Spectral density of mixtures of random density matrices for qubits

Abstract

We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of $n$ qubit density matrices randomly chosen from the Hilbert-Schmidt ensemble is never decreasing with the number $n$. We also get the exact value of the average squared fidelity. Some conjectures and open problems related to von Neumann entropy are also proposed.

arXiv comment21 pages, LaTex, 6 figures

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• Uncertainty regions of observables and state-independent uncertainty relations
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• Incompatibility probability of random quantum measurements
• Probability density functions of quantum mechanical observable uncertainties
• Quantifying coherence with quantum addition