Evolution equation for quantum coherence

The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures.

Scientific RepoRts | 6:29260 | DOI: 10.1038/srep29260 into  , Baumgratz et al. 10 presented the defining properties for an information-theoretic coherence measure C: (1) C(ρ) ≥ 0 for all states ρ, and C(δ) = 0 iff δ ∈  . (2) Monotonicity under the actions of Λ , C(ρ) ≥ C(Λ (ρ)). . There are several coherence measures satisfying the above conditions. They are the l 1 norm and relative entropy 10 , the Uhlmann fidelity 12 , the intrinsic randomness 14 , and the robustness of coherence 42 . In this work, we concentrate mainly on the l 1 norm of coherence, which is given by ρ ρ = ∑ ≠ C i j ( ) i j in the basis {|i〉 } i=1,…,d 10 , and will mention other coherence measures if necessary.
FR for quantum coherence. Consider a general d-dimensional state in the Hilbert space , with the density matrix , with f ijk (d ijk ) being the structure constants that are completely antisymmetric (symmetric) in all indices 43,44 . If one arranges = …  X u v { , , , 12 12 , , , } ii . Moreover, the notation i appeared in v jk is the imaginary unit.
For ρ represented as Eq. (1), can be derived as To investigate evolution equation of coherence, we suppose the system S of interest interacts with its environment E, then by considering S and E as a whole for which their evolution is unitary, the reduced density matrix for S is obtained by tracing out the environmental degrees of freedom, . In terms of the master equation description, the equation of motion of ρ can be written in a local-in-time form 31 with  being the Louville super-operator which may be time independent or time dependent. As it has been shown that for any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, one can always construct a linear map which gives ρ (the opposite case may not always be true), and the linear map can be expressed in the Kraus-type representations 45 . If the map  is completely positive and trace preserving (CPTP), then one can explicitly construct the Kraus operators {E μ } such that For convenience of later discussion, we turn to the Heisenberg picture to describe  via the map As an Hermitian operator  on  × d d can always be decom- can be further characterized by the transformation matrix T defined via  . By this classification scheme, different families of states are labeled by different unit vectors n, while states belong to the same family are characterized by a common n, and can be distinguished by different multiplicative factors χ (see Fig. 1). That is to say, ρn represents states with the characteristic vectors x along the same or completely opposite directions but possessing different lengths.
While ρn is fully described by χn, and the action of  on it can be written equivalently as the map: . Then as one can always make Q max ≥ 1 (otherwise, one can normalize it by multiplying a constant to it), we have the following lemma.
. This, together with Eq. (2), requires that all the nondiagonal elements of = ∑ µ µ µ † A E E must be zero.
obeys the FR (9). This corollary means that in addition to the usual completeness condition 31 , the FR (9) further requires ∑ µ µ µ † E E to be diagonal. We denote this kind of channels  F . Clearly, they include the ] as a special case. From a geometric perspective, Theorem 1 indicates that for all states of the same family ρn, namely, states with the characteristic vectors x along the same or opposite directions, their coherence dynamics measured by the l 1 norm can be represented qualitatively by that of the probe state ρp n , as the magnitude of This simplifies greatly the assessment of the decoherence process of an open system. Moreover, the FR (9) provides a strong link between amount of the coherence loss of a system and structures of the applied quantum channels. Particularly, as ρn with the vectors x along the same or opposite directions fulfill the same decoherence law, the approach adopted here may offer a route for better understanding the interplay between geometry of the state space and various aspects of its When ρn traverse a quantum channel  (right), their decoherence process can be described qualitatively by that of ρp n with the unit vector n (the bottommost golden one). quantum features. It might also provides a deeper insight into the effects of gate operation in quantum computing and experimental generation of coherent resources in noisy environments, as  ρ( ) n F can specify the actions of environments, of measurements, or of both on the states ρn.
When some restrictions are imposed on the quantum channels, the FR (9) can be further simplified.

Corollary 2. If a channel  yields
l l 1 1  holds for the family of states Clearly, its evolution is solely determined by the product of the initial coherence and a noise parameter |q(t)|.
There are many quantum channels satisfying the condition of Corollary 2. For instance, the Pauli channel PL  and Gell-Mann channel GM  given in ref. 41, and the generalized amplitude damping channel  GAD 31 . Notably, PL  covers the bit flip, phase flip, bit-phase flip, phase damping, and depolarizing channels which embody typical noisy sources in quantum information, while  GAD covers the structured reservoirs with Lorentzian and Ohmic-type spectral densities.
One can also construct quantum channel G  under the action of which obeys the FR (10) for arbitrary initial state. The Kraus operators describing  G are given by , while q and q 0 are time-dependent noisy parameters. Clearly, G  reduces to the depolarizing channel when q 0 = q.
N-qubit case. A general N-qubit state can be written as here,  σ = 0 2 , and σ 1,2,3 are the usual Pauli matrices, while j k takes the possible values of {0, 1, 2, 3} other than the special case j k = 0 for all k. In the Methods section, we have proved that for every family of the N-qubit states a m i j ij j , with a ij being determined by the transformation between {Y j } and {X i }: = ∑ X a Y i j ij j . This corollary generalizes the FR (9) for the N-qubit states. It shows that coherence of the evolved state under the actions of two cascaded channels F aux   is determined by the product of the coherence for the evolved probe state under the action of F  and the coherence for the generated state by aux  . As every Y j can always be decomposed as linear combinations of the generators {X i }, the above result applies also to the qudit states with d = 2 N . As an explicit example, the transformation between {Y j } and {X i } for N = 2 is given in the Methods section, from which aux  and {a ij } can be constructed directly.
Frozen coherence. By Theorem 1 we can also derive conditions on the quantum channel for which the l 1 norm of coherence is frozen. To elucidate this, we return to Eq. (9), from which one can see that   being orthogonal matrices, i.e.,  = T T ( ) r S T r S 2 , the l 1 norm of coherence for ρn will be frozen during the entire evolution.
The proof is given in Methods. It enables one to construct channels  for which the l 1 norm of coherence is frozen. As an explicit example, we consider the one-qubit case, with  being described by Then by Corollary 4, one can obtain that when ε i0 = ε i3 = 0, and ε ε , with Re(·) and Im(·) representing, respectively, the real and imaginary parts of a number, the l 1 norm of coherence will be frozen. There are a host of {ε ij } that fulfill the requirements, e.g., ε 01 = q(t), , ε k0 = ε k3 = 0, with k ∈ {1, 2, 3}, and q(t) contains the information on 's structure and its coupling with the system. Moreover, for certain special initial states, the freezing condition presented in Corollary 4 may be further relaxed. In fact, for ρn with certain n 2r−1 = 0 (or n 2r = 0), ). For instance, when considering the channel  PL 41 , the l 1 norm of coherence for ρn with n 2 = 0 is frozen during the entire evolution when q 1 = 1 (i.e., the bit flip channel). Similarly, for ρn with n 1 = 0, it is frozen when q 2 = 1 (i.e., the bit-phase flip channel). These are in facts the results obtained in ref. 21. Needless to say, when  = T T ( ) r S T r S 2 , the l 1 norm of coherence is also frozen for ρn with certain n 2r−1 = 0 or n 2r = 0.
Outlook. The FR (9) presented here can be of direct relevance to other issues of quantum theory. For example, the l 1 norm of coherence is a monotone of the entanglement-based coherence measure for one-qubit states 12 . Its logarithmic form ρ dC log [ ( )] l 2 1 is lower bounded by the relative entropy of coherence C r (ρ) which has a clear physical interpretation, while for arbitrary ρ has also been conjectured 46 . Further study shows that ρ C ( ) l 1 also bounds the robustness of coherence, i.e., 42 . It is also connected to the success probability of state discrimination in interference experiments 29 and the negativity of quantumness 21,47 . Thus, our results provide a route for inspecting the interrelations between decay behaviors of coherence, quantumness, and entanglement.
The FR also applies to other related coherence measures, as well as quantum correlations which are relevant to coherence. Some examples are as follows (see Methods section for their proof): (i) the coherence concurrence for one-qubit states 14 , and the trace norm coherence for one-qubit and certain qutrit states 13,46 ; (ii) the genuine quantum coherence (GQC) defined via the Schatten p-norm for all states 48 , which is related to quantum thermodynamics and the resource theory of asymmetry; (iii) the robustness of coherence for the one-qubit states and d-dimensional states with X-shaped density matrix, and its lower bound ρ ρ − ∆( ) 2 2 which is a measure of the GQC for all states 42 ; (iv) the K coherence defined based on the Wigner-Yanase skew information 11 , although it is problematic in the framework of coherence by Baumgratz et al. 49 , it may be a proper measure of the GQC 48 ; (v) the purity of a state which is complementary with quantum coherence 28 ; (vi) the geometric discord 50-54 and measurement-induced nonlocality 55,56 ; (vii) the maximum Bell-inequality violation 57 , and average fidelity of remote state preparation 58 and quantum teleportation 59 . All these manifest the universality of the FR formulated in this paper, and will certainly deepen our understanding of the already rich and appealing subject of quantum channels or the CPTP maps.
Recently, Jing et al. studied quantum speed limits to the rate of change of quantumness measured by the non-commutativity of the algebra of observables 60 . We note that the coherence quantifiers can also be considered as a measure of quantumness, but it is different from the notion of quantumness considered in ref. 60 and references therein, although they both characterize global quantum nature of a state, and are intimately related to quantum correlations such as discord. The coherence monotones characterize quantumness of a single state. It is basis dependent, and vanishes for the diagonal states. The quantumness based on the non-commutativity relations measures the relative quantumness of two states. It is basis independent, and vanishes only for the maximally mixed states. Of course, it is as well crucial to study evolution equation of it in future work.

Discussion
We have established a simple FR for the evolution equation of the l 1 norm of coherence, which is of practical relevance for assessing coherence loss of an open quantum system. For a general d-dimensional state, we determined condition such that this FR holds. The condition can be described as a restriction on the transformation matrix, or on the operator ∑ µ µ µ † E E , of the quantum channel. By introducing an auxiliary channel, we further presented a more general relation which applies to any N-qubit state. With the help of the FR, we have also determined a condition the transformation matrix should satisfy such that the l 1 norm of coherence for a general state is dynamically frozen, and constructed explicitly the desired channels for one-qubit states. Finally, we showed that the FR holds for many other related coherence and quantum correlation measures. We hope these results may help in understanding the interplay between structure of the quantum channel, geometry of the state space, and decoherence of an open system, as well as their combined effects on decay behaviors of various quantum correlations.   (16) and (17)

Proof of Corollar y 3. Suppose aux  is described by the Kraus operators
Then, by employing the anticommutation relation of the Pauli operators σ 1,2,3 , we obtain , and hence completes the proof.
Frozen coherence of one qubit. Suppose the required channel  is described by the Kraus operators , with i ∈ {0, 1, 2, 3}, and the values of  ε ∈ ij should satisfy certain constraints such that the requirement of Corollary 4 is satisfied. First, the completeness condition of the CPTP map, namely, where ε ⁎ ij represents conjugation of ε ij , and the notation i before ε i2 , Re(·), and Im(·) is the imaginary unit. Second, Corollary 4 requires T 10 = T 20 = 0, and T S to be a rectangular block diagonal matrix which corresponds to T 13 = T 23 = 0. This yields  By comparing Eqs (22) and (24), one can note that the equalities are satisfied when . Under these two constraints, Eq. (25) simplifies, respectively, to  T  T  T  T  T T  T T  1, 1, 0, and from Eqs (26) and (27), one can see that the third equality of Eq. (28) is always satisfied, while the first two equalities are equivalent. Therefore, to freeze the l 1 norm of coherence, ε ij should satisfy one of the following two conditions: (i) ε i0 = ε i3 = 0 for i ∈ {0, 1, 2, 3}, and  , the FR also holds as the optimal δ is given by Δ (ρ) 48 . (vii) For two-qubit states, the maximum Bell-inequality violation B max (ρ) 57 , remote state preparation fidelity F rsp (ρ) 58 , and N qt (ρ) which is a monotone of the average teleportation fidelity ρ ρ = + F N ( ) 1/2 ( )/6 qt qt 59 , are given by