# Based on

### Research project Uncertainty Relations and Quantum Entanglement

is based on our results presented in the following earlier publications.

is based on our results presented in the following earlier publications.

J. Phys. A 46, 272002 (12pp) (2013)

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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 R´enyi entropies describing probability distributions
associated with a given pure state expanded in eigenbases of two observables.
Obtained bounds are expressed in terms of the largest singular values of
submatrices of the unitary rotation matrix. Numerical simulations show that
for a generic unitary matrix of size N = 5, our bound is stronger than the wellknown
result of Maassen and Uffink (MU) with a probability larger than 98%.
We also show that the bounds investigated are invariant under the dephasing
and permutation operations. Finally, we derive a classical analogue of the MU
uncertainty relation, which is formulated for stochastic transition matrices.

JPhys. Rev. A 89, 052115 (2014)

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We analyze entropic uncertainty relations in a finite-dimensionalHilbert 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 [P. Coles and M. Piani, Phys. Rev. A 89, 022112 (2014)], which
are known to be stronger than the well-known result of Maassen and Uffink [H. Maassen and J. B. M. Uffink,
Phys. Rev. Lett. 60, 1103 (1988)]. Furthermore, we find a bound based on majorization techniques, which also
happens to be stronger than the recent results involving the largest singular values of submatrices of the unitary
matrix connecting both bases. The first set of bounds gives better results for unitary matrices close to the Fourier
matrix, while the second one provides a significant improvement in the opposite sectors. Some results derived
admit generalization to arbitrary mixed states, so that corresponding bounds are increased by the von Neumann
entropy of the measured state. The majorization approach is finally extended to the case of several measurements.

Phys. Rev. A 92, 032109, (2015)

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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 because
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, N = pk , and the number of measurements L = N + 1, the upper bound
for the average entropy becomes minimal for a collection of mutually unbiased bases. An analogous approach
is used to study entanglement with respect to L different splittings of a composite system linked by bipartite
quantum gates.We show that, for any two-qubit unitary gate U ∈ U(4) there exist states being mutually separable
or mutually entangled with respect to both splittings (related by U) of the composite system. The latter statement
follows from the fact that the real projective space RP3 ⊂ CP3 is nondisplaceable by a unitary transformation.
For L = 3 splittings the maximal sum of L entanglement entropies is conjectured to achieve its minimum for a
collection of three mutually entangled bases, formed by two mutually entangling gates.

J. Math. Phys. 57, 032204 (2016)

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We analyze entropic uncertainty relations for two orthogonal measurements on a
N-dimensional Hilbert space, performed in two generic bases. It is assumed that the
unitary matrix U relating both bases is distributed according to the Haar measure
on the unitary group. We provide lower bounds on the average Shannon entropy
of probability distributions related to both measurements. The bounds are stronger
than those obtained with use of the entropic uncertainty relation by Maassen and
Uffink, and they are optimal up to additive constants. We also analyze the case of
a large number of measurements and obtain strong entropic uncertainty relations,
which hold with high probability with respect to the random choice of bases. The
lower bounds we obtain are optimal up to additive constants and allow us to prove
a conjecture by Wehner and Winter on the asymptotic behavior of constants in
entropic uncertainty relations as the dimension tends to infinity. As a tool we develop
estimates on the maximum operator norm of a submatrix of a fixed size of a random
unitary matrix distributed according to the Haar measure, which are of independent
interest.