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.

Introduction

Quantum entanglement is an important resource in quantum information theory. Different from classical correlations, this restricted shareability of entanglement in multi-particle systems is known as monogamy property. The more entanglement shared between two parties implies the less entanglement shared with the rest of the system. Monogamy property plays a crucial role in quantum cryptography: which restricts the quantity of information captured by an eavesdropper about the secret key to be extracted[1-3]. Monogamy property has also been discussed in the device-independent quantum information processing[4], condensed matter physics[5] and black-hole physics[6, 7].

The study of monogamy property has a long history. The first monogamy relation was found by Coffman et al., who considered a three-qubit system ABC [8], and showed that the amount of entanglement (which is quantified by the squared concurrence) between A and B, plus the amount of entanglement between A and C, cannot be greater than the amount of entanglement between A and the pair BC. Further, Osborne et al. proved the squared concurrence follows a general monogamy inequality for the N-qubit system[1]. Monogamy inequalities for different entanglement measures have been noted, such as concurrence[9-12], entanglement of formation[13, 14], negativity[15-19], Rényi entropy entanglement[20, 21], and Tsallis entropy entanglement[22-24]. For the other physical resources, such as discord and steering, the monogamy property of them has also been discussed[25-28].

As dual to monogamy property, polygamy property in multi-particle systems has arised many interests by researchers[15, 19, 22, 29, 30]. Polygamy property was first provided by using the concurrence of assistance to quantify the distributed bipartite entanglement in multi-qubit systems[29, 30]. Polygamy property has also considered in many entanglement measures, such as Rényi entropy[20], Tsallis entropy[22, 31] and convex-roof extended negativity[19].

Unified (q, s)-entropy is an important entropic measure, which can be used in many areas of quantum information theory. In this paper, 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.

Results

This paper is organized as follows. In the first subsection, we recall the definition of unified (q, s)-entropy and discuss the properties of unified (q, s)-entropy entanglement. In the second subsection, we give our main results. We summarize our results in the third subsection.

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] for any q, s ≥ 0 such that and .

When s tends to 1, the unified (q, s)-entropy converges to Tsallis entropy T (ρ)[33]

When s tends to 0, the unified (q, s)-entropy converges to Rényi entropy R (ρ)[34]

When q tends to 1, the unified (q, s)-entropy converges to von Neumann entropy S(ρ)[35]

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 in the Hilbert space (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 , the unified (q, s)-entropy entanglement can be defined via the convex-roof extensionwhere the minimum is taken over all possible ensembles of ρ with and p  ≥ 0. It is straightforward to verify that if and only if is a separable state for .

When tends to 1, the unified -entropy entanglement becomes Tsallis entanglement[31]. When tends to 0, the unified -entropy entanglement becomes Rényi entanglement[20]. Especially, The unified -entropy entanglement becomes the entanglement of formation when tends to 1. The entanglement of formation is defined as[37, 38] where is the von Neumann entropy, the minimum is taken over all possible ensembles of ρ with and p  ≥ 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 aswhere the maximum is taken over all possible ensembles of ρ with and p  ≥ 0. To understand (q, s)-EOA better, consider a tripartite pure state shared among three parties referred to as Alice, Bob, and Charlie[39]. The entanglement supplier, Charlie, performs a measurement on his share of the tripartite state, which yields a known bipartite entangled state for Alice and Bob. Tracing over Charlie’s system yields the bipartite mixed state shared by Alice and Bob. Charlie’s aim is to maximize entanglement for Alice and Bob, and the maximum average entanglement he can create is the (q, s)-EOA.

Concurrence and concurrence of assistance

For any pure state in the Hilbert space , the concurrence is defined as[40] where .

For a mixed state ρ , the concurrence can be defined via the convex-roof extensionwhere the minimum is taken over all possible ensembles of ρ with and p  ≥ 0.

As a dual quantity to concurrence, the concurrence of assistance (COA) can be defined aswhere the maximum is taken over all possible ensembles of ρ with and p  ≥ 0.

Analytical formula for two-qubit states

For a two-qubit mixed state ρ , concurrence and COA are known to have analytic formula[30, 40] where λ being the eigenvalues, in decreasing order, of matrix .

In ref. 40, Wootters derived an analytical formula of entanglement of formation for a two-qubit mixed state ρ where is the binary entropy.

In ref. 36, Kim found has an analytical formula for a two-qubit mixed state, which can be expressed as a function of concurrence for q ≥ 1, 0 ≤ s ≤ 1 and qs ≤ 3where the function f (x) has the formwhere 0 ≤ x ≤ 1.

Main Results

In this section, we will provide our main results. First, we have following result:

Theorem 1. For any tripartite mixed state ρ in the Hilbert space , we havewhere q ≥ 1 and qs ≥ 1.

Proof: Let be an optimal decomposition of . That is

For any bipartite pure state , the unified (q, s)-entropy entanglement . In ref. 41, Rastegin proved that for any q ≥ 1 and qs ≥ 1, the unified (q, s)-entropy is subadditive, that is

Combining Eq. (18) with Eq. (19), we have

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 , the unified (q, s)-entropy entanglement .

We also have the following corollary:

Corollary 1. For any mixed state in the Hilbert space , we havewhere 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 . In particular, for any pure state , the unified (q, s)-entropy entanglement .

Example 1: Let’s consider the general GHZ state , where |α|2 + |β|2 = 1 and n ≥ 3. It’s easy to show that  = .

Example 2: Consider a four-qubit cluster state , which is a type of highly entangled state of four-qubit[42, 43]. The reduced states of are , thus , which is nonnegative for q ≥ 1 and qs ≥ 1.

We note that for any n-qubit mixed state , the polygamy relation holds:for 1 ≤ q ≤ 2 and −q 2 + 4q − 3 ≤ s ≤ 1[44]. Combining Eq. (17) with Eq. (22), we have

Corollary 2. For any multi-qubit mixed state , the following inequality holdswhere 1 ≤ q ≤ 2, s = 1.In this case, (q, s)-EOA becomes Tsallis entropy entanglement of assistance which has discussed in ref. 22.

Before our second main result, we have following lemma:

Lemma 1. For q = 2 and , the function f (x) in Eq. (16) satisfies Proof: For q ≥ 2, 0 ≤ s ≤ 1, and qs ≤ 3, we have [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 . This completes the proofs.

Next, the following result will provide a lower bound of unified (q, s)-entropy entanglement of , with respect to the bipartition between AB and :

Theorem 2. For any multi-qubit pure state in the Hilbert space, we havewhere q = 2 and .

Proof: Given a multi-qubit pure state , the unified (q, s)-entropy is subadditive for any q ≥ 1 and qs ≥ 1. Thus, the following equality holdswhich implies , and similarly, . Combine with the two equalities above, one obtain

From the definition of unified (q, s)-entropy entanglement of , with respect to the bipartition between AB and , we have

Note that for any pure state in a system, the following equality holds[45, 46] where ρ and ρ are the reduced matrices of state respectively.

For q = 2 and , we havewhere we have used Eq. (29) in the second equality, the third equality holds is due to lemma 1. Therefore,

Compare Eq. (30) with Eq. (31), it’s easy to see that

We also note thatwhere the first equality holds is due to the monogamy of concurrence[1] and f (x) is an increasing function for q ≥ 2, 0 ≤ s ≤ 1, and qs ≤ 3[36].

On the other hand, we have

Combine Eqs (32) and (33) with Eq. (34), we have

Putting Eq. (35) into Eq. (32), we obtain our result. Similarly, we haveThus, the proof is completed.

Theorem 2 shows a monogamy relation for a multi-qubit pure state . The lower bound of the unified (q, s)-entropy entanglement for can’t be less than the sum of the two-qubit entanglement between bipartitions of the system. In particular, if , the entanglement of is equal to the entanglement of . In this case, for . Theorem 2 becomes , which is a CKW-type monogamy relation[1, 8].

Example 3: Consider a pure state in the four-qubit system. for the range q = 2 and , we have , and . where i = 1, 2 and . Therefore, we can see saturates the inequality Eq. (25).

Example 4: Finally, let’s consider a general W state in the n-qubit system, where . The reduced state of subsystem A 1 A 2 iswhich implies . It’s also easy to show that the reduced state is separable, where . Thus . We find that the right side of the inequality Eq. (25) is  = 0. Which mean the inequality Eq. (25) holds for the general W state.

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 , a generalized monogamy relation is provided for unified (q, s)-entropy entanglement. This monogamy relation provides a lower bound for the bipartition in the multi-qubit system. In particular, if , 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.