezzling state then this depicts Definition \ref{['def:ve-ent-of-sim']}. (c) Entangled source simulation where the auxiliary state is required to be output approximately decoupled from the simulated state. This depicts Definition \ref{['def:emb-ent-of-sim']}.
• Figure 5: The three correlation of formations and their corresponding tasks: (a) Correlation of formation captures the amount of randomness for distributed source simulation. (b) Entanglement-assisted correlation of formation captures the amount of broadcasted randomness needed such that there exists a set of (possibly entangled) states $\{\widetilde{\rho}_{AC}^{x}\}_{x \in \mathcal{X}}$ to distribute so that the entire output is approximately indistinguishable from a distributed source simulation implementation of the target state. (c) Private correlation of formation measures the amount of randomness needed for a distributed source simulation protocol so that if the randomness were to be leaked it would be approximately indistinguishable from an exact distributed source simulation of the target state with leaked classical information.

Theorems & Definitions (134)

• Definition 1
• Definition 2
• Lemma 3
• Definition 4
• Theorem 5
• proof
• Remark 6
• Example 7: One-Shot DSS of Entangled State
• Theorem
• Theorem