Entropy of Quantum Measurements

• URL: http://arxiv.org/abs/2210.15738v2
• Date: Tue, 1 Nov 2022 16:53:27 GMT
• Title: Entropy of Quantum Measurements
• Authors: Stan Gudder
• Abstract summary: In Section2, we provide bounds on $S_a(rho )$ and show that if $a+b$ is an effect, then $S_a+b(rho )ge S_a(rho )+S_b(rho )$. In Section3, we employ $S_a(rho )$ to define the $rho$-entropy $S_A(rho )$ for an observable $A$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: If $a$ is a quantum effect and $\rho$ is a state, we define the $\rho$-entropy $S_a(\rho )$ which gives the amount of uncertainty that a measurement of $a$ provides about $\rho$. The smaller $S_a(\rho )$ is, the more information a measurement of $a$ gives about $\rho$. In Section~2, we provide bounds on $S_a(\rho )$ and show that if $a+b$ is an effect, then $S_{a+b}(\rho )\ge S_a(\rho )+S_b(\rho )$. We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their $\rho$-entropies. In Section~3, we employ $S_a(\rho )$ to define the $\rho$-entropy $S_A(\rho )$ for an observable $A$. We show that $S_A(\rho )$ directly provides the $\rho$-entropy $S_\iscript (\rho )$ for an instrument $\iscript$. We establish bounds for $S_A(\rho )$ and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider $\rho$-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.

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