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Sci Rep. 2016; 6: 29260.
Published online 2016 July 7. doi:  10.1038/srep29260
PMCID: PMC4935996

Evolution equation for quantum coherence

Abstract

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.

Quantum coherence, an embodiment of the superposition principle of states, lies at the heart of quantum mechanics, and is also a major concern of quantum optics1. Physically, coherence constitutes the essence of quantum correlations (e.g., entanglement2 and quantum discord3) in bipartite and multipartite systems which are indispensable resources for quantum communication and computation tasks. It also finds support in the promising subject of thermodynamics4,5,6,7,8 and quantum biology9.

Clarifying the decoherence mechanism of an noisy system is an important research direction of quantum mechanics. But due to the lack of rigorous coherence measures, studies in this subject were usually limited to the qualitative analysis. Sometimes, coherence behaviors were also analyzed indirectly via various quantum correlation measures3. However, coherence and quantum correlations are in fact different. Very recently, the characterization and quantification of quantum coherence from a mathematically rigorous and physically meaningful perspective has been achieved10. This sets the stage for quantitative analysis of coherence, which were carried out mainly around the identification of various coherence monotones11,12,13,14,15,16 and their calculation17. Some other progresses about coherence quantifiers include their connections with quantum correlations18,19,20, their behaviors in noisy environments21,22, their local and nonlocal creativity23,24, their distillation25,26 and the role they played in the fundamental issue of quantum mechanics27,28,29,30.

One major goal of quantum theory is to find effective ways of maintaining the amount of coherence within a system. The reason is twofold. First, coherence represents a basic feature of quantum states, and underpins all forms of quantum correlations1. Second, coherence itself is a precious resource for many new quantum technologies, but the unavoidable interaction of quantum devices with the environment often decoheres the input states and induces coherence loss, hence damage the superiority of these quantum technologies31.

Looking for general law determining the evolution equation of coherence can facilitate the design of effective coherence preservation schemes. Remarkably, the evolution equations for certain entanglement monotones (or their bounds)32,33,34,35,36,37,38,39,40 and geometric discords41 were found to obey the factorization relation (FR) for specific initial states. Then, it is natural to ask whether there exists similar FR for various coherence monotones. In this work, we aimed at solving this problem. We first classify the general d-dimensional states into different families, and then prove a FR which holds for them. By employing this FR, we further identified condition on the quantum channel for freezing coherence. We also showed that this FR applies to many other coherence and correlation measures. These results are hoped to add another facet to the already rich theory of decoherence, and shed light on revealing the interplay between structures of quantum channel and geometry of the state space, as well as how they determine quantum correlation behaviors of an open system.

Results

Coherence measures

By establishing rigorously the sets An external file that holds a picture, illustration, etc.
Object name is srep29260-m1.jpg of incoherent states which are diagonal in the reference basis {|iright angle bracket}i=1,…,d, and incoherent operations Λ specified by the Kraus operators {El} which map An external file that holds a picture, illustration, etc.
Object name is srep29260-m2.jpg into An external file that holds a picture, illustration, etc.
Object name is srep29260-m3.jpg, 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m4.jpg. (2) Monotonicity under the actions of Λ, C(ρ)  C(Λ(ρ)). (3) Monotonicity under selective incoherent operations on average, i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m5.jpg, where An external file that holds a picture, illustration, etc.
Object name is srep29260-m6.jpg, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m7.jpg is the probability of obtaining the outcome l. (4) Convexity, An external file that holds a picture, illustration, etc.
Object name is srep29260-m8.jpg, with pl  0 and An external file that holds a picture, illustration, etc.
Object name is srep29260-m9.jpg.

There are several coherence measures satisfying the above conditions. They are the l1 norm and relative entropy10, the Uhlmann fidelity12, the intrinsic randomness14, and the robustness of coherence42. In this work, we concentrate mainly on the l1 norm of coherence, which is given by An external file that holds a picture, illustration, etc.
Object name is srep29260-m10.jpg in the basis {|iright angle bracket}i=1,…,d10, and will mention other coherence measures if necessary.

FR for quantum coherence

Consider a general d-dimensional state in the Hilbert space An external file that holds a picture, illustration, etc.
Object name is srep29260-m11.jpg, with the density matrix

An external file that holds a picture, illustration, etc.
Object name is srep29260-m12.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m13.jpg is the d × d identity matrix, An external file that holds a picture, illustration, etc.
Object name is srep29260-m14.jpg, An external file that holds a picture, illustration, etc.
Object name is srep29260-m15.jpg, xi = Tr(ρXi), and Xi [proportional, variant] Ti. Here, {Ti} are generators of the Lie algebra SU(d). They can be represented by the d × d traceless Hermitian matrices which satisfy An external file that holds a picture, illustration, etc.
Object name is srep29260-m16.jpg, with fijk (dijk) being the structure constants that are completely antisymmetric (symmetric) in all indices43,44. If one arranges An external file that holds a picture, illustration, etc.
Object name is srep29260-m17.jpgAn external file that holds a picture, illustration, etc.
Object name is srep29260-m18.jpg, then

An external file that holds a picture, illustration, etc.
Object name is srep29260-m19.jpg

where j, k [set membership] {1, 2, …, d} with j < k, and l [set membership] {1, 2, …, d  1}. Clearly, {Xi} satisfy An external file that holds a picture, illustration, etc.
Object name is srep29260-m20.jpg. Moreover, the notation i appeared in vjk is the imaginary unit.

For ρ represented as Eq. (1), An external file that holds a picture, illustration, etc.
Object name is srep29260-m21.jpg can be derived as

An external file that holds a picture, illustration, etc.
Object name is srep29260-m22.jpg

where d0 = (d2  d)/2, and xl related to wl which is diagonal in the basis {|iright angle bracket}i=1,…,d do not contribute to An external file that holds a picture, illustration, etc.
Object name is srep29260-m23.jpg.

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, An external file that holds a picture, illustration, etc.
Object name is srep29260-m24.jpg. In terms of the master equation description, the equation of motion of ρ can be written in a local-in-time form31

An external file that holds a picture, illustration, etc.
Object name is srep29260-m25.jpg

with An external file that holds a picture, illustration, etc.
Object name is srep29260-m26.jpg 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m27.jpg (the opposite case may not always be true), and the linear map can be expressed in the Kraus-type representations45. If the map An external file that holds a picture, illustration, etc.
Object name is srep29260-m28.jpg is completely positive and trace preserving (CPTP), then one can explicitly construct the Kraus operators {Eμ} such that

An external file that holds a picture, illustration, etc.
Object name is srep29260-m29.jpg

where elements of An external file that holds a picture, illustration, etc.
Object name is srep29260-m30.jpg for An external file that holds a picture, illustration, etc.
Object name is srep29260-m31.jpg are given by An external file that holds a picture, illustration, etc.
Object name is srep29260-m32.jpg.

For convenience of later discussion, we turn to the Heisenberg picture to describe An external file that holds a picture, illustration, etc.
Object name is srep29260-m33.jpg via the map An external file that holds a picture, illustration, etc.
Object name is srep29260-m34.jpg, which gives An external file that holds a picture, illustration, etc.
Object name is srep29260-m35.jpg. As an Hermitian operator An external file that holds a picture, illustration, etc.
Object name is srep29260-m36.jpg on An external file that holds a picture, illustration, etc.
Object name is srep29260-m37.jpg can always be decomposed as An external file that holds a picture, illustration, etc.
Object name is srep29260-m38.jpg An external file that holds a picture, illustration, etc.
Object name is srep29260-m39.jpg, An external file that holds a picture, illustration, etc.
Object name is srep29260-m40.jpg can be further characterized by the transformation matrix T defined via

An external file that holds a picture, illustration, etc.
Object name is srep29260-m41.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m42.jpg, and here we denote by An external file that holds a picture, illustration, etc.
Object name is srep29260-m43.jpg. Clearly, T00 = 1, and T0j = 0 for j  1. This further gives An external file that holds a picture, illustration, etc.
Object name is srep29260-m44.jpg.

To present our central result, we first classify the states ρ into different families: An external file that holds a picture, illustration, etc.
Object name is srep29260-m45.jpg, with

An external file that holds a picture, illustration, etc.
Object name is srep29260-m46.jpg

and An external file that holds a picture, illustration, etc.
Object name is srep29260-m47.jpg is a unit vector in An external file that holds a picture, illustration, etc.
Object name is srep29260-m48.jpg, while χ is a parameter satisfying An external file that holds a picture, illustration, etc.
Object name is srep29260-m49.jpg as An external file that holds a picture, illustration, etc.
Object name is srep29260-m50.jpg. By this classification scheme, different families of states are labeled by different unit vectors An external file that holds a picture, illustration, etc.
Object name is srep29260-m51.jpg, while states belong to the same family are characterized by a common An external file that holds a picture, illustration, etc.
Object name is srep29260-m52.jpg, and can be distinguished by different multiplicative factors χ (see Fig. 1). That is to say, An external file that holds a picture, illustration, etc.
Object name is srep29260-m53.jpg represents states with the characteristic vectors An external file that holds a picture, illustration, etc.
Object name is srep29260-m54.jpg along the same or completely opposite directions but possessing different lengths.

Figure 1
States of the same family An external file that holds a picture, illustration, etc.
Object name is srep29260-m268.jpg are represented by the characteristic vectors An external file that holds a picture, illustration, etc.
Object name is srep29260-m269.jpg along the same or opposite directions (left).

While An external file that holds a picture, illustration, etc.
Object name is srep29260-m55.jpg is fully described by An external file that holds a picture, illustration, etc.
Object name is srep29260-m56.jpg, and the action of An external file that holds a picture, illustration, etc.
Object name is srep29260-m57.jpg on it can be written equivalently as the map: An external file that holds a picture, illustration, etc.
Object name is srep29260-m58.jpg, a measure Q may only be function of An external file that holds a picture, illustration, etc.
Object name is srep29260-m59.jpg, i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m60.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m61.jpg (α  d2  1). Then as one can always make Qmax  1 (otherwise, one can normalize it by multiplying a constant to it), we have the following lemma.

Lemma 1. For any quantum measure of An external file that holds a picture, illustration, etc.
Object name is srep29260-m62.jpg that can be factorized as An external file that holds a picture, illustration, etc.
Object name is srep29260-m63.jpg, and quantum channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m64.jpg that gives the map An external file that holds a picture, illustration, etc.
Object name is srep29260-m65.jpg, the FR

An external file that holds a picture, illustration, etc.
Object name is srep29260-m66.jpg

holds, where f(χ) and An external file that holds a picture, illustration, etc.
Object name is srep29260-m67.jpg are functionals of χ and An external file that holds a picture, illustration, etc.
Object name is srep29260-m68.jpg, respectively, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m69.jpg is the probe state, with χp solution of the equation An external file that holds a picture, illustration, etc.
Object name is srep29260-m70.jpg.

The proof is given in Methods. Equipped with this lemma, we are now in position to present our central result.

Theorem 1. If the transformation matrix elements Tk0 = 0 for k [set membership] {1, 2, …, d2  d}, then the evolution of An external file that holds a picture, illustration, etc.
Object name is srep29260-m71.jpg obeys the following FR

An external file that holds a picture, illustration, etc.
Object name is srep29260-m72.jpg

with An external file that holds a picture, illustration, etc.
Object name is srep29260-m73.jpg the probe state, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m74.jpg.

The proof is left to the Methods. Here, we further show an implication of it. As Tk0 = 0 for k [set membership] {1, 2, …, d2  d}, we have An external file that holds a picture, illustration, etc.
Object name is srep29260-m75.jpg, hence An external file that holds a picture, illustration, etc.
Object name is srep29260-m76.jpg. On the other hand, An external file that holds a picture, illustration, etc.
Object name is srep29260-m77.jpg. This, together with Eq. (2), requires that all the nondiagonal elements of An external file that holds a picture, illustration, etc.
Object name is srep29260-m78.jpg must be zero.

Corollary 1. If the operator An external file that holds a picture, illustration, etc.
Object name is srep29260-m79.jpg is diagonal, then the evolution of An external file that holds a picture, illustration, etc.
Object name is srep29260-m80.jpg obeys the FR (9).

This corollary means that in addition to the usual completeness condition An external file that holds a picture, illustration, etc.
Object name is srep29260-m81.jpg of the CPTP map31, the FR (9) further requires An external file that holds a picture, illustration, etc.
Object name is srep29260-m82.jpg to be diagonal. We denote this kind of channels An external file that holds a picture, illustration, etc.
Object name is srep29260-m83.jpg. Clearly, they include the unital channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m84.jpg [i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m85.jpg] as a special case.

From a geometric perspective, Theorem 1 indicates that for all states of the same family An external file that holds a picture, illustration, etc.
Object name is srep29260-m86.jpg, namely, states with the characteristic vectors An external file that holds a picture, illustration, etc.
Object name is srep29260-m87.jpg along the same or opposite directions, their coherence dynamics measured by the l1 norm can be represented qualitatively by that of the probe state An external file that holds a picture, illustration, etc.
Object name is srep29260-m88.jpg, as the magnitude of An external file that holds a picture, illustration, etc.
Object name is srep29260-m89.jpg equals the product of the initial coherence An external file that holds a picture, illustration, etc.
Object name is srep29260-m90.jpg and the evolved coherence An external file that holds a picture, illustration, etc.
Object name is srep29260-m91.jpg. 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m92.jpg with the vectors An external file that holds a picture, illustration, etc.
Object name is srep29260-m93.jpg 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 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m94.jpg can specify the actions of environments, of measurements, or of both on the states An external file that holds a picture, illustration, etc.
Object name is srep29260-m95.jpg.

When some restrictions are imposed on the quantum channels, the FR (9) can be further simplified.

Corollary 2. If a channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m96.jpg yields An external file that holds a picture, illustration, etc.
Object name is srep29260-m97.jpg for An external file that holds a picture, illustration, etc.
Object name is srep29260-m98.jpg (β  d2  d), with q(t) containing information on An external file that holds a picture, illustration, etc.
Object name is srep29260-m99.jpgs structure, then the FR

An external file that holds a picture, illustration, etc.
Object name is srep29260-m100.jpg

holds for the family of states An external file that holds a picture, illustration, etc.
Object name is srep29260-m101.jpg.

The proof of this corollary is direct. As An external file that holds a picture, illustration, etc.
Object name is srep29260-m102.jpg, the parameters An external file that holds a picture, illustration, etc.
Object name is srep29260-m103.jpg for An external file that holds a picture, illustration, etc.
Object name is srep29260-m104.jpg are given by An external file that holds a picture, illustration, etc.
Object name is srep29260-m105.jpg. Therefore, by Eq. (3) we obtain An external file that holds a picture, illustration, etc.
Object name is srep29260-m106.jpg. 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m107.jpg and Gell-Mann channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m108.jpg given in ref. 41, and the generalized amplitude damping channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m109.jpg31. Notably, An external file that holds a picture, illustration, etc.
Object name is srep29260-m110.jpg covers the bit flip, phase flip, bit-phase flip, phase damping, and depolarizing channels which embody typical noisy sources in quantum information, while An external file that holds a picture, illustration, etc.
Object name is srep29260-m111.jpg covers the structured reservoirs with Lorentzian and Ohmic-type spectral densities.

One can also construct quantum channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m112.jpg under the action of which An external file that holds a picture, illustration, etc.
Object name is srep29260-m113.jpg obeys the FR (10) for arbitrary initial state. The Kraus operators describing An external file that holds a picture, illustration, etc.
Object name is srep29260-m114.jpg are given by

An external file that holds a picture, illustration, etc.
Object name is srep29260-m115.jpg

with k [set membership] {1, …, d2  d}, and l [set membership] {d2  d + 1, …, d2  1}, while q and q0 are time-dependent noisy parameters. Clearly, An external file that holds a picture, illustration, etc.
Object name is srep29260-m116.jpg reduces to the depolarizing channel when q0 = q.

N-qubit case

A general N-qubit state can be written as An external file that holds a picture, illustration, etc.
Object name is srep29260-m117.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m118.jpg, and

An external file that holds a picture, illustration, etc.
Object name is srep29260-m119.jpg

here, An external file that holds a picture, illustration, etc.
Object name is srep29260-m120.jpg, and σ1,2,3 are the usual Pauli matrices, while jk takes the possible values of {0, 1, 2, 3} other than the special case jk = 0 for all k. In the Methods section, we have proved that for every family of the N-qubit states An external file that holds a picture, illustration, etc.
Object name is srep29260-m121.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m122.jpg being a given unit vector, one can construct an auxiliary channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m123.jpg such that An external file that holds a picture, illustration, etc.
Object name is srep29260-m124.jpg. This, together with Eq. (9), gives:

Corollary 3. For any N-qubit state ρN, there exists an auxiliary channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m125.jpg such that

An external file that holds a picture, illustration, etc.
Object name is srep29260-m126.jpg

with An external file that holds a picture, illustration, etc.
Object name is srep29260-m127.jpg, An external file that holds a picture, illustration, etc.
Object name is srep29260-m128.jpg, d0 = (4N  2N)/2, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m129.jpg, with aij being determined by the transformation between {Yj} and {Xi}: An external file that holds a picture, illustration, etc.
Object name is srep29260-m130.jpg.

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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m131.jpg is determined by the product of the coherence for the evolved probe state under the action of An external file that holds a picture, illustration, etc.
Object name is srep29260-m132.jpg and the coherence for the generated state by An external file that holds a picture, illustration, etc.
Object name is srep29260-m133.jpg. As every Yj can always be decomposed as linear combinations of the generators {Xi}, the above result applies also to the qudit states with d = 2N. As an explicit example, the transformation between {Yj} and {Xi} for N = 2 is given in the Methods section, from which An external file that holds a picture, illustration, etc.
Object name is srep29260-m134.jpg and {aij} can be constructed directly.

Frozen coherence

By Theorem 1 we can also derive conditions on the quantum channel for which the l1 norm of coherence is frozen. To elucidate this, we return to Eq. (9), from which one can see that An external file that holds a picture, illustration, etc.
Object name is srep29260-m135.jpg is frozen if the coherence of the probe state remains constant 1 during the evolution, i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m136.jpg. For later use, we denote by TS the submatrix of T consisting Tij with i ranging from 1 to d2  d and j from 1 to d2  1. Then by Theorem 1 and the reasoning in its proof, we obtain the fourth corollary.

Corollary 4. If Tk0 = 0 for k [set membership] {1, 2, …, d2  d}, and TS is a rectangular block diagonal matrix, with the main diagonal blocks

An external file that holds a picture, illustration, etc.
Object name is srep29260-m137.jpg

being orthogonal matrices, i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m138.jpg, the l1 norm of coherence for An external file that holds a picture, illustration, etc.
Object name is srep29260-m139.jpg will be frozen during the entire evolution.

The proof is given in Methods. It enables one to construct channels An external file that holds a picture, illustration, etc.
Object name is srep29260-m140.jpg for which the l1 norm of coherence is frozen. As an explicit example, we consider the one-qubit case, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m141.jpg being described by An external file that holds a picture, illustration, etc.
Object name is srep29260-m142.jpg, i [set membership] {0, 1, 2, 3} and An external file that holds a picture, illustration, etc.
Object name is srep29260-m143.jpg. Then by Corollary 4, one can obtain that when εi0 = εi3 = 0, andAn external file that holds a picture, illustration, etc.
Object name is srep29260-m144.jpgAn external file that holds a picture, illustration, etc.
Object name is srep29260-m145.jpg, or when εi1 = εi2 = 0, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m146.jpg An external file that holds a picture, illustration, etc.
Object name is srep29260-m147.jpg, with Re(·) and Im(·) representing, respectively, the real and imaginary parts of a number, the l1 norm of coherence will be frozen. There are a host of {εij} that fulfill the requirements, e.g., ε01 = q(t), An external file that holds a picture, illustration, etc.
Object name is srep29260-m148.jpg, εk1 = εk2 = 0, or ε00 = q(t), An external file that holds a picture, illustration, etc.
Object name is srep29260-m149.jpg, εk0 = εk3 = 0, with k [set membership] {1, 2, 3}, and q(t) contains the information on An external file that holds a picture, illustration, etc.
Object name is srep29260-m150.jpg’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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m151.jpg with certain n2r−1 = 0 (or n2r = 0), An external file that holds a picture, illustration, etc.
Object name is srep29260-m152.jpg simplifies to An external file that holds a picture, illustration, etc.
Object name is srep29260-m153.jpg (or An external file that holds a picture, illustration, etc.
Object name is srep29260-m154.jpg). For instance, when considering the channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m155.jpg41, the l1 norm of coherence for An external file that holds a picture, illustration, etc.
Object name is srep29260-m156.jpg with n2 = 0 is frozen during the entire evolution when q1 = 1 (i.e., the bit flip channel). Similarly, for An external file that holds a picture, illustration, etc.
Object name is srep29260-m157.jpg with n1 = 0, it is frozen when q2 = 1 (i.e., the bit-phase flip channel). These are in facts the results obtained in ref. 21. Needless to say, when An external file that holds a picture, illustration, etc.
Object name is srep29260-m158.jpg, the l1 norm of coherence is also frozen for An external file that holds a picture, illustration, etc.
Object name is srep29260-m159.jpg with certain n2r−1 = 0 or n2r = 0.

Outlook

The FR (9) presented here can be of direct relevance to other issues of quantum theory. For example, the l1 norm of coherence is a monotone of the entanglement-based coherence measure for one-qubit states12. Its logarithmic form An external file that holds a picture, illustration, etc.
Object name is srep29260-m160.jpg is lower bounded by the relative entropy of coherence Cr(ρ) which has a clear physical interpretation, while An external file that holds a picture, illustration, etc.
Object name is srep29260-m161.jpg for arbitrary ρ has also been conjectured46. Further study shows that An external file that holds a picture, illustration, etc.
Object name is srep29260-m162.jpg also bounds the robustness of coherence, i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m163.jpg42. It is also connected to the success probability of state discrimination in interference experiments29 and the negativity of quantumness21,47. 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 states14, and the trace norm coherence for one-qubit and certain qutrit states13,46; (ii) the genuine quantum coherence (GQC) defined via the Schatten p-norm for all states48, 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m164.jpg which is a measure of the GQC for all states42; (iv) the K coherence defined based on the Wigner-Yanase skew information11, although it is problematic in the framework of coherence by Baumgratz et al.49, it may be a proper measure of the GQC48; (v) the purity of a state which is complementary with quantum coherence28; (vi) the geometric discord50,51,52,53,54 and measurement-induced nonlocality55,56; (vii) the maximum Bell-inequality violation57, and average fidelity of remote state preparation58 and quantum teleportation59. 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 observables60. 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 l1 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 An external file that holds a picture, illustration, etc.
Object name is srep29260-m165.jpg, 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 l1 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.

Methods

Proof of Lemma 1. As An external file that holds a picture, illustration, etc.
Object name is srep29260-m166.jpg gives the map An external file that holds a picture, illustration, etc.
Object name is srep29260-m167.jpg, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m168.jpg fulfills An external file that holds a picture, illustration, etc.
Object name is srep29260-m169.jpg, we have

An external file that holds a picture, illustration, etc.
Object name is srep29260-m170.jpg

Hence, it is evident that An external file that holds a picture, illustration, etc.
Object name is srep29260-m171.jpg when An external file that holds a picture, illustration, etc.
Object name is srep29260-m172.jpg.

If Qmax  1, the equation An external file that holds a picture, illustration, etc.
Object name is srep29260-m173.jpg with respect to χp is always solvable as An external file that holds a picture, illustration, etc.
Object name is srep29260-m174.jpg. If Qmax < 1, one can normalize it by simply introducing a constant N such that An external file that holds a picture, illustration, etc.
Object name is srep29260-m175.jpg, with Q′ obeying the FR of Eq. (8).

Proof of Theorem 1. First, by using Eq. (3) and the fact that An external file that holds a picture, illustration, etc.
Object name is srep29260-m176.jpg, we obtain

An external file that holds a picture, illustration, etc.
Object name is srep29260-m177.jpg

which corresponds to An external file that holds a picture, illustration, etc.
Object name is srep29260-m178.jpg, with f(χ) = χ and An external file that holds a picture, illustration, etc.
Object name is srep29260-m179.jpg.

Second, when the transformation matrix elements Tk0 = 0 for k [set membership] {1, 2, …, d2  d}, we have

An external file that holds a picture, illustration, etc.
Object name is srep29260-m180.jpg

and therefore An external file that holds a picture, illustration, etc.
Object name is srep29260-m181.jpg.

From Eqs (16) and (17) one can see that both the l1 norm of coherence and the quantum channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m182.jpg fulfill the requirements of Lemma 1, and the probe state An external file that holds a picture, illustration, etc.
Object name is srep29260-m183.jpg, with χp being solution of the equation An external file that holds a picture, illustration, etc.
Object name is srep29260-m184.jpg, which can be solved as An external file that holds a picture, illustration, etc.
Object name is srep29260-m185.jpg. This completes the proof.

Proof of Corollary 3. Suppose An external file that holds a picture, illustration, etc.
Object name is srep29260-m186.jpg is described by the Kraus operators An external file that holds a picture, illustration, etc.
Object name is srep29260-m187.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m188.jpg. Then, by employing the anticommutation relation of the Pauli operators σ1,2,3, we obtain

An external file that holds a picture, illustration, etc.
Object name is srep29260-m189.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m190.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m191.jpg if vkμk(vk  μk) = 0, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m192.jpg otherwise. This formula is equivalent to An external file that holds a picture, illustration, etc.
Object name is srep29260-m193.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m194.jpg encoding the information of An external file that holds a picture, illustration, etc.
Object name is srep29260-m195.jpg.

To solve εμ, we define coefficient matrix An external file that holds a picture, illustration, etc.
Object name is srep29260-m196.jpg, and column vectors An external file that holds a picture, illustration, etc.
Object name is srep29260-m197.jpg, An external file that holds a picture, illustration, etc.
Object name is srep29260-m198.jpg, then An external file that holds a picture, illustration, etc.
Object name is srep29260-m199.jpg becomes An external file that holds a picture, illustration, etc.
Object name is srep29260-m200.jpg, hence ε can be derived as An external file that holds a picture, illustration, etc.
Object name is srep29260-m201.jpg, with c−1 denoting the inverse matrix of c. Finally, by choosing An external file that holds a picture, illustration, etc.
Object name is srep29260-m202.jpg, we obtain An external file that holds a picture, illustration, etc.
Object name is srep29260-m203.jpg, thus completes the proof.

The transformation between generators {Yj} for the two-qubit states and {Xi} for the qudit states with d = 4 are as follows:

An external file that holds a picture, illustration, etc.
Object name is srep29260-m204.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m205.jpg, and elements An external file that holds a picture, illustration, etc.
Object name is srep29260-m206.jpg of An external file that holds a picture, illustration, etc.
Object name is srep29260-m207.jpg are arranged with (j1j2) in the sequence (01), (02), (03), (10), (11), (12), (13), …, (33).

Proof of Corollary 4. As the submatrix TS is rectangular block diagonal, the elements Tij in the off-diagonal blocks are all zero. This, together with Tk0 = 0 for k [set membership] {1, 2, …, d2  d}, yields

An external file that holds a picture, illustration, etc.
Object name is srep29260-m208.jpg

for r [set membership] {1, 2, …, d0}. Moreover, the requirement that An external file that holds a picture, illustration, etc.
Object name is srep29260-m209.jpg yields

An external file that holds a picture, illustration, etc.
Object name is srep29260-m210.jpg

By using the above two equations, it is straightforward to see that An external file that holds a picture, illustration, etc.
Object name is srep29260-m211.jpg, and therefore from Eq. (16) we have An external file that holds a picture, illustration, etc.
Object name is srep29260-m212.jpg. This, together with Theorem 1, implies An external file that holds a picture, illustration, etc.
Object name is srep29260-m213.jpg, and hence completes the proof.

Frozen coherence of one qubit

Suppose the required channel An external file that holds a picture, illustration, etc.
Object name is srep29260-m214.jpg is described by the Kraus operators An external file that holds a picture, illustration, etc.
Object name is srep29260-m215.jpg, with i [set membership] {0, 1, 2, 3}, and the values of An external file that holds a picture, illustration, etc.
Object name is srep29260-m216.jpg should satisfy certain constraints such that the requirement of Corollary 4 is satisfied. First, the completeness condition of the CPTP map, namely, An external file that holds a picture, illustration, etc.
Object name is srep29260-m217.jpg31, requires

An external file that holds a picture, illustration, etc.
Object name is srep29260-m218.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m219.jpg represents conjugation of εij, and the notation i before εi2, Re(·), and Im(·) is the imaginary unit.

Second, Corollary 4 requires T10 = T20 = 0, and TS to be a rectangular block diagonal matrix which corresponds to T13 = T23 = 0. This yields

An external file that holds a picture, illustration, etc.
Object name is srep29260-m220.jpg

from which one can obtain

An external file that holds a picture, illustration, etc.
Object name is srep29260-m221.jpg

and

An external file that holds a picture, illustration, etc.
Object name is srep29260-m222.jpg

By comparing Eqs (22) and (24), one can note that the equalities are satisfied when εi0 = εi3 = 0, An external file that holds a picture, illustration, etc.
Object name is srep29260-m223.jpg, or when εi1 = εi2 = 0, An external file that holds a picture, illustration, etc.
Object name is srep29260-m224.jpg. Under these two constraints, Eq. (25) simplifies, respectively, to

An external file that holds a picture, illustration, etc.
Object name is srep29260-m225.jpg

and

An external file that holds a picture, illustration, etc.
Object name is srep29260-m226.jpg

Finally, the requirement that An external file that holds a picture, illustration, etc.
Object name is srep29260-m227.jpg, corresponds to

An external file that holds a picture, illustration, etc.
Object name is srep29260-m228.jpg

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 l1 norm of coherence, εij should satisfy one of the following two conditions:

(i) εi0 = εi3 = 0 for i [set membership] {0, 1, 2, 3}, and

An external file that holds a picture, illustration, etc.
Object name is srep29260-m229.jpg

(ii) εi1 = εi2 = 0 for i [set membership] {0, 1, 2, 3}, and

An external file that holds a picture, illustration, etc.
Object name is srep29260-m230.jpg

Other measures fulfilling the FR

(i) The coherence concurrence for the one-qubit states14, and the trace norm coherence for the one-qubit and certain qutrit states13,46, coincide with the l1 norm of coherence. Hence, the FR applies to them.

(ii) For the GQC measure An external file that holds a picture, illustration, etc.
Object name is srep29260-m231.jpg presented in ref. 48, we have

An external file that holds a picture, illustration, etc.
Object name is srep29260-m232.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m233.jpg denotes full dephasing of ρ in the basis {|iright angle bracket}i=1,…,d. Thus, An external file that holds a picture, illustration, etc.
Object name is srep29260-m234.jpg, with f(χ) = χ/2, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m235.jpg.

For the GQC measure An external file that holds a picture, illustration, etc.
Object name is srep29260-m236.jpg, the FR also holds as the optimal δ is given by Δ(ρ)48.

(iii) CRoC(ρ) for the one-qubit states and d-dimensional states with X-shaped density matrix, equals to the l1 norm of coherence, and thus the FR holds.

(iv) The K coherence is defined as An external file that holds a picture, illustration, etc.
Object name is srep29260-m237.jpg11. As An external file that holds a picture, illustration, etc.
Object name is srep29260-m238.jpg can be decomposed as An external file that holds a picture, illustration, etc.
Object name is srep29260-m239.jpg53, An external file that holds a picture, illustration, etc.
Object name is srep29260-m240.jpg is a function of An external file that holds a picture, illustration, etc.
Object name is srep29260-m241.jpg, i.e., An external file that holds a picture, illustration, etc.
Object name is srep29260-m242.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m243.jpg. Then by using [X0, K] = 0, we obtain

An external file that holds a picture, illustration, etc.
Object name is srep29260-m244.jpg

thus An external file that holds a picture, illustration, etc.
Object name is srep29260-m245.jpg, with f(χ) = χ2/2, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m246.jpg.

(v) For the quantifier An external file that holds a picture, illustration, etc.
Object name is srep29260-m247.jpg which is a monotonic function of the purity P(ρ) = Trρ2 of a state, we have the FR An external file that holds a picture, illustration, etc.
Object name is srep29260-m248.jpg, with ρp bing the probe state for which An external file that holds a picture, illustration, etc.
Object name is srep29260-m249.jpg.

(vi) The general form of geometric quantum correlation measure can be written as

An external file that holds a picture, illustration, etc.
Object name is srep29260-m250.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep29260-m251.jpg denotes the Schatten p-norm, and opt represents the optimization over some class An external file that holds a picture, illustration, etc.
Object name is srep29260-m252.jpg of the local measurements An external file that holds a picture, illustration, etc.
Object name is srep29260-m253.jpg. This definition covers the geometric discord50,51,52 and measurement-induced nonlocality55,56. For these measures, as An external file that holds a picture, illustration, etc.
Object name is srep29260-m254.jpg, we have

An external file that holds a picture, illustration, etc.
Object name is srep29260-m255.jpg

then by comparing with Lemma 1, we obtain f(χ) = (χ/2)p, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m256.jpg, i.e., the FR holds.

If ρ in Eq. (33) is replaced by An external file that holds a picture, illustration, etc.
Object name is srep29260-m257.jpg, then one obtains the Hellinger distance discord for p = 253,54. As An external file that holds a picture, illustration, etc.
Object name is srep29260-m258.jpg, with An external file that holds a picture, illustration, etc.
Object name is srep29260-m259.jpg and An external file that holds a picture, illustration, etc.
Object name is srep29260-m260.jpg being the sets of Hermitian operators which constitute the orthonormal operator bases for the Hilbert space An external file that holds a picture, illustration, etc.
Object name is srep29260-m261.jpg and An external file that holds a picture, illustration, etc.
Object name is srep29260-m262.jpg53, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m263.jpg, the FR also holds for it.

(vii) For two-qubit states, the maximum Bell-inequality violation Bmax(ρ)57, remote state preparation fidelity Frsp(ρ)58, and Nqt(ρ) which is a monotone of the average teleportation fidelity An external file that holds a picture, illustration, etc.
Object name is srep29260-m264.jpg59, are given by

An external file that holds a picture, illustration, etc.
Object name is srep29260-m265.jpg

where E1  E2  E3 are eigenvalues of the 3 × 3 matrix TT, and An external file that holds a picture, illustration, etc.
Object name is srep29260-m266.jpg. This gives An external file that holds a picture, illustration, etc.
Object name is srep29260-m267.jpg for i [set membership] {1, 2, 3}, which implies that all measures of Eq. (35) satisfy the requirement of Lemma 1.

Additional Information

How to cite this article: Hu, M.-L. and Fan, H. Evolution equation for quantum coherence. Sci. Rep. 6, 29260; doi: 10.1038/srep29260 (2016).

Acknowledgments

This work was supported by NSFC (Grant Nos 11205121, 91536108), New Star Project of Science and Technology of Shaanxi Province (Grant No. 2016KJXX-27), Doctoral Fund of XUPT (Grant No. ZL2015), and CAS (Grant No. XDB01010000).

Footnotes

Author Contributions M.-L.H. contributed the idea and performed the calculations. M.-L.H. and H.F. wrote the paper. All authors reviewed the manuscript and agreed with the submission.

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