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Complete complementarity relations for multipartite pure states

Complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For a complete incoherent state, it is known that wave and particle quantifiers can reach zero, and hence no information about the wave and particle aspects of the system can be obtained. This means that the information is being shared with another systems, and these correlations can be seen as responsible for the loss of purity of the quanton. In this paper, by exploring the purity of bi- and tri-partite pure quantum states, we show that it is possible to obtain complete complementarity relations. This procedure allows us to create a general framework for obtaining complete complementarity relations for a subsystem that belongs to an arbitrary multi-partite quantum system in a pure state. Besides, by some simple examples, we show that if the predictability measure is changed then the correlation measure must also be changed in order to obtain complete complementarity relations for pure cases.


I. INTRODUCTION
One of the most intriguing aspects of quantum mechanics is the wave-particle duality. This aspect is generally captured, in a qualitative way, by Bohr's complementarity principle [1]. It states that quantons [2] have characteristics that are equally real, but mutually exclusive. The wave-particle duality is the best known example of this principle, where, in a two-way interferometer, such as the Mach-Zehnder interferometer or the double-slit interferometer, the wave aspect is characterized by interference fringes, meanwhile the particle nature is given by the which-way information of the path along the interferometer, so that the complete knowledge of the path destroys the interference pattern and vice-versa. The first quantitative version of the wave-particle duality was explored by Wootters and Zurek [3], when they investigated interferometers in which one obtains incomplete which-way information by introducing pathdetecting device, and showed that a sharply interference pattern can still be retained. Later, this work was extended by Englert, who derived a wave-particle duality relation [4]. Also, using a different line of reasoning, Greenberger and Yasin [5], considering a two-beam interferometer, in which the intensity of each beam was not necessarily the same, defined a measure of path information, called predictability. Hence, if the quantum system passing through the beam-splitter have different probability of getting reflected in the two paths, one could predict the path information of the quantum system. This kind of reasoning was followed by Jaeger, Shimony and Vaidmann [6], and can be summarized by a simple complementarity relation where P is the predictability and V is the visibility of the interference pattern. It is worthwhile noticing that such aspects of a quantum system are not necessarily totally mutually exclusive. An experiment can provide partial information about the wave and particle nature of a quantum system, but the more information it gives about one aspect of the system, the less information the experiment can provide about the other. More recently, several steps have been taken towards the quantification of the wave-particle duality by many authors, such as Dürr's [7] and Englert et al.'s [8], that established criteria for checking the reliability of newly defined predictability measures and interference pattern quantifiers, and extended measures of the wave-particle aspects to discrete d-dimensional quantum systems. As well, with the development of the field of Quantum Information, it was suggested that the quantum coherence [9] would be a good generalization of the visibility measure [10][11][12][13]. Predictability is a measure of the knowledge about the quantum level in which a quanton is to be found. These levels can represent, besides the paths on a Mach-Zehnder interferometer, energy levels of an atom [14] or , more generally, population levels [15]. So far, many lines of reasoning were taken for quantifying the wave-particle properties of a quantum system [16][17][18][19][20]. Complementarity relations like the one in Eq. (1) are saturated only for pure, single-particle, quantum states. For mixed states, the left hand side is always less than one and can even reach zero for a maximally mixed state. Hence no information about the wave and particle aspects of the system can be obtained. As noticed by Jakob and Bergou [21], this lack of knowledge about the system is due to another intriguing quantum feature: entanglement [22,23]. This means that the information is being shared with another system and this kind of quantum correlation can be seen as responsible for the loss of purity of each subsystem such that, for pure maximally entangled states, it is not possible to obtain information about the local properties of the subsystems. As showed by these authors, the Concurrence [24] is recognized as the appropriate quantum correlation measure in a bipartite state of two qubits that completes relation (1). It is worth pointing out that this complete relation for two qubits was claimed to be experimentally confirmed recently [25]. Jakob and Bergou extended this idea for composite bipartite systems of arbitrary dimension [26,27], suggesting that there must exist a complementary relation between the information of the local properties of each subsystem and the entanglement of the composite system, and showed that I-Concurrence [28] is the measure of quantum correlation that completes relation (1) for composite bipartite pure states. Following the same reasoning, Hiesmayr and Huber [29] derived an operational entanglement measure for any multiparty system. However, it's known that entanglement is not the only quantum correlation existing in multipartite quantum systems [30][31][32][33][34]. For example, quantum discord is a type of quantum correlation that describes the incapacity of obtaining information from one interacting subsystem without perturbing it [35]. Quantum coherence in a composite system can be contained either locally or in the correlations between the subsystems. The portion of quantum coherence contained within correlations can be viewed as a kind quantum correlation, called correlated coherence [36,37]. Therefore a natural question one could ask, in the context of complementarity relations, is if entanglement measures are the only quantum correlation measures that complete relations like (1). In this article we show that the answer to this question is negative. By exploring some simple examples, we show that if one changes the predictability measure, one has to change also the correlation measure in order to obtain a complete complementarity relation for pure multipartite states. Also, we'll show that, by exploring the purity of bipartite quantum system, one can obtain a complete complementarity relation equivalent to obtained by [26,27], and a new one if we change the coherence measure. In addition, by exploring the purity of tripartite quantum system, one can obtain a complete complementarity relation with a new measure that quantifies the correlations between the subsystems, where the correlation measure is equivalent to the generalized concurrence for tripartite system obtained by [38], using exterior algebra. This procedure allow one to create a general framework for obtain complete complementarity relations for a subsystem that belongs to an arbitrary multipartite pure quantum system.
We organized the remainder of this article in the following manner. In Sec. II we explore the properties of bipartite pure quantum systems, and, using the relative entropy of coherence, we obtain complementarity relations equivalent to those reported in [26,27] and new ones. Also, exploring the purity of a tripartite quantum system, we obtain a new complete complementarity relation in Sec. III. Next, we obtain complete complementarity relation from the purity of an arbitrary multipartite quantum system in Sec. IV. In Sec. V, we show that in changing the predictability measure, one also has to change the correlation measure in order to obtain a complete complementarity relation for pure states. Finally, we give our conclusions in Sec. VI.

II. COMPLEMENTARITY RELATIONS FOR BIPARTITE PURE QUANTUM SYSTEMS
In this section, we will explore the properties of bipartite pure quantum systems and their respective subsystems A and B of dimension d A and d B , respectively. So, a general state of the system |Ψ A,B can be represented as vector in the composite Hilbert space be a local orthonormal basis for the spaces is a orthonormal basis for representing vectors in H A ⊗ H B . Therefore, an arbitrary state of a bipartite quantum system can be written as a ij |i, j A,B , or, equivalently, by the density operator [40,41] where ρ ij,kl = a ij a * kl . Meanwhile, the states of the subsystem A (B) are obtained by tracing over B (A): In general, the states of the subsystem A and B are not pure, which implies that some information of the subsystems is missing. It's easy to see that by exploring the properties of the density matrix of one of the subsytems. For example, once that A is mixed, we have 1 − Tr ρ 2 A ≥ 0, which implies in that can be written as a complementarity relation obtained in [20]: where is the Hilbert-Schmidt 1 quantum coherence [42]. It's worth noting that the information content missing of the system A is represented by the inequality in equation (5). From now on, we'll sometimes, for convenience, omit the upper limits of the summations. As pointed out by Jakob and Bergou [26,27], actually the missing information is being shared among the correlations with the subsystem B, and we can see that by exploring the purity of the bipartite quantum system, once that can be rewritten as a complete complementarity relation for both subsystems if we define the non-local quantum coherence of A (B), that's being shared with B(A), as The complementarity relations (8) and (9) are equivalent to the relations P 2 for pure states, obtained by Jakob and Bergou in [26,27], where C (n) , for all bipartite pure quantum systems.
1 Even though such measure of quantum coherence isn't a coherence monotone, it's a bona fide measure for the visibility of a quanton [20].
For any quantum state ρ A of dimension d A , the relative entropy of coherence is defined as [9] C re (ρ A ) = min where I is the set of all incoherent states, and S vn ( Once C re (ρ A ) ≤ S vn (ρ Adiag ), it's possible to obtain an incomplete complementarity relation from this inequality: with ii as a measure of the predictability, already defined in [8,20]. Such measure is only possible to define because we can interpret the diagonal elements of ρ A as a probability distribution, which is a consequence of the properties of ρ A [20]. The complementarity relation (14) is incomplete due the presence of correlations. However, ρ A is the subsystem of bipartite pure quantum system |Ψ A,B , which allow us to take S vn (ρ A ) as a measure of entanglement of the subsystem A with B [43]. So, it's possible to interpret Eq. (13) as a complete complementarity relation

III. COMPLEMENTARITY RELATION FOR TRIPARTITE PURE QUANTUM SYSTEMS
Following the same logic as in the previous section, a tripartite pure quantum system can be represented by be local orthonormal bases for the Hilbert spaces H A , H B , H C , respectively, so that represents a tripartite pure quantum system. The subsystem A, for example, is represented by the reduced density operator and similarly for the other subsystems. In general the state of the subsystem A is mixed, so, by exploring the properties of the reduced density matrix of A, one obtains an incomplete complementarity relation, just as before. Once the information content of A is being shared with B and C, it is natural to explore the purity of ρ A,B,C , 1 − Tr ρ 2 A,B,C = 0, which implies that This equation can be recast as , and C nl hs (ρ A|BC ) is the non-local coherence of A, shared with B and C, defined as Since C nl hs (ρ A|BC ) is equal to the linear entropy of A, the maximum of C nl We notice the following identities where, in the first identity we just rewrote the product of the diagonal elements using the fact that Tr ρ A,B,C = 1, while in the second identity we explored the purity of ρ A,B,C . So, which also shows that E M = 2C nl hs (ρ A|BC ), where E M is the generalized concurrence defined in [38] for the special case of a tripartite pure quantum system. Now, if the system A isn't correlated with B or C, then A must be pure, since the impurity of a system is attributed to the correlations with other systems. Hence, if ρ A,B,C = ρ A ⊗ ρ B,C , i.e., if ρ A,B,C is, at least, a bi-separable state, then ρ A is pure and C nl hs (ρ A|BC ) = 0. In addition, is easy to see that C nl hs (ρ A|BC ) ≥ 0 for any tripartite pure state: where it was used the fact that ρ A,B,C is pure, and, for dummy indices that were summed, one can write ρ ijk,ijk ρ lmn,lmn = 1 2 (ρ ijk,ijk ρ lmn,lmn + ρ imn,imn ρ ljk,ljk ).
Finally, complementarity relations like (19) can be obtained for the subsystem B and C exploring the purity of ρ A,B,C . Also, we'll see in the next section that, for some tripartite quantum states, C nl hs (ρ A|BC ) is related to some correlation measures already defined in the literature.
Without loss of generality, let's consider the state of the subsystem A 1 , which is obtained by tracing over the other subsystems, for which the Hilbert-Schmidt quantum coherence and the corresponding predictability measure are given by From these equations an incomplete complementarity relation, The purity condition (39) can be rewritten as a complementarity relation where the non-local quantum coherence of system A 1 , shared with A 2 , ..., A n , is defined as To show that E M = 2C hs (ρ A1|A2,...,An ), where E M is the generalized concurrence defined in [38], and 0 ≤ C hs (ρ A1|A2,...,An ) ≤ (d A1 − 1)/d A1 , where C hs (ρ A1|A2,...,An ) = 0 iff ρ A1,...,An = ρ A1 ⊗ ρ A2,...,An , and hence ρ A1 is pure, it's enough to notice that C hs (ρ A1|A2,...,An ) is equal to the linear entropy of A 1 : According to [36], with respect to the local reference bases , the correlated coherence for a bipartite quantum system is given by C c (ρ A,B ) := C(ρ A,B )−C(ρ A )−C(ρ B ) for some measure of coherence. As showed by the same authors, for the l 1 -norm C c l1 (ρ A,B ) ≥ 0 for any bipartite quantum state. Also, it's worth being pointed which implies that C l1 (ρ A ⊗ ρ B ) = C l1 (ρ A ) + C l1 (ρ B ), while for the relative entropy of coherence the equality is always satisfied, i.e., C c re (ρ A ⊗ρ B ) = 0 [37]. For the Hilbert-Schmidt measure of coherence, it turns out that C c hs (ρ A,B ) can be negative for some cases, for example, by considering a bipartite quantum system such that ρ A,B = ρ A ⊗ σ B , where σ B is an incoherent state. Besides that, if the off-diagonal elements of that reduced density matrices satisfy the following properties it is straightforward to show that C c hs (ρ A,B ) ≥ 0. For multipartite states, this will still be true if the reduced density matrices of the subsystems satisfy properties similar to (51) and (52). All the states considered in this article, except the last one, satisfy these conditions. Now, for a bipartite quantum system in the state |Ψ A,B = x |0, where C

(2)
A,B is the concurrence measure for entanglement [26,27], and P l1 (ρ) := d − 1 − j =k √ ρ jj ρ kk is a measure of predictability obtained in [20]. One can easily see that C (2) A,B = C c l1 (ρ A,B ). In Fig. 1, we plotted the different measures of predictability and correlation for comparison. As expected, the measures of predictability (and of correlation) reach their maximum and minimum values in the same point of the domain, although the maximum (and minimum) values of the function differs from one measure to another. However, the inflection point in the domain is not the same among the measures of predictability (and correlations), what is interesting once that inflection point represents the point in the domain where the predictability and the respectively measure of correlation reach the same value. Another example was given by Jacob and Bergou in [27], where they considered the following state with x ∈ [0, 1]. The predictability and concurrence measures defined in [27] are given by P 2 A,B ] 2 = 4/3. However, if we change the measure of predictability, the measure of correlation also has to be changed if one wants to obtain a complementarity relation that saturates. For P l1 (ρ A ) = P l1 (ρ B ) = 2(1 − x 2 /2 − 2x 2 (1 − x 2 )), the corresponding correlation measure is the l 1norm correlated coherence C c l1 (ρ A,B ) = 2( Another interesting fact is that if we use the non-normalized predictability measure defined by Roy and Qureshi in [19], it is not possible to obtain a complete complementarity relation, neither using the correlated coherence nor with the concurrence. In addition, by writing the state as Ψ = a 00 |0, 0 A,B + a 11 |1, 1 A,B + a 22 |2, 2 A,B such that a 00 = a 11 = x/ √ 2 and a 22 = √ 1 − x 2 , one can see that the off-diagonal elements of the density matrix related to C c l1 (ρ A,B ) and to C A,B are the same, the only difference between these measures is the form of the function: C c l1 (ρ A,B ) = 2(ρ 00,11 + ρ 00,22 + ρ 11,22 ), while C A,B = 2 ρ 2 00,11 + ρ 2 00,22 + ρ 2 11,22 . In this case, one correlation measure can't be written as a function of the other correlation measure. In Fig. 2, we plotted the different measures of predictability and correlation for comparison, including P vn and S vn defined in Sec. II. The analysis is the same as for the last example.

VI. CONCLUSIONS
By exploring the purity of multipartite pure quantum states, we showed that it is possible to obtain complementarity relations that characterize completely a quantum system. For the bipartite case, the complete complementarity relation obtained is equivalent to the complementarity relation proposed by Jakob and Bergou [26,27]. For the tripartite case, we obtained a new complete complementarity relation, with the correlation measure being equivalent to the generalized concurrence obtained in [38]. Also, in the same framework, it was possible to obtain a new complementarity relation, just by reinterpreting the definition of the relative entropy of coherence. Such a procedure of exploring the purity of the density matrix, allowed us to created a general framework to obtain such complementarity relations, and enabled us to generalize these relations for the multipartite case. Through simple examples, we showed that in changing the predictability measure one has to change the correlation measure in order to obtain a complete complementarity relation for pure cases. As quoted by Bohr [48]: "... evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects". However, to fully characterize a quanton, it is not a enough to consider its wave-particle aspect; one has also to regard its correlations with other systems.