Partial Recovery of Coherence Loss in Coherence-Assisted Transformation

Coherence-assisted transformation under incoherent operations is discussed. For transformation from the pure state to the mixed state, we show that the coherence loss can be partially recovered by adding auxiliary coherent states. First, we discuss the coherence-assisted transformation for qubit states and give the sufficient and necessary condition for the partial recovery of coherence loss, and the maximum of the recovery of coherence loss is also studied in this case. Second, the maximally coherent state can be obtained in the above recovery scheme, so we give the full characterization of obtaining the maximally coherent state in a qubit system. Finally, we show that the coherence-assisted transformation for arbitrary finite-dimensional main coherent states and low-dimensional auxiliary coherent states is always possible, and the coherence loss also can be partially recovered in these cases.

The resource theory of quantum coherence was first proposed by Baumgratz et al., and they established a rigorous framework of quantifying coherence [29].In the resource theory of quantum coherence, due to the interaction with the environment, the decoherence phenomenon occurs.Singh et al. proved that quantum chaos and diminishing of information about the mixed initial state favors the generation of quantum coherence through unitary evolution [30].Kurashvili et al. proved that nonunitary evolution leads to the generation of quantum coherence in some cases [31].From another point of view, according to the golden rule of quantum resource theory, the coherence of the state does not increase under free operations.Therefore, in this paper, we study the partial recovery of coherence loss in state transformations under free operations.So far, the state transformation problem for two pure coherent states has been studied extensively, Du et al. proposed the sufficient and necessary condition for the transformation from a pure coherent state to another pure coherent state under incoherent operations [32].Thus, we can determine whether the above state transformation can be realized.Suppose the above state transformation can be realized; it follows that the coherence of the state does not increase under incoherent operations: that is, the coherence loss in the state transformation is inevitable.For this case, Xing proposed a recovery scheme that adds an auxiliary system to the original system such that the coherence loss can be partially recovered in pure state transformations [33].The process of adding auxiliary systems to the original system and performing a joint incoherent operation on the two systems is called coherence-assisted transformation.When the coherence of the auxiliary coherent state increases, the whole process recovers from coherence loss.However, due to the existence of noise, most states are mixed states in practice.Therefore, in this article, we discuss the recovery of coherence loss from a pure state to a mixed state.
In this paper, we discuss coherence-assisted transformation under incoherent operations.For comparable main coherent states, i.e., a pure state that can be transformed to a mixed state under incoherent operations, we show that the coherence loss can be partially recovered by adding auxiliary coherent states.First, we consider the simplest coherence-assisted transformation, i.e., both the main coherent states and auxiliary coherent states are qubit states.We give the sufficient and necessary condition for the coherence loss that can be partially recovered in a coherence-assisted transformation.Thus, we can find all of the auxiliary coherent states that satisfy the above conditions.Moreover, for given main coherent states satisfying the condition in Proposition 1, we give the concrete auxiliary coherent state that can obtain the maximum of the recovery of coherence loss.Second, as a direct application of the above recovery scheme, we give the full characterization of obtaining a maximally coherent state in a qubit system.Finally, we show that if the arbitrary finite-dimensional main coherent states satisfy a strictly majorization relation, there exist two-dimensional auxiliary coherent states that can realize the above recovery scheme.
The paper is organized as follows.In Section 2, we introduce the preliminary knowledge of quantum coherence, including incoherent states, incoherent operations, the relative entropy of coherence and the necessary and sufficient condition for the transformation from a pure coherent state to a mixed coherent state under incoherent operations.In Section 3, we discuss the coherence-assisted transformation for qubit states and obtain some interesting conclusions.In Section 4, we give the full characterization for obtaining a two-dimensional maximally coherent state in the above recovery scheme.In Section 5, we show that the recovery for arbitrary finite-dimensional main coherent states and low-dimensional auxiliary coherent states is always possible.

Preliminary
In the resource theory of quantum coherence, we first need to understand incoherent states and incoherent operations [29].Let {|i } d i=1 be a fixed orthonormal basis in ddimensional Hilbert space; if the density matrix is diagonal in the basis, then the diagonal density matrix is called an incoherent state.The set of all incoherent states is denoted by I for any δ ∈ I, which can be written as δ = ∑ d i=1 δ i |i i|.Otherwise, it is called a coherent state.Incoherent operations (IOs) are defined as the set of completely positive and trace-preserving maps for which the Kraus operators {K l } take incoherent states to incoherent states, i.e., K l ρK † l /tr K l ρK † l ∈ I for all ρ ∈ I , where ∑ l K † l K l = I.In order to quantify the coherence of a quantum state, we need to choose a proper coherence measure.A proper coherence measure needs to satisfy four conditions [29], and one of the conditions is monotonicity, i.e., the coherence of the quantum state does not increase under incoherent operations.Here, we adopt the relative entropy of coherence to explain the corresponding results because it is easy to calculate.Notice that the results in the paper are also equally applicable to other proper coherence measures.The relative entropy of coherence is equal to the distillable coherence [34] and can be interpreted as the minimal amount of noise required for fully decohering a state [35]; it is defined as C r (ρ) = S ρ diag − S(ρ), where S(ρ) = −tr(ρ log ρ) is the von Neumann entropy of the quantum state, and ρ diag is the diagonal part of ρ.If the non-zero eigenvalues of ρ are {λ x } r x=1 , r = rank(ρ), the von Neumann entropy of the quantum state can be written as S(ρ) = − ∑ r x=1 λ x log λ x .Moreover, the coherence measure of the pure state is easy to calculate.Notice that if ρ is a pure state, C r (ρ) = S ρ diag .

Coherence-Assisted Transformation for Qubit States
In this part, we show that the coherence loss from a pure state to a mixed state can be partially recovered in coherence-assisted transformation.Coherence-assisted transformation [33] is the process of adding an auxiliary system to the ordinary coherence transformation.More specifically, during the state transformation for comparable main coherent states, we add auxiliary coherent states such that the whole transformation can still be realized under joint incoherent operations.Here, the initial and final auxiliary coherent states are different; such a transformation is called a coherence-assisted transformation.In a coherence-assisted transformation, when the coherence of the auxiliary coherent state increases, the coherence loss can be partially recovered.The following is a detailed description of the recovery scheme.
Suppose a pure coherent state ψ can be transformed to a mixed coherent state σ under incoherent operations, i.e., ψ IO − → σ.We add a pure auxiliary coherent state ω 1 and perform a joint incoherent operation on the two particles ψ and ω 1 such that the coherence- Then, the coherence loss can be partially recovered, and the recovered coherence loss is We can see that the coherence of the auxiliary coherent state is increased-that is, the reduced coherence of the initial main coherent state can be partially transformed to the final auxiliary coherent state-then, the coherence loss can be partially recovered, and the recovered coherence loss is ∆ = C r (ω 2 ) − C r (ω 1 ).This is due to the fact that the relative entropy of coherence satisfies additivity, i.e, C r (ρ ⊗ σ) = C r (ρ) + C r (σ) [34].
The core of the coherence-assisted transformation ψ ⊗ ω 1 IO − → σ ⊗ ω 2 is that we need to perform joint incoherent operations.In the following, we give the specific incoherent operations that are implemented.For the main coherent states ψ = |ψ ψ| and σ = ∑ m l=1 q l |φ l φ l | of dimension d 1 , auxiliary coherent states φ l ⊗ω 2 ; the proof of the result is similar to that found in the literature [36].First, according to the majorization relation satisfied on the right, we define an intermediate pure state then, there exists an incoherent operation Φ 1 such that Φ 1 (ψ ⊗ ω 1 ) = η ⊗ ω 2 .Next, for any 1 l m, define It is easy to check that the map Last, it is obvious that the composition of any two incoherent operations is still an incoherent operation, so Φ = Φ 2 • Φ 1 is an incoherent operation, and Let us give a concrete example: the following example makes this phenomenon of partial recovery of coherence loss in a coherence-assisted transformation more intuitive.

Example 1. Consider the states with the following form
It is easy to see that the squared coefficients of the above states satisfy the majorization relation j=1 q j λ φ j , i.e., (0.63, 0.37) ≺ (0.72, 0.28), so we can obtain ψ IO − → σ.At the same time, there exist auxiliary coherent states j=1 q j λ φ j ⊗ω 2 , i.e., (0.4032, 0.2368, 0.2268, 0.1332) ≺ (0.4176, 0.3024, 0.1624, 0.1176), so we can obtain ψ ⊗ ω 1 In above coherence-assisted transformation, let ω 1 be a given auxiliary coherent state; we can see that the choice of auxiliary coherent state ω 2 is not unique.As in Example 1, there exists another auxiliary coherent state . A natural question is how to find all auxiliary coherent states ω 2 , i.e., for given states (ψ, σ, ω 1 ) that satisfy ψ ).Let us start with the simplest case, in which the main coherent states and the auxiliary coherent states are both qubit states (d = 2).Let σ = q|φ 1 φ 1 | + (1 − q)|φ 2 φ 2 | be a pure state decomposition of σ; notice that we only consider mixed state σ of rank-2 in this part.Then, the main coherent states ψ = |ψ ψ| and σ have the following pure state decomposition Here the squared coefficients are arranged in non-increasing order, which mean 1 2 α 1, 1 2 β 1 1 and 1 2 β 2 1.In order to give our main result, we first give the following lemma.
First, we can obtain the squared coefficients of |ψ ⊗ |ω 1 , |φ 1 ⊗ |ω 2 and |φ 2 ⊗ |ω 2 : In order to find the three inequalities that satisfy the majorization relation λ ψ⊗ω 1 ≺ ∑ 2 j=1 q j λ φ j ⊗ω 2notice that the fourth equality is trivial-we need to sort the elements in Equation ( 4) in decreasing order and denote its elements by a (1)  a (2) a (3) a (4) , b (2) Second, we can obtain the first and third inequalities of the majorization relation a (1) qb Third, we need to determine the next-largest element of B 1 , B 2 in Equation ( 4).If Then, the following four cases can determine the second-and the third-largest elements in B 1 , B 2 : (i) After finding all the possibilities for B 1 and B 2 , we can calculate ∑ l i=1 b (i) 2 for all l = 1, . . ., 4.
The condition α β is implied in Equation (2).In this case we have Since α β, we have . At the same time, we have c < β 2 .Otherwise c β 2 > qd + (1 − q)β 2 ; this is in contradiction with the above second inequality (c qd + (1 − q)β 2 ).Combining the two conditions c < β 2 and β 1 < β < β 2 , we can divide the system into two parts: c β and β < c < β 2 .Similarly, for c β, we have . This means that the above inequality can be simplified.In this case, we have Same as in case (iii), in this case we have Combining Equations ( 7)-( 9), we obtain the second inequality of majorization relation To sum up, combining Equations (3), ( 5), ( 6) and ( 10), we can obtain Thus, the proof of the proposition is completed.
Notice that the condition c < max{β 1 , β 2 } of Proposition 1 is crucial in the recovery scheme.The condition indicates that if we want to recover the coherence loss in the above state transformation, the initial auxiliary coherent state ω 1 must have enough coherence.
Proposition 1 tells us that the partial recovery of coherence loss in qubit state transformation is always possible by choosing appropriate qubit auxiliary coherent states.At the same time, Proposition 1 is constructive: it provides us a way to find all auxiliary coherent states in the above recovery scheme.Here, we give a specific example to explain the effectiveness of the above proposition.As in Example 1, let 1  2 , αc β = max{0.5,0.56} = 0.56, so we find all auxiliary coherent , where 0.56 d < 0.64.
For given states (ψ, σ, ω 1 ) that satisfy ψ IO − → σ, we find all auxiliary coherent states ω 2 that can recover the coherence loss in a qubit system.We ask what kind of ω 2 can maximize the recovery of coherence loss.Since ω 1 is a given state, we have that C r (ω 1 ) is a fixed constant and denote it by m, so where ∆ is a decreasing function in 1 2 d 1.That is, in order to obtain the maximum recovery of coherence loss, we only need to find the smallest d among all the possibilities of ω 2 .As in Example 1, we obtain 0.56 d < 0.64; when we choose the final auxiliary coherent state as |ω 2 = √ 0.56|0 + √ 0.44|1 , we can obtain the maximum recovery of coherence loss ∆ = C r (ω 2 ) − C r (ω 1 ) ≈ 0.99 − 0.94 = 0.05.Now we ask what happens if ω 1 is not a given state, i.e., for given states (ψ, σ) that satisfy ψ
Proof.The condition ψ By Lemma 2, the proof of the proposition can be reduced to finding the condition that satisfies majorization relation λ ψ⊗ω ≺ ∑ r j=1 q j λ φ j ⊗Φ 2 with assumption (11).First, we can obtain the squared coefficients of |ψ ⊗ |ω , Notice that for |φ j ⊗ |Φ 2 , there only exist two different elements 1 2 β j and 1 2 1 − β j , where j , j = 1, . . ., r.Similarly, we need to sort the elements in A in decreasing order and denote its elements by a (1) a (2) a (3) a (4) .It is obvious that a (1) = αc, a (4) Second, we can obtain the first and third inequalities of the majorization relation Finally, we still need to calculate ∑ l i=1 a (i) to obtain the second inequality of the majorization relation.In Equation ( 12), a (1) ; by Lemma 3, we have a (1) + a (2) = max{α, c}.Then the second inequality of the majorization relation can be written as max{α, c} ∑ r j=1 q j β j = β.The condition α β is implied in Equation (11).In this case, we have To sum up, combining Equations ( 13)-( 15) and Thus, the proof of the proposition is completed.
The above proposition gives us a new way to obtain the maximally coherent state.
For the transformation ψ IO − → σ, as long as the parameter c of the auxiliary coherent state satisfies 1  2 ≤ c β 2α , we can prepare the maximally coherent state under joint incoherent operations.

Coherence-Assisted Transformation for Arbitrary Finite-Dimensional Main Coherent States and Low-Dimensional Auxiliary Coherent States
In Section 3, we obtain that the partial recovery of coherence loss in two-dimensional state transformations is always possible by choosing appropriate two-dimensional auxiliary coherent states.In fact, we can show that the coherence-assisted transformation for arbitrary finite dimensional main coherent states and two-dimensional auxiliary coherent states is always possible, and the coherence loss can be partially recovered simultaneously.Specifically, for given main coherent states ψ = |ψ ψ| and σ = ∑ r j=1 q j |φ j φ j | that satisfy ψ β ji |i , j = 1, . . ., r, let λ ψ = (α 1 , . . . ,α n ), λ φ j = φ j1 , . . ., φ jn ; then, the squared coefficients of the above states satisfy the majorization relation λ ψ ≺ ∑ r j=1 q j λ φ j .If all inequalities in above majorization relation

Discussion
In this paper, for the transformation from a pure state to a mixed state under incoherent operations, we add auxiliary coherent states such that the transformation can still be realized under joint incoherent operations; such a process is called a 'coherence-assisted transformation'.When the coherence of an auxiliary coherent state increases, the reduced coherence of the initial main coherent state can be partially transformed to the final auxiliary coherent state, so the coherence loss can be partially recovered.We first discuss the coherence-assisted transformation for qubit states and give the sufficient and necessary condition for the partial recovery of coherence loss.The maximum of the recovery of coherence loss is also studied in this case.We also give the sufficient and necessary condition for obtaining the maximally coherent state in a qubit system.If the parameter of the initial auxiliary coherent state satisfies a certain condition, we can obtain a twodimensional maximally coherent state.Furthermore, the coherence-assisted transformation for qubit states can be extended to the general case, i.e., arbitrary finite-dimensional main coherent states and low-dimensional auxiliary coherent states.In this case, we show that if the main coherent states satisfy a strictly majorization relation, there exist two-dimensional auxiliary coherent states that can realize the above recovery scheme.At the same, there are some open questions: What is the relation between the dimensionality of auxiliary coherent states and the amount of coherence recovery?For the transformation between two mixed states, what is the condition for the partial recovery of coherence loss?We hope that the results presented in this paper contribute to a better understanding of the resource theory of quantum coherence.

Figure 3 .
Figure 3.The amount of the recovery of coherence loss in Example 3.