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Majorization uncertainty relations for mixed quantum states

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Rudnicki,  Łukasz
Quantumness, Tomography, Entanglement, and Codes, Leuchs Division, Max Planck Institute for the Science of Light, Max Planck Society;

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Citation

Puchała, Z., Rudnicki, Ł., Krawiec, A., & Życzkowski, K. (in preparation). Majorization uncertainty relations for mixed quantum states.


Abstract
Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $\rho$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $\rho$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.