Cone-Restricted Information Theory

TL;DR

This work explores more deeply the idea of building information-theoretic quantities from different base cones and determines which results in quantum information theory rely upon the positive semidefinite cone and which can be generalized and relates this framework to general conic norms and their non-additivity.

Abstract

The max-relative entropy and the conditional min-entropy it induces have become central to one-shot information theory. Both may be expressed in terms of a conic program over the positive semidefinite cone. Recently, it was shown that the same conic program altered to be over the separable cone admits an operational interpretation in terms of communicating classical information over a quantum channel. In this work, we generalize this framework of replacing the cone to determine which results in quantum information theory rely upon the positive semidefinite cone and which can be generalized. We show the fully quantum Stein's lemma and asymptotic equipartition property break down if the cone exponentially increases in resourcefulness but never approximates the positive semidefinite cone. However, we show for CQ states, the separable cone is sufficient to recover the asymptotic theory, thereby drawing a strong distinction between the fully and partial quantum settings. We present parallel results for the extended conditional min-entropy. In doing so, we extend the notion of k-superpositive channels to superchannels. We also present operational uses of this framework. We first show the cone restricted min-entropy of a Choi operator captures a measure of entanglement-assisted noiseless classical communication using restricted measurements. We show that quantum majorization results naturally generalize to other cones. As a novel example, we introduce a new min-entropy-like quantity that captures the quantum majorization of quantum channels in terms of bistochastic pre-processing. Lastly, we relate this framework to general conic norms and their non-additivity. Throughout this work we emphasize the introduced measures' relationship to general convex resource theories. In particular, we look at both resource theories that capture locality and resource theories of coherence/Abelian symmetries.

Figures (7)

• Figure 1: Diagram of common uses of the relative max-entropy, its induced entropy, and related measures. Dotted arrows denote information-theoretic applications achieved via the limit of the regularized smoothed measure. Dotted lines represent uses in resource-theoretic settings. Zig-zag line represented a near equivalence under some choice of smoothing. A wavy line represents duality of measures. The standard arrow shows what measures induce others.
• Figure 2: Depiction of the operational interpretation of cone-restricted min-entropy. Alice encodes half of an entangled state and sends it through the channel to Bob. Bob then measures the entire state, but is restricted to measurements in the cone. The unnormalized success probability that Bob gets Alice's message is given by the restricted min-entropy.
• Figure 3: Geometric intuition of the restricted max-relative entropy measuring the distance from the maximally mixed state to a pure state. Viewing the path distance as representing the value of the measure, we see that as long as $\rho \in \mathcal{K}$, $D_{\max}^{\mathcal{K}}(\rho||\pi) = D_{\max}(\rho||\pi)$ as they are both flat lines. However, if $\rho \not \in \mathcal{K}$, the path distance outside of the cone is altered. The black dotted line represents that $\alpha I \in \mathrm{Relint}(\mathcal{K})$ for all $\alpha > 0$.
• Figure 4: Diagram summarizing under what cones the restricted measures recover the asymptotic results for various types of bipartite quantum states. The grey bar denotes the space of local unitary-invariant cones. $\mathrm{Ent}_{\lfloor \delta d_{A} \rfloor^{n}}$ represents that even if the entanglement rank cone grows exponentially in a fraction of $d_{A}$ captured by $\delta < 1$, then the AEP won't hold (Theorem \ref{['thm:anti-AEP']}).
• Figure 5: Here we depict the $\mathcal{K}-$communication value following Definition \ref{['def:k-cv']}. Note the only restriction is on the power of the measurement in terms of the cone $\mathcal{K}$ that it lives in.

Theorems & Definitions (170)

• Proposition 1
• Proposition 2: Stein's Lemma Chernoff-1952aHiai-1991a
• Proposition 3
• Definition 1
• Proposition 4
• Definition 2
• Proposition 5
• Definition 3
• Definition 4
• Proposition 6