Multiplicativity of completely bounded $p$-norms implies a strong converse for entanglement-assisted capacity (1310.7028v3)

Abstract: The fully quantum reverse Shannon theorem establishes the optimal rate of noiseless classical communication required for simulating the action of many instances of a noisy quantum channel on an arbitrary input state, while also allowing for an arbitrary amount of shared entanglement of an arbitrary form. Turning this theorem around establishes a strong converse for the entanglement-assisted classical capacity of any quantum channel. This paper proves the strong converse for entanglement-assisted capacity by a completely different approach and identifies a bound on the strong converse exponent for this task. Namely, we exploit the recent entanglement-assisted "meta-converse" theorem of Matthews and Wehner, several properties of the recently established sandwiched Renyi relative entropy (also referred to as the quantum Renyi divergence), and the multiplicativity of completely bounded $p$-norms due to Devetak et al. The proof here demonstrates the extent to which the Arimoto approach can be helpful in proving strong converse theorems, it provides an operational relevance for the multiplicativity result of Devetak et al., and it adds to the growing body of evidence that the sandwiched Renyi relative entropy is the correct quantum generalization of the classical concept for all $\alpha>1$.

• The paper establishes a strong converse for entanglement-assisted capacity by leveraging the multiplicativity of completely bounded p-norms.
• The authors adapt the Arimoto approach and employ the sandwiched Rényi relative entropy to derive robust error bounds.
• The findings imply exponential error growth when communication rates exceed channel capacity, strengthening quantum cryptographic security.

Multiplicativity of Completely Bounded $p$-Norms and Entanglement-Assisted Capacity

The paper by Manish K. Gupta and Mark M. Wilde presents a comprehensive paper on the strong converse theorem for the entanglement-assisted classical capacity of quantum channels, providing a novel approach distinct from earlier methods by Bennett et al. and Berta et al. The work demolishes traditional boundaries by leveraging the multiplicativity of completely bounded $p$-norms, alongside recent advances in quantum information theory such as the sandwiched Rényi relative entropy.

Central Contributions

The authors begin by tackling the fully quantum reverse Shannon theorem, which underpins the optimal rate required for simulating noisy quantum channels with classical communication. Turning this theorem on its head, Gupta and Wilde offer a new proof for the strong converse of entanglement-assisted capacity, identifying a bound on the strong converse exponent necessary for such a task. This approach exploits several cutting-edge elements, including the entanglement-assisted "meta-converse" theorem, properties of the sandwiched Rényi relative entropy, and the aforementioned multiplicativity results.

Novel Methodology

1. Utilization of Sandwiched Rényi Relative Entropy: The authors rely on the sandwiched Rényi relative entropy, particularly for the regime $\alpha > 1$. This entropy, proven to obey monotonicity inequalities and akin to the von Neumann relative entropy as $\alpha$ approaches 1, plays a pivotal role in the authors' framework.
2. Proof via the Arimoto Approach: The paper draws from the Arimoto approach, extrapolating it to demonstrate the strong converse theorem using generalized divergences. Arimoto’s method, traditionally effective for classical information theory, becomes a powerful tool in the quantum domain through Gupta and Wilde’s insightful adaptation.
3. Role of $p$-Norm Multiplicativity: Implementing the multiplicativity of completely bounded $p$-norms—a result from Devetak et al.—provides operational relevance to the entanglement-assisted capacity. The transformation of these technical norms reveals deep connections within quantum channel theory, reinforcing the paper’s argumentation.

Implications and Future Directions

The practical implications of a strong converse theorem for entanglement-assisted capacity stretch beyond theoretical pursuits, with potential applications in cryptographic security. Strong converse theorems ensure any communication rate exceeding the channel’s capacity results in exponentially increasing error probabilities, thereby reinforcing security protocols in quantum cryptography.

The paper contributes to the broader understanding that the sandwiched Rényi relative entropy could be the correct quantum analog to its classical counterpart, providing a more precise framework for future explorations into quantum channel capacities. Furthermore, the methodology and results of this paper could catalyze research into other areas of quantum information, such as the quantum capacity of degradable channels—a pressing and unresolved question in the field.

Gupta and Wilde effectively bridge components of quantum channel theory, broaden the understanding of quantum divergences, and set the stage for further rigorous inquiries into quantum communications' theoretical limits. This work substantiates the notion that even in quantum domains, classical methods and new mathematical properties can invigorate novel pathways, underscoring the significance of their multiplicative findings in quantum channel theory.