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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.
Highlights
Correlations between particles are central for many physical phenomena, ranging from phase transitions in condensed matter systems to applications like quantum metrology
While it is known that strong subadditivity (SSA) does not hold in general for the linear entropy [17], we show, using techniques from semidefinite programming (SDP), that the symmetrized version is true for three qubits
We showed how to combine methods from quantum mechanics, coding theory and semidefinite programming to obtain strict bounds on linear combinations of sector lengths for multi-qubit systems
Summary
Correlations between particles are central for many physical phenomena, ranging from phase transitions in condensed matter systems to applications like quantum metrology. As all correlation measures, invariant under local unitary transformations [3] They are expressible in terms of purities of the reduced states of a system, and as such, they can be experimentally characterized by randomized measurements on a single copy of the state [4]. We fully classify the set of admissible tuples of sector lengths for two- and three-qubit states by characterizing all bounds on linear combinations of the sector lengths. We show that in these cases, the admissible sector lengths form a convex polytope that can be characterized by few constraints One of these constraints can be viewed as a symmetrized version of strong subadditivity (SSA) of the linear entropy. We completely characterize the allowed sector length configurations by considering a symmetrized SSA for linear entropies for three-qubit systems. While it is known that SSA does not hold in general for the linear entropy [17], we show, using techniques from semidefinite programming (SDP), that the symmetrized version is true for three qubits
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