Entanglement monogamy in three qutrit systems

By introducing an arbitrary-dimensional multipartite entanglement measure, which is defined in terms of the reduced density matrices corresponding to all possible two partitions of the entire system, we prove that multipartite entanglement cannot be freely shared among the parties in both n-qubit systems and three-qutrit systems. Furthermore, our result implies that the satisfaction of the entanglement monogamy is related to the number of particles in the quantum system. As an application of three-qutrit monogamy inequality, we give a condition for the separability of a class of two-qutrit mixed states in a 3 ⊗ 3 system.

By introducing an arbitrary-dimensional multipartite entanglement measure, which is defined in terms of the reduced density matrices corresponding to all possible two partitions of the entire system, we prove that multipartite entanglement cannot be freely shared among the parties in both n-qubit systems and three-qutrit systems. Furthermore, our result implies that the satisfaction of the entanglement monogamy is related to the number of particles in the quantum system. As an application of three-qutrit monogamy inequality, we give a condition for the separability of a class of two-qutrit mixed states in a 3 ⊗ 3 system.
Quantum entanglement is an essential feature of quantum mechanics, which distinguishes the quantum from the classical world. Because of entanglement, different quantum systems can affect each other, even if there is no classical connection between the multiple quantum systems. So quantum entanglement can be used to perform a number of tasks which can not be completed in the classical mechanical system. Quantification of quantum entanglement plays an important role in quantum information processing and quantum computation [1][2][3][4][5] . The mathematical study of entanglement has become a very active field and has led to many operational and information theoretic insights.
Entanglement is monogamous, which was first discovered by tangle for three qubit systems in the seminal paper of Coffiman, Kundu and Wootters 6 . It describes the constraint on distributed entanglement among many parties. It is also a key ingredient in quantum cryptography security 7, 8 , statistical mechanics 9 , the foundations of quantum mechanics 10 and black-hole physics 11 . In addition to having a wide range of practical applications, monogamy has also profound theoretical significance, allowing simplified proofs of no-broadcasting bounds and constraints for qubit multitap channel capacities 12 .
The author stated in ref. 12 that the monogamy inequality in the condensed matter physics gives rise to the frustration effects observed in, e.g., Heisenberg antiferromagnets. The perfect ground state for an antiferromagnet would in fact consist of singlets between all interacting spins. However, as a particle can only share one unit of entanglement with all its neighbors, it will try to spread its entanglement in an optimal way with all its neighbors leading to a strongly correlated ground state. Such qualitative statements have been turned into quantitative ones in n-qubit systems through the square of the concurrence 12 , the square of the entanglement of formation 13 and the square of convex-roof extended negativity 14 , respectively.
Suppose that E is an entanglement measure for the multipartite system . Monogamous relation expressed in terms of inequalities can be represented as The proof of this theorem can be found in the Supplemental Material. For n-qudit pure state |ψ〉 in the system Consider the state = + + GHZ ( 100 010 001 ) 1 3 , we find that , which violates the monogamy inequality. Now we add a coefficient which is related to the particle number of systems in Eq. (3), and let . Thus, the following result follows immediately from Theorem 1.

Corollary 1.
For an 3-qubit system, N  satisfies the monogamy inequality. The above discussion implies that the concurrence itself does not satisfy monogamous relation. This, together with Corollary 1, shows that the satisfaction of entanglement monogamy characterized by an entanglement measure is generally related to the number of particles of the system.
Next we discuss the entanglement monogamy in three qutrit systems. Until now, no true entanglement measure has been proven to be monogamous for three-dimensional tripartite systems. Taking the square of concurrence as an example, an explicit counterexample showing the violation of the monogamy inequality in three-dimensional quantum systems is as follows 16 , . For an arbitrary pure state |Φ〉 AB , a discussion just as in ref. 16 , which means that the square of concurrence does not work for monogamy inequality on a three-qutrit system. Using the entanglement measure  M , it can be calculated that M . More generally, we will prove that the measure M  satisfies the monogamy inequality in a three-qutrit system. As a first step toward proving this inequality, we will now derive a computable formula for M  .

Discussions
The monogamy of entanglement characterized by the entanglement measure describes quantitatively the entanglement between quantum systems. Choosing the proper entanglement measure helps to reveal the nature of entanglement. The more system information reflected by an entanglement measure, the better it can describe the entanglement of the system. Through giving an entanglement measure which is related to the number of particles of the system, we prove that multipartite entanglements cannot be freely shared among the parties in both n-qubit systems and three-qutrit systems. Corollary 1 and the discussion perior to Corollary 1 imply that the satisfaction of entanglement monogamy characterized by an entanglement measure is generally connected with the number of particles of the system. For the state |Ψ〉 given before Lemma 1 in a three qutrit system ⊗ ⊗ , that is, the monogamy inequality holds, where N  is defined in Eq. (4). More generally, we conjecture that the entanglement measure N  satisfies the monogamy inequality in three qutrit systems. As a subsequent work, we will continue to discuss it.
In addition, the entanglement monogamy inequality gives an upper bound for the entanglement degree of two-qutrit mixed states, for which the general separability criteria and computable entanglement measures remain still open. In the Supplemental Material, by such an upper bound, a condition is given for the separability of a class of two-qutrit mixed states in a 3 ⊗ 3 system.

Methods
Proof of Theorem 2. Let |φ〉 ABC be a pure state in the three-qutrit system Using Lemma 1, we calculate the entanglement between the particle A and the particles BC, Next we estimate the entanglement between particles A and C. Let   → :  (i = 0, 1, 2) be three projections defined, respectively, by Thus, by Lemma 1, we obtain the entanglement degree of |τ j 〉 (j = 0, 1, 2), 2  3  cos  2  3  and   1 , , Similarly to the above discussion for  AC M , consider a decomposition ρ ψ ς ς   Let λ j1 , λ j2 and λ j3 be the three non-negative eigenvalues of s j ρ A (|τ j 〉) (j = 0, 1, 2), then