Monogamy of correlations and entropy inequalities in the Bloch picture

Abstract

We investigate monogamy of correlations and entropy inequalities in the Bloch representation. Here, both can be understood as direct relations between different correlation tensor elements and thus appear intimately related. To that end we introduce the split Bloch basis, that is particularly useful for representing quantum states with low dimensional support and thus amenable to purification arguments. Furthermore, we find dimension dependent entropy inequalities for the Tsallis 2-entropy. In particular, we present an analogue of the strong subadditivity and a quadratic entropy inequality. These relations are shown to be stronger than subadditivity for finite dimensional cases.

MaximumN-body correlations do not in general imply genuine multipartite entanglement

“…The systematic investigation of the properties of the Bloch representation for finite-dimensional multi-party quantum systems is a relatively recent subject [4][5][6][7][8][9][10][11][12][13][14][15], although many important results were found earlier, mostly relating specific features of the Bloch picture to the entanglement properties of the state (e.g., [16][17][18][19][20][21][22][23][24][25][26]). Currently much activity is devoted to working out the technical details and properties for an easier use of the Bloch representation to solving physics problems.…”

Joint Schmidt-type decomposition for two bipartite pure quantum states

“…As a direct consequence of these results we derive several interesting relations for the reductions of rank-1 operators. As the Schmidt decomposition, the existence of relations for one-party reductions, and the Bloch representation [15][16][17][18][19][20][21][22] of bipartite quantum states are intimately related concepts [19,21,23], we conclude our discussion by analyzing the most salient of these mathematical connections.…”

Exponentially many entanglement and correlation constraints for multipartite quantum states

“…[2,3] or to find new independent monogamy relations, see, e.g., Refs. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Furthermore, it was found that also correlations other than entanglement, such as nonlocality, may obey monogamy relations [21][22][23].…”

“…Note that the linear entropy is proportional to the Tsallis 2-entropy [29,48] and has a simple functional relation with the Rényi α-entropy for α = 2 [49]. To date, only few inequalities are known for the linear entropy [6,15,19,29,50].…”

“…(A8) we readily see the product property of the inverter on product states ρ prod = ρ S ⊗ ρ S c , which is at the origin of Eq. (19) in the main text:…”