**Proposition 4.**


For item (a), the non-negativity of *<sup>F</sup>*(*ρS*|<sup>Π</sup>, *U*) follows from inequality (16) in Proposition 2. The upper bound is evident in view of inequality (32).

For item (b), let *c* ∈ [0, 1], *ρS* and *σS* be two states with *ρS* = <sup>Φ</sup>*S*(*ρS*), *σS* = <sup>Φ</sup>*S*(*<sup>σ</sup>S*). Straightforward manipulation yields

$$\begin{aligned} &cF(\rho\_S|\Pi,\mathsf{II}) + (1-c)F(\sigma\_S|\Pi,\mathsf{II}) - F(c\rho\_S + (1-c)\sigma\_S|\Pi,\mathsf{II}) \\ &= c(1-c)\left(\mathsf{tr}((\rho\_S - \sigma\_S)^2) - \mathsf{tr}((\rho\_S' - \sigma\_S')^2)\right) \\ &\ge 0. \end{aligned}$$

The last inequality follows from

$$
\lambda (\rho\_S' - \sigma\_S') \nrightarrow \lambda (\rho\_S - \sigma\_S).
$$

We see that, on the one hand, *<sup>D</sup>*(*ρS*|<sup>Π</sup>, *U*) and *<sup>F</sup>*(*ρS*|<sup>Π</sup>, *U*) share some similar properties, and, on the other hand, they have different advantages and disadvantages. This is reminiscent of the comparison between the conventional variance and Fisher information.

#### **4. Influence on Environment Caused by System**

The interaction between the system and the environment is mutual. While we are focusing on the decoherence of the system caused by the environment, it may also be useful to investigate the influence on the environment caused by the system. In a formal fashion, this may also be interpreted as the decoherence of the environment caused by the system. Due to the asymmetry of the system–environment interaction, there are subtle differences between the influence on the environment caused by the system and that on the system caused by the environment.

From Equation (20), we obtain the final environment state

$$\rho\_E' = \text{tr}\_S \rho\_{SE}' = \sum\_{i=1}^d \text{tr}(\Pi\_i \rho\_S) \mathcal{U}\_{E\_i} \rho\_E \mathcal{U}\_{E\_i}^\dagger = \sum\_{i=1}^d p\_i \mathcal{U}\_{E\_i} \rho\_E \mathcal{U}\_{E\_i}^\dagger$$

after the system–environment interaction. Here *pi* = tr(*ρS*Π*i*) = *i*|*ρS*|*i*-. We denote the above operation as

$$\Phi\_E(\rho\_E) = \rho\_E' = \sum\_{i=1}^d p\_i \mathcal{U}\_{E\_i} \rho\_E \mathcal{U}\_{E\_i}^\dagger$$

which is a random unitary channel with Kraus operators √*piUEi* . Moreover, noting that *ρE* = *ρ<sup>E</sup>*1 ⊗ *ρ<sup>E</sup>*2 ⊗···⊗ *ρEd* , we have

$$J(\rho\_E, \Phi\_E) = 1 + \text{tr}\left(\sqrt{\rho\_E} \Phi\_E^\dagger(\sqrt{\rho\_E})\right) = 1 + \sum\_{i=1}^d p\_i \text{tr}(\sqrt{\rho\_E} \mathcal{U}\_{E\_i}^\dagger \sqrt{\rho\_E} \mathcal{U}\_{E\_i})$$

$$= 1 + \sum\_{i=1}^d p\_i \text{tr}(\sqrt{\rho\_{E\_i}} \mathcal{U}\_{E\_i}^\dagger \sqrt{\rho\_{E\_i}} \mathcal{U}\_{E\_i})\tag{33}$$

and

$$J(\rho\_{E'}', \Phi\_E) = 1 + \text{tr}\left(\sqrt{\rho\_E'} \Phi\_E^\dagger(\sqrt{\rho\_E'})\right) = 1 + \sum\_{i=1}^d p\_i \text{tr}\left(\sqrt{\rho\_E'} \mathcal{U}\_{E\_i}^\dagger \sqrt{\rho\_E'} \mathcal{U}\_{E\_i}\right)$$

The final state of the *i*-th sub-environment reads

$$\rho'\_{E\_i} = \text{tr}\_{E\_i} \rho'\_E = \text{tr}\_{E\_i} \left( \sum\_{i=1}^d p\_i \mathcal{U}\_{E\_i} \rho\_E \mathcal{U}\_{E\_i}^\dagger \right) \\ = p\_i \mathcal{U}\_{E\_i} \rho\_{E\_i} \mathcal{U}\_{E\_i}^\dagger + (1 - p\_i) \rho\_{E\_i}.$$

where the notation tr*E*<sup>2</sup>*i* denotes the partial trace over all sub-environments except for *Ei*. We denote the corresponding operation as the channel

$$\Phi\_{E\_i}(\rho\_{E\_i}) = \rho\_{E\_i}' = p\_i \mathcal{U}\_{E\_i} \rho\_{E\_i} \mathcal{U}\_{E\_i}^\dagger + (1 - p\_i)\rho\_{E\_i}$$

which is also a random unitary channel. The classicality of the environment can be evaluated as

$$J(\rho\_{\mathbb{E}\_i}, \Phi\_{\mathbb{E}\_i}) = 1 + \text{tr}\left(\sqrt{\rho\_{\mathbb{E}\_i}} \Phi\_{\mathbb{E}\_i}^\dagger(\sqrt{\rho\_{\mathbb{E}\_i}})\right) = 2 - p\_i + p\_i \text{tr}(\sqrt{\rho\_{\mathbb{E}\_i}} \mathcal{U}\_{\mathbb{E}\_i}^\dagger \sqrt{\rho\_{\mathbb{E}\_i}} \mathcal{U}\_{\mathbb{E}\_i}) \tag{34}$$

,

and

$$J(\rho'\_{E\_i\prime}, \Phi\_{E\_i}) = 1 + \text{tr}(\sqrt{\rho'\_{E\_i}} \Phi\_{E\_i}^\dagger(\sqrt{\rho'\_{E\_i}})) = 2 - p\_i + p\_i \text{tr}(\sqrt{\rho'\_{E\_i}} \mathcal{U}\_{E\_i}^\dagger \sqrt{\rho'\_{E\_i}} \mathcal{U}\_{E\_i}).$$

Comparing Equations (33) and (34), we ge<sup>t</sup>

$$(2 - J(\rho\_{E'} \Phi\_E) = \sum\_{i=1}^d \left(2 - J(\rho\_{E\_i'} \Phi\_{E\_i})\right)\_{i=1}$$

or equivalently,

$$I(\rho\_{E'}, \Phi\_E) = \sum\_{i=1}^d I(\rho\_{E\_{i'}}, \Phi\_{E\_i})\_{i'}$$

which shows a kind of additivity property, as intuitively expected since the initial environment is in a product state *ρE* = *ρ<sup>E</sup>*1 ⊗ *ρ<sup>E</sup>*2 ⊗···⊗ *ρEd* .

In terms of the classicality *J*(*ρ<sup>E</sup>*, <sup>Φ</sup>*E*) of the environment, we define the influence on the total environment caused by the system as

$$D(\rho\_E|\mathbf{U}, \Pi) = f(\rho\_{E'}^\prime \Phi\_E) - f(\rho\_{E'} \Phi\_E) \tag{35}$$

and the influence on the *i*-th sub-environment caused by the system as

$$D(\rho\_{E\_i}|\mathcal{U}, \Pi) = f(\rho\_{E\_{i'}}', \Phi\_{E\_i}) - f(\rho\_{E\_{i'}}, \Phi\_{E\_i}), \tag{36}$$

.

respectively. Compared with Equation (27), we have deliberately swapped the place of *U* and Π to indicate the difference of the reference channels. The influence on the environment caused by the system can be explicitly evaluated as

$$D(\rho\_E|\mathcal{U}, \Pi) = \sum\_{i=1}^d p\_i \text{tr}\left(\sqrt{\rho\_E'} \mathcal{U}\_{E\_i}^\dagger \sqrt{\rho\_E'} \mathcal{U}\_{E\_i} - \sqrt{\rho\_E} \mathcal{U}\_{E\_i}^\dagger \sqrt{\rho\_E} \mathcal{U}\_{E\_i}\right).$$

Similarly,

$$D(\rho\_{E\_i}|\mathcal{U}, \Pi) = p\_i \text{tr}\left(\sqrt{\rho\_{E\_i}^{\prime}} \mathcal{U}\_{E\_i}^{\dagger} \sqrt{\rho\_{E\_i}^{\prime}} \mathcal{U}\_{E\_i} - \sqrt{\rho\_{E\_i}} \mathcal{U}\_{E\_i}^{\dagger} \sqrt{\rho\_{E\_i}} \mathcal{U}\_{E\_i}\right)^{\dagger}$$

If we use the alternative quantifier of classicality *<sup>C</sup>*(*ρ*, K), then the classicality of the initial and final environment states with respect to the reduced environment channel can be evaluated as

$$\begin{aligned} \mathbb{C}(\rho\_E, \Phi\_E) &= 1 - \text{tr}(\rho\_E^2) + \sum\_{i=1}^d p\_i^2 (2 - p\_i d\_i) + d - 2 \\ \mathbb{C}(\rho\_{E'}^{'}, \Phi\_E) &= 1 - \text{tr}(\rho\_E^{'2}) + \sum\_{i=1}^d p\_i^2 (2 - p\_i d\_i) + d - 2. \end{aligned}$$

where *di* is the dimension of the *i*-th sub-environment. In terms of the classicality of the environment *<sup>C</sup>*(*ρ<sup>E</sup>*, <sup>Φ</sup>*E*), we have an alternative measure of influence on the total environment caused by the system as

$$F(\rho\_E|\mathbf{U}, \Pi) = \mathbb{C}(\rho\_{E'}'\Phi\_E) - \mathbb{C}(\rho\_{E'}\Phi\_E).$$

and the influence on the *i*-th environment caused by the system as

$$F(\rho\_{E\_i}|\mathbf{U}, \Pi) = \mathbb{C}(\rho\_{E\_{i'}}' \Phi\_{E\_i}) - \mathbb{C}(\rho\_{E\_{i'}} \Phi\_{E\_i})\_{\prime}$$

respectively. It turns out that

$$\begin{aligned} F(\rho\_E|\mathbf{U}, \Pi) &= \text{tr}(\rho\_E^2) - \text{tr}(\rho\_E'^2) \\ F(\rho\_{E\_i}|\mathbf{U}, \Pi) &= \text{tr}(\rho\_{E\_i}^2) - \text{tr}(\rho\_{E\_i}'^2) .\end{aligned}$$

These quantities of influence on the environment (caused by the system) may be compared with the quantifiers of decoherence of the system (caused by the environment), and they should be correlated due to the system-environment coupling.

#### **5. Illustrating Decoherence in Interferometry**

with

We illustrate the effectiveness of the quantifiers proposed in the preceding section with a two-path interferometer, as depicted in Figure 2. The system Hilbert space of interest here is effectively a qubit space with the two paths labeled as Π1 = |0-0| and Π2 = |1-1|. Let the initial system state (the path degree part of the physical state) be

$$\rho\_S = \frac{1}{2} \left( \mathbf{1} + \sum\_{i=1}^3 r\_j \sigma\_{\bar{\jmath}} \right) = \frac{1}{2} \begin{pmatrix} 1 + r\_3 & r\_1 - ir\_2 \\ r\_1 + ir\_2 & 1 - r\_3 \end{pmatrix} \tag{37}$$

with the Bloch vector *r* = (*<sup>r</sup>*1,*r*2,*r*3) ∈ R<sup>3</sup> satisfying |*r*| = *r*21 + *r*22 + *r*23 ≤ 1 and *σj* being the Pauli spin matrices. The eigenvalues of *ρS* are (1 ± |*r*|)/2. It can be directly evaluated that

$$
\sqrt{\rho\_S} = \frac{1}{2\sqrt{\gamma}} \begin{pmatrix} \gamma + r\_3 & r\_1 - ir\_2 \\ r\_1 + ir\_2 & \gamma - r\_3 \end{pmatrix}
$$

$$
\gamma = 1 + \sqrt{1 - |r|^2}.\tag{38}
$$

$$\begin{array}{c|c} \Pi\_{\mathcal{U}} = \Pi\_1 \otimes U\_{E\_1} + \Pi\_2 \otimes U\_{E\_2} \\ \rho\_S = \begin{array}{c} \text{Path } \Pi\_1 = |0\rangle(0| \\ \hline \end{array} \\ \rho\_{E\_1} = \begin{array}{c} \text{Path } \Pi\_1 = |0\rangle(0| \\ \hline \end{array} \\ \rho\_{E\_2} = |1\rangle(1| \\ \hline \end{array} \\ \rho\_{E\_2} = |1\rangle(1| \\ \hline \end{array} \\ \rho\_E' = \text{tr}\_S \rho\_{SE}'$$

**Figure 2.** Schematic illustration of decoherence induced by the array of path detectors *U* = {*UEi* : *i* = 1, 2} (serving as the environment of the system state *ρS*) attached to the collection of path Π = {<sup>Π</sup>*i* : *i* = 1, <sup>2</sup>}. Each path Π*i* is probed by a detector *UEi* . The initial system state is *ρS*, while the initial array of detector state is *ρE* = *ρE*1 ⊗ *ρE*2 . The combined initial state is *ρSE* = *ρS* ⊗ *ρE*. The system–detector coupling is via the combined unitary operator Π*U* = ∑<sup>2</sup>*i*=<sup>1</sup> Π*i* ⊗ *UEi* , and the final combined system is *<sup>ρ</sup>SE* = Π*U <sup>ρ</sup>SE*Π†*U* with final system state *ρS* = tr*ρSE*. The decoherence of *ρS* (with respect to Π) induced by the path detectors are quantified by *<sup>D</sup>*(*ρS*|<sup>Π</sup>, *U*) = *J*(*ρS*, Π) − *J*(*ρS*, Π) and *<sup>F</sup>*(*ρS*|<sup>Π</sup>, *U*) = *<sup>C</sup>*(*ρS*, Π) − *<sup>C</sup>*(*ρS*, <sup>Π</sup>), which are the increasing amount of classicality of the system state caused by the path detectors.

For a two-path interferometer with a detector attached to each path, let *ρEi* be the initial detector state attached to path *i*; the system and detector evolve under the controlled-*U* operation

$$
\Pi\_{\mathcal{U}} = \Pi\_1 \otimes \mathcal{U}\_{E\_1} + \Pi\_2 \otimes \mathcal{U}\_{E\_2}.\tag{39}
$$

 From an information-theoretic point of view, this controlled-*U* operation correlates the quantum system and the detector and leads to the combined final state

$$\begin{split} \rho'\_{SE} &= \Pi\_{\mathcal{U}} (\rho\_S \otimes \rho\_E) \Pi\_{\mathcal{U}}^{\dagger} \\ &= (\Pi\_1 \rho\_S \Pi\_1) \otimes (\mathcal{U}\_{E\_1} \rho\_E \mathcal{U}\_{E\_1}^{\dagger}) + (\Pi\_1 \rho\_S \Pi\_2) \otimes (\mathcal{U}\_{E\_1} \rho\_E \mathcal{U}\_{E\_2}^{\dagger}) \\ &+ (\Pi\_2 \rho\_S \Pi\_1) \otimes (\mathcal{U}\_{E\_2} \rho\_E \mathcal{U}\_{E\_1}^{\dagger}) + (\Pi\_2 \rho\_S \Pi\_2) \otimes (\mathcal{U}\_{E\_2} \rho\_E \mathcal{U}\_{E\_2}^{\dagger}). \end{split} \tag{40}$$

The final system state can be obtained by taking the partial trace over the detector as

$$\begin{split} \rho'\_{S} &= \text{tr}\_{E}\rho'\_{SE} \\ &= \Pi\_{1}\rho\_{S}\Pi\_{1} + \Pi\_{1}\rho\_{S}\Pi\_{2}\text{tr}(\mathcal{U}\_{E\_{1}}\rho\_{E}\mathcal{U}\_{E\_{2}}^{\dagger}) + \Pi\_{2}\rho\_{S}\Pi\_{1}\text{tr}(\mathcal{U}\_{E\_{2}}\rho\_{E}\mathcal{U}\_{E\_{1}}^{\dagger}) + \Pi\_{2}\rho\_{S}\Pi\_{2} \\ &= \frac{1}{2}\begin{pmatrix} 1 + r\_{3} & (r\_{1} - ir\_{2})\mathcal{V}^{\*} \\ (r\_{1} + ir\_{2})\mathcal{V} & 1 - r\_{3} \end{pmatrix} .\end{split}$$

with

$$\mathcal{V} = \text{tr}(\mathcal{U}\_{E\_2}\rho\_E \mathcal{U}\_{E\_1}^\dagger) = \text{tr}(\mathcal{U}\_{E\_2}\rho\_{E\_2}) \cdot \text{tr}(\rho\_{E\_1}\mathcal{U}\_{E\_1}^\dagger)$$

being a complex number. By the Cauchy–Schwarz inequality, we have

$$\begin{split} \left| \left| \mathcal{V} \right|^{2} = \left| \text{tr} \left( (\mathcal{U}\_{E\_{2}} \sqrt{\rho\_{E}}) (\mathcal{U}\_{E\_{1}} \sqrt{\rho\_{E}})^{\dagger} \right) \right|^{2} \\ \leq & \text{tr} \left( (\mathcal{U}\_{E\_{2}} \sqrt{\rho\_{E}}) (\mathcal{U}\_{E\_{2}} \sqrt{\rho\_{E}})^{\dagger} \right) \cdot \text{tr} \left( (\mathcal{U}\_{E\_{1}} \sqrt{\rho\_{E}}) (\mathcal{U}\_{E\_{1}} \sqrt{\rho\_{E}})^{\dagger} \right) \right| \\ = & 1. \end{split}$$

The eigenvalues of *ρS* = <sup>Φ</sup>*S*(*ρS*) are

$$\lambda\_1(\rho\_S') = \frac{1}{2} \left( 1 + \sqrt{r\_3^2 + (r\_1^2 + r\_2^2)|\mathcal{V}|^2} \right) \tag{41}$$

$$
\lambda\_2(\rho\_S') = \frac{1}{2} \left( 1 - \sqrt{r\_3^2 + (r\_1^2 + r\_2^2)|\mathcal{V}|^2} \right). \tag{42}
$$

Consequently, we see that

$$
\lambda\left(\rho\_S'\right) \prec \lambda\left(\rho\_S\right),
$$

as it should be by Proposition 2.

> Noting that

$$
\sqrt{\rho\_S'} = \frac{1}{2\sqrt{\gamma'}} \begin{pmatrix} \gamma' + r\_3 & (r\_1 - ir\_2)\mathcal{V}^\* \\ (r\_1 + ir\_2)\mathcal{V} & \gamma' - r\_3 \end{pmatrix} \tag{49}
$$

$$
\gamma' = \pm \sqrt{\frac{1}{1 + (\gamma' + r\_1)(111^2 - r\_2^2)}} \tag{40}
$$

where

$$\gamma' = 1 + \sqrt{1 - (r\_1^2 + r\_2^2)|\mathcal{V}|^2 - r\_{3'}^2} \tag{43}$$

we obtain

$$\begin{split} D(\rho\_S|\Pi, \Pi) &= J(\rho\_S', \Pi) - J(\rho\_S, \Pi) \\ &= \left(\frac{\gamma' + r\_3}{2\sqrt{\gamma'}}\right)^2 + \left(\frac{\gamma' - r\_3}{2\sqrt{\gamma'}}\right)^2 - \left(\frac{\gamma + r\_3}{2\sqrt{\gamma}}\right)^2 - \left(\frac{\gamma - r\_3}{2\sqrt{\gamma}}\right)^2 \\ &= \frac{1}{2}(\gamma' - \gamma)\left(1 - \frac{r\_3^2}{\gamma'\gamma}\right). \end{split} \tag{44}$$

Since *γ* ≥ *γ*, *γ* ≥ 1 ≥ *r*3, we see readily that the above quantity is non-negative. Moreover, because *γ* is a decreasing function of |V|2, from Equation (44) we see that *<sup>D</sup>*(*ρS*|<sup>Π</sup>, *U*) is a decreasing function of |V|2.

Similarly, we can evaluate

$$\begin{aligned} \mathcal{C}(\rho'\_{S'}\Pi) &= 1 - \frac{1}{4} \left( (1+r\_3)^2 + (r\_1^2 + r\_2^2)|\mathcal{V}|^2 + (1-r\_3)^2 + (r\_1^2 + r\_2^2)|\mathcal{V}|^2 \right) \\ &+ \frac{1}{4} \left( (1+r\_3)^2 + (1-r\_3)^2 \right) \\ &= 1 - \frac{1}{2} (r\_1^2 + r\_2^2)|\mathcal{V}|^2 \end{aligned}$$

from which we obtain

$$F(\rho\_{\mathcal{S}}|\Pi, \mathcal{U}) = \mathbb{C}(\rho\_{\mathcal{S}}', \Pi) - \mathbb{C}(\rho\_{\mathcal{S}}, \Pi) = \frac{1}{2}(r\_1^2 + r\_2^2)(1 - |\mathcal{V}|^2),\tag{45}$$

which is also apparently a decreasing function of |V|.

It can be easily verified that both the decoherence quantifiers *<sup>D</sup>*(*ρS*|<sup>Π</sup>, *U*) and *<sup>F</sup>*(*ρS*|<sup>Π</sup>, *U*) are decreasing functions of |V|, and achieve the minimal value 0 when |V| = 1, which corresponds to the situation when the detector does not obtain the path information. In this case, coherence is preserved, and there is no decoherence. This is consistent with our intuition since decoherence can be regarded as the washing out of interference, while *<sup>D</sup>*(*ρS*|<sup>Π</sup>, *U*) and *<sup>F</sup>*(*ρS*|<sup>Π</sup>, *U*) can be regarded as measures of path information leakage to the detectors (classical path information). The detectors, from which we can obtain the path information of the quantum system, would inevitably reduce the interference ability of the quantum system.

The quantity V = tr(*UE*2*<sup>ρ</sup>EU*† *E*1) arises naturally in at least two other contexts:

 (a) If we take *UE*1 = **1** and *UE*2 = *U*, then we come to the setup of Englert [34], in which |V| is the fringe visibility in the complementarity relation

$$|\mathcal{V}|^2 + \mathcal{D}^2 \le 1,$$

with D = 1 2 tr|*<sup>U</sup>ρEU*† − *ρE*| being the quantitative measure of distinguishability. In this context, V is also called the interference function.

(b) If we define the generalized variance of measuring any operator *X* in state *σ* as

$$V(\sigma, X) = \text{tr}\left(\sigma (X - \text{tr}(\sigma X)) (X - \text{tr}(\sigma X))^\dagger\right),$$

and consider the unitary operator *U*† *E*1*UE*2 = *U*† *E*1⊗ *UE*2 , then we have

$$\begin{split} \mathcal{V}(\rho\_{E'} \mathcal{U}\_{E\_1}^\dagger \mathcal{U}\_{E\_2}) &= \text{tr}\{\rho\_E(\mathcal{U}\_{E\_1}^\dagger \mathcal{U}\_{E\_2} - \text{tr}(\rho\_E \mathcal{U}\_{E\_1}^\dagger \mathcal{U}\_{E\_2}))(\mathcal{U}\_{E\_1}^\dagger \mathcal{U}\_{E\_2} - \text{tr}(\rho\_E \mathcal{U}\_{E\_1}^\dagger \mathcal{U}\_{E\_2}))^\dagger\} \\ &= 1 - |\text{tr}(\mathcal{U}\_{E\_2} \rho\_E \mathcal{U}\_{E\_1}^\dagger)|^2 \\ &= 1 - |\mathcal{V}|^2 \end{split}$$

which is a kind of measure of path detecting capability. The above relation immediately leads to

$$V(\rho\_{E'} \| \boldsymbol{I}\_{E\_1}^\dagger \boldsymbol{\mathcal{U}}\_{E\_2}) + |\mathcal{V}|^2 = 1,$$

which is apparently a complementary relation between the path information and fringe visibility. Furthermore, combined with Equation (45), we have

$$F(\rho\_S|\Pi, \mathcal{U}) = \frac{1}{2} (r\_1^2 + r\_2^2) V(\rho\_{E\_1} \mathcal{U}\_{E\_1}^\dagger \mathcal{U}\_{E\_2}) \,\prime$$

which relates the decoherence directly with the path-detecting information. This is consistent with our intuitive understanding of decoherence as the information leakage to the detectors (environment).

Now we make some comparison of our quantifiers of decoherence with existing ones. Since, in general, decoherence is also regarded as the establishment of correlations between the system and environment, it is expected that decoherence should be related to correlations, as quantified by the mutual information between the system and the environment. For simplicity, we consider the setup described by Figure 2 and assume that the initial system state and environment state are both pure. In this case, the final system–environment state *<sup>ρ</sup>SE* is pure since the coupling Π*U* is unitary. Consequently, the mutual information of the final system–environment state is

$$I(\rho\_{SE}') = S(\rho\_S') + S(\rho\_E') - S(\rho\_{SE}') = 2S(\rho\_S') = 2\left(-\lambda\_1(\rho\_S')\ln\lambda\_1(\rho\_S') - \lambda\_2(\rho\_S')\ln\lambda\_2(\rho\_S')\right),$$

where *<sup>λ</sup>j*(*ρ S*) are determined by Equations (41) and (42), and *<sup>S</sup>*(*σ*) = −tr *σ*ln *σ* is the von Neumann entropy of the state *σ*. Since the initial system state *ρS* defined by Equation (37) is pure, we have *r*2 1 + *r*2 2 + *r*2 3 = 1. Therefore by Equations (41), (42) and (45), we have

$$\begin{aligned} \lambda\_1(\rho\_S') &= \frac{1}{2} \left( 1 + \sqrt{1 - (r\_1^2 + r\_2^2)(1 - |\mathcal{V}|^2)} \right) = \frac{1}{2} \left( 1 + \sqrt{1 - 2F(\rho\_S|\Pi, l\mathcal{U})} \right), \\\lambda\_2(\rho\_S') &= \frac{1}{2} \left( 1 - \sqrt{1 - (r\_1^2 + r\_2^2)(1 - |\mathcal{V}|^2)} \right) = \frac{1}{2} \left( 1 - \sqrt{1 - 2F(\rho\_S|\Pi, l\mathcal{U})} \right). \end{aligned}$$

Now the mutual information can be expressed as

$$I(\rho\_{SE}') = 2H\left(\frac{1}{2}\left(1 + \sqrt{1 - 2F(\rho\_S|\Pi, \mathcal{U})}\right)\right),$$

where *<sup>H</sup>*(*p*) = −*p*ln*p* − (1 − *p*)ln(<sup>1</sup> − *p*) is the binary Shannon entropy function, 0 ≤ *p* ≤ 1. From the above equation, we see that the mutual information is monotonically related to the decoherence: when decoherence increases, the mutual information increases, which is consistent with the intuition that larger decoherence corresponds to larger amount of correlations established between the system and the environment (larger information leakage to the environment). Although the above result is proved for initial pure states and *<sup>F</sup>*(*ρS*|<sup>Π</sup>, *<sup>U</sup>*), the general cases concerning mixed initial states and the decoherence quantifier *<sup>D</sup>*(*ρS*|<sup>Π</sup>, *U*) are similar, but the calculations are more complicated. It will be also interesting to make a more comprehensive comparative studies between various quantities related to decoherence and correlations.

### **6. Summary**

In order to quantify decoherence induced by environment, we reviewed two quantifiers of classicality in a general setup of state–channel interaction by exploiting the Jordan symmetric product and a modified notion of variance. These quantifiers may be of independent interest in addressing the classical–quantum interplay. We also elucidated some simple ye<sup>t</sup> useful features of the decoherence channel (Hadamard channel).

Employing the above quantifiers of classicality, we introduced two quantifiers of decoherence induced by environment in the combined "system + environment" setup. These quantifiers have some nice properties and can be used to summarize the decoherence strength of an open system. Connections with complementarity were discussed. The results were illustrated via a two-path interferometer.

A natural approach to quantifying decoherence is via correlations between the system and the environment. There are various quantifiers for correlations such as the quantum mutual information, entanglement, quantum discord, measurement-induced disturbance, measurement-induced nonlocality, classical correlations, etc. In particular, decoherence is quantified from a decorrelating perspective in Refs. [106,107]. However, correlations are generally hard to evaluate. Our present approaches differ from the conventional approach to decoherence via correlations such as quantum mutual information. Our quantifiers of decoherence are relatively easier to calculate and have intimate relations with the Wigner–Yanase skew information, uncertainty, and the resource theory of coherence. This indicates certain operational significance of the quantities. However, it remains to further study the operational meaning of these quantifiers of decoherence and to investigate their implications for foundational issues and experimental practices.

For open quantum systems, apart from decoherence, another prominent characteristic is quantum Markovianity/non-Markovianity [108–116]. Although the classical Markovianity is uniquely defined and well understood, there is not a single universally accepted definition of quantum Markovianity. A host of quantum Markovianity-related concepts coexist, such as GKS–Lindblad master equations, distinguishability, divisibility, no-information backflow, monotonic decreasing in correlations, etc. However, just like decoherence is related to the decaying of off-diagonal entries of the density matrix, a general common feature of the various Markovianities is related to information loss and memoryless effects. This indicates that there are intimate relations between decoherence and Markovianity. We remark that the feature of decoherence as information monotonically flowing into the environment is deeply related to the Markovian approximation. In non-Markovian dynamics, in contrast to decoherence, recoherence may occur. The interplay and relations between decoherence and quantum Markovianity/non-Markovianity are worth further investigations.

**Author Contributions:** All authors contributed equally to the work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Key R&D Program of China, Grant No. 2020YFA0712700; the Fundamental Research Funds for the Central Universities, Grant No. FRF-TP-19-012A3; and the National Natural Science Foundation of China, Grant Nos. 11875317 and 61833010.

**Data Availability Statement:** Not appliable.

**Conflicts of Interest:** The authors declare no conflict of interest.
