Qudit Concurrence and Monogamy

It is well known that entanglement is one of the distinctive aspects of the quantum world. The measurement and quantification of entanglement is one of the major problems in understanding and applying this resource. This research aims to investigate entanglement measures and their properties in high-dimensional systems. One such property is entanglement monogamy, viz. quantum correlations are not shareable, different from the classical ones. On the other hand, the study of entanglement depends on the definitions of its measures. Here we examine the applicability of concurrence, an entanglement measure known for its effectiveness in the analysis of qubit entanglement. While the entanglement of two-level systems or qubits is widely understood, its extension to higher dimensions remains challenging. Our research establishes equivalence between different concurrence definitions for bipartite qubit pure states and extends them to qudits. Subsequently, we investigate the monogamous relations of these measurements to explore their characteristics. This approach might enhance our comprehension of quantum traits in qudit systems and advance our understanding of entanglement in broader quantum contexts.


Introduction
Entanglement is a characteristic property of quantum mechanics [1] that distinguishes the quantum world from the classical one, as demonstrated by Bell's inequalities [2].An important problem in entanglement is determining the level of entanglement in a composite quantum system and defining a measure that quantifies it.An entanglement measure is a monotonous function that maps a density matrix  to a positive real number, with a lower bound indicating separable states and an upper bound indicating maximally entangled states, while not increasing on average due to LOCC (Local Operations and Classical Communication) [3,4].
Quantum entanglement cannot be freely distributed among multiple parties.For example, in a multiparty qubit state, if two parties are maximally entangled, then neither of these two parties can share entanglement with other parties in the system simultaneously [5].The limitation on the sharing of entanglement in multipartite scenarios is known as the monogamy of entanglement (monogamy inequality) [6,7] ( | ) ≥ ( | ) + ( | ), where  represents the entanglement measure employed.
A fundamental and important problem in the context of the monogamy of entanglement is determining the monogamous property of an entanglement measure [8].There exist several

Pure state qubit concurrence
Concurrence and entanglement of formation in a system of two qubits are defined in [14,15].These definitions are further refined and extended by deriving monogamy inequality relations for tripartite qubits based on squared concurrence [5].Subsequently, monogamy inequalities are developed for multiqubit systems with expression [27]   (  1 | 2 …  ) ≥   (  1 | 2 ) + ⋯ +   (  1 |  ), where   is the entanglement measure.Concurrence indicates the correlation between two qubits, is monotonic, with a lower bound of 0 indicating separable states and an upper bound of 1 indicating maximally entangled states [5, [14][15].Therefore, it can be used as an entanglement measure.Equivalent definitions of concurrence for pure state qubits are provided as follows: i) References [5,15] define concurrence between parties  and , (  ), as where   represents generator associated with rotations around the -axis within the (2) group, and   =     is the reduced density matrix of qubit ; using the state yields results consistent with reference [30], as demonstrated by ii) Based on [31], concurrence (  ) as a function of the Bloch vector  ⃗ or  , is expressed as The Bloch vectors  ⃗ and  satisfying the conditions   = ( +   . ), where   and   are (2) group generators correspond to parties  and , respectively.iii) Furthermore, according to [32,33], concurrence (  ) is expressed as which is also equivalent to the previous definitions.Thus, all four concurrence formulas in equations (3-6) mentioned above are equivalent.iv) In [30], concurrence (  ) is expressed as where  1 ,  2 are the coefficients in the Schmidt decomposition [37][38][39] of |⟩ = Referring to the equivalent definitions above, qubit concurrence is limited to the range [0, 1] and is monotonous.However, when extended to the qutrit case, concurrence does not always have an upper bound of 1 [28][29]34].Questions about the upper bound of concurrence for higher-dimensional cases can be used to normalize the range of concurrence values to [0, 1], where   = 0 for separable states and   = 1 for maximally entangled states.

Qudit concurrence
In extending the dimension of concurrence for the pure state of two qudits, several assumptions and alternative definitions will be employed as follows: a) Qudit concurrence is defined as an extension of qubit concurrence [14][15] and expressed as a function of the linear combination coefficients   , denoted as   (  ).b) Qudit concurrence is expressed as a function of a "-dimensional Bloch vector"  ⃗⃗ , denoted as   ( ⃗⃗ ).
c) Qudit concurrence is expressed as a function of the trace of   2 , denoted as   ((  2 )).
d) Qudit concurrence is expressed as a function of the Schmidt decomposition coefficients   , denoted as   (  ).These definitions should be bounded, monotonic, and consistent with the qubit limit case.

Pure state qudit concurrence as a function of linear combination coefficients, 𝐶 𝐴𝐵 (𝛼 𝑖𝑗 )
For the first definition,   (  ), the concurrence of a  ×  system ( × -dimensional qudits) is given by [33] While for the   ×   system (  ×   -dimensional qudits), is provided by [34] which is obtained by extension to higher dimensions from the qubit case Equation ( 10) can be expressed as where  represents the group generator of  (2).The definition of concurrence in equation (11) can be extended to   ×   system by using the generators (  ) and (  ), respectively [34].For instance, the (  ) generators   ,  = 1, … where | ̃⟩ = (  ⨂  )| * ⟩.Thus, equation ( 9) is obtained.For   > 2 and   > 2, equation (9) needs to be modified so that   is bounded within [0, 1], becoming where  = min(  ,   ).Hence,   in [5] is modified to where It can be shown that equation ( 8) is equal to equation (13) for   =   .On the other hand, qubit concurrence satisfies equation (3) and is equivalent to the other definitions.However, in the extension to higher dimensions, concurrence as a function of det   is no longer equivalent to the other definitions.Therefore, an alternative function is needed that is equivalent to the definition of concurrence in higher dimensions and reduces to det   in the qubit limit case.For example, the expression in equation (13), which reduces to det   in the qubit scenario.We refer to where  ̂ is given by and the number of ).In qubit case, pdet  ̂= (det  ̂)2 = det   , therefore equation (3) becomes   = 2√pdet  ̂.

Pure state qudit concurrence as a function of "𝑑-dimensional Bloch vector", 𝐶 𝐴𝐵 (𝑤 ⃗⃗ )
The density operator of 2-qutrits,   , can be expanded in terms of 3-dimensional Bloch vectors,   and   , as mentioned in [31,40] as follows where   ,   , and   are defined as so that   =  where   = 0 for separable states and   = 1 for maximally entangled states, which is an extension from the qubit case to qutrit [31].Explicitly,   for a pure state of 2-qutrits has a value of If  3 = 0 and  3 = 0, then   will reduce to the concurrence of 2-qubits but with a different factor.The difference in factors is due to the difference in the definition of the qutrit Bloch vector   = √3 2 (    ⨂), compared to the qubit Bloch vector   = (    ⨂).To understand these coefficients, a general formula in higher dimensions will be formulated.Equation (20) can be extended to   ×   dimensions by writing   as follows where The Bloch vectors in equation ( 22) satisfy   =  22) with a pure state   is provided in the appendix.Furthermore, by introducing the "diagonal determinant" function as follows the concurrence of a pure state of two qudits   in   ×   dimensions becomes where  = min(  ,   ), and || = { ||,  A = min(  ,   ) ||,   = min(  ,   ) .Evaluating concurrence using vectors in higher dimensions will produce different factors compared to equation (13).Since any state must be Hermitian-positive definite with (  ) = 1; decomposition in equation ( 21) applies to all density matrix   , both pure and mixed.However, the concurrence in equation (24) does not hold for mixed states.The concurrence for mixed states will be given in convex roof extension in subsection 3.5.= .In the more specific case, for a pure state where the dimensions of the two qudits are equal,   =   = , we obtain || 2 = || 2 , and  = .Thus, equation ( 24) can be written as   = √1 − || 2 = √1 − || 2 .Therefore, by substituting the respective components of the Bloch vector   or   , we obtain where For example, the concurrence   has been calculated for a 3 × 3 (qutrit-qutrit) system yielding result in equation (20), and for a 2 × 2 (qubit-qubit) system resulting in equation (4).
It is important to note that while Bloch vectors uniquely defined for 1-partite states, they are not unique for 2-partite states.For example, Bell states have the same Bloch vector, which is the zero vector, and are equivalent to   = 1 (maximally entangled).When considering 2-partite systems, the Bloch vector only indicates the magnitude of concurrence (entanglement) between the two parties.

Pure state qudit concurrence as a function of 𝑡𝑟(𝜌 𝐴
2 ),   ((  2)) The concurrence of a pure state system in   ×   dimensions can be expressed as a function of (  2 ), as follows [32] where   is a positive constant, with   = 1 for qubit.
For the pure state   of a   ×   system we obtain Thus, the concurrence becomes where  = min(  ,   ).For mixed states, the convex roof extension in subsection 3.5 is used.

Mixed state qudit concurrence, 𝐶 𝐴𝐵 𝑚𝑖𝑥
A mixed state   can be decomposed into , where normalized pure state |  ⟩  represents -th term of pure state decomposition of   .Thus, the concurrence for a mixed state qubit case is given by the convex roof extension in [41,42] and extended to qudit case as follows where concurrence of pure state |  ⟩  is given by ).

Concurrence and monogamy relation
Based on the formulation of concurrence that can be used for all dimensions, as provided by equations ( 13), ( 24), (32), and ( 42), the concurrence expressed in (42) has been proven by [43] to satisfy the property of monotonicity [3,[44][45], thus making it a valid measure of entanglement.In this paper, after establishing the monotonic property of concurrence, an attempt will be made to explore its monogamous property, whether it satisfies monogamy inequalities or not.
Take the partial trace of   , to obtain   =     ; iii.
Identify whether   is a pure state or a mixed state; iv.
If   is a pure state, use equation ( 13); v.
If   is a mixed state, use the convex roof extension in equation ( 43), then for each pure state decomposition, apply equation (13).The partitioning of the subsystems affects the value of  used in equation (13).For example, in the case of 3 × 2 × 2, the partition | leads to   = 3 and   = 4. Thus,  = min(  ,   ) = 3.Similarly, for other partitioning like | and | leads to  = 2.
Monogamy inequality for pure state   is derived by constructing   and   and then summing   and   and transforming the right-hand side into  | 2 .Using equation (14) gives where  = min(  ,   ) and  ̃ is defined in equation (15), and by the same method we obtain inequality for   2 .Therefore where  ′ = min(  ,   ).Meanwhile, Therefore, by symmetry we obtain where  ′′ = min(  ,   are referred to minimum of average over all decomposition of   ,   , and   , respectively. Inequality (48) will reduced to inequality (1) with squared concurrence as the entanglement measure for   =   =   = 2. Inequality (48) employing qudit concurrence in equation ( 13) is not satisfied by two counterexamples [28,29], and generally, it is not fulfilled for high dimensions.To address this issue, a modification of the extended monogamy inequality (48) is required.One specific modification (albeit trivial), namely by using for  ≥ 2, results in The choice of  is very trivial and requires further investigation.Suppose  is chosen as  = max(  ,   ,   ), we obtain equation (52) satisfied by counterexamples [28,29].For counterexample [28], with |  ⟩ =   = √8/9,   = √8/9, and   = 1/3.Therefore, both of the counterexamples satisfy equation (52), but not equation (48).Another possible approach to address the validity issue of the extended monogamy inequality (48) is to employ tightening monogamy [46] or polygamy inequalities program [47], thus altering the utilized measure (from squared concurrence to α-concurrence) or introducing a new measure such as concurrence of assistance, respectively.Furthermore, when considering qutrits, with states with cyclic permutations (for example |⟩ = In principle, monogamous relations through other definitions of concurrence and entanglement measures can be obtained, and further investigation is needed to ascertain whether these monogamous relations are equivalent to equation (48).Similarly, monogamous relations for multipartite scenarios can, in principle, be derived using methods as in [27].Therefore, both of these topics require further investigation.

Conclusions
In conclusion, the study of concurrence in quantum systems has provided valuable insights and challenges when extended to higher dimensions.The established formulas for qubit concurrence, which values constrained within the range [0, 1], serve as a foundational framework.To address issues related to concurrence range violations [34] and monogamy inequality in higher dimensions [28][29], we have However, extending monogamous relations to higher dimensions cannot be achieved straightforwardly in the same manner as establishing qubit monogamous relations.Simply applying the usual method does not result in extended monogamy inequalities.The extension itself produces inequalities that, in general, cannot be satisfied in higher dimensions.To make it applicable in higher dimensions, a modification of the extended monogamy inequality (48) is necessary, such as employing  = max(  ,   ,   ) in equation ( 52).Another potential solution to address the validity concern of the extended monogamy inequality (48) is to apply either tightening monogamy (by modifying squared concurrence to -concurrence) [46] or polygamy inequalities program (by introducing concurrence of assistance) [47].
The extension of monogamous relations for qudits using other concurrence definitions and entanglement measures is a promising avenue for future research, with the potential to reveal further connections.Additionally, exploring the concept of monogamy in multi-partite scenarios following existing works [27] is an exciting prospect.In the realm of qutrits, intriguing patterns have emerged in states involving cyclic permutations, warranting further investigation into such states and other combinations in higher dimensions.Furthermore, in the discussion of concurrence as a function of the Bloch vector, it has also been demonstrated that every density matrix   can be decomposed as in equation ( 21) with the associated Bloch vector provided in equation (22).Overall, the study of concurrence continues to be a rich field of research, offering both challenges and opportunities for a deeper understanding of entanglement in quantum systems.