Practical Quantum Broadcasting

Abstract

Incorporating sample efficiency, by requiring the number of states consumed by broadcasting does not exceed that of a naive prepare-and-distribute strategy, gives rise to the no practical quantum broadcasting theorem. To navigate this limitation, we introduce approximate and probabilistic virtual broadcasting and derive analytic expressions for their optimal sample complexity overheads. Allowing deviations at the receivers restores sample efficiency even in the 1-to-2 approximate setting, whereas probabilistic protocols obey a stronger no-go theorem that excludes all sample efficient 1-to-2 implementations for arbitrary dimension and success probability. Rather counterintuitive, this obstruction does not persist at larger receiver numbers: for qubit systems, practical 1-to-6 virtual broadcasting becomes attainable. These results elevate sample complexity from a technical constraint to a defining operational principle, opening an unexplored route to the efficient distribution of quantum information.

Figures (11)

• Figure 1: (Color online) Broadcasting Protocols. (a) Broadcasting: Alice prepares independent copies of the input state and distributes them to the receivers, Bob and Claire. The corresponding process produces outputs whose marginals exactly reproduce the input state. (b) Approximate Broadcasting: The process generates outputs whose marginals approximate the input state up to a prescribed error. (c) Probabilistic Broadcasting: Broadcasting succeeds with a generally non-unit probability.
• Figure 2: (Color online) Practical Approximate Virtual Broadcasting. Sample complexity overhead for approximate virtual broadcasting as a function of the tolerated errors on the receivers, shown for the 1-to-2 and 1-to-3 settings in panels (a) and (c), respectively. The green shaded regions in panels (b) and (d) delineate parameter regimes where the broadcasting rate exceeds that of the naive prepare-and-distribute strategy, and the protocols are therefore sample efficient.
• Figure 3: (Color online) Broadcasting Across Scales (a) SE rules out 1-to-2 PBC for arbitrary success probability and system dimension. In contrast, for qubit systems, 1-to-6 virtual broadcasting satisfies SE, even in the deterministic limit. (b) Comparison of sample requirements for the naive prepare-and-distribute protocol (red) and PBC with unit success probability (blue). Once the number of receivers exceeds six, the protocol enters the SE regime.
• Figure 4: (Color online) Partition of the Feasible Region. Constraints \ref{['ln:hyperbolic_1']} and \ref{['ln:hyperbolic_2']} define connected regions $C_1$ (see Eq. \ref{['eq:Set_C_1']}) and $C_2$ (see Eq. \ref{['eq:Set_C_2']}) in the $(z,x)$-plan, highlighted by the colored areas in the figure.
• Figure 5: (Color online) Plot of $f(\theta)$. The figure plots $f(\theta)$ defined in Eq. \ref{['eq:def_f']} as a function of $\theta$. The maximum always occurs in the regime $\cosh{\theta}\leqslant d$.

Theorems & Definitions (27)

• Lemma 1
• Lemma 2: ABC UC Optimality
• Theorem 1: ABC Optimal Sample Complexity
• Corollary 1: 1-to-2 Practical ABC
• Lemma 3: Local Success Probability
• Lemma 4: Global Success Probability
• Lemma 5: PBC UC Optimality
• Theorem 2: PBC Optimal Sample Complexity
• Corollary 2: No 1-to-2 Practical PBC
• Corollary 3: 1-to-6 Practical BC