NCN Research Grant Maestro 7
Uncertainty Relations and Quantum Entanglement
Number 2015/18/A/ST2/00274 realised in 2016-2021

National Science Center Jagiellonian University National Science Center
Uncertainty relations Multipartite entanglement

Based on

Research project Uncertainty Relations and Quantum Entanglement

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

Uncertainty relations

[U1] Z. Puchała, Ł. Rudnicki, K. Życzkowski

Majorization entropic uncertainty relations

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.

[U2] Ł. Rudnicki, Z. Puchała, K. Życzkowski

Strong majorization entropic uncertainty relations

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.

[U3] Z. Puchała, Ł. Rudnicki, K. Chabuda, M. Paraniak and K. Życzkowski

Certainty relations, mutual entanglement and non-displacable manifolds

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.

[U4] R. Adamczak, R. Latała, Z. Puchała, K. Życzkowski

Asymptotic entropic uncertainty relations

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.