{We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size N in L orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For L=2 the maximal average entropy saturates at logN as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for L>=3 measurements. In the case of a prime power dimension

}, doi = {10.1103/PhysRevA.92.032109}, url = {https://doi.org/10.1103/PhysRevA.92.032109}, author = {Zbigniew Pucha{\l}a and {\L}. Rudnicki and K. Chabuda and M. Paraniak and Karol {\.Z}yczkowski} } @article {iitisid_0593, title = {Constructive entanglement test from triangle inequality}, journal = {J. Phys. A: Math. Theor.}, volume = {47}, number = {42}, year = {2014}, note = {arXiv:1211.2306}, pages = {424035}, abstract = {We derive a simple lower bound on the geometric measure of entanglement for mixed quantum states in the case of a general multipartite system. The main ingredient of the presented derivation is the triangle inequality applied to the root infidelity distance in the space of density matrices. The obtained bound leads to entanglement criteria with a straightforward interpretation. The proposed criteria provide an experimentally accessible, powerful entanglement test. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to {\textquoteright}50 years of Bell{\textquoteright}s theorem{\textquoteright}.

}, doi = {10.1088/1751-8113/47/42/424035}, url = {https://doi.org/10.1088/1751-8113/47/42/424035}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0656, title = {Strong Majorization Entropic Uncertainty Relations}, journal = {Phys. Rev. A}, volume = {89}, number = {5}, year = {2014}, note = {arXiv:1402.0129}, pages = {052115}, abstract = {We analyze entropic uncertainty relations in a finite dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani [ArXiv:1307.4265], which are known to be stronger than the well known result of Maassen and Uffink. Furthermore, we find a novel bound based on majorization techniques, which also happens to be stronger than the recent results involving largest singular values of submatrices of the unitary matrix connecting both bases. The firsts set of new bounds give better results for unitary matrices close to the Fourier matrix, while the second one works better in the opposite sectors. Some results derived admit generalization to arbitrary mixed states and the bounds are increased by the von Neumann entropy of the measured state

}, doi = {10.1103/PhysRevA.89.052115}, url = {https://doi.org/10.1103/PhysRevA.89.052115}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0562, title = {Entropic trade-off relations for quantum operations}, journal = {Phys. Rev. A}, volume = {87}, number = {3}, year = {2013}, note = {arXiv:1206.2536 doi: 10.1103/PhysRevA.87.032308}, pages = {032308}, abstract = {Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel Φ. We prove that for any map acting on an N-dimensional quantum system the sum of both entropies is not smaller than lnN. For any bistochastic map this lower bound reads 2lnN. We investigate also the corresponding R{\'e}nyi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.}, author = {Wojciech Roga and Zbigniew Pucha{\l}a and {\L}. Rudnicki and Karol {\.Z}yczkowski} } @article {iitisid_0617, title = {Majorization entropic uncertainty relations}, journal = {J. Phys. A: Math. Theor.}, volume = {46}, year = {2013}, note = {arXiv:1304.7755}, pages = {272002}, abstract = {Entropic uncertainty relations in a finite dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of Renyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables. The bounds obtained are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix. For a generic unitary matrix of size N = 5 the bound obtained is stronger than the one of Maassen and Uffink (MU) with probability larger than 98\%, and this ratio increases with N. We show also that the bounds investigated are invariant for unitary matrices equivalent up to dephasing and permutation and derive a classical analogue of the MU uncertainty relation formulated for stochastic transition matrices.}, author = {Zbigniew Pucha{\l}a and {\L}. Rudnicki and Karol {\.Z}yczkowski} } @article {iitisid_0589, title = {Collectibility for Mixed Quantum States}, journal = {Phys. Rev. A}, volume = {86}, number = {6}, year = {2012}, note = {arXiv:1211.0573 doi:10.1103/PhysRevA.86.062329}, pages = {062329}, abstract = {Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for pure states only [ Phys. Rev. Lett. 107 150502 (2011)], is extended here for mixed quantum states. We define collectibility for any mixed states of a multipartite system. Deriving bounds for collectibility for positive partially transposed states of given purity provides insight into the structure of entangled quantum states. In the case of two qubits the application of complementary measurements and coincidence based detections leads to a test of entanglement of pseudopure states.}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} }