Characterizing quantum states via sector lengths

Abstract

Correlations in multiparticle systems are constrained by restrictions from quantum mechanics. A prominent example for these restrictions are monogamy relations, limiting the amount of entanglement between pairs of particles in a three-particle system. A powerful tool to study correlation constraints is the notion of sector lengths. These quantify, for different k, the amount of k-partite correlations in a quantum state in a basis-independent manner. We derive tight bounds on the sector lengths in multi-qubit states and highlight applications of these bounds to entanglement detection, monogamy relations and the n-representability problem. For the case of two-and three qubits we characterize the possible sector lengths completely and prove a symmetrized version of strong subadditivity for the linear entropy.In this section, we prove Proposition 3 from the main text:Proposition 3. For all qubit states of n parties with n ≥ 4, it holds that n j=2 A 2 (ρ 1j ) ≤ n − 1.Proof. We prove the claim for n = 4 first. In this case we distribute all Pauli operators whose expectation values contribute to the bipartite sector lengths into anticommuting sets,such that in each set all operators pairwise anticommute. Here, XX11 means again X ⊗ X ⊗ 1 ⊗ 1. For any anticommuting set M , it holds that m∈M m 2 ≤ 1 [35,37,38]. The sets are chosen such that

Entanglement characterization using quantum designs

“…Though the latter approaches have proven useful for experimental implementations in trapped ion experiments [24,27], it is important to note that the knowledge of all moments corresponding to different sectors, i.e., subsets of the involved parties, gives more insight into the entanglement properties of the underlying state. For the case of second moments, this follows from the relation between the respective moments and socalled sector lengths which have been independently under investigation in the context of entanglement detection [28][29][30]. Similar insights have been reached in Ref.…”

“…It is important to note that one can gain more information about ρ by considering also moments of smaller qubit sectors, i.e., the respective reduced states, as has been investigated for t = 2 in Refs. [24][25][26][27][28][29][30]. However, in the remainder of this manuscript we will focus on the characterization of multipartite entanglement based on full N -qubit moments (2).…”

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

“…We note that Eqs. (14) elucidate the role of the N -sector for pure states: All sector lengths have to be adjusted so as to obey the k-purity relations (k = 1 . .…”

“…(13) and Eqs. (14) for k = 0, 1, 2, and increasingly tedious algebra. N = 2 : We have d 2 = 1 + S 1 + S 2 , so that…”

“…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.…”