Tsallis entropy, q -expectation value, and constraints on three-party quantum correlations

Abstract

We show that the mutually exclusive nature of classical and quantum correlations distributed in multi-party quantum systems can be characterized in terms of q-expectation. Using Tsallis-q entropy and q-expectation, we first provide generalized definitions of classical and quantum correlations, and establish their trade-off relations in three-party quantum systems of arbitrary dimension with respect to q-expectation for q ≥ 1. We also provide equivalence conditions for monogamy and polygamy inequalities of quantum entanglement and quantum discord distributed in three-party quantum systems of arbitrary dimension with respect to q-expectation for q ≥ 1.

Weighted polygamy inequalities of multiparty q-expected quantum entanglement

“…which is the q-expectation value of the q-logarithm. For a bipartite pure state |ψ⟩ AB , the q-expected entanglement was defined as [29]:…”

“…The concavity of Tsallis q-entropy guarantees the nonnegativity of Tsallis q-difference for q ⩾ 1, and when q tends to 1, it converges to the Holevo quantity. For a bipartite quantum state ρ AB , its one-way classicalq-correlation (q-CC) [29] was defined as the maximum Tsallis-q difference:…”

“…over all probability ensembles E of ρ A induced by rank-1 measurements on subsystem B. Moreover, for any three-party pure state |ψ⟩ ABC , with its two-party reduced density matrices ρ AB = Tr C |ψ⟩ ABC ⟨ψ|, ρ AC = Tr B |ψ⟩ ABC ⟨ψ| and ρ A = Tr B C|ψ⟩ ABC ⟨ψ|, the q-expected entanglement was defined as [26,29]:…”