1. Introduction
The pivotal and indispensable role of complex numbers in quantum mechanics is undeniable, exerting profound and widespread influence. Complex numbers are important in building the framework of quantum mechanics. By using complex numbers, quantum states and wave functions can accurately describe the behavior of microscopic particles [1-3]. In the dynamic evolution of quantum mechanics [4], complex numbers are particularly crucial. As an integral part of the Schrödinger equation [5-11], the introduction of the imaginary unit 'i' not only accurately depicts the evolutionary track of the wave function over time [12-18], but also endows essential phase information [19-21]. This phase information holds immeasurable value in comprehending unique quantum phenomena like interference [22-24] and diffraction [25-27], while also laying a solid theoretical foundation for the advancement of cutting-edge technologies such as quantum computing [28-30] and quantum communication [31-33]. In terms of experiments, the application of complex numbers in quantum mechanics has been thoroughly validated. By measuring the real and imaginary parts of the wave function [34-42], the accuracy of quantum theory prediction is verified. This further reveals the mysteries of the quantum world. Therefore, we can gain a more comprehensive grasp of the operational laws of the microscopic realm by studying the imaginary numbers.
In light of the pivotal role of complex numbers in quantum mechanics, Hickey and Gour [43] proposed a landmark approach in 2018, treating the imaginary component of quantum states as a resource for investigation and subsequently constructing a comprehensive framework, in which free states and free operations have been elucidated. Many imaginarity quantifiers, such as the trace norm of imaginarity [43], the fidelity of imaginarity [44], the weight of imaginarity [45], the relative entropy of imaginarity [45], the robustness of imaginarity [46], the l1 norm of imaginarity [47], the geometric-like imaginarity [48], have been introduced and studied. The relationship between the relative entropy of imaginarity and the l1 norm of imaginarity and the quantum state order under different quantum channels have been revealed [47].
The above imaginarity measures of a quantum state ρ are defined by taking the minimum over all free states σ for some distance measures between states ρ and σ. It is also possible to define the measures by the distance from a quantum state to its own conjugate, such as the Tsallis relative entropy of imaginarity [49], the α-z Rényi relative entropy imaginarity [50] and the Tsallis relative operator entropy of imaginarity [50]. The problem of imaginarity of Gaussian states [49,51] and Kirkwood-Dirac imaginarity have also been considered [52].
In manipulating the imaginarity resource, the main question is can any two given quantum states be converted into each other via free operations? Sufficient and necessary conditions for the validity of conversion under the circumstance of any two pure states [43] and any two qubit states [46] have been obtained. Moreover, the imaginarity resource in distributed scenarios has been discussed [53]. Like the coherence resource [54,55], the freezing imaginarity of quantum state [56] and the nonlocal advantages of imaginarity [57] have been investigated. The connections between coherence and imaginarity have also been explored [58,59].
The remainder of this paper is arranged as follows. In section 2 , we review the fundamental concepts of the quantum imaginary resource theory. In section 3 , we propose two imaginarity monotones induced by the unified-(α, β) relative entropy and explore their properties. In section 4 , we give some examples to illustrate our results. The conclusion is provided in section 5 .
2. Preliminaries
We denote by ${ \mathcal H }$f a d-dimensional Hilbert space, ${\{| j\rangle \}}_{j=0}^{d-1}$ a fixed orthonormal basis in ${ \mathcal H }$, ${ \mathcal D }({ \mathcal H })$ as the set of density operators (quantum states) ρ acting on ${ \mathcal H }$ and ρ* the conjugate of ρ. We recall the framework of the imaginarity resource theory.
Definition 1 [43] Let ρ be a quantum state on ${ \mathcal H }$. ρ is a free state or real state, if
$\begin{eqnarray}\rho =\displaystyle \sum _{jk}{\rho }_{jk}\,| j\rangle \langle k| ,\end{eqnarray}$
where ${\rho }_{jk}\in {\mathbb{R}}$ for arbitrary j,k ∈ {0, 1, …, d-1}. We denote the set of all real states by ${ \mathcal F }$. It can be easily obtained that $\rho \in { \mathcal F }$ if and only if ρ is symmetric, that is, ρT=ρ, where ρT denotes the transpose of ρ.Definition 2 [43]Let λ be a quantum operation, ρ be a quantum state and ${\rm{\Lambda }}(\rho )={\sum }_{j}\,{K}_{j}\rho {K}_{j}^{\dagger }$. λ is a free operation or real operation if
$\begin{eqnarray}\langle m| {K}_{j}| n\rangle \in {\mathbb{R}},\end{eqnarray}$
for arbitrary j and m, n ∈ {0, 1, …, d-1}.Definition 3 [43] An imaginarity measure is a functional M: ${ \mathcal D }({ \mathcal H })\to [0,\infty )$ satisfying the following properties:
(M1) Nonnegativity. M(ρ) ≥ 0, M(ρ)=0 if and only if $\rho \in { \mathcal F }$.
(M2) Imaginarity monotonicity. M(λ(ρ)) ≤ M(ρ) whenever λ is a free operation.
(M3) Strong imaginarity monotonicity. M(ρ) ≥ ∑j pj M(ρj), where ${p}_{j}={\rm{tr}}({K}_{j}\rho {K}_{j}^{\dagger })$, ${\rho }_{j}={K}_{j}\rho {K}_{j}^{\dagger }/{p}_{j}$ with free Kraus operators Kj.
(M4) Convexity. ∑j pj M(ρj) ≥ M(∑j pj ρj) for any set of states {ρj} and any pj ≥ 0 with ∑j pj =1.
Besides, the following property (M5) has also been introduced.
(M5) Additivity. M(pρ1 ⊕ (1-p)ρ2)=pM(ρ1)+(1-p)M(ρ2), where p ∈ (0, 1).
It has been proven that (M2) and (M5) are equivalent to (M3) and (M4) [45]. We will call M an imaginarity monotone if it satisfies (M1), (M2) and (M4).
Recently, many imaginarity measures have been proposed. The relative entropy of imaginarity is defined by [45]
$\begin{eqnarray}{M}_{r}(\rho )=S\left[\frac{1}{2}(\rho +{\rho }^{{\rm{T}}})\right]-S(\rho ),\end{eqnarray}$
Tsallis relative entropy of imaginarity is defined by [49] $\begin{eqnarray}{M}_{T,u}(\rho )=1-{\rm{tr}}\left[{\rho }^{u}{({\rho }^{* })}^{1-u}\right],\end{eqnarray}$
where u ∈ (0, 1) and α-z-Rényi relative entropy of imaginarity is defined by [50] $\begin{eqnarray}{M}_{\alpha ,z}^{R}(\rho )=1-{f}_{\alpha ,z}(\rho ,{\rho }^{* }),\end{eqnarray}$
where fα,z(ρ, σ)=tr${\left({\sigma }^{\frac{1-\alpha }{2z}}{\rho }^{\frac{\alpha }{z}}{\sigma }^{\frac{1-\alpha }{2z}}\right)}^{z}$ and $0\lt \max \{\alpha ,1-\alpha \}\leqslant z\lt 1$. We denote ${M}_{\alpha ,z}^{R}(\rho )$ by ${M}_{\alpha }^{R}(\rho )$ when α=z.The unified (α, β)-relative entropy has been introduced in [60].
Definition 4 [60] For any α ∈ [0, 1] and any $\beta \in {\mathbb{R}}$, the unified (α, β)-relative entropy is defined by
$\begin{eqnarray}{D}_{\alpha }^{\beta }(\rho | | \sigma )=\left\{\begin{array}{ll}{H}_{\alpha }^{\beta }(\rho | | \sigma ) & \,\rm{if}\,\,0\leqslant \alpha \lt 1,\beta \ne 0,\\ {H}_{\alpha }(\rho | | \sigma ) & \,\rm{if}\,\,0\leqslant \alpha \lt 1,\beta =0,\\ {H}^{\alpha }(\rho | | \sigma ) & \,\rm{if}\,\,0\leqslant \alpha \lt 1,\beta =1,\\ {}_{\frac{1}{\alpha }}H(\rho | | \sigma ) & \,\rm{if}\,\,0\lt \alpha \lt 1,\beta =\frac{1}{\alpha },\\ H(\rho | | \sigma ) & \,\rm{if}\,\alpha =1,\end{array}\right.\end{eqnarray}$
where $\begin{eqnarray}{H}_{\alpha }^{\beta }(\rho | | \sigma )=\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha }))}^{\beta }-1],\end{eqnarray}$
$\begin{eqnarray}{H}_{\alpha }(\rho | | \sigma )=\frac{1}{(\alpha -1)}{\mathrm{log}}_{2}({\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })),\end{eqnarray}$
$\begin{eqnarray}{H}^{\alpha }(\rho | | \sigma )=\frac{1}{(\alpha -1)}[{\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })-1],\end{eqnarray}$
$\begin{eqnarray}{}_{\alpha }H(\rho | | \sigma )=\frac{1}{(\alpha -1)}[{\left({\rm{tr}}({\rho }^{\frac{1}{\alpha }}{\sigma }^{1-\frac{1}{\alpha }})\right)}^{\alpha }-1],\end{eqnarray}$
$\begin{eqnarray}H(\rho | | \sigma )={\rm{tr}}(\rho {\mathrm{log}}_{2}\rho )-{\rm{tr}}(\rho {\mathrm{log}}_{2}\sigma ).\end{eqnarray}$
It can be easily obtained that ${H}_{\alpha }^{\beta }(\rho | | \sigma )$ reduces to the Rényi relative entropy, Tsallis relative entropy and relative entropy when β →0, β=1 and α →1, respectively.
In order to facilitate the subsequent proofs of the theorems, we present the following lemmas.
Lemma 1. [60] The unified (α, β)-relative entropy satisfies the following properties for any quantum states ρ, σ and any α ∈ (0, 1) and β ∈ (0, 1]:
(i) ${D}_{\alpha }^{\beta }(\rho | | \sigma )\geqslant 0$ and ${D}_{\alpha }^{\beta }(\rho | | \sigma )=0$ if ρ=σ.
(ii) ${D}_{\alpha }^{\beta }({ \mathcal E }(\rho )| | { \mathcal E }(\sigma ))\leqslant {D}_{\alpha }^{\beta }(\rho | | \sigma )$ whenever ${ \mathcal E }$ is a quantum operation.
(iii) ${D}_{\alpha }^{\beta }({\sum }_{j}\,{\lambda }_{j}{\rho }_{j}| | {\sum }_{j}\,{\lambda }_{j}{\sigma }_{j})\leqslant {\sum }_{j}\,{\lambda }_{j}{D}_{\alpha }^{\beta }({\rho }_{j}| | {\sigma }_{j})$.
(iv) For any states ρ and σ on H1 ⨂ H2, ${D}_{\alpha }^{\beta }({\rho }_{1}| | {\sigma }_{1})\leqslant {D}_{\alpha }^{\beta }(\rho | | \sigma )$, where ρ1 and σ1 are the partial traces of ρ and σ on H1, respectively.
(v) ${D}_{\alpha }^{\beta }(\rho | | \sigma )$ is increasing with respect to α ∈ (0, 1) for fixed 0 < β≤1.
(vi) ${D}_{\alpha }^{\beta }(\rho | | \sigma )$ is decreasing with respect to β ∈ (0, 1] for fixed 0 < α < 1.
Lemma 2. [49] For any real operation ${ \mathcal E }$ and any state ρ, it holds that
$\begin{eqnarray}{ \mathcal E }({\rho }^{* })={[{ \mathcal E }(\rho )]}^{* }.\end{eqnarray}$
The following lemma plays an important role in the proof of equation (24 ) in Example 1, and the proof of it is given in appendix A.
Lemma 3. Let $A\gt \sqrt{{B}^{2}+{C}^{2}}$, B2+C2 > 0, α ∈ (0, 1) and $f(x,\theta )=A[{x}^{1-\alpha }+{(1-x)}^{1-\alpha }]+(B\sin \theta +C\cos \theta )[{(1-x)}^{1-\alpha }\,-{x}^{1-\alpha }],$ where $x\in [0,\frac{1}{2}]$, θ ∈ [0, 2π]. Then f(x, θ) attains its maximum value at $({x}_{0},{\theta }_{0})=\left(\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1},\arcsin \frac{B}{\sqrt{{B}^{2}+{C}^{2}}}\right)$.
Finally, lemma 4 is crucial in proving item (i) of theorem 9, and the proof is given in appendix B .
Lemma 4.Suppose that σρ and στ are the quantum state that reaches the minimum in equation (20 ) when the input states are ρ and τ, respectively. Then pσρ ⊕ (1-p)στ is a quantum state achieving the minimum in equation (20 ) when the input state is pρ ⊕ (1-p)τ.
3. Imaginarity monotones based on the unified (α, β)-relative entropy
In this section, we mainly investigate two different imaginarity monotones induced by the unified (α, β)-relative entropy, along with its properties.
Now, we define the first imaginarity quantifier induced by unified (α, β)-relative entropy as
$\begin{eqnarray}{M}_{\alpha ,\beta }^{H}(\rho )=\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{({\rho }^{* })}^{1-\alpha }))}^{\beta }-1],\end{eqnarray}$
where α ∈ (0, 1), β ∈ (0, 1].Theorem 1. ${M}_{\alpha ,\beta }^{H}(\rho )$ defined by equation (
Proof.Due to Lemma 1(i), we know that ${M}_{\alpha ,\beta }^{H}(\rho )\,={D}_{\alpha }^{\beta }(\rho | | {\rho }^{* })\geqslant 0$ and
$\begin{eqnarray}{M}_{\alpha ,\beta }^{H}(\rho )=0\iff \rho ={\rho }^{* }\iff \rho \in { \mathcal F }.\end{eqnarray}$
Thus, ${M}_{\alpha ,\beta }^{H}(\rho )$ satisfies (M1). From Lemma 1(ii), we know ${D}_{\alpha }^{\beta }(\rho | | \sigma )$ is nonincreasing under any quantum operation when α ∈ (0, 1) and β ∈ (0, 1]. Let ${ \mathcal E }$ be any real operation. Combining Lemma 1(ii) and lemma 2, we have
$\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }^{H}({ \mathcal E }(\rho )) & =& \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({ \mathcal E }{(\rho )}^{\alpha }{({({ \mathcal E }(\rho ))}^{* })}^{1-\alpha }))}^{\beta }-1]\\ & =& \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({ \mathcal E }{(\rho )}^{\alpha }{({ \mathcal E }({\rho }^{* }))}^{1-\alpha }))}^{\beta }-1]\\ & =& {D}_{\alpha }^{\beta }({ \mathcal E }(\rho )| | { \mathcal E }({\rho }^{* }))\\ & \leqslant & {D}_{\alpha }^{\beta }(\rho | | {\rho }^{* })\\ & =& \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{({\rho }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & =& {M}_{\alpha ,\beta }^{H}(\rho ),\end{array}\end{eqnarray*}$
which implies that (M2) holds.Let pn ≥ 0 with ∑npn=1 and ${\rho }_{n}\in { \mathcal D }({ \mathcal H }).$ From Lemma 1(iii), we get
$\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }^{H}\left(\displaystyle \sum _{n}{p}_{n}{\rho }_{n}\right) & =& {D}_{\alpha }^{\beta }\left(\displaystyle \sum _{n}{p}_{n}{\rho }_{n}| | \displaystyle \sum _{n}{p}_{n}{\rho }_{n}^{* }\right)\\ & \leqslant & \displaystyle \sum _{n}{p}_{n}{D}_{\alpha }^{\beta }({\rho }_{n}| | {\rho }_{n}^{* })\\ & =& {\sum }_{n}{p}_{n}{M}_{\alpha ,\beta }^{H}({\rho }_{n}).\end{array}\end{eqnarray*}$
Hence, (M4) is proved. Therefore, ${M}_{\alpha ,\beta }^{H}(\rho )$ is an imaginarity monotone. This completes the proof.Furthermore, when ρ=∣ψ⟩⟨ψ∣ is a pure state, we have
$\begin{eqnarray}\displaystyle \begin{array}{rcl}{M}_{\alpha ,\beta }^{H}(\rho ) & =& {M}_{\alpha ,\beta }^{H}(| \psi \rangle \langle \psi | )=\frac{1}{(\alpha -1)\beta }\\ & & \times \,[({\rm{tr}}(| \psi \rangle \langle \psi | {\psi }^{* }\rangle \langle {\psi }^{* }| ){)}^{\beta }-1]\\ & =& \frac{1}{(\alpha -1)\beta }[| \langle \psi | {\psi }^{* }\rangle {| }^{2\beta }-1].\end{array}\end{eqnarray}$
In order to measure super-fidelity and sub-fidelity, a quantum network-based experimental scheme has been designed, in which ∣$\Psi$⟩12 is the program state as a mean value of σz of the measured controlling qubit (see figure 4 in [61]). By setting ∣$\Psi$⟩12=∣0⟩∣0⟩ one gets ${\rm{tr}}({\rho }_{1}{\rho }_{2})$ [61]. Noting that ${\rm{tr}}({\rho }_{1}{\rho }_{2})=| \langle \psi | {\psi }^{* }\rangle {| }^{2}$ when ρ1=∣ψ⟩⟨ψ∣ and ρ2=∣ψ*⟩⟨ψ*∣, it can be seen that the quantity in equation (15 ), i.e., ${M}_{\alpha ,\beta }^{H}(\rho )$ in the pure state case, can be measured using the scheme proposed in [61].
From equation (13 ), we can deduce that when β=1, ${M}_{\alpha ,\beta }^{H}(\rho )=\frac{1}{1-\alpha }{M}_{T,u}(\rho )$. Excluding the influence of coefficients, ${M}_{\alpha ,\beta }^{H}(\rho )$ reduces to the imaginarity measure in [49]. In addition, ${M}_{\alpha ,\beta }^{H}(\rho )$ has many elegant properties.
Theorem 2.For any quantum state ρ on ${ \mathcal H }$, and α ∈ (0, 1), β ∈ (0, 1], we have $\frac{1}{1-\alpha }{M}_{1-\alpha ,\beta }^{H}(\rho )=\frac{1}{\alpha }{M}_{\alpha ,\beta }^{H}(\rho ).$
Proof.Note that
$\begin{eqnarray*}\displaystyle \begin{array}{rcl}{M}_{1-\alpha ,\beta }^{H}(\rho ) & =& \frac{1}{-\alpha \beta }[{({\rm{tr}}({\rho }^{1-\alpha }{({\rho }^{* })}^{\alpha }))}^{\beta }-1]\\ & =& \frac{1-\alpha }{\alpha }\cdot \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{1-\alpha }{({\rho }^{* })}^{\alpha }))}^{\beta }-1]\\ & =& \frac{1-\alpha }{\alpha }{M}_{\alpha ,\beta }^{H}({\rho }^{* })=\frac{1-\alpha }{\alpha }{M}_{\alpha ,\beta }^{H}(\rho ).\end{array}\end{eqnarray*}$
So Theorem 2 holds. This completes the proof. Theorem 3.(Superadditivity under direct sum) For any quantum state ρ1, ρ2 on ${ \mathcal H }$, p ∈ [0, 1] and α ∈ (0, 1), β ∈ (0, 1], we have ${M}_{\alpha ,\beta }^{H}(p{\rho }_{1}\oplus (1-p){\rho }_{2})\geqslant p{M}_{\alpha ,\beta }^{H}({\rho }_{1})+(1-p){M}_{\alpha ,\beta }^{H}({\rho }_{2})$.
Proof.Direct calculation shows that
$\begin{eqnarray*}\begin{array}{rcl} & & {M}_{\alpha ,\beta }^{H}(p{\rho }_{1}\displaystyle \oplus (1-p){\rho }_{2})\\ & & \quad =\frac{1}{(\alpha -1)\beta }\\ & & \quad \quad \times [({\rm{tr}}({(p{\rho }_{1}\displaystyle \oplus (1-p){\rho }_{2})}^{\alpha }\\ & & \quad \quad \times {(p{\rho }_{1}^{* }\displaystyle \oplus (1-p){\rho }_{2}^{* })}^{1-\alpha }){)}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[({\rm{tr}}(p{\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha }\\ & & \quad \quad \,\displaystyle \oplus (1-p){\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha }){)}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[(p{\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha })\\ & & \quad \quad +\,(1-p){\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha }){)}^{\beta }-1],\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}p{M}_{\alpha ,\beta }^{H}({\rho }_{1})+(1-p){M}_{\alpha ,\beta }^{H}({\rho }_{2})\\ =\,\frac{1}{(\alpha -1)\beta }[p{({\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha }))}^{\beta }-p]\\ \,+\frac{1}{(\alpha -1)\beta }[(1-p){({\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha }))}^{\beta }-(1-p)]\\ =\,\frac{1}{(\alpha -1)\beta }[p{({\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha }))}^{\beta }\\ \,+(1-p){({\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha }))}^{\beta }-1].\end{array}\end{eqnarray*}$
Due to the concavity of f(x)=xβ, we obtain
$\begin{eqnarray*}\begin{array}{l}{(p{\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha })+(1-p){\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha }))}^{\beta }\\ \geqslant p{({\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha }))}^{\beta }+(1-p){({\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha }))}^{\beta }.\end{array}\end{eqnarray*}$
Therefore, ${M}_{\alpha ,\beta }^{H}(p{\rho }_{1}\oplus (1-p){\rho }_{2})\geqslant p{M}_{\alpha ,\beta }^{H}({\rho }_{1})+(1-p){M}_{\alpha ,\beta }^{H}({\rho }_{2})$. This completes the proof.Corollary 1. For α ∈ (0, 1) and β ∈ (0, 1], ${M}_{\alpha ,\beta }^{H}(\rho )$ satisfies the additivity under direct sum if and only if β=1.
Proof. From the proof of Theorem 3, it follows that the equality in equation (3 ) holds if ${\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha })={\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha })$ or β=1. However, ${\rm{tr}}({\rho }_{1}^{\alpha }{({\rho }_{1}^{* })}^{1-\alpha })={\rm{tr}}({\rho }_{2}^{\alpha }{({\rho }_{2}^{* })}^{1-\alpha })$ can not be true for any quantum states ρ1 and ρ2. Therefore, the equality in equation (3 ), and thus the additivity for ${M}_{\alpha ,\beta }^{H}(\rho )$ holds if β=1. This completes the proof.
Theorem 4.For quantum state ρ on ${{ \mathcal H }}_{1}\otimes {{ \mathcal H }}_{2}$, and α ∈ (0, 1), β ∈ (0, 1], we have ${M}_{\alpha ,\beta }^{H}({\rho }_{1})\leqslant {M}_{\alpha ,\beta }^{H}(\rho )$, where ρ1 is the partial trace of ρ on ${{ \mathcal H }}_{1}$.
Proof. Note that ${\rho }_{1}^{* }$ is the partial trace of ρ* on H1 if ρ1 is the partial trace of ρ on H1. Using Lemma 1(iv), we obtain ${D}_{\alpha }^{\beta }({\rho }_{1}| | {\rho }_{1}^{* })\leqslant {D}_{\alpha }^{\beta }(\rho | | {\rho }^{* })$, that is, ${M}_{\alpha ,\beta }^{H}({\rho }_{1})\leqslant {M}_{\alpha ,\beta }^{H}(\rho )$. This completes the proof.
Theorem 5. (Subadditivity under tensor product) For any quantum state ρ, τ on ${{ \mathcal H }}_{A}$ and ${{ \mathcal H }}_{B}$, respectively, and α ∈ (0, 1), β ∈ (0, 1], we have
$\begin{eqnarray}{M}_{\alpha ,\beta }^{H}(\rho \otimes \tau )\leqslant {M}_{\alpha ,\beta }^{H}(\rho )+{M}_{\alpha ,\beta }^{H}(\tau ).\end{eqnarray}$
Proof. Note that
$\begin{eqnarray*}\begin{array}{rcl} & & {M}_{\alpha ,\beta }^{H}(\rho \displaystyle \otimes \tau )\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({(\rho \displaystyle \otimes \tau )}^{\alpha }{({(\rho \displaystyle \otimes \tau )}^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({(\rho \displaystyle \otimes \tau )}^{\alpha }{({\rho }^{* }\displaystyle \otimes {\tau }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}(({\rho }^{\alpha }\displaystyle \otimes {\tau }^{\alpha })({({\rho }^{* })}^{1-\alpha }\displaystyle \otimes {({\tau }^{* })}^{1-\alpha })))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{({\rho }^{* })}^{1-\alpha }){\rm{tr}}({\tau }^{\alpha }{({\tau }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{({\rho }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad \quad \times [{({\rm{tr}}({\tau }^{\alpha }{({\tau }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad \quad +\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{({\rho }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad \quad +\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\tau }^{\alpha }{({\tau }^{* })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad =(\alpha -1)\beta {M}_{\alpha ,\beta }^{H}(\rho ){M}_{\alpha ,\beta }^{H}(\tau )\\ & & \quad \quad +{M}_{\alpha ,\beta }^{H}(\rho )+{M}_{\alpha ,\beta }^{H}(\tau ).\end{array}\end{eqnarray*}$
Since ${M}_{\alpha ,\beta }^{H}(\rho )$, ${M}_{\alpha ,\beta }^{H}(\tau )\geqslant 0$, and α ∈ (0, 1), β ∈ (0, 1], we obtain ${M}_{\alpha ,\beta }^{H}(\rho \otimes \tau )\leqslant {M}_{\alpha ,\beta }^{H}(\rho )+{M}_{\alpha ,\beta }^{H}(\tau )$. This completes the proof. Remark 1. (1) From the proof of Theorem 5, it can be seen that the equality in equation (16 ) holds if ${M}_{\alpha ,\beta }^{H}(\rho )=0$ or ${M}_{\alpha ,\beta }^{H}(\tau )=0$, i.e., if $\rho \in { \mathcal F }$ or $\tau \in { \mathcal F }$. In this case, equation (16 ) reduces to ${M}_{\alpha ,\beta }^{H}(\rho \otimes \tau )={M}_{\alpha ,\beta }^{H}(\tau )$ or ${M}_{\alpha ,\beta }^{H}(\rho \otimes \tau )={M}_{\alpha ,\beta }^{H}(\rho )$, which implies ancillary independence of the quantifier to some extent.
(2) It is worth pointing out that equation (16 ) does not hold for any bipartite quantum state ρab. To see this, consider the bipartite state ρab=(∣00⟩⟨00∣-i∣00⟩⟨11∣ +i∣11⟩⟨00∣+∣11⟩⟨11∣)/2. It follows that ${\rho }^{a}={\rho }^{b}=\frac{I}{2}\in { \mathcal F }$. Then ${M}_{\alpha ,\beta }^{H}({\rho }^{a})={M}_{\alpha ,\beta }^{H}({\rho }^{b})=0$ and ${M}_{\alpha ,\beta }^{H}({\rho }^{ab})\gt 0$, which implies that ${M}_{\alpha ,\beta }^{H}({\rho }^{ab})\gt {M}_{\alpha ,\beta }^{H}({\rho }^{a})+{M}_{\alpha ,\beta }^{H}({\rho }^{b})$.
(3) Note that the subadditivity under tensor product is an important property, and it holds for the Tsallis relative entropy of imaginarity, which has not been mentioned in [49]. In comparison, the property in Theorem 5 has not been discussed for the trace norm of imaginarity, the fidelity of imaginarity, the weight of imaginarity, the robustness of imaginarity, the l1 norm of imaginarity and the geometric-like imaginarity in previous literatures.
Theorem 6. For any α ∈ (0, 1) and β ∈ (0, 1], the imaginarity monotone ${M}_{\alpha ,\beta }^{H}(\rho )$ satisfies the following monotonicity:
(1) ${M}_{{\alpha }_{1},\beta }^{H}(\rho )\leqslant {M}_{{\alpha }_{2},\beta }^{H}(\rho )$ for fixed β when α1≤α2.
(2) ${M}_{\alpha ,{\beta }_{1}}^{H}(\rho )\geqslant {M}_{\alpha ,{\beta }_{2}}^{H}(\rho )$ for fixed α when β1≤β2.
Proof. (1) From Lemma 1(v), we have ${D}_{{\alpha }_{1}}^{\beta }(\rho | | \sigma )\leqslant {D}_{{\alpha }_{2}}^{\beta }(\rho | | \sigma )$ when β > 0 and α1 ≤ α2. Letting σ=ρ*, we have ${M}_{{\alpha }_{1},\beta }^{H}(\rho )\leqslant {M}_{{\alpha }_{2},\beta }^{H}(\rho ).$
(2) From Lemma 1(vi), we have ${D}_{\alpha }^{{\beta }_{1}}(\rho | | \sigma )\geqslant {D}_{\alpha }^{{\beta }_{2}}(\rho | | \sigma )$ when β1 ≤ β2. Letting σ=ρ*, we have ${M}_{\alpha ,{\beta }_{1}}^{H}(\rho )\geqslant {M}_{\alpha ,{\beta }_{2}}^{H}(\rho ).$ This completes the proof.
Theorem 7. For any quantum state ρ, the order of ${M}_{\alpha ,z}^{R}$, ${M}_{\alpha }^{R}$, ${M}_{\alpha }^{T}$ and ${M}_{\alpha ,\beta }^{H}$ is as follows:
$\begin{eqnarray}{M}_{\alpha }^{R}(\rho )\leqslant {M}_{\alpha ,z}^{R}(\rho )\leqslant {M}_{\alpha }^{T}(\rho )\leqslant {M}_{\alpha ,\beta }^{H}(\rho ).\end{eqnarray}$
Proof. From [50], we know that 18 -19 ), we obtain equation (17 ). This completes the proof.
$\begin{eqnarray}{M}_{\alpha }^{R}(\rho )\leqslant {M}_{\alpha ,z}^{R}(\rho )\leqslant {M}_{\alpha }^{T}(\rho ).\end{eqnarray}$
Using Theorem 6(2), we have $\begin{eqnarray}{M}_{\alpha }^{T}(\rho )={M}_{\alpha ,1}^{H}(\rho )\leqslant {M}_{\alpha ,\beta }^{H}(\rho ).\end{eqnarray}$
From equations (In quantum coherence theory [62-65], a common approach for constructing a coherence quantifier for a quantum state ρ is to minimize a distance measure D (ρ, σ) over all incoherent states σ. A similar scheme can be used in quantum imaginary resource theory. Now, we define the second imaginarity monotone induced by unified (α, β)-relative entropy as
$\begin{eqnarray}{M}_{\alpha ,\beta }^{E}(\rho )=\mathop{\min }\limits_{\sigma \in { \mathcal F }}\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha }))}^{\beta }-1],\end{eqnarray}$
where α ∈ (0, 1) and β ∈ (0, 1].Theorem 8. ${M}_{\alpha ,\beta }^{E}(\rho )$ defined by equation (20 ) is an imaginarity monotone.
Proof. From Lemma 1(i), we can easily obtain that ${M}_{\alpha ,\beta }^{E}(\rho )$ satisfies (M1). For (M2), it follows from Lemma 1(ii) that
$\begin{eqnarray*}{D}_{\alpha }^{\beta }({ \mathcal E }(\rho )| | { \mathcal E }(\sigma ))\leqslant {D}_{\alpha }^{\beta }(\rho | | \sigma ),\end{eqnarray*}$
holds for any real operation ${ \mathcal E }$, which yields that $\begin{eqnarray*}\mathop{\min }\limits_{\sigma \in { \mathcal F }}{D}_{\alpha }^{\beta }({ \mathcal E }(\rho )| | { \mathcal E }(\sigma ))\leqslant \mathop{\min }\limits_{\sigma \in { \mathcal F }}{D}_{\alpha }^{\beta }(\rho | | \sigma ),\end{eqnarray*}$
Therefore, we have $\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }^{E}({ \mathcal E }(\rho )) & =& \mathop{\min }\limits_{\sigma \in { \mathcal F }}\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({ \mathcal E }{(\rho )}^{\alpha }{\sigma }^{1-\alpha }))}^{\beta }-1]\\ & \leqslant & \mathop{\min }\limits_{\sigma \in { \mathcal F }}\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({ \mathcal E }{(\rho )}^{\alpha }{({ \mathcal E }(\sigma ))}^{1-\alpha }))}^{\beta }-1]\\ & =& \mathop{\min }\limits_{\sigma \in { \mathcal F }}{D}_{\alpha }^{\beta }({ \mathcal E }(\rho )| | { \mathcal E }(\sigma ))\\ & \leqslant & \mathop{\min }\limits_{\sigma \in { \mathcal F }}{D}_{\alpha }^{\beta }(\rho | | \sigma )\\ & =& \mathop{\min }\limits_{\sigma \in { \mathcal F }}\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha }))}^{\beta }-1]\\ & =& {M}_{\alpha ,\beta }^{E}(\rho ).\end{array}\end{eqnarray*}$
Let pn ≥ 0, ∑npn=1 and ${\rho }_{n}\in { \mathcal D }({ \mathcal H }).$ Then we have
$\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }^{E}\left({\sum }_{n}{p}_{n}{\rho }_{n}\right) & =& \mathop{\min }\limits_{\sigma \in { \mathcal F }}{D}_{\alpha }^{\beta }\left({\sum }_{n}{p}_{n}{\rho }_{n}| | \sigma \right)\\ & \leqslant & {D}_{\alpha }^{\beta }\left({\sum }_{n}{p}_{n}{\rho }_{n}| | {\sum }_{n}{p}_{n}{\sigma }_{n}\right)\\ & \leqslant & {\sum }_{n}{p}_{n}{D}_{\alpha }^{\beta }({\rho }_{n}| | {\sigma }_{n})\\ & =& {\sum }_{n}{p}_{n}{M}_{\alpha ,\beta }^{E}({\rho }_{n}),\end{array}\end{eqnarray*}$
where ${M}_{\alpha ,\beta }^{E}({\rho }_{n})={D}_{\alpha }^{\beta }({\rho }_{n}| | {\sigma }_{n})$. This implies (M4) holds for ${M}_{\alpha ,\beta }^{E}(\rho )$. This completes the proof.From Theorem 8, we can derive that when α →1, ${M}_{\alpha ,\beta }^{E}(\rho )={M}_{r}(\rho )$ and Theorem 2 degrades to the imaginarity measure induced by relative entropy in [45]. In addition, ${M}_{\alpha ,\beta }^{E}(\rho )$ also has many properties similar to ${M}_{\alpha ,\beta }^{H}(\rho )$.
Theorem 9. For any α ∈ (0, 1) and β ∈ (0, 1], ${M}_{\alpha ,\beta }^{E}(\rho )$ satisfies the following properties:
(1) (Superadditivity under direct sum) For any quantum state ρ1, ρ2 on ${ \mathcal H }$ and p ∈ [0, 1], we have${M}_{\alpha ,\beta }^{E}(p{\rho }_{1}\oplus (1-p){\rho }_{2})\geqslant p{M}_{\alpha ,\beta }^{E}({\rho }_{1})+(1-p){M}_{\alpha ,\beta }^{E}({\rho }_{2})$.
(2) ${M}_{\alpha ,\beta }^{E}(\rho )$ satisfies the additivity under direct sum if and only if β=1.
(3) For quantum state ρ on ${{ \mathcal H }}_{1}\otimes {{ \mathcal H }}_{2}$, we have ${M}_{\alpha ,\beta }^{E}({\rho }_{1})\leqslant {M}_{\alpha ,\beta }^{E}(\rho )$, where ρ1 is the partial trace of ρ on ${{ \mathcal H }}_{1}$.
(4) (Subadditivity under tensor product) For any quantum state ρ, τ on ${{ \mathcal H }}_{A}$ and ${{ \mathcal H }}_{B}$, respectively, we have${M}_{\alpha ,\beta }^{E}(\rho \otimes \tau )\leqslant {M}_{\alpha ,\beta }^{E}(\rho )+{M}_{\alpha ,\beta }^{E}(\tau )$.
(5) ${M}_{\alpha ,{\beta }_{1}}^{E}(\rho )\geqslant {M}_{\alpha ,{\beta }_{2}}^{E}(\rho )$ for fixed α when β1≤β2.
Proof. (1) Suppose that σ1 and σ2 are the quantum state that reaches the minimum in equation (20 ) when the input states are ρ1 and ρ2, respectively. Utilizing Lemma 4, we obtain
$\begin{eqnarray*}\begin{array}{l}p{M}_{\alpha ,\beta }^{E}({\rho }_{1})+(1-p){M}_{\alpha ,\beta }^{E}({\rho }_{2})\\ =\,\frac{1}{(\alpha -1)\beta }[p{({\rm{tr}}({\rho }_{1}^{\alpha }{({\sigma }_{1})}^{1-\alpha }))}^{\beta }\\ +\,(1-p){({\rm{tr}}({\rho }_{2}^{\alpha }{({\sigma }_{2})}^{1-\alpha }))}^{\beta }-1]\\ \leqslant \,\frac{1}{(\alpha -1)\beta }[(p{\rm{tr}}({\rho }_{1}^{\alpha }{({\sigma }_{1})}^{1-\alpha })\\ +\,(1-p){\rm{tr}}({\rho }_{2}^{\alpha }{({\sigma }_{2})}^{1-\alpha }){)}^{\beta }-1]\\ =\,\,\frac{1}{(\alpha -1)\beta }[({\rm{tr}}({(p{\rho }_{1}\displaystyle \oplus (1-p){\rho }_{2})}^{\alpha }\\ {(p{\sigma }_{1}\displaystyle \oplus (1-p){\sigma }_{2})}^{1-\alpha }){)}^{\beta }-1]\\ =\,{M}_{\alpha ,\beta }^{E}(p{\rho }_{1}\displaystyle \oplus (1-p){\rho }_{2}),\end{array}\end{eqnarray*}$
where the inequality is true because f(x)=xβ is a convex function. So item (1) holds. (2) Imitating the proof of corollary 1, item (2) can be easily obtained, and so we omit it here.
(3) Let $\tilde{\sigma }$ be a quantum state achieving the minimum in equation (20 ), ${\tilde{\sigma }}_{1}$ be the partial trace of $\tilde{\sigma }$ on ${{ \mathcal H }}_{1}$ and σ1 be any real state on ${{ \mathcal H }}_{1}$. Using Lemma 1(iv), we obtain
$\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }^{E}(\rho ) & =& \mathop{\min }\limits_{\sigma \in { \mathcal F }}\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha }))}^{\beta }-1]\\ & =& \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\tilde{\sigma }}^{1-\alpha }))}^{\beta }-1]\\ & =& D(\rho | | \tilde{\sigma })\\ & \geqslant & D({\rho }_{1}| | {\tilde{\sigma }}_{1})\\ & =& \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }_{1}^{\alpha }{\tilde{\sigma }}_{1}^{1-\alpha }))}^{\beta }-1]\\ & \geqslant & \mathop{\min }\limits_{{\sigma }_{1}\in { \mathcal F }}\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }_{1}^{\alpha }{\sigma }_{1}^{1-\alpha }))}^{\beta }-1]\\ & =& {M}_{\alpha ,\beta }^{E}({\rho }_{1}),\end{array}\end{eqnarray*}$
that is, ${M}_{\alpha ,\beta }^{E}({\rho }_{1})\leqslant {M}_{\alpha ,\beta }^{E}(\rho )$. Hence, item (3) is true.(4) Suppose that σρ and στ are the quantum state that reaches the minimum value in equation (20 ) when the input states are ρ and τ, respectively. Then we obtain
$\begin{eqnarray*}\begin{array}{rcl} & & {M}_{\alpha ,\beta }^{E}(\rho \displaystyle \otimes \tau )\\ & & \quad =\mathop{\min }\limits_{\sigma \in { \mathcal F }}\frac{1}{(\alpha -1)\beta }\\ & & \quad \quad \times [{({\rm{tr}}({(\rho \displaystyle \otimes \tau )}^{\alpha }{\sigma }^{1-\alpha }))}^{\beta }-1]\\ & & \quad \leqslant \mathop{\min }\limits_{{\sigma }_{a},{\sigma }_{b}\in { \mathcal F }}\frac{1}{(\alpha -1)\beta }\\ & & \,\times \,[{({\rm{tr}}({(\rho \displaystyle \otimes \tau )}^{\alpha }{({\sigma }_{a}\displaystyle \otimes {\sigma }_{b})}^{1-\alpha }))}^{\beta }-1]\\ & & \quad \leqslant \frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({(\rho \displaystyle \otimes \tau )}^{\alpha }{({\sigma }_{\rho }\displaystyle \otimes {\sigma }_{\tau })}^{1-\alpha }))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}(({\rho }^{\alpha }\displaystyle \otimes {\tau }^{\alpha })({\sigma }_{\rho }^{1-\alpha }\displaystyle \otimes {\sigma }_{\tau }^{1-\alpha })))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }_{\rho }^{1-\alpha }){\rm{tr}}({\tau }^{\alpha }{\sigma }_{\tau }^{1-\alpha }))}^{\beta }-1]\\ & & \quad =\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }_{\rho }^{1-\alpha }))}^{\beta }-1][{({\rm{tr}}({\tau }^{\alpha }{\sigma }_{\tau }^{1-\alpha }))}^{\beta }-1]\\ & & \quad \quad +\,\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\rho }^{\alpha }{\sigma }_{\rho }^{1-\alpha }))}^{\beta }-1]\\ & & \quad \quad +\,\frac{1}{(\alpha -1)\beta }[{({\rm{tr}}({\tau }^{\alpha }{\sigma }_{\tau }^{1-\alpha }))}^{\beta }-1]\\ & & \quad =(\alpha -1)\beta {M}_{\alpha ,\beta }^{E}(\rho ){M}_{\alpha ,\beta }^{E}(\tau )+{M}_{\alpha ,\beta }^{E}(\rho )+{M}_{\alpha ,\beta }^{E}(\tau ).\end{array}\end{eqnarray*}$
Since ${M}_{\alpha ,\beta }^{E}(\rho )$, ${M}_{\alpha ,\beta }^{E}(\tau )\geqslant 0$, and α ∈ (0, 1), β ∈ (0, 1], we obtain ${M}_{\alpha ,\beta }^{E}(\rho \otimes \tau )\leqslant {M}_{\alpha ,\beta }^{E}(\rho )+{M}_{\alpha ,\beta }^{E}(\tau )$, which implies that item (4) holds.(5) Suppose that $\hat{\sigma }$ is a quantum state reaching the minimum value in the definition of ${M}_{\alpha ,{\beta }_{1}}^{E}(\rho )$, that is,
$\begin{eqnarray*}{M}_{\alpha ,{\beta }_{1}}^{E}(\rho )=\frac{1}{(\alpha -1){\beta }_{1}}[{({\rm{tr}}({\rho }^{\alpha }{\hat{\sigma }}^{1-\alpha }))}^{{\beta }_{1}}-1].\end{eqnarray*}$
Then it is obvious that $\begin{eqnarray*}{M}_{\alpha ,{\beta }_{2}}^{E}(\rho )=\frac{1}{(\alpha -1){\beta }_{2}}[{({\rm{tr}}({\rho }^{\alpha }{\hat{\sigma }}^{1-\alpha }))}^{{\beta }_{2}}-1].\end{eqnarray*}$
Using Lemma 1(vi), we have $\begin{eqnarray*}{M}_{\alpha ,{\beta }_{1}}^{E}(\rho )={D}_{\alpha }^{{\beta }_{1}}(\rho \parallel \hat{\sigma })\geqslant {D}_{\alpha }^{{\beta }_{2}}(\rho \parallel \hat{\sigma })={M}_{\alpha ,{\beta }_{2}}^{E}(\rho ).\end{eqnarray*}$
Therefore, item (5) is derived. This completes the proof.Remark 2. It bears noting that ${M}_{{\alpha }_{1},\beta }^{E}(\rho )\leqslant {M}_{{\alpha }_{2},\beta }^{E}(\rho )$ for fixed β when α1≤α2 is not true, since when α varies, the quantum state σ that minimizes the quantity in equation (20 ) may also change accordingly, making it impossible to compare the values of ${M}_{{\alpha }_{1},\beta }^{E}(\rho )$ and ${M}_{{\alpha }_{2},\beta }^{E}(\rho )$.
4. Examples
In this section, we investigate the imaginarity monotones ${M}_{\alpha ,\beta }^{H}(\rho )$ and ${M}_{\alpha ,\beta }^{E}(\rho )$ under some special quantum states.
Example 1. Consider a qubit state ρ expressed in Bloch representation as 13 ), we have 20 ), ${M}_{\alpha ,\beta }^{E}(\rho )$ reads
$\begin{eqnarray}\rho =\frac{1}{2}({I}+\overrightarrow{r}\cdot \overrightarrow{\sigma })=\frac{1}{2}\left(\begin{array}{cc}1+{r}_{3} & {r}_{1}-{\rm{i}}{r}_{2}\\ {r}_{1}+{\rm{i}}{r}_{2} & 1-{r}_{3}\end{array}\right),\end{eqnarray}$
where $\overrightarrow{r}=({r}_{1},{r}_{2},{r}_{3})$ is a real vector with $r=| \overrightarrow{r}| =\sqrt{{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\leqslant 1$ and $\overrightarrow{\sigma }=({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})$ with ${\sigma }_{x}=\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right)$, ${\sigma }_{y}=\left(\begin{array}{cc}0 & -{\rm{i}}\\ {\rm{i}} & 0\end{array}\right)$and ${\sigma }_{z}=\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right)$. If $\rho \in { \mathcal F }$, we obtain ${M}_{\alpha ,\beta }^{H}(\rho )=0$. Otherwise, it follows from [49], we know that $\begin{eqnarray}\displaystyle \begin{array}{rcl}{\rm{tr}}[{\rho }^{\alpha }{({\rho }^{* })}^{1-\alpha }] & =& \frac{1}{2{r}^{2}}\left\{(1-r)\left[{\left(r-\frac{{r}_{2}^{2}}{r-{r}_{3}}\right)}^{2}+\frac{{r}_{1}^{2}{r}_{2}^{2}}{{(r-{r}_{3})}^{2}}\right]\right.\\ & & +(1+r)\left[{\left(r-\frac{{r}_{2}^{2}}{r+{r}_{3}}\right)}^{2}\right.+\left.\frac{{r}_{1}^{2}{r}_{2}^{2}}{{(r+{r}_{3})}^{2}}\right]\\ & & \left.+{r}_{2}^{2}[{(1-r)}^{\alpha }{(1+r)}^{1-\alpha }+{(1-r)}^{1-\alpha }{(1+r)}^{\alpha }]\Space{0ex}{4.25ex}{0ex}\right\}.\end{array}\end{eqnarray}$
Therefore, by equation ( $\begin{eqnarray}\displaystyle \begin{array}{l}{M}_{\alpha ,\beta }^{H}(\rho )=\frac{1}{(\alpha -1)\beta }\left\{\frac{1}{{2}^{\beta }{r}^{2\beta }}\left\{(1-r)\right.\right.\\ \quad \times \,\left[{\left(r-\frac{{r}_{2}^{2}}{r-{r}_{3}}\right)}^{2}+\frac{{r}_{1}^{2}{r}_{2}^{2}}{{(r-{r}_{3})}^{2}}\right]\\ \quad +\,(1+r)\left[{\left(r-\frac{{r}_{2}^{2}}{r+{r}_{3}}\right)}^{2}\right.\\ \quad \left.+\,\frac{{r}_{1}^{2}{r}_{2}^{2}}{{(r+{r}_{3})}^{2}}\right]\\ \quad +\,{r}_{2}^{2}[{(1-r)}^{\alpha }{(1+r)}^{1-\alpha }+{(1-r)}^{1-\alpha }\\ \quad \left.{\left.\times \,{(1+r)}^{\alpha }]\right\}}^{\beta }-1\right\},\end{array}\end{eqnarray}$
for all α ∈ (0, 1) and β ∈ (0, 1], and by equation ( $\begin{eqnarray}\begin{array}{l}{M}_{\alpha ,\beta }^{E}(\rho )=\frac{1}{(\alpha -1)\beta }\\ \left\{\frac{1}{{2}^{\beta }}\left[{\left(1-\frac{1}{{\left(\frac{(r-\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1-r)}^{\alpha }+(r+\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1+r)}^{\alpha }}{(r+\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1-r)}^{\alpha }+(r-\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1+r)}^{\alpha }}\right)}^{\frac{1}{\alpha }}+1}\right)}^{1-\alpha }\right.\right.\\ \times \left(\left(1-\frac{\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}}{r}\right){\left(\frac{1-r}{2}\right)}^{\alpha }\right.\\ \quad +\left.\left(1+\frac{\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}}{r}\right){\left(\frac{1+r}{2}\right)}^{\alpha }\right)\\ \left.+\left(\left(1+\frac{\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}}{r}\right){\left(\frac{1-r}{2}\right)}^{\alpha }\right.\quad +\left(1-\frac{\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}}{r}\right){\left(\frac{1+r}{2}\right)}^{\alpha }\right)\\ \left.{\left.\times {\left(\frac{1}{{\left(\frac{(r-\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1-r)}^{\alpha }+(r+\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1+r)}^{\alpha }}{(r+\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1-r)}^{\alpha }+(r-\sqrt{{r}_{1}^{2}+{r}_{3}^{2}}){(1+r)}^{\alpha }}\right)}^{\frac{1}{\alpha }}+1}\right)}^{1-\alpha }\right]}^{\beta }-1\right\},\end{array}\end{eqnarray}$
for all α ∈ (0, 1) and β ∈ (0, 1].The proof of equation (24 ) is given in appendix C , and the surfaces of ${M}_{\alpha ,\beta }^{H}(\rho )$ and ${M}_{\alpha ,\beta }^{E}(\rho )$ in Equations (23 ) and (24 ) are plotted in figure 1.
Figure 1. The variations of ${M}_{\alpha ,\beta }^{H}(\rho )$ and ${M}_{\alpha ,\beta }^{E}(\rho )$ in equation ( |
Remark 3. From appendix
$\begin{eqnarray*}\sigma =\bar{\sigma }=\frac{1}{2}\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right),\end{eqnarray*}$
for any state $\begin{eqnarray*}\tilde{\rho }=\frac{1}{2}\left(\begin{array}{cc}1 & -{r}_{2}{\rm{i}}\\ {r}_{2}{\rm{i}} & 1\end{array}\right),\end{eqnarray*}$
and $\begin{eqnarray*}{\rm{tr}}({\tilde{\rho }}^{\alpha }{\bar{\sigma }}^{1-\alpha })=\frac{1}{2}\left[{(1-{r}_{2})}^{\alpha }+{(1+{r}_{2})}^{\alpha }\right].\end{eqnarray*}$
By equation ( $\begin{eqnarray*}\begin{array}{rcl}{\rm{tr}}({\tilde{\rho }}^{\alpha }{({\tilde{\rho }}^{* })}^{1-\alpha }) & =& \frac{1}{2}\left[{(1+{r}_{2})}^{1-\alpha }{(1-{r}_{2})}^{\alpha }\right.\\ & & \left.+{(1+{r}_{2})}^{\alpha }{(1-{r}_{2})}^{1-\alpha }\right].\end{array}\end{eqnarray*}$
Let $\begin{eqnarray*}\begin{array}{l}M({r}_{2},\alpha )={\rm{tr}}({\tilde{\rho }}^{\alpha }{({\tilde{\rho }}^{* })}^{1-\alpha })-{\rm{tr}}({\tilde{\rho }}^{\alpha }{\bar{\sigma }}^{1-\alpha })\\ \quad =\,\frac{1}{2}\{[{(1+{r}_{2})}^{1-\alpha }-1]{(1-{r}_{2})}^{\alpha }\\ \quad +\,[{(1-{r}_{2})}^{1-\alpha }-1]{(1+{r}_{2})}^{\alpha }\}.\end{array}\end{eqnarray*}$
When r2 ≠ ± 1, it can be shown that $\begin{eqnarray*}\begin{array}{l}{M}_{{r}_{2}}^{{\prime} }({r}_{2},\alpha )=\frac{1}{2}\left\{(1-\alpha )\left[\frac{{(1-{r}_{2})}^{\alpha }}{{(1+{r}_{2})}^{\alpha }}-\frac{{(1+{r}_{2})}^{\alpha }}{{(1-{r}_{2})}^{\alpha }}\right]\right.\left.-\alpha \left[\frac{{(1+{r}_{2})}^{1-\alpha }-1}{{(1-{r}_{2})}^{1-\alpha }}-\frac{{(1-{r}_{2})}^{1-\alpha }-1}{{(1+{r}_{2})}^{1-\alpha }}\right]\right\}\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}{M}_{\alpha }^{{\prime} }({r}_{2},\alpha )=\frac{1}{2}\left\{{(1+{r}_{2})}^{1-\alpha }{(1-{r}_{2})}^{\alpha }\mathrm{ln}\frac{1-{r}_{2}}{1+{r}_{2}}\right.\\ \quad +{(1+{r}_{2})}^{\alpha }{(1-{r}_{2})}^{1-\alpha }\mathrm{ln}\frac{1+{r}_{2}}{1-{r}_{2}}\\ \quad \left.-{(1-{r}_{2})}^{\alpha }\mathrm{ln}(1-{r}_{2})-{(1+{r}_{2})}^{\alpha }\mathrm{ln}(1+{r}_{2})\right\}.\end{array}\end{eqnarray*}$
Letting ${M}_{{r}_{2}}^{{\prime} }({r}_{2},\alpha )=0$ and ${M}_{\alpha }^{{\prime} }({r}_{2},\alpha )=0$, we get r2=0. Noting that ${M}_{{r}_{2}}^{{\prime} }({r}_{2},\alpha )$ is symmetric about r2=0, ${M}_{{r}_{2}}^{{\prime} }(0,\alpha )=0$ and $M(\frac{1}{2},\alpha )=\frac{1}{2}({3}^{\alpha }+{3}^{1-\alpha })-{\left(\frac{1}{2}\right)}^{\alpha }-{\left(\frac{3}{2}\right)}^{\alpha }\leqslant M(0,\alpha )=0$ for any α ∈ (0, 1), we obtain that (0, α) is a maximum value point. When r2=± 1, M(-1, α)=M(1, α)=-2α-1 < 0. Hence, M(r2, α)≤0, that is, ${M}_{\alpha ,\beta }^{H}(\tilde{\rho })\geqslant {M}_{\alpha ,\beta }^{E}(\tilde{\rho })$ for any α ∈ (0, 1) and β ∈ (0, 1]. We have not found a proof of ${M}_{\alpha ,\beta }^{H}(\rho )\geqslant {M}_{\alpha ,\beta }^{E}(\rho )$ for any α ∈ (0, 1), β ∈ (0, 1] and any qubit state ρ, however, numerical experiments shows that it might be true, and it seems difficult to find a counterexample. It is also conjectured that ${M}_{\alpha ,\beta }^{H}(\rho )\geqslant {M}_{\alpha ,\beta }^{E}(\rho )$ for any α ∈ (0, 1), β ∈ (0, 1] and any quantum state ρ.
Example 2. Consider the modified Werner state
$\begin{eqnarray}{\rho }_{{\rm{w}}}=\left(\begin{array}{cccc}\frac{1-k}{4} & 0 & 0 & 0\\ 0 & \frac{1+k}{4} & \frac{k}{2}{\rm{i}} & 0\\ 0 & -\frac{k}{2}{\rm{i}} & \frac{1+k}{4} & 0\\ 0 & 0 & 0 & \frac{1-k}{4}\end{array}\right),\end{eqnarray}$
where k ∈ [0, 1]. Note that ρw is a pure state when k=1, ${\rho }_{{\rm{w}}}\in { \mathcal F }$ when k=0 and the eigenvalues of ρw are ${w}_{1}=\frac{1-k}{4}$ and ${w}_{2}=\frac{1+3k}{4}$. By equation ( $\begin{eqnarray}\begin{array}{l}{M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})=\frac{1}{(\alpha -1)\beta }\left\{\left[\frac{1}{32}({w}_{1}^{1-\alpha }{w}_{2}^{\alpha }+{w}_{1}^{\alpha }{w}_{2}^{1-\alpha }\right.\right.\\ \quad \left.{\left.\,-\,8({w}_{2}-{w}_{1})+24)\right]}^{\beta }-1\right\}\\ \,=\,\frac{1}{(\alpha -1)\beta }\left\{\left[\frac{1}{8}({(3k+1)}^{\alpha }{(1-k)}^{1-\alpha }\right.\right.\\ \quad \left.{\left.\,+\,{(3k+1)}^{1-\alpha }{(1-k)}^{\alpha }-2k+6)\right]}^{\beta }-1\right\},\end{array}\end{eqnarray}$
for all α ∈ (0, 1) and β ∈ (0, 1].And the linear entropy (mixedness) of the modified Werner state ρw is $L({\rho }_{{\rm{w}}})=1-{\rm{tr}}{\rho }_{{\rm{w}}}^{2}=3{w}_{1}(1+{w}_{2}-{w}_{1})=\frac{3}{4}(1-{k}^{2})$. We plot the variation of ${M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})$ and L(ρw) with k in figure 2. Interestingly, ${M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})$ is increasing with respect to k while L(ρw) is decreasing with respect to k. ${M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})$ reaches its maximum value of $\frac{1}{(\alpha -1)\beta }(\frac{1}{{2}^{\beta }}-1)$ for any fixed α and β when k=1, in which case L(ρw) reaches its minimum value of 0, while ${M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})$ reaches its minimum value of 0 when k=0 for any fixed α and β, in which case the linear entropy (mixedness) L(ρw) attains its maximum value of $\frac{3}{4}$.
Figure 2. The variations of ${M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})$ and L(ρw) with k for fixed α and β. Red solid line fixes $\alpha =\beta =\frac{1}{2}$ in equation ( |
Then consider the modified isotropic state
$\begin{eqnarray}{\rho }_{{\rm{iso}}}=\left(\begin{array}{cccc}\frac{1}{6}(2F+1) & 0 & 0 & \frac{1}{6}(4F-1){\rm{i}}\\ 0 & \frac{1}{3}(1-F) & 0 & 0\\ 0 & 0 & \frac{1}{3}(1-F) & 0\\ -\frac{1}{6}(4F-1){\rm{i}} & 0 & 0 & \frac{1}{6}(2F+1)\end{array}\right),\end{eqnarray}$
where F ∈ [0, 1]. Swapping the position of basis ∣00⟩⟨11∣ with ∣01⟩⟨10∣ and the position of basis ∣11⟩⟨00∣ with ∣10⟩⟨01∣, and setting $F=\frac{3k+1}{4}$, we have ρiso=ρw, i.e, ρiso and ρw are essentially the same state, and we have ${M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{iso}}})={M}_{\alpha ,\beta }^{H}({\rho }_{{\rm{w}}})$ when $F=\frac{3k+1}{4}$.5. Conclusions
We have proposed two classes of imaginarity monotones induced by the unified (α, β)-relative entropy. The first class of imaginarity monotone ${M}_{\alpha ,\beta }^{H}(\rho )$ is defined by setting σ=ρ*. We have investigated several properties of ${M}_{\alpha ,\beta }^{H}(\rho )$, including superadditivity under direct, subadditivity under tensor product and monotonicity, among others. The second class of imaginarity monotone ${M}_{\alpha ,\beta }^{E}(\rho )$ is defined by minimizing the unified (α, β)-relative entropy over all free states. The properties of ${M}_{\alpha ,\beta }^{E}(\rho )$ exhibit a high degree of similarity to those of ${M}_{\alpha ,\beta }^{H}(\rho )$. Furthermore, we have presented the analytical expressions of ${M}_{\alpha ,\beta }^{H}(\rho )$ for several special quantum states. For ${M}_{\alpha ,\beta }^{E}(\rho )$, we have focused on deriving its analytical expression for qubit states, which differs significantly from the one of ${M}_{\alpha ,\beta }^{H}(\rho )$ under the qubit state case.
Acknowledgments
The authors would like to express their sincere gratitude to the anonymous referees for their useful suggestions which greatly improved this paper. The authors would also like to thank Prof. Jianwei Xu and Lin Zhang for the fruitful discussions. This work was supported by National Natural Science Foundation of China (Grant No. 12161056) and Natural Science Foundation of Jiangxi Province (Grant No. 20232ACB211003).
Appendix A. Proof of Lemma 3
We first prove that $({x}_{0},{\theta }_{0})=\left(\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1},\arcsin \frac{B}{\sqrt{{B}^{2}+{C}^{2}}}\right)$ is the unique extreme point of f(x, θ). In fact, it can be shown that
$\begin{eqnarray*}\begin{array}{rcl}{f}_{x}^{{\prime} }(x,\theta ) & =& A(1-\alpha )\left(\frac{1}{{x}^{\alpha }}-\frac{1}{{(1-x)}^{\alpha }}\right)\\ & & -(B\sin \theta +C\cos \theta )(1-\alpha )\left(\frac{1}{{(1-x)}^{\alpha }}+\frac{1}{{x}^{\alpha }}\right),\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}{f}_{\theta }^{{\prime} }(x,\theta )=(B\cos \theta -C\sin \theta )\\ \times [{(1-x)}^{1-\alpha }-{x}^{1-\alpha }].\end{array}\end{eqnarray*}$
Letting ${f}_{x}^{{\prime} }(x,\theta )=0$ and ${f}_{\theta }^{{\prime} }(x,\theta )=0$, we obtain that ${x}_{0}=\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}$, $\sin {\theta }_{0}=\frac{B}{\sqrt{{B}^{2}+{C}^{2}}}$ and $\cos {\theta }_{0}=\frac{C}{\sqrt{{B}^{2}+{C}^{2}}}$, or ${x}_{1}=\frac{1}{{\left(\frac{A-\sqrt{{B}^{2}+{C}^{2}}}{A+\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}$, $\sin {\theta }_{0}=-\frac{B}{\sqrt{{B}^{2}+{C}^{2}}}$ and $\cos {\theta }_{0}=-\frac{C}{\sqrt{{B}^{2}+{C}^{2}}}$. Since $A-\sqrt{{B}^{2}+{C}^{2}}\gt 0$ and ${x}_{0}\in [0,\frac{1}{2}]$, ${x}_{1}\notin [0,\frac{1}{2}]$. Therefore, (x0, θ0) is the unique extreme point of f(x, θ).
Then we prove that (x0, θ0) is the maximum value point of f(x, θ). Direct calculation shows that
$\begin{eqnarray*}\begin{array}{l}{f}_{xx}^{{\prime\prime} }(x,\theta )=-\alpha (1-\alpha )\left[(A-B\sin \theta -C\cos \theta )\frac{1}{{x}^{\alpha +1}}\right.\\ \quad \left.+\,(A+B\sin \theta +C\cos \theta )\frac{1}{{(1-x)}^{\alpha +1}}\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{f}_{x\theta }^{{\prime\prime} }(x,\theta )=-(1-\alpha )(B\cos \theta -C\sin \theta )\\ \quad \times \,\left[\frac{1}{{(1-x)}^{\alpha }}+\frac{1}{{x}^{\alpha }}\right],\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{l}{f}_{\theta \theta }^{{\prime\prime} }(x,\theta )=-(B\sin \theta +C\cos \theta )\\ \quad [{(1-x)}^{1-\alpha }-{x}^{1-\alpha }].\end{array}\end{eqnarray*}$
It can be seen that ${f}_{xx}^{{\prime\prime} }({x}_{0},{\theta }_{0})\lt 0$, ${f}_{x\theta }^{{\prime\prime} }({x}_{0},{\theta }_{0})=0$ and ${f}_{\theta \theta }^{{\prime\prime} }({x}_{0},{\theta }_{0})\lt 0$. Therefore, ${[{f}_{x\theta }^{{\prime\prime} }({x}_{0},{\theta }_{0})]}^{2}-{f}_{xx}^{{\prime\prime} }({x}_{0},{\theta }_{0}){f}_{\theta \theta }^{{\prime\prime} }({x}_{0},{\theta }_{0})\lt 0$. Noting that f(x, θ) is a continuous function, we conclude that (x0, θ0) is the maximum value point of f(x, θ). This completes the proof.
Appendix B. Proof of Lemma 4
First of all, since σρ and ${\sigma }_{\tau }\in { \mathcal F }$, we have $p{\sigma }_{\rho }\oplus (1-p){\sigma }_{\tau }\in { \mathcal F }$. Suppose that 20 ) when the input state is pρ ⊕ (1-p)τ. This completes the proof.
$\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }^{E}(p\rho \displaystyle \oplus (1-p)\tau ) & =& \frac{1}{(\alpha -1)\beta }\\ & & \times \,[{({\rm{tr}}({(p\rho \displaystyle \oplus (1-p)\tau )}^{\alpha }{\sigma }_{0}^{1-\alpha }))}^{\beta }-1].\end{array}\end{eqnarray*}$
Let $\begin{eqnarray*}{\sigma }_{0}^{1-\alpha }=\left(\begin{array}{cc}{\boldsymbol{S}} & {\boldsymbol{U}}\\ {\boldsymbol{V}} & {\boldsymbol{W}}\end{array}\right).\end{eqnarray*}$
Then we have $\begin{eqnarray*}\begin{array}{l}{\rm{tr}}({(p\rho \displaystyle \oplus (1-p)\tau )}^{\alpha }{\sigma }_{0}^{1-\alpha })\,\\ \quad =\,\rm{tr}\,\left[\left(\begin{array}{cc}{(p\rho )}^{\alpha } & {\boldsymbol{0}}\\ {\boldsymbol{0}} & {((1-p)\tau )}^{\alpha }\end{array}\right)\left(\begin{array}{cc}{\boldsymbol{S}} & {\boldsymbol{U}}\\ {\boldsymbol{V}} & {\boldsymbol{W}}\end{array}\right)\right]\\ \quad =\,{\rm{tr}}({\boldsymbol{S}}{(p\rho )}^{\alpha }+{\boldsymbol{W}}{((1-p)\tau )}^{\alpha }).\end{array}\end{eqnarray*}$
Therefore, ${\rm{tr}}[{(p\rho \oplus (1-p)\tau )}^{\alpha }{\sigma }_{0}^{1-\alpha }]$ attains its maximum value when ${\boldsymbol{S}}={(p{\sigma }_{\rho })}^{1-\alpha }$ and ${\boldsymbol{W}}={((1-p){\sigma }_{\tau })}^{1-\alpha }$, which is independent of U and V, that is, when $\begin{eqnarray*}{\sigma }_{0}^{1-\alpha }=\left(\begin{array}{cc}{(p{\sigma }_{\rho })}^{1-\alpha } & {\boldsymbol{0}}\\ {\boldsymbol{0}} & {((1-p){\sigma }_{\tau })}^{1-\alpha }\end{array}\right).\end{eqnarray*}$
Thus, σ0=pσρ ⊕ (1-p)στ is a quantum state that reaches the minimum value in equation(Appendix C. Proof of equation (24 )
Let λ1,2=(1 ∓ r)/2 be the eigenvalues of a qubit state ρ, where $r=| \overrightarrow{r}| $. For any qubit free state σ, we assume its Bloch vector is $\overrightarrow{s}=({s}_{1},0,{s}_{3})$, and the eigenvalues of σ are μ1,2=(1 ∓ s)/2, where $s=| \overrightarrow{s}| $. If r=0, it is obvious that ${M}_{\alpha ,\beta }^{E}(\rho )=0$. Now we consider the case of r ≠ 0. It follows from [66] that C3 ) we have
$\begin{eqnarray}{\rho }^{\alpha }=\left(\begin{array}{cc}\frac{{\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha }}{2}+\frac{{r}_{3}({\lambda }_{2}^{\alpha }-{\lambda }_{1}^{\alpha })}{2r} & \frac{(-{r}_{1}+{\rm{i}}{r}_{2})({\lambda }_{1}^{\alpha }-{\lambda }_{2}^{\alpha })}{2r}\\ \frac{(-{r}_{1}-{\rm{i}}{r}_{2})({\lambda }_{1}^{\alpha }-{\lambda }_{2}^{\alpha })}{2r} & \frac{{\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha }}{2}-\frac{{r}_{3}({\lambda }_{2}^{\alpha }-{\lambda }_{1}^{\alpha })}{2r}\end{array}\right),\end{eqnarray}$
and $\begin{eqnarray}{\sigma }^{1-\alpha }=\left(\begin{array}{cc}\frac{{\mu }_{1}^{1-\alpha }+{\mu }_{2}^{1-\alpha }}{2}+\frac{{s}_{3}({\mu }_{2}^{1-\alpha }-{\mu }_{1}^{1-\alpha })}{2s} & \frac{-{s}_{1}({\mu }_{1}^{1-\alpha }-{\mu }_{2}^{1-\alpha })}{2s}\\ \frac{-{s}_{1}({\mu }_{1}^{1-\alpha }-{\mu }_{2}^{1-\alpha })}{2s} & \frac{{\mu }_{1}^{1-\alpha }+{\mu }_{2}^{1-\alpha }}{2}-\frac{{s}_{3}({\mu }_{2}^{1-\alpha }-{\mu }_{1}^{1-\alpha })}{2s}\end{array}\right),\end{eqnarray}$
where s ≠ 0. It implies that $\begin{eqnarray}\begin{array}{rcl}{\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha }) & =& \frac{1}{2}({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha })({\mu }_{1}^{1-\alpha }+{\mu }_{2}^{1-\alpha })\\ & & +\frac{{r}_{3}}{2r}({\lambda }_{2}^{\alpha }-{\lambda }_{1}^{\alpha })\cdot \frac{{s}_{3}}{s}({\mu }_{2}^{1-\alpha }-{\mu }_{1}^{1-\alpha })\\ & & +\frac{{r}_{1}}{2r}({\lambda }_{2}^{\alpha }-{\lambda }_{1}^{\alpha })\cdot \frac{{s}_{1}}{s}({\mu }_{2}^{1-\alpha }-{\mu }_{1}^{1-\alpha }).\end{array}\end{eqnarray}$
Letting $A=\frac{1}{2}({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha })$, $B=\frac{{r}_{1}}{2r}({\lambda }_{2}^{\alpha }-{\lambda }_{1}^{\alpha })$ and $C\,=\frac{{r}_{3}}{2r}({\lambda }_{2}^{\alpha }-{\lambda }_{1}^{\alpha })$, ${s}_{1}=c\sin \theta $ and ${s}_{3}=c\cos \theta $, where c ∈ [0, 1] and θ ∈ [0, 2π] and $x=\frac{1-c}{2}$, by equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha }) & =& A({x}^{1-\alpha }+{(1-x)}^{1-\alpha })\\ & & +(B\sin \theta +C\cos \theta )[{(1-x)}^{1-\alpha }-{x}^{1-\alpha }].\end{array}\end{eqnarray}$
And when s=0, $\begin{eqnarray}{\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })=\frac{{\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha }}{{2}^{1-\alpha }}.\end{eqnarray}$
If B2 + C2=0, we have ${\rm{tr}}{({\rho }^{\alpha }{\sigma }^{1-\alpha })}_{{\rm{\max }}}={\rm{\max }}\{A({x}^{1-\alpha }+{(1-x)}^{1-\alpha }),({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha })/{2}^{1-\alpha }\}$. Direct calculation shows that ${\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })$ achieves its maximum value of 2αA when s1=s3=0. In this case, ${M}_{\alpha ,\beta }^{E}(\rho )\,=\frac{1}{(\alpha -1)\beta }({2}^{\alpha \beta }{A}^{\beta }-1)$.
If B2+C2 > 0, we have ${\left(\frac{1}{2}\right)}^{1-\alpha }({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha })$ $\leqslant {\rm{\max }}\{A({x}^{1-\alpha }\,+{(1-x)}^{1-\alpha })+(B\sin \theta +C\cos \theta )[{(1-x)}^{1-\alpha }-{x}^{1-\alpha }]\}$. So ${\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })$ achieves its maximum value when s ≠ 0. Note that $x\in [0,\frac{1}{2}]$, $A\gt \sqrt{{B}^{2}+{C}^{2}}$, B2+C2 > 0 and α ∈ (0, 1). It follows from Lemma 3 that ${\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })$ reaches its maximum value at $({x}_{0},{\theta }_{0})=\left(\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1},\arcsin \frac{B}{\sqrt{{B}^{2}+{C}^{2}}}\right)$ and the maximum value is
$\begin{eqnarray}\begin{array}{rcl}{\rm{tr}}{({\rho }^{\alpha }{\sigma }^{1-\alpha })}_{\max } & =& (A+\sqrt{{B}^{2}+{C}^{2}})\\ & & \times {\left(1-\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}\right)}^{1-\alpha }\\ & & +(A-\sqrt{{B}^{2}+{C}^{2}}){\left(\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}\right)}^{1-\alpha },\end{array}\end{eqnarray}$
when s ≠ 0.Since α ∈ (0, 1) and β ∈ (0, 1], it is obvious that ${M}_{\alpha ,\beta }^{E}(\rho )$ reaches its minimum value when ${\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })$ reaches its maximum value. By equation (C6 ) and equation (20 ), we obtain equation (24 ). When ${x}_{0}=\frac{1}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}$, we have ${c}_{0}=1-2{x}_{0}=1-\frac{2}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}$, which implies that ${\rm{tr}}({\rho }^{\alpha }{\sigma }^{1-\alpha })$ reaches its maximum value, and thus equation (20 ) reaches its minimum value for qubit free state when ${s}_{1}={c}_{0}\sin {\theta }_{0}=\left(1-\frac{2}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}\right)\frac{B}{\sqrt{{B}^{2}+{C}^{2}}}$ and ${s}_{3}={c}_{0}\cos {\theta }_{0}=-\left(1-\frac{2}{{\left(\frac{A+\sqrt{{B}^{2}+{C}^{2}}}{A-\sqrt{{B}^{2}+{C}^{2}}}\right)}^{\frac{1}{\alpha }}+1}\right)\frac{C}{\sqrt{{B}^{2}+{C}^{2}}}$. Note that the value of ${\rm{tr}}{({\rho }^{\alpha }{\sigma }^{1-\alpha })}_{{\rm{\max }}}$ when B2+C2=0 is equal to the value of ${\rm{tr}}{({\rho }^{\alpha }{\sigma }^{1-\alpha })}_{{\rm{\max }}}$ when B2+C2 ≠ 0. This completes the proof.