11,960 research outputs found
{Improved Bounds on Fourier Entropy and Min-entropy}
Given a Boolean function , the Fourier distribution assigns probability to . The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal constant C>0 such that , where is the Shannon entropy of the Fourier distribution of and is the total influence of . 1) We consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture. This asks if , where is the min-entropy of the Fourier distribution. We show , where is the minimum parity certificate complexity of . We also show that for every , we have , where is the approximate spectral norm of . As a corollary, we verify the FMEI conjecture for the class of read- s (for constant ). 2) We show that , where is the average unambiguous parity certificate complexity of . This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture. We show that a weaker version of FEI already implies Mansour's conjecture: is ?, where are the 0- and 1-certificate complexities of , respectively. 3) We study what FEI implies about the structure of polynomials that 1/3-approximate a Boolean function. We pose a conjecture (which is implied by FEI): no "flat" degree- polynomial of sparsity can 1/3-approximate a Boolean function. We prove this conjecture unconditionally for a particular class of polynomials
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 in orthogonal bases. Lower bounds lead
to novel entropic uncertainty relations, while upper bounds allow us to
formulate universal certainty relations. For the maximal average entropy
saturates at as there exists a mutually coherent state, but certainty
relations are shown to be nontrivial for measurements. In the case of
a prime power dimension, , and the number of measurements , the
upper bound for the average entropy becomes minimal for a collection of
mutually unbiased bases. Analogous approach is used to study entanglement with
respect to different splittings of a composite system, linked by bi-partite
quantum gates. We show that for any two-qubit unitary gate there exist states being mutually separable or mutually
entangled with respect to both splittings (related by ) of the composite
system. The latter statement follows from the fact that the real projective
space is non-displacable. For
splittings the maximal sum of entanglement entropies is conjectured to
achieve its minimum for a collection of three mutually entangled bases, formed
by two mutually entangling gates
Entropic uncertainty relations - A survey
Uncertainty relations play a central role in quantum mechanics. Entropic
uncertainty relations in particular have gained significant importance within
quantum information, providing the foundation for the security of many quantum
cryptographic protocols. Yet, rather little is known about entropic uncertainty
relations with more than two measurement settings. In this note we review known
results and open questions.Comment: 12 pages, revte
Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory
Uncertainty relations capture the essence of the inevitable randomness
associated with the outcomes of two incompatible quantum measurements.
Recently, Berta et al. have shown that the lower bound on the uncertainties of
the measurement outcomes depends on the correlations between the observed
system and an observer who possesses a quantum memory. If the system is
maximally entangled with its memory, the outcomes of two incompatible
measurements made on the system can be predicted precisely. Here, we obtain a
new uncertainty relation that tightens the lower bound of Berta et al., by
incorporating an additional term that depends on the quantum discord and the
classical correlations of the joint state of the observed system and the
quantum memory. We discuss several examples of states for which our new lower
bound is tighter than the bound of Berta et al. On the application side, we
discuss the relevance of our new inequality for the security of quantum key
distribution and show that it can be used to provide bounds on the distillable
common randomness and the entanglement of formation of bipartite quantum
states.Comment: v1: Latex, 4 and half pages, one fig; v2: 9 pages including 4-page
appendix; v3: accepted into Physical Review A with minor change
Optimality of entropic uncertainty relations
The entropic uncertainty relation proven by Maassen and Uffink for arbitrary
pairs of two observables is known to be non-optimal. Here, we call an
uncertainty relation optimal, if the lower bound can be attained for any value
of either of the corresponding uncertainties. In this work we establish optimal
uncertainty relations by characterising the optimal lower bound in scenarios
similar to the Maassen-Uffink type. We disprove a conjecture by Englert et al.
and generalise various previous results. However, we are still far from a
complete understanding and, based on numerical investigation and analytical
results in small dimension, we present a number of conjectures.Comment: 24 pages, 10 figure
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