Transformations of superpositions by means of incoherent operations

In this paper we study how the coherence of a superposition of pure states is related with the coherence of its components. We consider two pure initial states and two pure final coherent states, such that the former ones cannot be transformed into the latter ones by means of incoherent transformations. In this situation, we analyze conditions for the existence of superpositions of the initial states that can be transformed into superpositions of the final states. In particular, we consider superpositions formed by quantum states belonging to orthogonal subspaces. By appealing to the majorization theory, we obtain necessary and sufficient conditions for such transformations to be possible. Finally, we provide some examples that illustrate the difference between the obtained conditions and the necessary criterion based on the relative entropy of coherence.

1 is the incoherent basis 1 . In this way, incoherent operations map incoherent states into incoherent states. We will use the notation IO ⟩ ⟩ |Ψ → |Φ to mean that the state |Ψ〉 can be transformed into the state |Φ〉 using an incoherent operation.
It has been shown that the necessary and sufficient conditions for the transformation |Ψ〉 → |Φ〉 , 1}, with the symbol ↓ indicating that the components of the probability vectors are sorted in a decreasing order (see, e.g., ref. 8 for an introduction to majorization theory and refs. 9,10 for a comprehensive review about its applications on quantum information). Notice that relation (1) can be seen as the analogous of the celebrated Nielsen's theorem 11 for quantum coherence.
Since coherence is a consequence of the superposition principle, it is important to understand how the coherence of a superposition of coherent states is related with the coherence of the superposed states. Some progress has been made in this direction. For example, for a given superposition of two coherent states | 〉 a and | 〉 b of the form α α |Ψ〉 = | 〉 + | 〉 a b 1 2 , it has been recently investigated the relation among the coherence of the superposition |Ψ〉 C( ) and the coherence of the superposed states | 〉 C a ( ) and | 〉 C b ( ) 12 . In particular, lower and upper bounds of |Ψ⟩ C( ) in terms of | 〉 C a ( ) and | 〉 C b ( ) for several measures of quantum coherence, like the relative entropy of coherence and the  1 -norm of coherence, have been recently obtained in refs. [12][13][14] . ) be two arbitrary superpositions. We want to characterize the values of α 1 , α 2 , β 1 and β 2 for which the transformation |Ψ〉 → |Φ〉 IO is possible. Let Ψ p( ) and Φ p( ) be the probability vectors associated with the superpositions |Ψ〉 and |Φ〉 in the incoherent basis, respectively. According to relation (1), the transformation |Ψ〉 → |Φ〉 IO is possible if and only if the probability vectors Ψ p( ) and Φ p( ) satisfy the majorization relation, i.e.,  Φ Ψ p p ( ) ( ). Let us confine our problem to the case of superpositions of quantum states that belong to orthogonal subspaces. This case is particularly interesting because quantum states from orthogonal subspaces play an important role in quantum information and encoding 15 . More precisely, we consider 1 . Notice that this situation was considered when studying the coherence of superpositions in ref. 12 , and also in the case of entanglement of superposition of bipartite systems in refs. 16,17 . As we are interested in the probability vectors Ψ p( ) and Φ p( ), and the states | 〉 a , | 〉 b and | 〉 c , | 〉 d belong to orthogonal subspaces, we can assume without loss of generality that all coefficients are positive or zero, i.e., ≥ a b c d , , , 0 For the same reason, we can also assume that α α = , therefore the initial and final superpositions have the form Under these constrains, the first non-trivial case for which there are pure states satisfying (2) and (3) appears when both states |c〉 and |d〉 have two non-null coefficients in the incoherent basis. Because, otherwise, |c〉 and |d〉 are incoherent states and conditions (2) and (3) cannot be satisfied. For this to be possible the dimension of the Hilbert space has to be more than 3. We are going to restrict our analysis to this first case, i.e., we consider a Hilbert space of dimension equal to 4. For the states |a〉 and |b〉 there are only two possible situations: (1) they have three and one non-null coefficients in the incoherent basis, or (2) they have two non-null coefficients in the incoherent basis.
In case (1), we can assume without loss of generality that | 〉 = | 〉 a 0 and | = | . Let us denote this case as . The states |c〉 and |d〉 have two possible options: . However, the case (2.ii) can be transformed into the case (2.i) by means of the simple incoherent operation | 〉 ↔ | 〉 1 2 IO . Therefore, we only have to consider the case (2.i). Let us denote this case as → (2, 2) (2, 2) IO . For these two cases we will obtain necessary and sufficient conditions for coefficients α and β in order to allow the transformation under study. Case: ). Notice that we can sort the coefficients in this way without loss of generality, since if this is not the case, we can perform a simple Scientific RepoRtS | (2020) 10:8245 | https://doi.org/10.1038/s41598-020-63661-w www.nature.com/scientificreports www.nature.com/scientificreports/ permutation (which is an incoherent operation) in order to obtain the expected order. For instance, let us assume that ≤ c d, then applying the permutations | 〉 ↔ | 〉 0 2 and | 〉 ↔ | 〉 1 3, one obtains Clearly, since the state |a〉 is an incoherent state, it satisfies the condition (2). On the other hand, the coherent state |b〉 satisfies the condition (3) if and only if > b c 1 . Therefore, the coefficients have to satisfy the following inequalities In this case, the initial and final superpositions have the following form Therefore, the probability vectors in the incoherent basis associated with the superpositions |Ψ〉 and |Φ〉 are In addition, we assume that α ≥ 1/2, which implies that p(Ψ) is already sorted in a decreasing order. Regarding the vector p(Φ), there are in principle six different ways of sorting its components. However, we only consider the case in which p(Φ) is already sorted. This restriction is equivalent to consider β ≥ − +  Under these constraints we obtain the following proposition:  (2) and (3).
Taking into account the previous scenario, we have the following results: In this case, the coherent states | 〉 a , | 〉 b , | 〉 c , | 〉 d have the following form with ≥ > a b 0, ≥ > c d 0 and ≥ a b c d , , , 1/2. As in the previous case, we can sort the coefficients in this way without loss of generality.
In order to satisfy the conditions (2) and (3), the coefficients a b c d , , , have to satisfy the following constraints: > a c d max{ , } and > b c d max{ , }. Therefore, the coefficients have to satisfy In this case, the initial and final superpositions have the following form and Therefore, the probability vectors in the incoherent basis associated with the superpositions |Ψ〉 and |Φ〉 are In addition, we assume that p(Ψ) and p(Φ) are already sorted in a decreasing order, which is equivalent to ) . In this case, the majorization condition Under these constraints we obtain the following proposition: (2) and (3).

be coherent states as in Eqs. (18)-(21) that satisfy the conditions
be arbitrar y super positions, such that .
With respect to the scenario described above, we obtain the following results: comparison with the relative entropy of coherence criterion. A necessary condition for the transformation |Ψ〉 → |Φ〉 IO can be obtained using the relative entropy of coherence criterion: IO re re ⟩ ⟩ ⟹ ⟩ ⟩ with ρ C ( ) re the relative entropy of coherence, which in the case of pure states reduces to is the Shannon entropy of the probability vector associated with the state |Ψ〉 in the incoherent basis. More generally, we can use any coherence monotone measure to formulate alternative necessary criteria (see, e.g., ref. 3 ).
In Fig. 1(a), we consider particular pure states | 〉 a , | 〉 b , | 〉 c and | 〉 d as in Eqs. c 0 8 and = . d 0 79822. These quantum states satisfy the inequalities given in (9). For this case, we plot the difference between the relative entropy of coherence of the superpositions |Ψ〉 and |Φ〉, i.e., , for β . ≤ ≤ 0 8 1 and α . ≤ ≤ 0 5 1. In addition, we plot the region of α and β where the transition |Ψ〉 → |Φ〉 IO is allowed, according to the results of Proposition 1. In Fig. 1(b), we consider another set of pure

Discussion
Quantum coherence is not only a fundamental notion of quantum mechanics, but also a useful quantum resource used in quantum information processing. Since coherence is a consequence of the superposition principle, it is relevant to understand how the coherence of a superposition of coherent states is related with the coherence of the superposed states.
In this work, we have considered two pure initial states and two pure final coherent states, such that the former ones cannot be transformed into the latter ones by means of incoherent transformations. In this situation, we have analyzed the conditions for the existence of superpositions of the initial states that can be transformed into superpositions of the final states. In particular, we have considered superpositions of quantum states belonging to orthogonal subspaces. By appealing to the majorization theory, we have obtained the necessary and sufficient conditions so that the transformations under consideration are possible.
For the initial superposition state we have considered two cases: (1) The initial states is a superposition of two states, each one with two non-null coefficients in the fixed basis. (2) The initial states is a superposition of two states, with one and three non-null coefficients in the fixed basis, respectively. For the final state, we have considered a superposition of two coherent states, each one with two non-null coefficients in the fixed basis. In Propositions 1 and 2, we have obtained necessary and sufficient conditions for such transformations to be possible.
Finally, we have provided two examples that illustrate the difference between the conditions obtained in Propositions 1 and 2, and the necessary criterion based on the relative entropy of coherence.