1 Introduction

Quantum resource theories (QRTs) approach plays a significant role in quantum information theory, in particular, to determine the measures for evaluating physical resources [1,2,3,4]. QRTs are defined by constraints that characterize a set of free operations that do not generate a resource and the corresponding set of free states that are devoid of the resource. These restrictions may arise from either fundamental conservation laws or practical restrictions resulting from the difficulty of performing quantum operations. In other words, a strong framework for studying various quantum phenomena can be provided by QRTs. Using this perspective to study quantum systems is natural since their characteristic quantum features can be destroyed due to processes such as decoherence. Taking into account that the structure of QRTs is very general, i.e., the great freedom in the definition of free states and free operations, QRTs can be applied to study many different branches of quantum physics such as entanglement [5, 6], quantum reference frames and asymmetry [7], quantum thermodynamics [8], non-locality [9], non-Markovianity [10], and quantum coherence and superposition [11,12,13,14,15,16,17]. According to a common structure in all QRTs, one can see similarities and connections between different QRTs in terms of available resource measures and resource reversibility [1, 18,19,20,21,22]. On the other hand, there are cases where multi-resources are required to accomplish a specific task, so attempts have been made to combine the different QRTs as resource theory of thermodynamics that it is a mixture of the purity theory and the asymmetry theory [21].

Quantum coherence is one of the most prominent features of quantum systems, and it is an important physical resource in many quantum information processes as it can be provided at a specific cost and consumed to accomplish useful tasks [11,12,13,14,15,16,17]. In the resource theory of coherence, the diagonal states in a reference basis are chosen as the free states, which are known as incoherent states, and the incoherent operations that cannot generate the coherence are known as free operations [4]. So far, many efforts have been accomplished to understand this phenomenon and its relation with other quantum resources, such as quantum entanglement, quantum magic, and quantum discord. [23,24,25,26,27,28,29,30].

The quantumness of open systems is extremely fragile due to the inevitable interactions with their surrounding environment, which leads to different noisy quantum channels [31]. It is very important to know the ability of these channels to change physical resources such as quantum coherence [32]; hence, the quantum resource theory can help us to understand how the quantum properties of a system change under such evolutions. For this reason, in this paper, we investigate the coherence of quantum channels within the framework of the resource theory. To achieve this aim, we use Choi–Jamiolkowski isomorphism [31] and introduce coherence-breaking channels (CBCs) as free operations and their corresponding Choi states as free states [33]. In our framework, the quantum-incoherent relative entropy of coherence (\(\mathcal {QI}\) REC) of quantum channels is equivalent to the \(\mathcal {QI}\) REC of their corresponding Choi states [25, 34]. We show that it consists of two parts: the relative entropy of coherence (REC) of open system and the basis-dependent quantum asymmetric discord [35,36,37], where the former is zero for both quantum unital channels and quantum-incoherent channels. On the other hand, one can prove the \(\mathcal {QI}\) REC of channel is decreasing for divisible quantum-incoherent channels and it can be a witness of non-Markovianity for quantum-incoherent channels. Also, we demonstrate for qubit channels, the REC can be equivalent to the REC of their corresponding Choi states and the basis-dependent quantum symmetric discord can never be more than it. Our results will provide new light for a better understanding of the relationship between the coherence of channels and quantum correlations, and the coherence of a quantum channels can be considered as the quantumness of that.

We emphasize our formalism can be extended to the resource theory of entanglement for quantum channels. So, observing the analogical similarity between CBC and entanglement breaking channels (EBC) [38], one can define these operations as free channels and their corresponding Choi states as free states. In light of this, one can able to define the relative entropy of entanglement (REE) for quantum channels in the context of QRT [39].

The paper is organized as follows. In Sect. 2, a review of the quantum channels and the Choi representation is provided. In Sect. 3, we briefly study the resource theory of coherence. We define the \(\mathcal {QI}\) REC of quantum channels within the framework of the quantum resource theory in Sect. 4. We focus on the coherence of qubit channels in Sect. 5. To illustrate the coherence of channel, two examples are investigated in Sect. 6. The paper concludes in Sect. 7.

2 Quantum channels

A quantum channel is a linear map that satisfies the completely positive and trace-preserving (CPTP) conditions [31]. It can be shown that such mapping admits dilations as following form

$$\begin{aligned} \Lambda (\rho ) =\hbox {Tr}_\mathrm{E}[U_\mathrm{SE}(\rho _\mathrm{S}\otimes \rho _\mathrm{E})U^{\dagger }_\mathrm{SE}], \end{aligned}$$
(1)

where \(\rho _\mathrm{S}\) and \(\rho _\mathrm{E}\) are state of an open system and its environment, respectively, and \(U_\mathrm{SE}\) denotes the unitary time evolution operator of the total system. On the other hand, the CPTP map \(\Lambda \) can be represented by Kraus form [31], and is shown by

$$\begin{aligned} \Lambda (\rho ) =\sum _{i} (K_i)^\dagger K_{i}, \end{aligned}$$
(2)

where \(K_{i}\) are Kraus operators that \(\sum K^{\dagger }_{i} K_{i}=I\). Note that the quantum channels can have several Kraus representations.

In this work, we focus on another representation of the quantum channel where it is known as the Choi matrix. According to Choi–Jamiolkowski isomorphism, any CPTP map can be related to a density matrix of the composite system AS in which A is an auxiliary system with the same dimension d as S. The Choi state of the channel \(\Lambda \) is defined as

$$\begin{aligned} \Omega _{\Lambda }=(I_{A}\otimes \Lambda _{S})(\vert \Psi \rangle \langle \Psi \vert ), \end{aligned}$$
(3)

where \(\vert \Psi \rangle =\frac{1}{\sqrt{d}}\sum _{i} \vert ii\rangle \) is a maximally entangled state of SA. It is important to emphasize that there is a unique Choi state for every quantum channel includes all the channel’s information [31]. Hence, we claim the coherence of any channel can be determined by the coherence of its corresponding Choi state.

To define the coherence of a channel, we review the resource theory of coherence in the next section.

3 The resource theory of coherence

As stated in the Introduction, quantum coherence is a physical property that is used as a resource for quantum systems. Hence, the quantitative determination of coherence for quantum systems has been extensively studied [13, 40,41,42,43]. Let us consider the Hilbert space \(\mathcal {H}\) with fixed basis \({\vert i \rangle }_{i=0,\ldots ,d-1}\), then an incoherent state is defined as \(\delta =\sum ^{d-1}_{i}\delta _{i}\vert \ i\rangle \langle \ i\vert \). The set of incoherent states is be denoted by \(\mathcal {I}\) and \(\vert \psi \rangle =\frac{1}{\sqrt{d}}\sum ^{d-1}_{i} e^{i \theta } \vert i\rangle \) is a maximally coherent state where \(\theta _{i}\) is an arbitrary phase. A completely positive and trace preserving map \(\Lambda \) is maximally incoherent (MIO) if \(\Lambda (\delta ) \in \mathcal {I}\) for any state \(\delta \in \mathcal {I}\) [44]. Meanwhile, an incoherent operation (IO) has a Kraus representation such that \(\frac{K_{i}\delta K^{\dagger }_{i}}{\text {Tr}(K_{i}\delta K^{\dagger }_{i})} \in \mathcal {I}\) for all n and \(\delta \in \mathcal {I}\) [4]. By this restriction, the Kraus operators can be in the form \(K_{i}=\sum ^{d-1}_{j=0}c_{ij}\vert d_{i}(j)\rangle \langle j\vert \) for any incoherent operation, which are incoherent and \(d_{i}(j)\) is a function of the index j and \(c_{ij}\) are coefficients [45]. If \(d_{i}(j)\) is a permutation or one to one, then \(K^{\dagger }_{i}\) is also incoherent as well as \(K_{i}\). It is explicit from the definition that \(\hbox {IO}\subseteq \hbox {MIO}\). In resource theory of coherence, the incoherent states and the incoherent operations are known as free states and free operations, respectively. A measure for quantum coherence of state \(\rho \) is characterized by a function \(C(\rho )\) which satisfies the following properties [4]:

  1. i.

    \(C(\rho )\ge 0\), for any \(\rho \) and \(C(\delta )=0\) if only if \(\delta \in \mathcal {I}\);

  2. ii.

    The coherence cannot increase under MIO map \(\Lambda \), i.e., \(C(\Lambda (\rho ))\le C(\rho )\);

  3. iii.

    For every \(\Lambda \in \hbox {IO}\) with Kraus representation \(\lbrace K_{i}\rbrace \), the coherence is non-increasing on average under selective measurement, i.e., \(\sum _{i}p_{i}C(\rho _{i})\le C(\rho )\), where \(\rho _{i}=\frac{K_{i}\rho K^{\dagger }_{i}}{\text {Tr}(K_{i}\rho K^{\dagger }_{i})}\);

  4. iv.

    The coherence cannot increase by mixing quantum states, i.e., \(C(\sum _{i}p_{i}\rho _{i})\le \sum _{i}p_{i}C(\rho _{i})\).

One of the measures that satisfy all the above requirements is the REC defined by [4]

$$\begin{aligned} C_{r}(\rho )=\min _{\delta \in {\mathcal {I}}}S(\rho \Vert \delta )=S(\rho ^{d})-S(\rho ), \end{aligned}$$
(4)

where \(S(\rho \Vert \delta )=\hbox {tr}[\rho (\hbox {Log}{\rho }-\hbox {Log}{\delta })]\) is the relative entropy [31], and the diagonal part of \(\rho \) in the reference basis \(\lbrace \vert i\rangle \rbrace \) is \(\rho ^{d}=\sum _{i}\vert \ i\rangle \langle \ i\vert \rho \vert \ i\rangle \langle \ i\vert \).

In the next section, we use the resource theory and introduce the \(\mathcal {QI}\) REC of channels.

4 The resource theory of coherence for quantum channels

In the previous section, we introduced the resource theory of coherence for states. Here, we intend to define, in the context of the resource theory, a measure to determine the coherence of quantum channels. To achieve this purpose, we have to define the set of free states and free operations.

\(\mathbf{Free} operations: \) The free operations cannot generate resources, e.g., in resource theory of entanglement the LOCC are considered as free operations [5, 6, 46]. In this paper, we consider coherence-breaking channels CBCs as free operations [33]. A quantum-incoherent channel \(\Lambda \) is called coherence-breaking if \(\Lambda (\rho )\) is an incoherent state for any state \(\rho \). The set of all CBCs denoted by \(S_\mathrm{cbc}\). A coherence-breaking channel kills any coherence present in the state, and this is our motivation in this work.

\(\mathbf{Free} states: \) The Choi states corresponding to the free operations are regarded as the free states. The set of free states can be introduced by the following form

$$\begin{aligned} \mathcal {F}=\lbrace \Omega _{\Phi }\mid ~\Phi \in S_\mathrm{cbc}\rbrace . \end{aligned}$$
(5)

The corresponding Choi states of all coherence-breaking channels have the following form [33]

$$\begin{aligned} \Omega _{\Phi }=\sum _{i}\lambda _{i}\rho _{i}\otimes \vert i\rangle \langle i\vert . \end{aligned}$$
(6)

It is clear that \(\mathcal {F}\) is a set of quantum-incoherent states and \((\Lambda \otimes I) \Omega _{\Phi }\in \mathcal {F}\) for every state \(\Omega _{\Phi }\in \mathcal {F}\) and \(\Lambda \in {S_\mathrm{cbc}}\). By using the Choi–Jamiolkowski isomorphism, we define the \(\mathcal {QI}\) REC of channels as [25, 34]

$$\begin{aligned} C_\mathrm{QI}(\Lambda )=C_{r}\left( \Omega ^{A\vert S}_{\Lambda }\right) =\min _{\Omega _{\Phi }\in \mathcal {F}}S(\Omega _{\Lambda }\Vert \Omega _{\Phi }), \end{aligned}$$
(7)

with the minimization taken over the set of \(\mathcal {F}\). Applying Theorem 2 in [47], \(C_\mathrm{QI}(\Lambda )\) can also be written as

$$\begin{aligned} C_\mathrm{QI}(\Lambda )=S(\Delta ^{S}(\Omega _{\Lambda }))-S(\Omega _{\Lambda }), \end{aligned}$$
(8)

in which \(\Delta ^{S}(\Omega _{\Lambda })=\sum _{i}(I\otimes \vert i\rangle \langle i\vert )\Omega _{\Lambda }(I\otimes \vert i\rangle \langle i\vert )\). Notice that the relative entropy of Choi states is monotonicity decreasing under the local quantum-incoherent operations.

In the following, we are going to obtain the relationship between the \(\mathcal {QI}\) REC of channel and quantum correlations. For this purpose, we regard the basis-dependent quantum asymmetric discord [35,36,37]

$$\begin{aligned} D^{A\vert S}(\Omega _{\Lambda })=I(\Omega _{\Lambda })-I[\Delta ^{S}(\Omega _{\Lambda })], \end{aligned}$$
(9)

where \(I(\rho _{AS})=S(\rho _{S})+S(\rho _{A})-S(\rho _{AS})\) is mutual information [31]. With the help of Eq. (9), it is straightforward to obtain the following equality

$$\begin{aligned} C_\mathrm{QI}(\Lambda )=C_{r}(\rho _{S})+D^{A\vert S}(\Omega _{\Lambda }). \end{aligned}$$
(10)

where \(C_\mathrm{QI}(\Lambda )\) compose of the REC of open system and the quantum asymmetric discord. Equation (10) tells us that \(C_\mathrm{QI}(\Lambda )\ge D^{A\vert S}(\Omega _{\Lambda })\), or in other words, the quantum asymmetric discord can never exceed the \(\mathcal {QI}\) REC for any quantum channels.

By taking partial trace over the ancilla A, one can obtain the state of the system S as

$$\begin{aligned} \rho _{S}=\hbox {Tr}_{A}(\Omega _{\Lambda })=\frac{1}{d}\sum _{i}K_{i}K^{\dagger }_{i}. \end{aligned}$$
(11)

Suppose \(\Lambda \) is a quantum unital channel, i.e., \(\sum _{i}K_{i}K^{\dagger }_{i}=\sum _{i}K^{\dagger }_{i}K_{i}=I\), so we conclude \(C_r(\rho _{S})\) is zero for any quantum unital channel.

The state of the system, Eq. (11), for an incoherent quantum channel has the following form

$$\begin{aligned} \rho _{S}=\frac{1}{d}\sum ^{d-1}_{l=0}\sum _{i}c_{il}c^{*}_{il}\vert d_{i}(l)\rangle \langle d_{i}(l)\vert , \end{aligned}$$
(12)

where the Kraus operator is \(K_{i}=\sum ^{d-1}_{j=0}c_{ij}\vert d_{i}(j)\rangle \langle j\vert \). Since \(\rho _{S}\) in the above equation is an incoherent state then \(C_{r}(\rho _{S})\) is also zero. So, the \(\mathcal {QI}\) REC of the unital and the incoherent channels equals to \(D^{A\vert S}(\Omega _{\Lambda })\). Besides, we prove the \(\mathcal {QI}\) REC is decreasing for divisible incoherent channels in the following proposition.

Proposition

If \(\Lambda \) is a divisible incoherent channel then \(C_\mathrm{QI}(\Lambda )\) is decreasing.

Proof

Assume \(\Lambda \) is a divisible incoherent channel, i.e., \(\Lambda _{t+\epsilon ,0}=\Lambda _{t+\epsilon ,t}\Lambda _{t,0}\). Using the facts that the \(\mathcal {QI}\) REC in Eq. (7) is reduced under local incoherent channels, then we have \(C^{A\vert S}_{r}(( I\otimes \Lambda )(\rho _{AS}))\le C^{A\vert S}_{r}(\rho _{AS})\). So, one can write

$$\begin{aligned} C^{A\vert S}_{r}(\rho _{AS}(t+\epsilon ))= & {} C^{A\vert S}_{r}((I\otimes \Lambda _{t+\epsilon ,0})(\rho _{AS}(0))),\nonumber \\= & {} C^{A\vert S}_{r}((I\otimes \Lambda _{t+\epsilon ,t}\Lambda _{t,0})(\rho _{AS}(0)))\nonumber \\= & {} C^{A\vert S}_{r}((I\otimes \Lambda _{t+\epsilon ,t})(\rho _{AS}(t)))\nonumber \\\le & {} C^{A\vert S}_{r}(\rho _{AS}(t)), \end{aligned}$$
(13)

The inequality (13) shows that the coherence of the channel is decreasing and the proof is complete. \(\square \)

According to the above proposition, the \(\mathcal {QI}\) REC can be regarded as a witness for the non-Markovianity of quantum incoherent channels.

Now one can ask a question: what is the relation between the \(\mathcal {QI}\) REC and the quantumness of a channel? Taking into account Eq. (7), it is clear as long as the \(\mathcal {QI}\) REC of a channel is reduced, the channel will be closer to a CBC. This means that the coherence of the output state of the channel is also decreased; thus, the output state becomes more classical. Therefore, the \(\mathcal {QI}\) REC of channel can be considered as a measure of the quantumness of the channel. It is important to say that we interpret the quantumness, the amount of quantumness remains in the state of the system during the evolution, and this is different from the channel’s ability to create quantum correlations.

5 Coherence of qubit channels

In this section, we consider the evolution of a single qubit under a quantum channel. An arbitrary qubit is expressed as \(\rho =\frac{1}{2}(I+\mathbf {r}\cdot \varvec{\sigma })\) , where \(\mathbf {r}\) is the 3-dimensional Bloch vector with \(\mathbf {r} \in R^{3}(\vert \mathbf {r} \vert \le 1)\) and \(\sigma _{i}\) are Pauli matrices. A qubit channel, \(\Lambda \), can be represented by a \(4\times 4\) matrix in the following form [48, 49]

$$\begin{aligned} \mathbf {F}= \left( \begin{array}{cc} 1&{}\quad \mathbf {0} \\ \varvec{\tau } &{}\quad T \\ \end{array} \right) , \end{aligned}$$
(14)

where, T is a \(3\times 3\) real matrix and \(\varvec{\tau }\) and \(\mathbf {0}\) are 3-dimensional column and row vectors, respectively. Then, we have

$$\begin{aligned} \Lambda (\rho )=\frac{1}{2}[I+(T\mathbf {r}+\varvec{\tau })\cdot \varvec{\sigma }]. \end{aligned}$$
(15)

It is worthwhile to note that any qubit channel can be written as [48, 49]

$$\begin{aligned} \mathbf {F}= \left( \begin{array}{cccc} 1&{}\quad 0&{}\quad 0&{}\quad 0 \\ \tau ^{\prime }_{1}&{}\quad \lambda _{1}&{}\quad 0&{}\quad 0 \\ \tau ^{\prime }_{2}&{}\quad 0&{}\quad \lambda _{2}&{}\quad 0 \\ \tau ^{\prime }_{3}&{}\quad 0&{}\quad 0&{}\quad \lambda _{3}\\ \end{array} \right) , \end{aligned}$$
(16)

where \(\lambda \)’s are singular values of the matrix T. For all coherence-breaking qubit channels, the matrix \(\mathbf {F}\) is in the following form [33]

$$\begin{aligned} \mathbf {F_{\Phi }}= \left( \begin{array}{cccc} 1&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ \tau _{3}&{}\quad 0&{}\quad 0&{}\quad \lambda _{3}\\ \end{array} \right) , \end{aligned}$$
(17)

and the corresponding Choi matrix is

$$\begin{aligned} \Omega _{\Phi }= \left( \begin{array}{cccc} 1+\tau _{3}-\lambda _{3}&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1+\tau _{3}+\lambda _{3}&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1-\tau _{3}+\lambda _{3}&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 1-\tau _{3}-\lambda _{3}\\ \end{array} \right) . \end{aligned}$$
(18)

Here,\(\Omega _{\Phi }\) is an incoherent matrix. Therefore, for qubit channels in the form of Eq. (17), the coherence of the channel coincides with the REC of its corresponding Choi state

$$\begin{aligned} C_{r}(\Lambda )=C_{r}(\Omega _{\Lambda })=\min _{\Omega _{\Phi }\in \mathcal {F}}S(\Omega _{\Lambda }\Vert \Omega _{\Phi }), \end{aligned}$$
(19)

and regarding Eq. (4), we have

$$\begin{aligned} C_{r}(\Lambda )=S\left( \Omega ^{d}_{\Lambda }\right) -S(\Omega _{\Lambda }). \end{aligned}$$
(20)

Also, by using the basis-dependent quantum symmetric discord [35,36,37]

$$\begin{aligned} D(\Omega _{\Lambda })=I(\Omega _{\Lambda })-I\left[ \left( \Omega ^{d}_{\Lambda }\right) \right] , \end{aligned}$$
(21)

The coherence of qubit channel gets the following form

$$\begin{aligned} C_{r}(\Lambda )=C_{r}(\rho _{S})+C_{r}(\rho _{A})+D(\Omega _{\Lambda }). \end{aligned}$$
(22)

The above equation tells us that the quantum symmetric discord \(D(\Omega _{\Lambda })\)can never exceed the coherence \(C_{r}(\Lambda )\) for qubit channels.

6 Examples

In this section, the coherence of the channel will be illustrated by means two examples.

6.1 Amplitude damping channel

Here, we calculate the coherence for an amplitude damping channel with Kraus operators \(K_{0}=\vert 0\rangle \langle 0\vert +\sqrt{1-p}\vert 1\rangle \langle 1\vert \) and \(K_{1}=\sqrt{p}\vert 0\rangle \langle 1\vert \) [31]. The corresponding Choi matrix for this channel is given by

$$\begin{aligned} \Omega _{\Lambda ^{AD}}= \left( \begin{array}{cccc} \frac{1}{2}&{}\quad 0&{}\quad 0&{}\quad \frac{\sqrt{1-p}}{2} \\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad \frac{p}{2}&{}\quad 0 \\ \frac{\sqrt{1-p}}{2}&{}\quad 0&{}\quad 0&{}\quad \frac{1-p}{2}\\ \end{array} \right) , \end{aligned}$$
(23)

and the coherence of the channel will be

$$\begin{aligned} C_{r}\left( \Lambda ^{AD}\right) =\frac{p-1}{2}\log {\left( \frac{1-p}{2}\right) }+\frac{2-p}{2}\log {\left( \frac{2-p}{2}\right) }+\frac{1}{2}. \end{aligned}$$
(24)

The behavior of the coherence of the channel (dashed red line) of the corresponding Choi state in terms of pis plotted in Fig. 1. From the figure, one can see that the amount of coherence is reduced by increasing the parameter p from 0 to 1 and it is very close to the quantumness \(Q_{C}(\Lambda ^{AD})\) which is defined in [30].

Fig. 1
figure 1

(Color online) Plot of \(C_\mathrm{QI}(\Lambda ^{AD})\) as a function of the parameter p, for amplitude damping channel

Full size image

6.2 Phase covariant channel

Here, we consider a general model of qubit dynamics which includes dephasing, dissipation, and heating effects [50,51,52]. The time-local master equation for this evolution is given by

$$\begin{aligned} \frac{\text {d}\rho }{\text {d}t}= & {} -i (\omega +h(t))[\sigma _{z},\rho ]+\frac{\gamma _{z}(t)}{2}(\sigma _{z}\rho \sigma _{z}-\rho )\nonumber \\&+\frac{\gamma _{1}(t)}{2}(\sigma _{+}\rho \sigma _{-}-\frac{1}{2}\lbrace \sigma _{-}\sigma _{+}\rho \rbrace )\nonumber \\&+\frac{\gamma _{2}(t)}{2}(\sigma _{-}\rho \sigma _{+}-\frac{1}{2}\lbrace \sigma _{+}\sigma _{-},\rho \rbrace ),\nonumber \\ \end{aligned}$$
(25)

where \(\gamma _{i}(t) (i= 1,2,z)\) are time-dependent decay rates, \(\sigma _{\pm }\) are the raising and lowering operators of the qubit, \(\sigma _{z}\) is the Pauli spin operator in the z-direction and h(t) time-dependent frequency shift, and \(\omega \) is the transition frequency of the qubit. It should be noted that the decay rates are time-dependent functions which can be negative at some times. The Eq. (25) is the most general time-local master equations for a qubit which indicates a phase covariant transform. The Choi matrix for such a transformation will be in the following form

$$\begin{aligned} {\Omega _{\Lambda _{\omega }}= \left( \begin{array}{cccc} \frac{1+\kappa (t)+\eta _{\parallel }(t)}{4}&{}\quad 0&{}\quad 0&{}\quad \frac{\eta _{\perp }(t)}{2} \\ 0&{}\quad \frac{1-\kappa (t)-\eta _{\parallel }(t)}{4}&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad \frac{1+\kappa (t)-\eta _{\parallel }(t)}{4}&{}\quad 0 \\ \frac{\eta _{\perp }(t)}{2}&{}\quad \quad 0&{}\quad 0&{}\quad \frac{1-\kappa (t)+\eta _{\parallel }(t)}{4}\\ \end{array} \right) }, \end{aligned}$$
(26)

with

$$\begin{aligned} \kappa (t)= & {} - e^{-\Gamma (t)}(1+2G(t))+1,\nonumber \\ \eta _{\parallel }(t)= & {} e^{-\Gamma (t)},\nonumber \\ \eta _{\perp }(t)= & {} e^{-\Gamma (t)/2-\Gamma _{z}(t)},\nonumber \\ \end{aligned}$$
(27)

where the terms in the above equations are defined as

$$\begin{aligned} \Gamma (t)= & {} \int ^{t}_{0}\hbox {d}t^{\prime }\left( \frac{\gamma _{1}(t^{\prime })}{2}+\frac{\gamma _{2}(t^{\prime })}{2}\right) ,\nonumber \\ \Gamma _{z}(t)= & {} \int ^{t}_{0}\hbox {d}t^{\prime }\gamma _{z}(t^{\prime }),\nonumber \\ G(t)= & {} \int ^{t}_{0}\hbox {d}t^{\prime }e^{\Gamma (t^{\prime })}\frac{\gamma _{2}(t^{\prime })}{2},\nonumber \\ \end{aligned}$$
(28)

It is clear from that \(C_{r}(\rho _{S})=0\). The coherence of channel \(\Lambda _{\omega }\) can be evaluated as

$$\begin{aligned} C_{r}(\Lambda _{\omega })= & {} -\frac{1\pm \kappa (t)+\eta _{\parallel }(t)}{4}\log {\frac{1\pm \kappa (t)+\eta _{\parallel }(t)}{4}}\nonumber \\&+\frac{1+\eta _{\parallel }(t)\pm \sqrt{\kappa ^{2}(t)+4\eta _{\perp }^{2}(t)}}{4} \times \log {\frac{1+\eta _{\parallel }(t)\pm \sqrt{\kappa ^{2}(t)+4\eta _{\perp }^{2}(t)}}{4}}.\nonumber \\ \end{aligned}$$
(29)

Now let us assume both environment thermal and dephasing are at the same temperature T. Also, we ignore the effect of the Lamb shift corrections of the first term. The decay rates of heating and dissipation reservoir are \(\gamma _{1}(t)/2=Nf(t)\) and \(\gamma _{2}(t)/2=(N+1)f(t)\), respectively, where N is the mean number of thermal photons. The function f(t) depends on the form of the reservoir spectral density, and for a Lorentzian spectrum, it is expressed as [53]

$$\begin{aligned} f(t)=\hbox {Re}\left\{ \frac{\dot{c}(t)}{c(t)}\right\} , \end{aligned}$$
(30)

with

$$\begin{aligned} c(t)=e^{-\frac{t}{2}}\left[ \cosh {\left( \frac{\text {d}t}{2}\right) }+\frac{1}{d}\sinh {\left( \frac{\text {d}t}{2}\right) }\right] c(0), \end{aligned}$$
(31)

where \(d=\sqrt{1-2R}\), and \(R=\gamma _{0}/\lambda \) is a dimensionless positive number, in which \(\gamma _{0}\) is an effective coupling constant and \(\lambda \) is the width of the spectral density of the environment. For \(R<1/2\) (weak coupling), the dynamics is divisible (Markovian), while for \(R>1/2\) (strong coupling), it becomes non-divisible (non-Markovian).

We consider the spectral density \(J(\omega )=\alpha (\omega ^{s}/\omega ^{s-1}_{c})e^{-\omega /\omega _{c}}\) for pure dephasing dynamic, where \(\omega _{c}\) is the cutoff frequency, s is the Ohmicity parameter, and \(\alpha \) is the coupling constant. In this case, the decay rate for the dephasing channel is determined by [54,55,56]

$$\begin{aligned} \gamma _{z}(t)=\int \hbox {d}\omega J(\omega )\coth {\left( \frac{\hbar \omega }{2k_{B}T}\right) }\frac{\sin {\left( \omega t\right) }}{\omega }. \end{aligned}$$
(32)

The memory time of the dephasing environment can be defined by \(1/\omega _{c}\). To characterize the relation between the cutoff frequency of the dephasing environment and the width of the spectral density of the thermal reservoir, one can introduce a new parameter \(\beta =\omega _{c}/\lambda \). In dephasing dynamics, the non-Markovianity, \(\gamma _{z}(t)<0\), occurs whenever \(s>s_\mathrm{crit}(T=0) = 2\) [57]. Hence, the dynamics of the whole system can be determined by two parameters R and s.

Fig. 2
figure 2

(Color online) Dynamics of the coherence \(C_{r}(\Lambda _{\omega })\) as a function of \(\lambda t\). a the weak-coupling (Markovian) regime, \(R=.01\) and \(s=0.5\); b the strong-coupling (non-Markovian) regime, \(R=10\) and \(s=3.5\)

Full size image

In this paper, we take \(T=0\); thus, the dephasing rate \(\gamma _{z}(t)\) is independent of temperature and heating and dissipation rates are zero and f(t), respectively. Then, one can obtain the following expressions

$$\begin{aligned} \Gamma (t)= & {} -\mathfrak {R}(\ln {[u(t)]}),\nonumber \\ \Gamma _{z}(t)= & {} \frac{\alpha }{s-1}\widetilde{\Gamma }(s)\left( 1-\left( 1+\omega ^{2}_{c}t^{2}\right) \right. \nonumber \\&\left. \times [\cos {(s\arctan {(\omega _{c}t)})}+\omega _{c}t\sin {(s\arctan {(\omega _{c}t}))}]\right) ,\nonumber \\ \kappa (t)= & {} -(1-\exp {[\mathfrak {R}(\ln {[u(t)]})]}),\nonumber \\ \end{aligned}$$
(33)

where \(u(t)=\lbrace c(t)/c(0)\rbrace ^{2}\) and \(\widetilde{\Gamma }(s)\) is the Euler gamma function.

At this point, we now return to the coherence problem of the phase covariant channel in Eq. (29). The dynamical behavior of coherence for quantum channel \(\Lambda _{\omega }\) as a function of \(\lambda t\) is shown in Fig. 2. We assume that the memory time of the dephasing and thermal environments is the same, i.e., we choose the parameter \(\beta =1\). In Fig. 2a, the evolution is Markovian since we have \(s=0.5\) and \(R=0.01\). The coherence of channel is plotted for \(s=3.5\) and \(R=10\) in Fig. 2b that the dynamics is non-Markovian. As can be seen, the coherence of channel is decreasing for the Markovian regime over time, while in non-Markovian regime the coherence of channel damply oscillates and increase in some time intervals, which this behavior is due to non-divisibility of the dynamics.

7 Conclusion

In this paper, we have shown the quantum-incoherent relative entropy of coherence (\(\mathcal {QI}\) REC) of a quantum channel is equivalent to the \(\mathcal {QI}\) REC of its Choi state. We have used the Choi–Jamiolkowski isomorphism, within the framework of QRT, to define it, where the coherence-breaking channels (CBCs) are considered as free operations and their corresponding Choi states as free states. It is demonstrated that the \(\mathcal {QI}\) REC consists of two parts: the REC of open system and the basis-dependent quantum asymmetric discord that the former is zero for both the quantum unital and quantum-incoherent channels. Also, it is shown that the \(\mathcal {QI}\) REC is decreasing for any divisible quantum-incoherent channel and it can be considered as a witness of the non-Markovianity for incoherent channels. Also, we have proposed that the coherence of the channel can be regarded as the measure of the quantumness of that. Finally, we found that the coherence of qubit channels can coincide with the REC of their corresponding Choi states and the basis-dependent quantum symmetric discord can never exceed the coherence. Ultimately, it is worthwhile to note that our results could open a new way to better understand the relationship between quantum coherence as a physical resource and other quantum resources, such as quantum correlations.