Entanglement distribution in multi-particle systems in terms of unified entropy

We investigate the entanglement distribution in multi-particle systems in terms of unified (q, s)-entropy. We find that for any tripartite mixed state, the unified (q, s)-entropy entanglement of assistance follows a polygamy relation. This polygamy relation also holds in multi-particle systems. Furthermore, a generalized monogamy relation is provided for unified (q, s)-entropy entanglement in the multi-qubit system.

Unified (q, s)-entropy entanglement and unified (q, s)-entropy entanglement of assistance. Given a quantum state ρ in the Hilbert space . The unified (q, s)-entropy is defined as 32 q s q s , for any q, s ≥ 0 such that ≠ q 1 and ≠ s 0. When s tends to 1, the unified (q, s)-entropy converges to Tsallis  When q tends to 1, the unified (q, s)-entropy converges to von Neumann entropy S(ρ) 35 ρ ρ ρ = − . → S T r lim ( ) l n (4) q q s 1 , Because the limits exist in the case of q → 1 and s → 0, we will use q = 1 and s = 1 to represent the limits in this paper. Now, let's consider the entanglement in terms of the unified (q, s)-entropy. For any pure state ψ AB in the Hilbert space H H A B ⊗ (it's does not matter for the sizes of subsystem A and B), the unified (q, s)-entropy entanglement is defined as 36 for any q, s ≥ 0. For a mixed state ρ AB , the unified (q, s)-entropy entanglement can be defined via the convex-roof extension where the minimum is taken over all possible ensembles ψ if and only if ρ AB is a separable state for ≥ q s , 0. When s tends to 1, the unified q s ( , )-entropy entanglement becomes Tsallis entanglement 31 . When s tends to 0, the unified q s ( , )-entropy entanglement becomes Rényi entanglement 20 . Especially, The unified q s ( , )-entropy entanglement becomes the entanglement of formation when q tends to 1. The entanglement of formation is defined as 37,38 i is the von Neumann entropy, the minimum is taken over all possible ensembles ψ and p i ≥ 0. As a dual quantity to the unified (q, s)-entropy entanglement, the unified (q, s)-entropy entanglement of assistance ((q, s)-EOA) can be defined as For a mixed state ρ AB , the concurrence can be defined via the convex-roof extension where the minimum is taken over all possible ensembles ψ and p i ≥ 0. As a dual quantity to concurrence, the concurrence of assistance (COA) can be defined as Analytical formula for two-qubit states. For a two-qubit mixed state ρ AB , concurrence and COA are known to have analytic formula 30,40 f AB has an analytical formula for a two-qubit mixed state, which can be expressed as a function of concurrence AB  for q ≥ 1, 0 ≤ s ≤ 1 and qs ≤ 3 where the function f q,s (x) has the form where 0 ≤ x ≤ 1.
Main Results. In this section, we will provide our main results. First, we have following result: where q ≥ 1 and qs ≥ 1.
( ) . In ref. 41, Rastegin proved that for any q ≥ 1 and qs ≥ 1, the unified (q, s)-entropy is subadditive, that is Thus, the proof is completed. Theorem 1. Shows a simple but interesting polygamy relation of (q, s)-EOA in a tripartite quantum system. The upper bound of (q, s)-EOA of A|BC can't be greater than the sum of (q, s)-EOA of B|AC and (q, s)-EOA of C|AB. In particular, for a tripartite pure state ψ A BC , the unified (q, s)-entropy entanglement . We also have the following corollary: Corollary 1. For any mixed state ρ where q ≥ 1 and qs ≥ 1. Corollary 1. Shows a constrained relationship of (q, s)-EOA in the multi-particle system, and gives an upper bound of (q, s)-EOA of  A A A n 1 2 . In particular, for any pure state ψ , which is a type of highly entangled state of four-qubit 42,43 . The reduced states of C 4 are ρ ρ ρ ρ , which is nonnegative for q ≥ 1 and qs ≥ 1.
We note that for any n-qubit mixed state ρ  AC C n 1 , the polygamy relation holds:  , the function f q,s (x) in Eq. (16) satisfies  36 . On the other hand, for 1 ≤ q ≤ 2 and 0 ≤ s ≤ 1, we have 44 . The equality holds if and only if q = 2 and ≤ ≤ s 1 1 2 . This completes the proofs. Next, the following result will provide a lower bound of unified (q, s)-entropy entanglement of ψ  AB C C n 1 , with respect to the bipartition between AB and  C C n 1 : Theorem 2. For any multi-qubit pure state ψ , , n n 1 1 , and similarly, , , n n 1 1 . Combine with the two equalities above, one obtain Note that for any pure state ψ ABC in a ⊗ ⊗ d 2 2 system, the following equality holds 45,46 ABC a AB AC where ρ AB and ρ AC are the reduced matrices of state ψ ABC respectively. For q = 2 and ≤ ≤ s 1 1 2 , we have where we have used Eq. (29) in the second equality, the third equality holds is due to lemma 1. Therefore,  We also note that where the first equality holds is due to the monogamy of concurrence 1 and f q,s (x) is an increasing function for q ≥ 2, 0 ≤ s ≤ 1, and qs ≤ 3 36 .
On the other hand, we have

( )
. It's also easy to show that the reduced state . We find that the right side of the inequality Eq. (25) is

Conclusion
Unified (q, s)-entropy is an important generalized entropy in quantum information theory. Many entropies such as Tsallis entropy, Rényi entropy, and von Neumann entropy can be seen as a special case for unified (q, s)-entropy.
In this paper, we have investigated the entanglement distribution in multi-particle systems in terms of unified (q, s)-entropy. We find that for any tripartite mixed state, the (q, s)-EOA follows a polygamy relation for q ≥ 1 and qs ≥ 1. This polygamy relation provides an upper bound for the bipartition A|BC, which also holds in multi-particle systems. Furthermore, for q = 2 and ≤ ≤ s 1 1 2 , a generalized monogamy relation is provided for unified (q, s)-entropy entanglement. This monogamy relation provides a lower bound for the bipartition , the generalized monogamy relation becomes a CKW-type monogamy relation.
Both monogamy property and polygamy property are fundamental properties of multipartite entangled states. We have studied the properties above in detail, and provided a two-parameters entropy function to study the entanglement distribution. We believe our result provides a useful methodology to understand the entanglement distribution of multi-particle entanglement.