75 research outputs found
The operational meaning of min- and max-entropy
We show that the conditional min-entropy Hmin(A|B) of a bipartite
state rho_AB is directly related to the maximum achievable overlap
with a maximally entangled state if only local actions on the B-part
of rho_AB are allowed. In the special case where A is classical, this
overlap corresponds to the probability of guessing A given B. In a
similar vein, we connect the conditional max-entropy Hmax(A|B) to the
maximum fidelity of rho_AB with a product state that is completely
mixed on A. In the case where A is classical, this corresponds to the
security of A when used as a secret key in the presence of an
adversary holding B. Because min- and max-entropies are known to
characterize information-processing tasks such as randomness
extraction and state merging, our results establish a direct
connection between these tasks and basic operational problems. For
example, they imply that the (logarithm of the) probability of
guessing A given B is a lower bound on the number of uniform secret
bits that can be extracted from A relative to an adversary holding B
The Operational Meaning of Min- and Max-Entropy
In this paper, we show that the conditional min-entropy of a bipartite state is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the -part of are allowed. In the special case where is classical, this overlap corresponds to the probability of guessing given . In a similar vein, we connect the conditional max-entropy to the maximum fidelity of with a product state that is completely mixed on . In the case where is classical, this corresponds to the security of when used as a secret key in the presence of an - adversary holding . Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing given is a lower bound on the number of uniform secret bits that can be extracted from relative to an adversary holding
Min- and Max-Entropy in Infinite Dimensions
We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional settin
Bottleneck Problems: Information and Estimation-Theoretic View
Information bottleneck (IB) and privacy funnel (PF) are two closely related
optimization problems which have found applications in machine learning, design
of privacy algorithms, capacity problems (e.g., Mrs. Gerber's Lemma), strong
data processing inequalities, among others. In this work, we first investigate
the functional properties of IB and PF through a unified theoretical framework.
We then connect them to three information-theoretic coding problems, namely
hypothesis testing against independence, noisy source coding and dependence
dilution. Leveraging these connections, we prove a new cardinality bound for
the auxiliary variable in IB, making its computation more tractable for
discrete random variables.
In the second part, we introduce a general family of optimization problems,
termed as \textit{bottleneck problems}, by replacing mutual information in IB
and PF with other notions of mutual information, namely -information and
Arimoto's mutual information. We then argue that, unlike IB and PF, these
problems lead to easily interpretable guarantee in a variety of inference tasks
with statistical constraints on accuracy and privacy. Although the underlying
optimization problems are non-convex, we develop a technique to evaluate
bottleneck problems in closed form by equivalently expressing them in terms of
lower convex or upper concave envelope of certain functions. By applying this
technique to binary case, we derive closed form expressions for several
bottleneck problems
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