Hamming weight and tight constraints of multi-qubit entanglement in terms of unified entropy

We establish a characterization of multi-qubit entanglement constraints in terms of non-negative power of entanglement measures based on unified-(q, s) entropy. Using the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubit entanglement based on the αth-power of unified-(q, s) entanglement for α ≥ 1. For 0 ≤ β ≤ 1, we establish a class of tight polygamy inequalities of multi-qubit entanglement in terms of the βth-power of unified-(q, s) entanglement of assistance. Thus our results characterize the monogamy and polygamy of multi-qubit entanglement for the full range of non-negative power of unified entanglement.

for a three-qubit quantum state ρ ABC with its two-qubit reduced density matrices ρ AB = tr C ρ ABC and ρ AC = tr B ρ ABC , where τ(ρ A|BC ) is the bipartite entanglement between subsystems A and BC, quantified by tangle and τ(ρ A|B ) and τ(ρ A|C ) are the tangle between A and B and between A and C, respectively 6 . The tangle of a bipartite pure state AB ψ is defined as ( ) 2(1 tr ) is the reduced density matrix of AB ψ onto the subsystem A. For a bipartite mixed state ρ AB , its tangle is defined as τ ρ τψ where the minimum is taken over all possible pure-state decompositions of ρ ψ ψ = ∑ p AB i i i AB i 6 . Later, three-qubit CKW inequality was generalized for arbitrary multi-qubit systems 7 and some cases of multi-party, higher-dimensional quantum systems more than qubits in terms of various bipartite entanglement measures [8][9][10][11] .
Using the assisted entanglement that is a dual amount to bipartite entanglement measures, a dually monogamous (thus polygamous) property of multi-party entanglement was also established; for a three-qubit state ρ ABC , a polygamy inequality was proposed as is the tangle of assistance whose maximum is taken over all possible pure-state decompositions of ρ ψ ψ = ∑ p AB i i i AB i

Results
Unified entropy and multi-qubit entanglement constraints. For q, s ≥ 0 with q ≠ 1 and s ≠ 0, unified-(q, s) entropy of a quantum state ρ is defined as 24,25 , q s q s , Although unified-(q, s) entropy has a singularity at s = 0, it converges to Rényi-q entropy as s tends to 0 26,27 . We also note that unified-(q, s) entropy converges to Tsallis-q entropy 28 when s tends to 1, and for any nonnegative s, unified-(q, s) entropy converges to von Neumann entropy as q tends to 1, → Using unified-(q, s) entropy in Eq. (1), a two-parameter class of bipartite entanglement measures was introduced; for a bipartite pure state AB ψ , its unified-(q, s) entanglement (UE) 11 is with respect to the bipartition between A 1 and A 2  A N , and Later, it was shown that unified entropy can also be used to establish a class of polygamy inequalities of multi-qubit entanglement 15 ; for 1 ≤ q ≤ 2 and −q 2 + 4q − 3 ≤ s ≤ 1, we have with respect to the bipartition between A 1 and A 2  A N , and E ( ) Tight monogamy constraints of multi-qubit entanglement in terms of unified entanglement.
In this section, we establish a class of tight monogamy inequalities of multi-qubit entanglement using the α'th power of UE. Before we present our main results, we first provide some notations, definitions and a lemma, which are useful throughout this paper.
For any nonnegative integer j whose binary expansion is where ≤ j n log 2 and j i ∈ {0, 1} for i = 0, …, n − 1, we can always define a unique binary vector associated with j, which is defined as  29 . We also provide the following lemma whose proof is easily obtained by some straightforward calculus.
Now we provide our first result, which states that a class of tight monogamy inequalities of multi-qubit entanglement can be established using the α-powered UE and the Hamming weight of the binary vector related with the distribution of subsystems.
Proof. Without loss of generality, we may assume that the ordering of the qubit subsystems B 0 , …, B N−1 satisfies We first prove Inequality (17) for the case that N = 2 n , a power of 2, by using mathematical induction on n, and extend the result for any positive integer N.
For n = 1 and a three-qubit state ρ AB B 0 1 with two-qubit rduced density matrices ρ AB 0 and AB 1 ρ , we have where Inequalities (12) and (15) implies which recovers Inequality (17) for n = 1. (17) is true for N = 2 n−1 with n ≥ 2, and consider the case that N = 2 n . For an  From the induction hypothesis, we have

Now let us assume Inequality
Moreover, the last summation in Inequality (23) is also a summation of 2 n−1 terms starting from j = 2 n−1 to j = 2 n − 1. Thus, (after possible indexing and reindexing subsystems, if necessary) the induction hypothesis also leads us to Inequalities (23), (24) and (25) recover Inequality (17) for N = 2 n . Now let us consider an arbitrary positive integer N and a (N We first note that we can always consider a power of 2, which is an upper bound of N, that is, 0 ≤ N ≤ 2 n for some n. We also consider a (2 n + 1)-qubit state which is a product of .
AB AB j j ρ Γ = for each j = 0, …, N − 1. Thus, Inequality (27) together with Eqs (28) and (29) leads us to Here we would like to remark that, for a selective choice of q and s as well as α, Inequality (14) still holds for the counterexample of CKW inequality in Eq. (32); we first note that the two-qutrit reduced density matrix ρ AB of ψ ABC in Eq. (32) has a spectral decomposition, By the Hughston-Jozsa-Wootters (HJW) theorem 30 , any pure state ensemble of ρ AB can be realized as a superposition of x AB , y AB and z AB . Moreover, it is also straightforward to check that for arbitrary pure states |φ〉 AB = c 1 |x〉 AB + c 2 |y〉 AB + c 3 |z〉 AB with |c 1 | 2 + |c 2 | 2 + |c 3 | 2 = 1, its reduced density matrix σ A = tr B |φ〉 AB 〈φ| has the same spectrum { } , , 0 . Thus we have For the choice of s = 0 and q = 3, the unified-(q, s) entropy in Eq. (1) is reduced to , where the symmetry of ψ ABC under the permutation of subsystems, regardless of the global phase, also guarantees E 3,0 (ρ A|C ) = 1. (For calculation simplicity, we used the logarithmic function based on 2 throughout this paper, which does not affect on the validity of monogamy and polygamy inequalities). Now, we have for any α ≥ 4, which shows that Inequality (14) still holds for the counterexample of CKW inequality in Eq. (32) for a selective choice of q and s as well as α.
We also note that an analogous argument can be made to show the validity of Inequality (14) for the other counterexample of CKW inequality in ⊗ ⊗ 3 2 2 quantum systems 8 . Thus Theorem 2 provides us with a new class of tight monogamy inequalities of multi-qubit entanglement even without any concrete counterexample in higher-dimensional quantum systems more that qubits.
The following theorem shows that Inequality (14) of Theorem 2 can be even improved to be a tighter inequality with some condition on two-qubit entanglement; Theorem 3. For α ≥ 1, q ≥ 2, 0 ≤ s ≤ 1, qs ≤ 3 and any multi-qubit state Proof. Due to Inequality (16), it is enough to show Now let us assume the validity of Inequality (39) for any positive integer less than N. For a multi-qubit state where Inequality (12) and the condition in Inequality (38) lead Inequality (40) to where the second inequality is due to the induction hypothesis, and this complete the theorem. □ For any nonnegative integer j and its corresponding binary vector j → , the Hamming weight for any α ≥ 1. In other words, Inequality (14) in Theorem 2 can be made to be even tighter as Inequality (37) of Theorem 3 for any multi-qubit state Tight polygamy constraints of multi-qubit entanglement in terms of unified entanglement of assistance. As a dual property to the Inequality (14) of Theorem 2, we provide a class of polygamy inequalities of multi-qubit entanglement in terms of powered UEoA.
Proof. Without loss of generality, we assume the ordering of the qubit subsystems B 0 , …, B N−1 satisfying for each j = 0, …, N − 2. Moreover, due to the monotonicity of the function f(x) = x β for 0 ≤ β ≤ 1 and the UEoA-based multi-qubit polygamy inequality in (9), we have The proof method is similar to that of Theorem 2; we first prove Inequality (48) for the case that N = 2 n by using mathematical induction on n, and generalize the result to any positive integer N. For n = 1 and a three-qubit state ρ AB B where the ordering of subsystems in Inequality (46) and Inequality (13) Because each summation on the right-hand side of Inequality (52) is a summation of 2 n−1 terms, the induction hypothesis assures that  To illustrate the tightness of Inequality (45) in Theorem 4, let us first recall the general polygamy inequality of entanglement in arbitrary-dimensional multi-party quantum systems 16 ; For q tends to 1, the unified-(q, s) entanglement is reduced to EoA as in Eq. (7), therefore the marginal UEoA from Inequality (45) for three-qubit, W-state when q = 1 and β = Thus Inequality (45) is generally tighter than Inequality (58), which also delivers better bounds to characterize the W-class type three-party entanglement by means of bipartite ones.
We further note that an analogous argument for the improvement of monogamy inequalities from Theorem 2 to Theorem 3 can also be applied to Inequality (45) of Theorem 4 for a tighter class of polygamy inequalities with some condition on two-qubit entanglement of assistance. We also remark that the class of monogamy and polygamy inequalities established here hold in a tighter way than other multi-qubit entanglement inequalities provided so far. Thus our results can provide an efficient way of characterizing entanglement shareability and distribution among the multi-party quantum systems without any known counterexample even in higher-dimensional systems more than qubits.