1. Introduction
The Gottesman–Knill theorem indicates that classical computers can efficiently simulate any quantum computation composed solely of Clifford gates and preparation/measurement of stabilizer states [1–4]. Nonstabilizerness, therefore, plays a crucial role in demonstrating quantum advantage. Veitch et al first introduced the stabilizer resource theory in quantum computation and proposed two magic monotones which are widely employed now [5, 6]. At the same time, several applications of magic resource theory have been extensively studied, such as magic state distillation [7–13], which has already been studied in entanglement theory [14–19].
In the past decades, magic resource theory has rapidly developed. Many quantifiers of magic of quantum states, for instance, the relative entropy of magic and the mana [5], the robustness of magic [20, 21], min-relative entropy and max-relative entropy of magic [22], the thauma [13], stabilizer rank [23], stabilizer extent [24], the stabilizer Rényi entropy [25, 26] and the Lp-norm magic [27, 28], have been proposed, each of which has its own merits. In addition, the quantification of magic for quantum channels has attracted much attention in recent years. Wang et al [13] first introduced the thauma of quantum states and then applied the same idea to propose the thauma of quantum channels [29]. Saxena and Gour [30] studied the magic resource of multi-qubit quantum channels via the generalized robustness and the min-relative entropy. Seddon and Campbell [31] presented channel robustness and magic capacity in n-qubit systems. Recently, Li and Luo [32] employed the channel-state duality and L1-norm magic [28] to propose a new magic quantifier of channel, which is well-defined in all dimensions.
Quantum entropy and quantum divergence are foundational concepts with pervasive applications in resource theory. The concept of generalized quantum entropy was introduced in [33], while the unified $\left(\alpha ,\beta \right)$-relative entropy was proposed in [34], which has been applied to induce resource monotones in different resource theories such as coherence [35] and imaginarity [36]. The uncertainty relations for unified $\left(\alpha ,\beta \right)$-relative entropy of coherence have also been investigated extensively [35, 37]. The quantum Jensen–Shannon divergence was originated in [38], and it has been proved that the square root of the quantum Jensen–Shannon divergence is a true metric on the quantum state space [39]. Moreover, the quantum Jensen–Shannon divergence can naturally yield monotones to quantify magic [40] and imaginarity [41]. In this work, we utilize the quantum (α, β) entropy and the quantum $\left(\alpha ,\beta \right)$-relative entropy to define two versions of quantum (α, β) Jensen–Shannon divergence, and further propose the magic monotones induced by them.
In magic resource theory, a natural question arises as to how we can quantify the power of a quantum gate for generating magic resource. Zhu et al [42] studied amortized magic in terms of the stabilizer Rényi entropy, and revealed that magic generating power can be enhanced by prior nonstabilizerness of input states while considering the α-stabilizer Rényi entropy but this does not hold for the case of robustness of magic [21] or stabilizer extent [24]. When we focus on which quantum gate is optimal for generating magic resource, the answer provided by numerous studies is the T-gate, according to various magic quantifiers under certain conditions [40, 43–45].
The remainder of this work is structured as follows. In section 2 , we present a brief review of the stabilizer formalism and the framework of magic resource theory. In section 3 , we define quantum (α, β) Jensen–Shannon divergence from two distinct ways and derive some properties of them. Specifically, we exhibit a relationship between them, from which a similar relationship between the corresponding magic quantifiers in section 4 follows. In section 4 , we define two magic quantifiers via quantum (α, β) Jensen–Shannon divergence and prove that our magic quantifiers are both pure-state stabilizer monotones. Besides, we obtain a few desirable properties and compare our magic quantifiers with the robustness of magic. In section 5 , we show that as in the case of the stabilizer Rényi entropy, the magic generating power can also be enhanced by prior magic of input states when considering quantum (α, β) Jensen–Shannon divergence of magic. In section 6 , we give three detailed examples. Finally, we conclude our work with a summary in section 7 . The detailed proofs of our results are provided in the Appendix .
2. Preliminaries
In this section, we review the stabilizer formalism, the framework of magic resource theory and some basic properties of the quantum (α, β) entropy and the quantum $\left(\alpha ,\beta \right)$-relative entropy.
Let us first clarify some notations which we will use in this paper. Let ${ \mathcal H }$ be a d-dimensional Hilbert space with standard computational basis ${\left\{| i\rangle \right\}}_{i=0}^{d-1}$ and ${{ \mathcal H }}_{{d}^{n}}$ denotes the composite Hilbert space ${{ \mathcal H }}^{\otimes n}$. Denote the set of all density operators by ${ \mathcal D }({ \mathcal H })$, the set of all unitary operators by ${ \mathcal U }({ \mathcal H })$, and the set of all stabilizer states and pure stabilizer states on ${ \mathcal H }$ by ${{ \mathcal S }}_{d}$ and ${ \mathcal P }{{ \mathcal S }}_{d}$, respectively. We conventionally use ρ, σ, τ to represent quantum states and $| \psi \rangle ,| \psi ^{\prime} \rangle ,| \phi \rangle $ to represent pure states in ${ \mathcal D }({ \mathcal H })$. Moreover, ∥ · ∥1 denotes the trace distance, i.e. $\parallel \rho -\sigma {\parallel }_{1}=\frac{1}{2}{\rm{Tr}}| \rho -\sigma | $.
2.1. Stabilizer formalism
Let ${{\mathbb{Z}}}_{d}$ be the ring of integers modulo d and ${{\mathbb{Z}}}_{d}\times {{\mathbb{Z}}}_{d}$ be the direct product of ${{\mathbb{Z}}}_{d}$ and ${{\mathbb{Z}}}_{d}$ which can be regarded as the discrete phase space. The shift and boost operators are defined as [46]
$\begin{eqnarray*}\begin{array}{r}X=\displaystyle \sum _{j=0}^{d-1}| j+1\rangle \langle j| ,Z=\displaystyle \sum _{j=0}^{d-1}{\omega }^{j}| j\rangle \langle j| ,\end{array}\end{eqnarray*}$
respectively, where $\omega ={\,\rm{e}\,}^{\frac{2\pi {\rm{i}}}{d}}$. And the discrete Heisenberg-Weyl operators are defined as [47] $\begin{eqnarray*}\begin{array}{r}{T}_{u}={\tau }^{-{u}_{1}{u}_{2}}{Z}^{{u}_{1}}{X}^{{u}_{2}},\end{array}\end{eqnarray*}$
where $\tau ={\,\rm{e}\,}^{\frac{(d+1)\pi {\rm{i}}}{d}},u=({u}_{1},{u}_{2})\in {{\mathbb{Z}}}_{d}\times {{\mathbb{Z}}}_{d}$.The Clifford operators constitute the set defined as [5]
$\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal C }}_{d} & = & \left\{U\in { \mathcal U }({ \mathcal H }):\forall u\in {{\mathbb{Z}}}_{d}\times {{\mathbb{Z}}}_{d},\exists u^{\prime} \in {{\mathbb{Z}}}_{d}\times {{\mathbb{Z}}}_{d},\right.\\ & & \left.\theta \in {\mathbb{R}},\,{\rm{s.t.}}\,U{T}_{u}{U}^{\dagger }={\,\rm{e}\,}^{{\rm{i}}\theta }{T}_{u^{\prime} }\right\},\end{array}\end{eqnarray*}$
and the set of all pure stabilizer states is defined as [5] $\begin{eqnarray*}\begin{array}{r}{ \mathcal P }{{ \mathcal S }}_{d}=\left\{V| 0\rangle :V\in {{ \mathcal C }}_{d}\right\},\end{array}\end{eqnarray*}$
while the set of all stabilizer states is the convex hull of ${ \mathcal P }{{ \mathcal S }}_{d}$, i.e. [5], $\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal S }}_{d} & = & \left\{\rho \in { \mathcal D }({ \mathcal H }):\rho =\displaystyle \sum _{j}{p}_{j}{\rho }_{j},{\left\{{\rho }_{j}\right\}}_{j}\in { \mathcal P }{{ \mathcal S }}_{d},\right.\\ & & \left.{p}_{j}\geqslant 0,\displaystyle \sum _{j}{p}_{j}=1\right\}.\end{array}\end{eqnarray*}$
A quantum state is a magic state if it is not a stabilizer state.A stabilizer operation is any map from $\rho \in { \mathcal D }\left({{ \mathcal H }}_{{d}^{n}}\right)$ to $\sigma \in { \mathcal D }\left({{ \mathcal H }}_{{d}^{m}}\right)$ composed from the following operations [5]
(i) Clifford unitaries, ρ → VρV† where $V\in {{ \mathcal C }}_{d}$.
(ii) Composition with stabilizer states, ρ → ρ ⨂ σ where $\sigma \in {{ \mathcal S }}_{d}$.
(iii) Computational basis measurement on the final qudit, $\rho \to \frac{\left({\mathbb{I}}\otimes | i\rangle \langle i| \right)\rho \left({\mathbb{I}}\otimes | i\rangle \langle i| \right)}{\,\rm{Tr}\,\left(\rho {\mathbb{I}}\otimes | i\rangle \langle i| \right)}$ with probability ${\rm{Tr}}\,\left(\rho {\mathbb{I}}\otimes | i\rangle \langle i| \right)$, where ${\mathbb{I}}$ is the identity operator on ${ \mathcal D }\left({{ \mathcal H }}_{{d}^{n-1}}\right)$.
(iv) Partial trace of the final qudit, $\rho \to {\,\rm{Tr}\,}_{n}\left(\rho \right)$.
(v) The above quantum operations conditioned on the outcomes of measurements or classical randomness.
2.2. The framework of magic resource theory
Two important ingredients of a resource theory are free states and free operations. In magic resource theory, free states and free operations refer to stabilizer states and stabilizer operations. We first recall the framework of the quantification of magic resource for quantum states.
A functional ${ \mathcal M }$ that maps ${ \mathcal D }({ \mathcal H })$ to [0, + ∞) is called a magic measure if it satisfies [48]
(i) Faithfulness: ${ \mathcal M }(\rho )=0$ if and only if $\rho \in {{ \mathcal S }}_{d}$.
(ii) Monotonicity: ${ \mathcal M }({ \mathcal E }(\rho ))\leqslant { \mathcal M }(\rho )$ for any stabilizer operation ${ \mathcal E }$.
(iii) Convexity: ${ \mathcal M }({\sum }_{j}{p}_{j}{\rho }_{j})\leqslant {\sum }_{j}{p}_{j}{ \mathcal M }({\rho }_{j})$ for any ${\left\{{\rho }_{j}\right\}}_{j}\in { \mathcal D }({ \mathcal H })$ and probability distribution ${\left\{{p}_{j}\right\}}_{j}$.
(iv) Strong monotonicity: ${ \mathcal M }(\rho )\geqslant {\sum }_{j}{p}_{j}{ \mathcal M }({\rho }_{j})$ for any stabilizer operation ${ \mathcal E }$ satisfying ${ \mathcal E }(\rho )={\sum }_{j}{p}_{j}{\rho }_{j}$.
If ${ \mathcal M }$(·) only satisfies properties (i) and (ii), we say that ${ \mathcal M }$(·) is a magic monotone. And ${ \mathcal M }$(·) is called a pure-state stabilizer monotone if it satisfies [26]
${({\rm{i}})}^{{\prime} }$ Faithfulness: ${ \mathcal M }(| \psi \rangle \langle \psi | )=0$ if and only if $| \psi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}$.
${({\rm{ii}})}^{{\prime} }$ Monotonicity: ${ \mathcal M }(| \phi \rangle \langle \phi | )\leqslant { \mathcal M }(| \psi \rangle \langle \psi | )$ for any stabilizer operation ${ \mathcal E }$ satisfying ${ \mathcal E }(| \psi \rangle \langle \psi | )=| \phi \rangle \langle \phi | $.
A good magic quantifier should be at least a pure-state stabilizer monotone.
2.3. The quantum (α, β) entropy and the quantum $\left(\alpha ,\beta \right)$-relative entropy
The quantum (α, β) entropy is defined as [33]
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{S}_{\alpha ,\beta }\left(\rho \right)=\frac{1}{(1-\alpha )\beta }\left[{\left({\rm{Tr}}({\rho }^{\alpha })\right)}^{\beta }-1\right],\\ \,\,\alpha \in (0,1)\cup (1,+\infty ),\\ \,\,\beta \in (-\infty ,0)\cup (0,+\infty ),\end{array}\end{array}\end{eqnarray*}$
and the quantum $\left(\alpha ,\beta \right)$-relative entropy is defined as [34] $\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{D}_{\alpha ,\beta }\left(\rho \parallel \sigma \right)=\frac{1}{(1-\alpha )\beta }\left[1-{\left({\rm{Tr}}\left({\rho }^{\alpha }{\sigma }^{1-\alpha }\right)\right)}^{\beta }\right],\\ \alpha \in (0,1)\cup (1,+\infty ),\\ \beta \in (-\infty ,0)\cup (0,+\infty ).\end{array}\end{array}\end{eqnarray*}$
Throughout this paper, we follow the convention that for any density operator $\rho \in { \mathcal D }({ \mathcal H })$, ρ−1 is only evaluated on its support, i.e. we retain the zero eigenvalues of ρ−1, thus it is reasonable to define Dα,β(ρ∥σ) for α ∈ (1, + ∞).
The basic properties of Sα,β(ρ) and ${D}_{\alpha ,\beta }\left(\rho \parallel \sigma \right)$ are summarized in the following two lemmas, which will be useful in the next section.
Let ${\left\{{\rho }_{j}\right\}}_{j}\in { \mathcal D }({ \mathcal H })$, and ${\left\{{p}_{j}\right\}}_{j}$ be a probability distribution. Then we have
(i) (Concavity) For α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{S}_{\alpha ,\beta }\left(\displaystyle \sum _{j}^{}{p}_{j}{\rho }_{j}\right)\geqslant \displaystyle \sum _{j}^{}{p}_{j}{S}_{\alpha ,\beta }\left({\rho }_{j}\right),\end{array}\end{eqnarray*}$
and for α ∈ (1, + ∞), β ∈ [1, + ∞), it holds that $\begin{eqnarray*}\begin{array}{rcl}\displaystyle \sum _{j}^{}{p}_{j}{S}_{\alpha ,\beta }({\rho }_{j}) & \leqslant & {S}_{\alpha ,\beta }\left(\displaystyle \sum _{j}^{}{p}_{j}{\rho }_{j}\right)\\ & \leqslant & \displaystyle \sum _{j}^{}{p}_{j}{S}_{\alpha ,\beta }\left({\rho }_{j}\right)+{F}_{\alpha }^{\beta }\left({\left\{{p}_{j}\right\}}_{j}\right),\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{r}{F}_{\alpha }^{\beta }\left({\left\{{p}_{j}\right\}}_{j}\right)=\frac{1}{(1-\alpha )\beta }\left(\displaystyle \sum _{j}^{}{p}_{j}^{\alpha \beta }-1\right).\end{array}\end{eqnarray*}$
(ii) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0) ∪ (0, +∞), it holds that
$\begin{eqnarray*}\begin{array}{rcl}{S}_{\alpha ,\beta }\left(\rho \displaystyle \otimes \sigma \right) & = & {S}_{\alpha ,\beta }\left(\rho \right)+{S}_{\alpha ,\beta }\left(\sigma \right)\\ & & +(1-\alpha )\beta {S}_{\alpha ,\beta }\left(\rho \right){S}_{\alpha ,\beta }\left(\sigma \right).\end{array}\end{eqnarray*}$
(iii) (Unitary invariance) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0) ∪ (0, +∞) and any unitary operator $U\in { \mathcal U }({ \mathcal H })$, it holds that
$\begin{eqnarray*}\begin{array}{r}{S}_{\alpha ,\beta }\left(\rho \right)={S}_{\alpha ,\beta }\left(U\rho {U}^{\dagger }\right).\end{array}\end{eqnarray*}$
(iv) (Lipschitz continuity) For α ∈ (1, +∞), β ∈ [1, +∞), it holds that
$\begin{eqnarray*}\begin{array}{r}| {S}_{\alpha ,\beta }(\rho )-{S}_{\alpha ,\beta }(\sigma )| \leqslant \frac{\alpha }{\alpha -1}\parallel \rho -\sigma {\parallel }_{1}.\end{array}\end{eqnarray*}$
Let ${\left\{{\rho }_{j}\right\}}_{j},{\left\{{\sigma }_{j}\right\}}_{j}\in { \mathcal D }({ \mathcal H })$, and ${\left\{{p}_{j}\right\}}_{j}$ be a probability distribution. Then we have
(i) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0) ∪ (0, +∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{D}_{\alpha ,\beta }\left(\rho \parallel \sigma \right)\geqslant 0,\end{array}\end{eqnarray*}$
and the equality holds if and only if ρ = σ. (ii) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0) ∪ (0, +∞), it holds that
$\begin{eqnarray*}\begin{array}{l}{D}_{\alpha ,\beta }\left({\rho }_{1}\displaystyle \otimes {\rho }_{2}\parallel {\sigma }_{1}\displaystyle \otimes {\sigma }_{2}\right)={D}_{\alpha ,\beta }\left({\rho }_{1}\parallel {\sigma }_{1}\right)+{D}_{\alpha ,\beta }\left({\rho }_{2}\parallel {\sigma }_{2}\right)\\ \quad +(\alpha -1)\beta {D}_{\alpha ,\beta }\left({\rho }_{1}\parallel {\sigma }_{1}\right){D}_{\alpha ,\beta }\left({\rho }_{2}\parallel {\sigma }_{2}\right).\end{array}\end{eqnarray*}$
(iii) (Monotonicity) For any quantum operation ${ \mathcal E }$ and α ∈ (0, 1), β ∈ (−∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{D}_{\alpha ,\beta }\left({ \mathcal E }(\rho )\parallel { \mathcal E }(\sigma )\right)\leqslant {D}_{\alpha ,\beta }\left(\rho \parallel \sigma \right).\end{array}\end{eqnarray*}$
(iv) (Unitary invariance) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0) ∪ (0, +∞) and any unitary operator $U\in { \mathcal U }({ \mathcal H })$, it holds that
$\begin{eqnarray*}\begin{array}{r}{D}_{\alpha ,\beta }\left(U\rho {U}^{\dagger }\parallel U\sigma {U}^{\dagger }\right)={D}_{\alpha ,\beta }\left(\rho \parallel \sigma \right).\end{array}\end{eqnarray*}$
(v) (Joint convexity) For α ∈ (0, 1), β ∈ (−∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{D}_{\alpha ,\beta }\left(\displaystyle \sum _{j}^{}{p}_{j}{\rho }_{j}\parallel \displaystyle \sum _{j}^{}{p}_{j}{\sigma }_{j}\right)\leqslant \displaystyle \sum _{j}^{}{p}_{j}{D}_{\alpha ,\beta }\left({\rho }_{j},{\sigma }_{j}\right).\end{array}\end{eqnarray*}$
3. Quantum (α, β) Jensen–Shannon divergence
In this section, we present the definition and some properties of the quantum (α, β) Jensen–Shannon divergence.
The quantum Jensen–Shannon divergence is defined as [38]
$\begin{eqnarray*}\begin{array}{rcl}J(\rho ,\sigma ) & = & \frac{1}{2}\left[D\left(\rho \parallel \frac{\rho +\sigma }{2}\right)+D\left(\sigma \parallel \frac{\rho +\sigma }{2}\right)\right]\\ & = & S\left(\frac{\rho +\sigma }{2}\right)-\frac{1}{2}S(\rho )-\frac{1}{2}S(\sigma ),\end{array}\end{eqnarray*}$
where $S(\rho )=-{\rm{Tr}}\rho {\mathrm{log}}\rho $ is the von Neumann entropy of ρ and $D(\rho \parallel \sigma )={\rm{Tr}}(\rho {\mathrm{log}}\rho -\rho {\mathrm{log}}\sigma )$ is the relative entropy between ρ and σ.Motivated by the above concepts, we now define the quantum (α, β) Jensen–Shannon divergence as follows.
For α ∈ (0, 1) ∪(1, +∞), β ∈ (−∞, 0) ∪ (0, +∞), we define two kinds of quantum (α, β) Jensen–Shannon divergence as
$\begin{eqnarray}{\,J}_{\alpha ,\beta }(\rho ,\sigma )={S}_{\alpha ,\beta }\left(\frac{\rho +\sigma }{2}\right)-\frac{1}{2}{S}_{\alpha ,\beta }\left(\rho \right)-\frac{1}{2}{S}_{\alpha ,\beta }\left(\sigma \right),\end{eqnarray}$
and $\begin{eqnarray}{J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma )=\frac{1}{2}\left[{D}_{\alpha ,\beta }\left(\rho \parallel \frac{\rho +\sigma }{2}\right)+{D}_{\alpha ,\beta }\left(\sigma \parallel \frac{\rho +\sigma }{2}\right)\right].\end{eqnarray}$
Equations (
For any α ∈ (0, 1) ∪ (1, 2), β ∈ (−∞, 0) ∪(0, +∞), it holds that
$\begin{eqnarray}{J}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right)={J}_{2-\alpha ,\beta }^{{\prime} }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right).\end{eqnarray}$
We leave the proof of proposition 1 in appendix A.1 .
Based on lemma 1 and lemma 2 , the properties of Jα,β(ρ, σ) and ${J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma )$ can be derived as follows.
Let ${\rho }_{1},{\rho }_{2},{\sigma }_{1},{\sigma }_{2}\in { \mathcal D }({ \mathcal H })$, ${\left\{| {\psi }_{j}\rangle \right\}}_{j}$, ${\left\{| {\phi }_{j}\rangle \right\}}_{j}$ are pure states on ${ \mathcal H }$, and ${\left\{{p}_{j}\right\}}_{j}$ is a probability distribution. Then we have
(i) It holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }(\rho ,\sigma )\geqslant 0\end{array},\end{eqnarray*}$
for α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, 1) or α ∈ (1, + ∞), β ∈ [1, + ∞), and $\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }(\rho ,\sigma )\leqslant \frac{1}{(1-\alpha )\beta }\left({2}^{1-\alpha \beta }-1\right)\end{array},\end{eqnarray*}$
for α ∈ (1, +∞), β ∈ [1, +∞). (ii) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0)∪(0, +∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }(\rho \displaystyle \otimes \tau ,\sigma \displaystyle \otimes \tau )=\left[1+(1-\alpha )\beta {S}_{\alpha ,\beta }(\tau )\right]{J}_{\alpha ,\beta }(\rho ,\sigma ).\end{array}\end{eqnarray*}$
Specifically, when τ is a pure state, we have Jα,β(ρ ⨂ τ, σ ⨂ τ) = Jα,β(ρ, σ). (iii) (Unitary invariance) For α ∈ (0, 1) ∪ (1, +∞), β ∈ (−∞, 0)∪(0, +∞) and any unitary operator $U\in { \mathcal U }({ \mathcal H })$, it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }(U\rho {U}^{\dagger },U\sigma {U}^{\dagger })={J}_{\alpha ,\beta }(\rho ,\sigma ).\end{array}\end{eqnarray*}$
(iv) (Symmetry) For α ∈ (0, 1) ∪ (1, +∞), β ∈ ( − ∞, 0)∪(0, +∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }(\rho ,\sigma )={J}_{\alpha ,\beta }(\sigma ,\rho ).\end{array}\end{eqnarray*}$
(v) (Lipschitz continuity) For α ∈ (1, + ∞), β ∈ [1, + ∞), Jα,β(ρ, σ) is Lipschitz continuous in the first or second entry, i.e.
$\begin{eqnarray*}\begin{array}{r}| {J}_{\alpha ,\beta }({\rho }_{1},\sigma )-{J}_{\alpha ,\beta }({\rho }_{2},\sigma )| \leqslant \frac{\alpha }{\alpha -1}\parallel {\rho }_{1}-{\rho }_{2}{\parallel }_{1},\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{r}| {J}_{\alpha ,\beta }(\rho ,{\sigma }_{1})-{J}_{\alpha ,\beta }(\rho ,{\sigma }_{2})| \leqslant \frac{\alpha }{\alpha -1}\parallel {\sigma }_{1}-{\sigma }_{2}{\parallel }_{1}.\end{array}\end{eqnarray*}$
We leave the proof of theorem 1 in appendix A.2 .
Let ${\left\{{\rho }_{j}\right\}}_{j},{\left\{{\sigma }_{j}\right\}}_{j}\in { \mathcal D }({ \mathcal H })$, and ${\left\{{p}_{j}\right\}}_{j}$ be a probability distribution. Then we have
(i) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma )\geqslant 0,\end{array}\end{eqnarray*}$
and the equality holds if and only if ρ = σ. (ii) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }^{{\prime} }(\rho \displaystyle \otimes \tau ,\sigma \displaystyle \otimes \tau )={J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma ).\end{array}\end{eqnarray*}$
(iii) (Monotonicity) For any quantum operation ${ \mathcal E }$ and α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }^{{\prime} }\left({ \mathcal E }(\rho ),{ \mathcal E }(\sigma )\right)\leqslant {J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma ).\end{array}\end{eqnarray*}$
(iv) (Unitary invariance) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0)∪(0, + ∞) and any unitary operator $U\in { \mathcal U }({ \mathcal H })$, it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }^{{\prime} }(U\rho {U}^{\dagger },U\sigma {U}^{\dagger })={J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma ).\end{array}\end{eqnarray*}$
(v) (Joint convexity) For α ∈ (0, 1), β ∈ ( − ∞, 0) ∪(0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }^{{\prime} }\left(\displaystyle \sum _{j}^{}{p}_{j}{\rho }_{j},\displaystyle \sum _{j}^{}{p}_{j}{\sigma }_{j}\right)\leqslant \displaystyle \sum _{j}^{}{p}_{j}{J}_{\alpha ,\beta }^{{\prime} }\left({\rho }_{j},{\sigma }_{j}\right).\end{array}\end{eqnarray*}$
(vi) (Symmetry) For α ∈ (0, 1) ∪ (1, ∞), β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma )={J}_{\alpha ,\beta }^{{\prime} }(\sigma ,\rho ).\end{array}\end{eqnarray*}$
We leave the proof of theorem 2 in appendix A.3 .
By applying proposition
$\begin{eqnarray}{J}_{\alpha ,\beta }\left({ \mathcal E }\left(| \psi \rangle \langle \psi | \right),{ \mathcal E }(| \phi \rangle \langle \phi | )\right)\leqslant {J}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right).\end{eqnarray}$
4. Quantum (α, β) Jensen–Shannon divergence of magic
In this section, we present the definition and some properties of the quantum (α, β) Jensen–Shannon divergence of magic, and compare our magic quantifiers with other existing ones.
4.1. Two new stabilizer monotones
We start by defining two magic quantifiers for pure states via the quantum (α, β) Jensen–Shannon divergence.
For α ∈ (0, 1) ∪(1, + ∞),β ∈ ( − ∞, 0)∪(0, + ∞), define
$\begin{eqnarray}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)=\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{J}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right),\end{eqnarray}$
$\begin{eqnarray}{m}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)=\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{J}_{\alpha ,\beta }^{{\prime} }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right).\end{eqnarray}$
Next we use convex roof construction to define the quantum (α, β) Jensen–Shannon divergence of magic for mixed states.
For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0)∪(0, + ∞), we define two kinds of quantum (α, β) Jensen–Shannon divergence of magic as
$\begin{eqnarray}{M}_{\alpha ,\beta }\left(\rho \right)=\mathop{\min }\limits_{\left\{({p}_{j},| {\psi }_{j}\rangle )\right\}}\displaystyle \sum _{j}^{}{p}_{j}{M}_{\alpha ,\beta }\left(| {\psi }_{j}\rangle \langle {\psi }_{j}| \right),\end{eqnarray}$
$\begin{eqnarray}{m}_{\alpha ,\beta }\left(\rho \right)=\mathop{\min }\limits_{\left\{({p}_{j},| {\psi }_{j}\rangle )\right\}}\displaystyle \sum _{j}^{}{p}_{j}{m}_{\alpha ,\beta }\left(| {\psi }_{j}\rangle \langle {\psi }_{j}| \right),\end{eqnarray}$
where $\rho ={\sum }_{j}^{}{p}_{j}| {\psi }_{j}\rangle \langle {\psi }_{j}| $ are all pure-state decompositions of ρ.From the proof of proposition 1 , we can see that
$\begin{eqnarray}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)=\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\frac{1}{(1-\alpha )\beta }\left[{\left({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha }\right)}^{\beta }-1\right],\end{eqnarray}$
and $\begin{eqnarray}{m}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)=\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\frac{1}{(1-\alpha )\beta }\left[1-{\left({\lambda }_{1}^{2-\alpha }+{\lambda }_{2}^{2-\alpha }\right)}^{\beta }\right],\end{eqnarray}$
where ${\lambda }_{1}=\frac{1+| \langle \psi | \phi \rangle | }{2}$ and ${\lambda }_{2}=\frac{1-| \langle \psi | \phi \rangle | }{2}$ are the eigenvalues of $\frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}$. This implies that $\begin{eqnarray}{m}_{\alpha ,\beta }\left(\rho \right)={M}_{2-\alpha ,\beta }\left(\rho \right),\end{eqnarray}$
for α ∈ (0, 1) ∪ (1, 2), β ∈ ( − ∞, 0) ∪ (0, + ∞).Both of the quantities in equations (
Next we give some properties of ${M}_{\alpha ,\beta }\left(\rho \right)$ and ${m}_{\alpha ,\beta }\left(\rho \right)$.
Let ${\left\{{\rho }_{j}\right\}}_{j}\in { \mathcal D }({ \mathcal H })$, and ${\left\{{p}_{j}\right\}}_{j}$ be a probability distribution. Then we have
(i) (Faithfulness) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{M}_{\alpha ,\beta }\left(\rho \right)\geqslant 0,\end{array}\end{eqnarray*}$
and the equality holds if and only if $\rho \in {{ \mathcal S }}_{d}$. (ii) (Invariance under Clifford operations) For any Clifford operator $V\in {{ \mathcal C }}_{d}$ and α ∈ (0, 1) ∪ (1, + ∞),β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{M}_{\alpha ,\beta }\left(V\rho {V}^{\dagger }\right)={M}_{\alpha ,\beta }\left(\rho \right).\end{array}\end{eqnarray*}$
(iii) (Convexity) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{M}_{\alpha ,\beta }\left(\displaystyle \sum _{j}^{}{p}_{j}{\rho }_{j}\right)\leqslant \displaystyle \sum _{j}^{}{p}_{j}{M}_{\alpha ,\beta }\left({\rho }_{j}\right).\end{array}\end{eqnarray*}$
(iv) (Monotonicity) For α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1] and any stabilizer operation ${ \mathcal E }$ which obeys ${ \mathcal E }(| \psi \rangle \langle \psi | )=| \phi \rangle \langle \phi | $, it holds that
$\begin{eqnarray*}\begin{array}{r}{M}_{\alpha ,\beta }\left(| \phi \rangle \langle \phi | \right)\leqslant {M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right).\end{array}\end{eqnarray*}$
(v) For α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{M}_{\alpha ,\beta }\left(\rho \displaystyle \otimes \sigma \right)\geqslant {M}_{\alpha ,\beta }\left(\rho \right),\end{array}\end{eqnarray*}$
and the equality holds if $\sigma \in {{ \mathcal S }}_{d}$. (vi) (Lipschitz continuity) For α ∈ (1, + ∞), β ∈ [1, + ∞), Mα,β(ρ) is Lipschitz continuous for pure states, i.e.
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}| {M}_{\alpha ,\beta }(| {\psi }_{1}\rangle \langle {\psi }_{1}| )-{M}_{\alpha ,\beta }(| {\psi }_{2}\rangle \langle {\psi }_{2}| )| \\ \quad \leqslant \frac{\alpha }{\alpha -1}| | | {\psi }_{1}\rangle \langle {\psi }_{1}| -| {\psi }_{2}\rangle \langle {\psi }_{2}| | {| }_{1}.\end{array}\end{array}\end{eqnarray*}$
We leave the proof of theorem 3 in appendix A.4 .
By imitating the proof of theorem 3 and utilizing theorem 2 , we can also prove that mα,β exhibits the following desirable properties.
Let ${\left\{{\rho }_{j}\right\}}_{j}\in { \mathcal D }({ \mathcal H })$, and ${\left\{{p}_{j}\right\}}_{j}$ be a probability distribution. Then we have
(i) (Faithfulness) For α ∈ (0, 1) ∪ (1, + ∞),β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{m}_{\alpha ,\beta }\left(\rho \right)\geqslant 0,\end{array}\end{eqnarray*}$
and the equality holds if and only if $\rho \in {{ \mathcal S }}_{d}$. (ii) (Invariance under Clifford operations) For any Clifford operator $V\in {{ \mathcal C }}_{d}$ and α ∈ (0, 1)∪(1, + ∞),β ∈ ( − ∞, 0)∪(0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{m}_{\alpha ,\beta }\left(V\rho {V}^{\dagger }\right)={m}_{\alpha ,\beta }\left(\rho \right).\end{array}\end{eqnarray*}$
(iii) (Convexity) For α ∈ (0, 1) ∪ (1, + ∞),β ∈ ( − ∞, 0)∪(0, + ∞) and ρ = ∑jpjρj, convexity holds, i.e.
$\begin{eqnarray*}\begin{array}{r}{m}_{\alpha ,\beta }\left(\displaystyle \sum _{j}^{}{p}_{j}{\rho }_{j}\right)\leqslant \displaystyle \sum _{j}^{}{p}_{j}{m}_{\alpha ,\beta }\left({\rho }_{j}\right).\end{array}\end{eqnarray*}$
(iv) (Monotonicity) For α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, 1] and any stabilizer operation ${ \mathcal E }$ which satisfying ${ \mathcal E }(| \psi \rangle \langle \psi | )=| \phi \rangle \langle \phi | $, it holds that
$\begin{eqnarray*}\begin{array}{r}{m}_{\alpha ,\beta }\left(| \phi \rangle \langle \phi | \right)\leqslant {m}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right).\end{array}\end{eqnarray*}$
(v) For α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{m}_{\alpha ,\beta }\left(\rho \displaystyle \otimes \sigma \right)\geqslant {m}_{\alpha ,\beta }\left(\rho \right),\end{array}\end{eqnarray*}$
and the equality holds if $\sigma \in {{ \mathcal S }}_{d}$.The above two theorems show that the two versions of quantum (α, β) Jensen–Shannon divergence of magic are both pure-state stabilizer monotones with suitable parameter ranges which will be very helpful while dealing with some pure-state issues. Since Mα,β(ρ) and mα,β(ρ) share similar properties and are closely related, we may only focus on discussing Mα,β(ρ) in the rest of our work.
We first present two obvious lemmas which are easy to verify.
The function
$\begin{eqnarray}g(x,\alpha ,\beta )=\frac{{x}^{\beta }-1}{(1-\alpha )\beta }\end{eqnarray}$
satisfies (i) g(x, α, β) is strictly monotonically increasing for α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, + ∞) and strictly monotonically decreasing for α ∈ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞) with respect to x ∈ (0, + ∞);
(ii) g(x, α, β) is strictly convex with respect to x ∈ (0, + ∞) for α ∈ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, 1).
The function
$\begin{eqnarray}f(\lambda ,\alpha )={\lambda }^{\alpha }+{(1-\lambda )}^{\alpha }\end{eqnarray}$
is monotonically decreasing with respect to $\lambda \in (\frac{1}{2},1)$ for α ∈ (0, 1) and monotonically increasing with respect to $\lambda \in (\frac{1}{2},1)$ for α ∈ (1, + ∞).It seems a little difficult to calculate our magic monotones even for pure states. However, we can simplify the calculation of equation (5 ) by means of the following proposition.
For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞), we have
$\begin{eqnarray}\begin{array}{rcl}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right) & = & \frac{1}{(1-\alpha )\beta }\left\{\left[{\left(\frac{1+| {c}_{\psi }| }{2}\right)}^{\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1-| {c}_{\psi }| }{2}\right)}^{\alpha }\right]}^{\beta }-1\right\},\end{array}\end{eqnarray}$
where $| {c}_{\psi }| ={{\rm{\max }}}_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}| \langle \psi | \phi \rangle | $.We leave the proof of proposition 2 in appendix A.5 . From the above proposition, we can directly determine ${M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)$ by calculating ∣cψ∣, which may be easily calculated for any pure state ∣ψ⟩ in qubit or qutrit systems.
4.2. Relationship with other magic quantifiers
In this subsection, we compare the relationship between our magic quantifier Mα,β with other existing magic quantifiers.
The robustness of magic of ρ is defined as [21]
$\begin{eqnarray}R(\rho )=\mathop{\inf }\limits_{\sigma \in {{ \mathcal S }}_{d}}\left\{s\geqslant 0| \frac{\rho +s\sigma }{1+s}\in {{ \mathcal S }}_{d}\right\},\end{eqnarray}$
and the min-relative entropy of magic is defined by [22] $\begin{eqnarray}{D}_{{\rm{\min }}}(\rho )=\mathop{\min }\limits_{\sigma \in {{ \mathcal S }}_{d}}\left\{-{\rm{log}}\left[{\rm{Tr}}\left({\pi }_{\rho }\sigma \right)\right]\right\},\end{eqnarray}$
where πρ is the projector onto the support of ρ. These two quantifiers exhibit the following relationship [22] $\begin{eqnarray}{D}_{{\rm{\min }}}(\rho )\leqslant {\mathrm{log}}\left(1+R(\rho )\right).\end{eqnarray}$
When ρ = ∣ψ⟩⟨ψ∣ is a pure state, we have ${D}_{{\rm{\min }}}\left(| \psi \rangle \langle \psi | \right)=-{\rm{log}}F\left(| \psi \rangle \langle \psi | \right)$, where
$\begin{eqnarray}F\left(| \psi \rangle \langle \psi | \right)=\mathop{\max }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}| \langle \psi | \phi \rangle {| }^{2}\end{eqnarray}$
is the stabilizer fidelity [51], which implies that $\begin{eqnarray}\frac{1}{1+R\left(| \psi \rangle \langle \psi | \right)}\leqslant F\left(| \psi \rangle \langle \psi | \right).\end{eqnarray}$
In view of the monotonicity of classical Tsallis α-entropy with respect to α ∈ (1, + ∞) and the monotonicity of classical unified $\left(\alpha ,\beta \right)$-entropy with respect to β ∈ (1, + ∞), we have the following two lemmas.
For λ ∈ [0, 1],
$\begin{eqnarray}{H}_{\alpha }\left\{\left(\lambda ,1-\lambda \right)\right\}=\frac{1}{1-\alpha }\left[{\lambda }^{\alpha }+{\left(1-\lambda \right)}^{\alpha }-1\right]\end{eqnarray}$
is monotonically decreasing with respect to α ∈ (1, + ∞) and monotonically increasing with respect to α ∈ (0, 1).For λ ∈ [0, 1],
$\begin{eqnarray}{H}_{\alpha ,\beta }\left\{\left(\lambda ,1-\lambda \right)\right\}=\frac{1}{(1-\alpha )\beta }\left\{{\left[{\lambda }^{\alpha }+{\left(1-\lambda \right)}^{\alpha }\right]}^{\beta }-1\right\}\end{eqnarray}$
is monotonically decreasing in α ∈ (1, + ∞) and monotonically increasing in α ∈ (0, 1) with respect to β ∈ (1, + ∞).The relationship between the quantum (α, β) Jensen–Shannon divergence of magic and the robustness of magic can be established as follows.
For α ∈ (0, 1) ∪ (1, + ∞), λ ∈ [1/2, 1], β ∈ (1, + ∞), it holds that
$\begin{eqnarray}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)\leqslant {t}_{0}-\frac{1}{1+R\left(| \psi \rangle \langle \psi | \right)},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{r}{t}_{0}={(2{\lambda }_{0}-1)}^{2}+H\left\{\left({\lambda }_{0},1-{\lambda }_{0}\right)\right\},\end{array}\end{eqnarray*}$
and λ0 is the solution of equation λ = (1 − λ)162λ−1.We leave the proof of proposition 3 in appendix A.6 .
5. Magic generating power of quantum gates via quantum (α, β) Jensen–Shannon divergence
In this section, we discuss magic generating power of quantum gates via quantum (α, β) Jensen–Shannon divergence.
For any $U\in { \mathcal U }({ \mathcal H })$, we define the magic generating power of U based on Mα,β as
$\begin{eqnarray}{{ \mathcal M }}_{\alpha ,\beta }(U)=\mathop{\max }\limits_{\rho \in {{ \mathcal S }}_{d}}\,{M}_{\alpha ,\beta }(U\rho {U}^{\dagger }).\end{eqnarray}$
${{ \mathcal M }}_{\alpha ,\beta }(U)$ has the following properties.
For any $U,{U}_{1},{U}_{2}\in { \mathcal U }({ \mathcal H })$, we have
(i) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞), it holds that
$\begin{eqnarray*}\begin{array}{r}{{ \mathcal M }}_{\alpha ,\beta }(U)\geqslant 0,\end{array}\end{eqnarray*}$
and equality holds if and only if $U\in {{ \mathcal C }}_{d}$. (ii) For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞) and any ${V}_{1},{V}_{2}\in {{ \mathcal C }}_{d}$, it holds that
$\begin{eqnarray*}\begin{array}{r}{{ \mathcal M }}_{\alpha ,\beta }(U)={M}_{\alpha ,\beta }({V}_{1}U{V}_{2}).\end{array}\end{eqnarray*}$
(iii) For α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1], it holds that
$\begin{eqnarray*}\begin{array}{r}{{ \mathcal M }}_{\alpha ,\beta }({U}_{1}\displaystyle \otimes {U}_{2})\geqslant {{ \mathcal M }}_{\alpha ,\beta }({U}_{1}).\end{array}\end{eqnarray*}$
We leave the proof of theorem 5 in appendix A.7 .
Thus we can also simplify the calculation of equation (23 ) by the following proposition.
For α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞) and any $U\in { \mathcal U }({ \mathcal H })$, we have
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal M }}_{\alpha ,\beta }(U) & = & \frac{1}{(1-\alpha )\beta }\left\{\left[{\left(\frac{1+| {C}_{U}| }{2}\right)}^{\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1-| {C}_{U}| }{2}\right)}^{\alpha }\right]}^{\beta }-1\right\},\end{array}\end{eqnarray}$
where $\begin{eqnarray}| {C}_{U}| =\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\mathop{\max }\limits_{| \psi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}| \langle \psi | U| \phi \rangle | .\end{eqnarray}$
We leave the proof of proposition 4 in appendix A.8 . Proposition 4 reveals that ${{ \mathcal M }}_{\alpha ,\beta }(U)$ is totally determined by ∣CU∣.
By proving lemma 7 , we can apply proposition 4 to give proposition 5 .
The function
$\begin{eqnarray}w(x,\alpha )={{\rm{\cos }}}^{2\alpha }x+{{\rm{\sin }}}^{2\alpha }x\end{eqnarray}$
satisfies (i) w(x, α) is strictly monotonically decreasing for α ∈ (1, 2) with respect to $x\in (0,\frac{\pi }{4})$.
(ii) w(x, α) is strictly concave for α ∈ (1, 2) with respect to $x\in (0,\frac{\pi }{16}]$.
We leave the proof of lemma 7 in appendix A.9 .
In qubit systems, for α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1), there exists some ${U}_{0}\in { \mathcal U }({ \mathcal H })$ and input state ∣ψ0⟩ such that
$\begin{eqnarray}{M}_{\alpha ,\beta }({U}_{0}| {\psi }_{0}\rangle )-{M}_{\alpha ,\beta }(| {\psi }_{0}\rangle )\gt {{ \mathcal M }}_{\alpha ,\beta }({U}_{0}).\end{eqnarray}$
We leave the proof of proposition 5 in Appendix A.10 . This result shows that the initial nonstabilizerness in the input state can enhance the magic generating power when considering ${{ \mathcal M }}_{\alpha ,\beta }$ with α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1) in qubit systems.
6. Examples
In this section, we give three detailed examples to illustrate our results.
Note that in qubit systems, ${ \mathcal P }{{ \mathcal S }}_{2}\,=\left\{| 0\rangle ,| 1\rangle ,| +\rangle ,| -\rangle ,| +{\rm{i}}\rangle ,| -{\rm{i}}\rangle \right\}$, and any pure state ∣ψ⟩ can be represented as $| \psi \rangle =| {\psi }_{\theta ,\phi }\rangle ={\rm{\cos }}\frac{\theta }{2}| 0\rangle +{\,\rm{e}\,}^{{\rm{i}}\phi }{\rm{\sin }}\frac{\theta }{2}| 1\rangle $ with (θ, φ) ∈ [0, π] × (0, 2π]. Then we have
$\begin{eqnarray}\begin{array}{rcl}{M}_{\alpha ,\beta }\left(| {\psi }_{\theta ,\phi }\rangle \langle {\psi }_{\theta ,\phi }| \right) & = & \frac{1}{(1-\alpha )\beta }\left\{\left[{\left(\frac{1+| {q}_{\max }| }{2}\right)}^{\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1-| {q}_{\max }| }{2}\right)}^{\alpha }\right]}^{\beta }-1\right\},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{r}\begin{array}{l}| {q}_{\max }| ={\rm{\max }}\left\{\left|{\rm{\cos }}\frac{\theta }{2}\right|,\left|{\rm{\sin }}\frac{\theta }{2}\right|,\sqrt{\frac{1+{\rm{\sin }}\theta {\rm{\cos }}\phi }{2}},\right.\\ \left.\sqrt{\frac{1-{\rm{\sin }}\theta {\rm{\cos }}\phi }{2}},\sqrt{\frac{1+{\rm{\sin }}\theta \sin \phi }{2}},\sqrt{\frac{1-{\rm{\sin }}\theta {\rm{\sin }}\phi }{2}}\right\}.\end{array}\end{array}\end{eqnarray}$
Thus we obtain $\begin{eqnarray}\begin{array}{rcl}{M}_{1/2,2}\left(| {\psi }_{\theta ,\phi }\rangle \langle {\psi }_{\theta ,\phi }| \right) & = & \left[\left(\sqrt{\frac{1+| {q}_{\max }| }{2}}\right.\right.\\ & & \left.{\left.+\sqrt{\frac{1-| {q}_{\max }| }{2}}\right)}^{2}-1\right].\end{array}\end{eqnarray}$
We depict $| {q}_{\max }| $ and ${M}_{1/2,2}\left(| {\psi }_{\theta ,\phi }\rangle \langle {\psi }_{\theta ,\phi }| \right)$ in figure 1. At the same time, we can show that Mα,β(ρ) are bounded in qubit systems with α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞).
We first introduce a vital class of magic states which are called T-type states in qubit systems, i.e.
$\begin{eqnarray}| {T}_{j,k}\rangle ={\rm{\cos }}\frac{{\theta }_{j}}{2}| 0\rangle +{\,\rm{e}\,}^{{\rm{i}}{\phi }_{k}}{\rm{\sin }}\frac{{\theta }_{j}}{2}| 1\rangle ,\end{eqnarray}$
where ${\theta }_{0}={\rm{\arccos }}\left(\frac{1}{\sqrt{3}}\right)\in (0,\frac{\pi }{2}),{\theta }_{1}=\pi -{\theta }_{0}$ and ${\phi }_{k}\,=\frac{(2k+1)\pi }{4},j=0,1,k=0,1,2,3$. Direct calculations show that in qubit systems, we have
$\begin{eqnarray}\frac{1+\sqrt{\frac{3+\sqrt{3}}{6}}}{2}\leqslant | {q}_{\max }| \leqslant 1,\end{eqnarray}$
and the upper bound holds iff $\rho \in {{ \mathcal S }}_{d}$, while the lower bound holds iff ρ is a T-type state ∣Tj,k⟩ with j = 0, 1, k = 0, 1, 2, 3. Using lemma
$\begin{eqnarray}\begin{array}{rcl}0\leqslant {M}_{\alpha ,\beta }(| {\psi }_{\theta ,\phi }\rangle \langle {\psi }_{\theta ,\phi }| ) & \leqslant & \frac{1}{(1-\alpha )\beta }\left\{\left[{\left(\frac{1+\sqrt{\frac{3+\sqrt{3}}{6}}}{2}\right)}^{\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1-\sqrt{\frac{3+\sqrt{3}}{6}}}{2}\right)}^{\alpha }\right]}^{\beta }-1\right\}.\end{array}\end{eqnarray}$
Suppose that the optimal decomposition of ${M}_{\alpha ,\beta }\left(\rho \right)$ is $\left\{\left({p}_{j},| {\psi }_{j}\rangle \right)\right\}$. Then we have
$\begin{eqnarray}\begin{array}{rcl}0\leqslant {M}_{\alpha ,\beta }\left(\rho \right) & = & \displaystyle \sum _{j}{p}_{j}{M}_{\alpha ,\beta }\left(| {\psi }_{j}\rangle \langle {\psi }_{j}| \right)\\ & \leqslant & \frac{1}{(1-\alpha )\beta }\left\{\left[{\left(\frac{1+\sqrt{\frac{3+\sqrt{3}}{6}}}{2}\right)}^{\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1-\sqrt{\frac{3+\sqrt{3}}{6}}}{2}\right)}^{\alpha }\right]}^{\beta }-1\right\}.\end{array}\end{eqnarray}$
The lower bound in equation (
Figure 1. The surfaces of $| {q}_{{\rm{\max }}}| $ and ${M}_{1/2,2}\left(| {\psi }_{\theta ,\phi }\rangle \langle {\psi }_{\theta ,\phi }| \right)$ with the variation of θ ∈ [0, π] and φ ∈ [0, 2π), respectively. |
Consider qutrit systems and note that
$\begin{eqnarray*}\begin{array}{rcl}{ \mathcal P }{{ \mathcal S }}_{3} & = & \left\{| 0\rangle ,| 1\rangle ,| 2\rangle \right\}\\ & & \cup \left\{\frac{| 0\rangle +{\omega }^{j}| 1\rangle +{\omega }^{k}| 2\rangle }{\sqrt{3}}:\omega ={\,\rm{e}\,}^{\frac{2\pi }{3}{\rm{i}}},j,k=0,1,2\right\}.\end{array}\end{eqnarray*}$
The qutrit T state is defined as $| T\rangle \,=\frac{1}{\sqrt{3}}\left({\,\rm{e}\,}^{\frac{2\pi }{9}{\rm{i}}}| 0\rangle +| 1\rangle +{\,\rm{e}\,}^{\frac{-2\pi }{9}{\rm{i}}}| 2\rangle \right).$ Then by direct calculations, we have
$\begin{eqnarray*}\begin{array}{rcl}| \langle m| T\rangle | & = & \frac{1}{\sqrt{3}},\\ | \langle {\psi }_{j,k}| T\rangle | & = & | {\,\rm{e}\,}^{\frac{2\pi }{9}{\rm{i}}}+{\,\rm{e}\,}^{-\frac{2\pi j}{3}{\rm{i}}}+{\,\rm{e}\,}^{\left(-\frac{2\pi }{9}-\frac{2\pi k}{3}\right){\rm{i}}}| ,\\ m,j,k & = & 0,1,2,\end{array}\end{eqnarray*}$
where $| {\psi }_{j,k}\rangle =\frac{| 0\rangle +{\omega }^{j}| 1\rangle +{\omega }^{k}| 2\rangle }{\sqrt{3}},j,k=0,1,2.$ Thus we obtain $\begin{eqnarray*}\begin{array}{rcl}| {c}_{\max }| & = & {\rm{\max }}\left\{| \langle m| T\rangle | ,| \langle {\psi }_{j,k}| T\rangle | ,m,j,k=0,1,2\right\}\\ & = & | \langle {\psi }_{0,0}| T\rangle | =| \langle {\psi }_{0,2}| T\rangle | =| \langle {\psi }_{2,2}| T\rangle | \\ & = & \frac{1}{3}\left[1+{\rm{\cos }}\left(\frac{2\pi }{9}\right)\right].\end{array}\end{eqnarray*}$
By proposition
$\begin{eqnarray}\begin{array}{rcl}{M}_{\alpha ,\beta }\left(| T\rangle \langle T| \right) & = & \frac{1}{(1-\alpha )\beta }\left\{\left[{\left(\frac{2}{3}+\frac{{\rm{\cos }}\left(\frac{2\pi }{9}\right)}{6}\right)}^{\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1}{3}-\frac{{\rm{\cos }}\left(\frac{2\pi }{9}\right)}{6}\right)}^{\alpha }\right]}^{\beta }-1\right\},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{m}_{\alpha ,\beta }\left(| T\rangle \langle T| \right) & = & \frac{1}{(1-\alpha )\beta }\left\{1-\left[{\left(\frac{2}{3}+\frac{{\rm{\cos }}\left(\frac{2\pi }{9}\right)}{6}\right)}^{2-\alpha }\right.\right.\\ & & \left.{\left.+{\left(\frac{1}{3}-\frac{{\rm{\cos }}\left(\frac{2\pi }{9}\right)}{6}\right)}^{2-\alpha }\right]}^{\beta }\right\},\end{array}\end{eqnarray}$
for α ∈ (0, 1) ∪ (1, 2), β ∈ ( − ∞, 0) ∪ (0, + ∞). We depict ${M}_{\alpha ,\beta }\left(| T\rangle \langle T| \right)$ and ${m}_{\alpha ,\beta }\left(| T\rangle \langle T| \right)$ in equations (
Figure 2. The red surface represents ${M}_{\alpha ,\beta }\left(| T\rangle \langle T| \right)$ and the blue surface represents ${m}_{\alpha ,\beta }\left(| T\rangle \langle T| \right)$ with α ∈ (0, 1) ∪ (1, 2) and β ∈ (−20, 0) ∪ (0, 20). |
According to the proof of proposition
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{M}_{\alpha ,\beta }({T}^{1/4}| {\psi }_{0}\rangle )-{M}_{\alpha ,\beta }(| {\psi }_{0}\rangle )\\ \quad =\frac{1}{(1-\alpha )\beta }\left\{{\left[{\cos }^{2\alpha }\left(\frac{3\pi }{64}\right)+{\sin }^{2\alpha }\left(\frac{3\pi }{64}\right)\right]}^{\beta }\right.\\ \qquad -\left.{\left[{\cos }^{2\alpha }\left(\frac{\pi }{32}\right)+{\sin }^{2\alpha }\left(\frac{\pi }{32}\right)\right]}^{\beta }\right\},\end{array}\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal M }}_{\alpha ,\beta }({T}^{1/4}) & = & \frac{1}{(1-\alpha )\beta }\left\{\left[{\cos }^{2\alpha }\left(\frac{\pi }{64}\right)\right.\right.\\ & & \left.{\left.+{\sin }^{2\alpha }\left(\frac{\pi }{64}\right)\right]}^{\beta }-1\right\},\end{array}\end{eqnarray*}$
for α ∈ (1, 2) and β ∈ (−5, 0) ∪ (0, 1), where T is a qubit T-gate, and $| {\psi }_{0}\rangle =\frac{1}{\sqrt{2}}\left(| 0\rangle +{\,\rm{e}\,}^{\frac{\pi }{8}{\rm{i}}}| 1\rangle \right)$ in qubit systems. We depict Mα,β(T1/4∣ψ0⟩) − Mα,β(∣ψ0⟩) and ${{ \mathcal M }}_{\alpha ,\beta }({T}^{1/4})$ with α ∈ (1, 2) and β ∈ (− 5, 0) ∪ (0, 1) in figure 3. It shows that Mα,β(T1/4∣ψ0⟩) − Mα,β(∣ψ0⟩) is always larger than ${{ \mathcal M }}_{\alpha ,\beta }({T}^{1/4})$ in this parameter range.
Figure 3. The blue surface represents Mα,β(T1/4∣ψ0⟩) − Mα,β(∣ψ0⟩) and the red surface represents ${{ \mathcal M }}_{\alpha ,\beta }({T}^{1/4})$ with α ∈ (1, 2) and β ∈ (−5, 0) ∪ (0, 1). |
7. Summary
We have proposed two pure-state stabilizer monotones which can be calculated on low-dimensional systems for pure states. We have showed that the quantifier Mα,β(ρ) is bounded in qubit systems with α ∈ (0, 1) ∪ (1, + ∞) and β ∈ (−∞, 0) ∪ (0, + ∞), and the lower bound holds if and only if $\rho \in {{ \mathcal S }}_{d}$, while the upper bound holds if and only if ρ is a T-type state. We have also compared our magic quantifiers with other quantifiers, showing that Mα,β is controlled by some function of the robustness of magic R for pure states with α ∈ (0, 1) ∪ (1, + ∞), β ∈ (1, + ∞). As an application, we have proved that the initial nonstabilizerness in the input state can enhance the magic generating power in qubit systems when considering ${{ \mathcal M }}_{\alpha ,\beta }$ for α ∈ (1, 2), β ∈ (−∞, 0) ∪ (0, 1). Our results may shed some new light on the study of quantification of the magic at the level of quantum states, and may offer new perspective in the research of magic resource theory.
Author contributions
Linmao Wang wrote the main manuscript text and Zhaoqi Wu supervised and revised the manuscript. All authors reviewed the manuscript.
Competing interests
The authors declare no competing interests.
Appendix
Here we give the proof of propositions 1 –5 , theorems 1 –5 and lemma 7 .
A.1. Proof of proposition 1
Applying the spectral decomposition, we obtain
$\begin{eqnarray*}\begin{array}{r}\frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}={\lambda }_{1}| {\xi }_{1}\rangle \langle {\xi }_{1}| +{\lambda }_{2}| {\xi }_{2}\rangle \langle {\xi }_{2}| ,\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{rcl}{\lambda }_{1} & = & \frac{1+| \langle \psi | \phi \rangle | }{2},\,\,{\lambda }_{2}=\frac{1-| \langle \psi | \phi \rangle | }{2},\\ | {\xi }_{1}\rangle & = & \frac{| \psi \rangle +{\,\rm{e}\,}^{{\rm{i}}\theta }| \phi \rangle }{\sqrt{2(1+| s| )}},\,\,| {\xi }_{2}\rangle =\frac{| \psi \rangle -{\,\rm{e}\,}^{{\rm{i}}\theta }| \phi \rangle }{\sqrt{2(1-| s| )}},\\ s & = & \langle \phi | \psi \rangle ,\,\,s=| s| {\,\rm{e}\,}^{{\rm{i}}\theta },\theta \in {\mathbb{R}}.\end{array}\end{eqnarray*}$
Hence we have $\begin{eqnarray}\begin{array}{l}{J}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right)\\ ={S}_{\alpha ,\beta }\left(\frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}\right)\\ -\frac{1}{2}{S}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)\\ -\frac{1}{2}{S}_{\alpha ,\beta }\left(| \phi \rangle \langle \phi | \right)\\ ={S}_{\alpha ,\beta }\left(\frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}\right)\\ ={H}_{\alpha ,\beta }\left\{\left(\frac{1+| \langle \psi | \phi \rangle | }{2},\frac{1-| \langle \psi | \phi \rangle | }{2}\right)\right\}\\ =\frac{1}{(1-\alpha )\beta }\left[{\left({\lambda }_{1}^{\alpha }+{\lambda }_{2}^{\alpha }\right)}^{\beta }-1\right],\end{array}\end{eqnarray}$
where $\begin{eqnarray}{H}_{\alpha ,\beta }(\left\{{p}_{i}\right\})=\frac{1}{(1-\alpha )\beta }\left[{\left(\displaystyle \sum _{j}^{}{p}_{i}^{\alpha }\right)}^{\beta }-1\right]\end{eqnarray}$
is the unified (α, β)-entropy [52].Noting that A3 ) and (A4 ), we obtain A1 ) and (A5 ). Thus we complete the proof. □
$\begin{eqnarray*}\begin{array}{r}| \langle \psi | {\xi }_{i}\rangle {| }^{2}=| \langle \phi | {\xi }_{i}\rangle {| }^{2}={\lambda }_{i},i=1,2,\end{array}\end{eqnarray*}$
we have $\begin{eqnarray}\begin{array}{l}{D}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \parallel \frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}\right)\\ =\frac{1}{(1-\alpha )\beta }\left\{1-{\left[{\rm{Tr}}\left((| \psi \rangle \langle \psi | {)}^{\alpha }{\left(\frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}\right)}^{1-\alpha }\right)\right]}^{\beta }\right\}\\ =\frac{1}{(1-\alpha )\beta }\left[1-{\left({\lambda }_{1}^{1-\alpha }| \langle \psi | {\xi }_{1}\rangle {| }^{2}+{\lambda }_{2}^{1-\alpha }| \langle \psi | {\xi }_{2}\rangle {| }^{2}\right)}^{\beta }\right]\\ =\frac{1}{(1-\alpha )\beta }\left[1-{\left({\lambda }_{1}^{2-\alpha }+{\lambda }_{2}^{2-\alpha }\right)}^{\beta }\right],\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{D}_{\alpha ,\beta }\left(| \phi \rangle \langle \phi | \parallel \frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}\right)\\ =\frac{1}{(1-\alpha )\beta }\left\{1-{\left[{\rm{Tr}}\left((| \phi \rangle \langle \phi | {)}^{\alpha }{\left(\frac{| \psi \rangle \langle \psi | +| \phi \rangle \langle \phi | }{2}\right)}^{1-\alpha }\right)\right]}^{\beta }\right\}\\ =\frac{1}{(1-\alpha )\beta }\left[1-{\left({\lambda }_{1}^{1-\alpha }| \langle \phi | {\xi }_{1}\rangle {| }^{2}+{\lambda }_{2}^{1-\alpha }| \langle \phi | {\xi }_{2}\rangle {| }^{2}\right)}^{\beta }\right]\\ =\frac{1}{(1-\alpha )\beta }\left[1-{\left({\lambda }_{1}^{2-\alpha }+{\lambda }_{2}^{2-\alpha }\right)}^{\beta }\right].\end{array}\end{eqnarray}$
Combining equations ( $\begin{eqnarray}{J}_{\alpha ,\beta }^{{\prime} }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right)=\frac{1}{(1-\alpha )\beta }\left[1-{\left({\lambda }_{1}^{2-\alpha }+{\lambda }_{2}^{2-\alpha }\right)}^{\beta }\right],\end{eqnarray}$
and thus we get $\begin{eqnarray*}\begin{array}{r}{J}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right)={J}_{2-\alpha ,\beta }^{{\prime} }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right),\end{array}\end{eqnarray*}$
for all α ∈ (0, 1) ∪ (1, 2) and β ∈ ( − ∞, 0) ∪ (0, + ∞) by comparing equations (A.2. Proof of theorem 1
(i)–(iii) can be directly proved by applying lemma 1 , and (iv) is obvious from the definition of Jα,β(ρ, σ), so we only prove item (v).
For α ∈ (1, + ∞), β ∈ [1, + ∞), we have 1 ), the second inequality comes from lemma 1 (iv), and the second equality follows from the linearity of the trace distance.
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}| {J}_{\alpha ,\beta }({\rho }_{1},\sigma )-{J}_{\alpha ,\beta }({\rho }_{2},\sigma )| \\ =\left|{S}_{\alpha ,\beta }\left(\frac{{\rho }_{1}+\sigma }{2}\right)-{S}_{\alpha ,\beta }\left(\frac{{\rho }_{2}+\sigma }{2}\right)\right.\\ \left.+\frac{1}{2}{S}_{\alpha ,\beta }({\rho }_{2})-\frac{1}{2}{S}_{\alpha ,\beta }({\rho }_{1})\right|\\ \leqslant \left|{S}_{\alpha ,\beta }\left(\frac{{\rho }_{1}+\sigma }{2}\right)-{S}_{\alpha ,\beta }\left(\frac{{\rho }_{2}+\sigma }{2}\right)\right|\\ +\frac{1}{2}\left|{S}_{\alpha ,\beta }({\rho }_{2})-{S}_{\alpha ,\beta }({\rho }_{1})\right|\\ \leqslant \frac{\alpha }{\alpha -1}{\parallel \frac{{\rho }_{1}-{\rho }_{2}}{2}\parallel }_{1}+\frac{\alpha }{2(\alpha -1)}{\parallel {\rho }_{2}-{\rho }_{1}\parallel }_{1}\\ =\frac{\alpha }{\alpha -1}{\parallel {\rho }_{1}-{\rho }_{2}\parallel }_{1},\end{array}\end{array}\end{eqnarray*}$
where the first equality follows from equation (In a similar manner, we can prove that $| {J}_{\alpha ,\beta }(\rho ,{\sigma }_{1})\,-{J}_{\alpha ,\beta }(\rho ,{\sigma }_{2})| \leqslant \frac{\alpha }{\alpha -1}\parallel {\sigma }_{1}-{\sigma }_{2}{\parallel }_{1}$ for α ∈ (1, + ∞), β ∈ [1, + ∞). □
A.3. Proof of theorem 2
We can prove (iii)–(v) by lemma 2 (iii)–(v) respectively and (vi) can be obtained by definition, so we only prove (i) and (ii).
For item (i), by lemma 2 (i), we have ${J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma )\geqslant 0$ for α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞) and the inequality saturates if and only if
$\begin{eqnarray*}\begin{array}{r}{D}_{\alpha ,\beta }\left(\rho \parallel \frac{\rho +\sigma }{2}\right)=0={D}_{\alpha ,\beta }\left(\sigma \parallel \frac{\rho +\sigma }{2}\right),\end{array}\end{eqnarray*}$
which is equivalent to ρ = σ.For item (ii), we have 2 ), the second equality follows from lemma 2 (ii), and the third holds due to Dα,β(τ∥τ) = 0. □
$\begin{eqnarray*}\begin{array}{l}{J}_{\alpha ,\beta }^{{\prime} }(\rho \displaystyle \otimes \tau ,\sigma \displaystyle \otimes \tau )\\ =\frac{1}{2}\left[{D}_{\alpha ,\beta }\left(\rho \displaystyle \otimes \tau \parallel \frac{\rho +\sigma }{2}\displaystyle \otimes \tau \right)\right.\\ +\left.{D}_{\alpha ,\beta }\left(\sigma \displaystyle \otimes \tau \parallel \frac{\rho +\sigma }{2}\displaystyle \otimes \tau \right)\right]\\ =\frac{1}{2}\left[{D}_{\alpha ,\beta }\left(\rho \parallel \frac{\rho +\sigma }{2}\right)+(\alpha -1)\beta \right.\\ \left.{D}_{\alpha ,\beta }\,\left(\rho \parallel \frac{\rho +\sigma }{2}\right){D}_{\alpha ,\beta }\left(\tau \parallel \tau \right)\right]\\ +\frac{1}{2}\left[{D}_{\alpha ,\beta }\left(\sigma \parallel \frac{\rho +\sigma }{2}\right)\right.\\ +\left.(\alpha -1)\beta {D}_{\alpha ,\beta }\left(\sigma \parallel \frac{\rho +\sigma }{2}\right){D}_{\alpha ,\beta }\left(\,{,}(,\tau \parallel \tau \right)\Space{0ex}{3.25ex}{0ex}\right]\\ =\frac{1}{2}\left[{D}_{\alpha ,\beta }\left(\rho \parallel \frac{\rho +\sigma }{2}\right)+{D}_{\alpha ,\beta }\left(\sigma \parallel \frac{\rho +\sigma }{2}\right)\right]\\ ={J}_{\alpha ,\beta }^{{\prime} }(\rho ,\sigma ),\end{array}\end{eqnarray*}$
with α ∈ (0, 1) ∪ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞), where the first and last equality comes from equation (A.4. Proof of theorem 3
We only prove item (iv) and (vi), the other properties can be easily derived by imitating the proof of proposition 1 in [40].
For item (iv), suppose that α ∈ (1, 2), β ∈ ( − ∞, 1]. Then we have 5 ) and the fact that any stabilizer operation ${ \mathcal E }$ maps pure states to pure states, the first inequality is true since stabilizer operation preserves stabilizer states and the second inequality follows from remark 2 .
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{M}_{\alpha ,\beta }\left({ \mathcal E }\left(| \psi \rangle \langle \psi | \right)\right)\\ =\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{J}_{\alpha ,\beta }\left({ \mathcal E }\left(| \psi \rangle \langle \psi | \right),| \phi \rangle \langle \phi | \right)\\ \leqslant \mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{J}_{\alpha ,\beta }\left({ \mathcal E }\left(| \psi \rangle \langle \psi | \right),{ \mathcal E }(| \phi \rangle \langle \phi | )\right)\\ \leqslant \mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{J}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | ,| \phi \rangle \langle \phi | \right)\\ ={M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right),\end{array}\end{array}\end{eqnarray*}$
where the first equality holds by using equation (For item (vi), suppose that α ∈ (1, + ∞), β ∈ [1, + ∞). Then we have 1 (v). Similarly, we can obtain that Mα,β(∣ψ2⟩⟨ψ2∣) − Mα,β(∣ψ1⟩⟨ψ1∣)≤ $\frac{\alpha }{\alpha -1}\parallel | {\psi }_{1}\rangle \langle $ψ1∣ − ∣ψ2⟩⟨ψ2∣∥1, and thus we have
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{M}_{\alpha ,\beta }(| {\psi }_{1}\rangle \langle {\psi }_{1}| )-{M}_{\alpha ,\beta }(| {\psi }_{2}\rangle \langle {\psi }_{2}| )\\ =\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{J}_{\alpha ,\beta }\left(| {\psi }_{1}\rangle \langle {\psi }_{1}| ,| \phi \rangle \langle \phi | \right)\\ -{J}_{\alpha ,\beta }\left(| {\psi }_{2}\rangle \langle {\psi }_{2}| ,| {\phi }_{2}\rangle \langle {\phi }_{2}| \right)\\ \leqslant {J}_{\alpha ,\beta }\left(| {\psi }_{1}\rangle \langle {\psi }_{1}| ,| {\phi }_{2}\rangle \langle {\phi }_{2}| \right)\\ -{J}_{\alpha ,\beta }\left(| {\psi }_{2}\rangle \langle {\psi }_{2}| ,| {\phi }_{2}\rangle \langle {\phi }_{2}| \right)\\ \leqslant | {J}_{\alpha ,\beta }\left(| {\psi }_{1}\rangle \langle {\psi }_{1}| ,| {\phi }_{2}\rangle \langle {\phi }_{2}| \right)\\ -{J}_{\alpha ,\beta }\left(| {\psi }_{2}\rangle \langle {\psi }_{2}| ,| {\phi }_{2}\rangle \langle {\phi }_{2}| \right)| \\ \leqslant \frac{\alpha }{\alpha -1}\parallel | {\psi }_{1}\rangle \langle {\psi }_{1}| -| {\psi }_{2}\rangle \langle {\psi }_{2}| {\parallel }_{1},\end{array}\end{array}\end{eqnarray*}$
where the first equality holds because we assume that ∣φ2⟩ reaches the minimum in the definition of Mα,β(∣ψ2⟩⟨ψ2∣), the last inequality comes from theorem $\begin{eqnarray*}\begin{array}{r}\begin{array}{l}| {M}_{\alpha ,\beta }(| {\psi }_{1}\rangle \langle {\psi }_{1}| )-{M}_{\alpha ,\beta }(| {\psi }_{2}\rangle \langle {\psi }_{2}| )| \\ \quad \leqslant \frac{\alpha }{\alpha -1}\parallel | {\psi }_{1}\rangle \langle {\psi }_{1}| -| {\psi }_{2}\rangle \langle {\psi }_{2}| {\parallel }_{1},\end{array}\end{array}\end{eqnarray*}$
which completes the proof. □A.5. Proof of proposition 2
For α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, + ∞), we have 3 (i) and the third follows from lemma 4 .
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)\\ =\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\left\{\frac{1}{(1-\alpha )\beta }\left[\left({\left(\frac{1+| \langle \phi | \psi \rangle | }{2}\right)}^{\alpha }\right.\right.\right.\\ \left.\left.{\left.+{\left(\frac{1-| \langle \phi | \psi \rangle | }{2}\right)}^{\alpha }\right)}^{\beta }-1\right]\right\}\\ =\frac{1}{(1-\alpha )\beta }\left\{\left[\mathop{\min }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\left({\left(\frac{1+| \langle \phi | \psi \rangle | }{2}\right)}^{\alpha }\right.\right.\right.\\ \left.{\left.\left.+{\left(\frac{1-| \langle \phi | \psi \rangle | }{2}\right)}^{\alpha }\right)\right]}^{\beta }-1\right\}\\ =\frac{1}{(1-\alpha )\beta }\left\{{\left[{\left(\frac{1+| {c}_{\psi }| }{2}\right)}^{\alpha }+{\left(\frac{1-| {c}_{\psi }| }{2}\right)}^{\alpha }\right]}^{\beta }-1\right\},\end{array}\end{array}\end{eqnarray*}$
the first equality follows from the definition of ${M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)$, the second equality comes from lemma For α ∈ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞), we only need to replace min with max in the second equality to obtain the same conclusion. Thus we complete the proof. □
A.6. Proof of proposition 3
We only consider the case in which α ∈ (1, + ∞), while the proof is similar when α ∈ (0, 1). Denote
$\begin{eqnarray*}\begin{array}{r}h(\lambda ,\alpha ,\beta )=\frac{1}{(1-\alpha )\beta }\left\{{\left[{\lambda }^{\alpha }+{\left(1-\lambda \right)}^{\alpha }\right]}^{\beta }-1\right\}.\end{array}\end{eqnarray*}$
For α ∈ (1, + ∞), by applying lemma 6 , we have 5 and proposition 3 in [40], we obtain
$\begin{eqnarray*}\begin{array}{r}\left[{(2\lambda -1)}^{2}+h(\lambda ,\alpha ,\beta )\right]\leqslant \mathop{\mathrm{lim}}\limits_{\beta \to {1}^{+}}\left[{(2\lambda -1)}^{2}+h(\lambda ,\alpha ,\beta )\right].\end{array}\end{eqnarray*}$
Then by lemma $\begin{eqnarray*}\begin{array}{r}\begin{array}{l}\mathop{\mathrm{lim}}\limits_{\beta \to {1}^{+}}\left[{(2\lambda -1)}^{2}+h(\lambda ,\alpha ,\beta )\right]\\ \leqslant \mathop{\mathrm{lim}}\limits_{\alpha \to {1}^{+}}\mathop{\mathrm{lim}}\limits_{\beta \to {1}^{+}}\left[{(2\lambda -1)}^{2}+h(\lambda ,\alpha ,\beta )\right]\leqslant {t}_{0},\end{array}\end{array}\end{eqnarray*}$
which yields $\begin{eqnarray*}\begin{array}{r}\left[{(2\lambda -1)}^{2}+h(\lambda ,\alpha ,\beta )\right]\leqslant {t}_{0}.\end{array}\end{eqnarray*}$
From the definition of ${M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)$ and $F\left(| \psi \rangle \langle \psi | \right)$, we obtain
$\begin{eqnarray*}\begin{array}{r}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)+F\left(| \psi \rangle \langle \psi | \right)\leqslant {t}_{0},\end{array}\end{eqnarray*}$
thus $\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)+\frac{1}{1+R\left(| \psi \rangle \langle \psi | \right)}\\ \quad \leqslant {M}_{\alpha ,\beta }\left(| \psi \rangle \langle \psi | \right)+F\left(| \psi \rangle \langle \psi | \right)\leqslant {t}_{0},\end{array}\end{array}\end{eqnarray*}$
which completes the proof. □A.7. Proof of theorem 5
Items (i) and (ii) are obvious from the definition, so we only prove item (iii).
For α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1], we have 23 ), the first inequality is true since ${{ \mathcal S }}_{d}\otimes {{ \mathcal S }}_{d}\subseteq {{ \mathcal S }}_{{d}^{2}}$, and the second inequality follows from theorem 3 (v).
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{{ \mathcal M }}_{\alpha ,\beta }({U}_{1}\displaystyle \otimes {U}_{2})\\ =\mathop{\max }\limits_{\rho \in {{ \mathcal S }}_{{d}^{2}}}{M}_{\alpha ,\beta }\left[({U}_{1}\displaystyle \otimes {U}_{2})\rho ({U}_{1}^{\dagger }\displaystyle \otimes {U}_{2}^{\dagger })\right]\\ \geqslant \mathop{\max }\limits_{{\rho }_{1},{\rho }_{2}\in {{ \mathcal S }}_{d}}{M}_{\alpha ,\beta }\left[({U}_{1}\displaystyle \otimes {U}_{2})\left({\rho }_{1}\displaystyle \otimes {\rho }_{2}\right)({U}_{1}^{\dagger }\displaystyle \otimes {U}_{2}^{\dagger })\right]\\ =\mathop{\max }\limits_{{\rho }_{1},{\rho }_{2}\in {{ \mathcal S }}_{d}}{M}_{\alpha ,\beta }\left[\left({U}_{1}{\rho }_{1}{U}_{1}^{\dagger }\right)\displaystyle \otimes \left({U}_{2}{\rho }_{2}{U}_{2}^{\dagger }\right)\right]\\ \geqslant \mathop{\max }\limits_{{\rho }_{1}\in {{ \mathcal S }}_{d}}{M}_{\alpha ,\beta }\left({U}_{1}{\rho }_{1}{U}_{1}^{\dagger }\right)\\ ={{ \mathcal M }}_{\alpha ,\beta }\left({U}_{1}\right),\end{array}\end{array}\end{eqnarray*}$
where the first equality holds due to equation (A.8. Proof of proposition 4
For α ∈ (0, 1), β ∈ ( − ∞, 0) ∪ (0, + ∞), we have 3 (i) and lemma 4 .
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{{ \mathcal M }}_{\alpha ,\beta }(U)\\ =\mathop{\max }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}{M}_{\alpha ,\beta }(U| \phi \rangle \langle \phi | {U}^{\dagger })\\ =\mathop{\max }\limits_{| \phi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\mathop{\min }\limits_{| \psi \rangle \in { \mathcal P }{{ \mathcal S }}_{d}}\left\{\frac{1}{(1-\alpha )\beta }\left[\left({\left(\frac{1+| \langle \psi | U| \phi \rangle | }{2}\right)}^{\alpha }\right.\right.\right.\\ \left.\left.{\left.+{\left(\frac{1-| \langle \psi | U| \phi \rangle | }{2}\right)}^{\alpha }\right)}^{\beta }-1\right]\right\}\\ =\frac{1}{(1-\alpha )\beta }\left\{{\left[{\left(\frac{1+| {C}_{U}| }{2}\right)}^{\alpha }+{\left(\frac{1-| {C}_{U}| }{2}\right)}^{\alpha }\right]}^{\beta }-1\right\},\end{array}\end{array}\end{eqnarray*}$
where the first equality follows from the fact that ${{ \mathcal S }}_{d}$ is a convex set, the second equation comes from the definition, and the last follows from lemma It can be easily verified that the conclusion also holds for α ∈ (1, + ∞), β ∈ ( − ∞, 0) ∪ (0, + ∞). Hence we complete the proof. □
A.9. Proof of lemma 7
(i) For $\alpha \in (1,2),x\in (0,\frac{\pi }{4})$, it follows that
$\begin{eqnarray*}\begin{array}{rcl}\frac{\partial w(x,\alpha )}{\partial x} & = & 2\alpha \cdot {\rm{\sin }}x\cdot {\rm{\cos }}x\\ & & \times \left({{\rm{\sin }}}^{2\alpha -2}x-{{\rm{\cos }}}^{2\alpha -2}x\right)\lt 0.\end{array}\end{eqnarray*}$
(ii) For $\alpha \in (1,2),x\in (0,\frac{\pi }{16}]$, we obtain
$\begin{eqnarray*}\begin{array}{rcl}\frac{{\partial }^{2}w(x,\alpha )}{\partial {x}^{2}} & = & 2\alpha \left\{(2\alpha -1)[{\sin }^{2\alpha -2}x{\cos }^{2}x\right.\\ & & \left.+{\cos }^{2\alpha -2}x{\sin }^{2}x]-({\sin }^{2\alpha }x+{\cos }^{2\alpha }x)\right\}.\end{array}\end{eqnarray*}$
Let $t={\tan }^{2}x\in \left(0,{t}_{\max }\right]$, where ${t}_{\max }={\tan }^{2}\frac{\pi }{16}$. Then we have
$\begin{eqnarray*}\begin{array}{r}\frac{{\partial }^{2}w(x,\alpha )}{\partial {x}^{2}}=G(t,\alpha )=(2\alpha -1)\left({t}^{\alpha -1}+t\right)-\left({t}^{\alpha }+1\right).\end{array}\end{eqnarray*}$
We next show that G(t, α) < 0 for all $t\in (0,{t}_{\max }]$ and α ∈ (1, 2).By taking the partial derivative of G(t, α) with respect to t, we obtain
$\begin{eqnarray*}\begin{array}{rcl}\frac{\partial G(t,\alpha )}{\partial t} & = & (2\alpha -1)(\alpha -1){t}^{\alpha -2}+2\alpha -1-\alpha {t}^{\alpha -1}\\ & \geqslant & (2\alpha -1)(\alpha -1)+2\alpha -1-\alpha {t}^{\alpha -1}\\ & = & (2\alpha -1)(\alpha -1)+\left[\alpha (2-{t}^{\alpha -1})-1\right]\gt 0.\end{array}\end{eqnarray*}$
Then for all α ∈ (1, 2), $G{(t,\alpha )}_{\max }\,=$ $G({t}_{\max },\alpha )\,=$ $(2\alpha -1)\left({t}_{\max }^{\alpha -1}+{t}_{\max }\right)-$ $\left({t}_{\max }^{\alpha }+1\right).$ So we only need to prove $G({t}_{\max },\alpha )\lt 0$ for all α ∈ (1, 2).In fact, by taking the first and second derivative of $G({t}_{\max },\alpha )$, we have
$\begin{eqnarray*}\begin{array}{rcl}G^{\prime} ({t}_{\max },\alpha ) & = & 2\left({t}_{\max }^{\alpha -1}+{t}_{\max }\right)\\ & & +{\mathrm{ln}}{t}_{\max }\left[(2\alpha -1){t}_{\max }^{\alpha -1}-{t}_{\max }^{\alpha }\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{r}{G}^{{\prime\prime} }({t}_{\max },\alpha )={t}_{\max }^{\alpha -1}{\mathrm{ln}}{t}_{\max }\left[(2\alpha -1-{t}_{\max }){\mathrm{ln}}{t}_{\max }+4\right].\end{array}\end{eqnarray*}$
Noting that $\begin{eqnarray*}\begin{array}{r}{G}^{{\prime\prime} }({t}_{\max },\alpha )\gt 0\,\,{\rm{for}}\,\,\alpha \in \left(\frac{1+{t}_{\max }}{2}-\frac{2}{{\mathrm{ln}}{t}_{\max }},2\right),\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{r}{G}^{{\prime\prime} }({t}_{\max },\alpha )\lt 0\,\,{\rm{for}}\,\,\alpha \in \left(1,\frac{1+{t}_{\max }}{2}-\frac{2}{{\mathrm{ln}}{t}_{\max }}\right),\end{array}\end{eqnarray*}$
we have $\begin{eqnarray*}\begin{array}{rcl}G^{\prime} {({t}_{\max },\alpha )}_{\max } & = & \max \left\{G^{\prime} ({t}_{\max },1),G^{\prime} ({t}_{\max },2)\right\}\\ & = & G^{\prime} ({t}_{\max },2)\approx -0.2205\lt 0,\end{array}\end{eqnarray*}$
and so $\begin{eqnarray*}\begin{array}{r}G({t}_{\max },\alpha )\lt \mathop{\mathrm{lim}}\limits_{\alpha \to {0}^{+}}G({t}_{\max },\alpha )=0\end{array},\end{eqnarray*}$
for all α ∈ (1, 2). Thus we complete the proof. □A.10. Proof of proposition 5
Consider the unitary gate
$\begin{eqnarray*}\begin{array}{r}{U}_{0}={T}^{1/4}=\left[\begin{array}{cc}1 & 0\\ 0 & {\,\rm{e}\,}^{\frac{\pi }{16}{\rm{i}}}\end{array}\right],\end{array}\end{eqnarray*}$
and the input state $| {\psi }_{0}\rangle =\frac{1}{\sqrt{2}}\left(| 0\rangle +{\,\rm{e}\,}^{\frac{\pi }{8}{\rm{i}}}| 1\rangle \right).$By proposition 1 and proposition 4 , we obtain
$\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal M }}_{\alpha ,\beta }({T}^{1/4}) & = & \frac{1}{(1-\alpha )\beta }\\ & & \times \left\{{\left[{\cos }^{2\alpha }\left(\frac{\pi }{64}\right)+{\sin }^{2\alpha }\left(\frac{\pi }{64}\right)\right]}^{\beta }-1\right\},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{M}_{\alpha ,\beta }({T}^{1/4}| {\psi }_{0}\rangle )=\frac{1}{(1-\alpha )\beta }\\ \,\times \,\left\{{\left[{\cos }^{2\alpha }\left(\frac{3\pi }{64}\right)+{\sin }^{2\alpha }\left(\frac{3\pi }{64}\right)\right]}^{\beta }-1\right\},\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{M}_{\alpha ,\beta }(| {\psi }_{0}\rangle ) & = & \frac{1}{(1-\alpha )\beta }\\ & & \times \left\{{\left[{\cos }^{2\alpha }\left(\frac{\pi }{32}\right)+{\sin }^{2\alpha }\left(\frac{\pi }{32}\right)\right]}^{\beta }-1\right\}.\end{array}\end{eqnarray*}$
Denote 3 and lemma 7 , respectively.
$\begin{eqnarray*}\begin{array}{rcl}N(x,\alpha ,\beta ) & = & \frac{1}{(1-\alpha )\beta }\left[{\left({{\rm{\cos }}}^{2\alpha }x+{{\rm{\sin }}}^{2\alpha }x\right)}^{\beta }-1\right]\\ & = & g\left[w(x,\alpha ),\alpha ,\beta \right],\end{array}\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}K(x,\alpha ,\beta ) & = & N\left(x+\frac{\pi }{32},\alpha ,\beta \right)-N(x,\alpha ,\beta )\\ & & -N\left(\frac{\pi }{32},\alpha ,\beta \right),\end{array}\end{eqnarray*}$
where g(x, α, β) and w(x, α) are defined in lemma Thus we have
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}{M}_{\alpha ,\beta }({T}^{1/4}| {\psi }_{0}\rangle )-{M}_{\alpha ,\beta }(| {\psi }_{0}\rangle )-{{ \mathcal M }}_{\alpha ,\beta }({T}^{1/4})\\ \quad =K\left(\frac{\pi }{64},\alpha ,\beta \right).\end{array}\end{array}\end{eqnarray*}$
Now we prove that $K(\frac{\pi }{64},\alpha ,\beta )\gt 0$ for all α ∈ (1, 2) and β ∈ ( − ∞, 0) ∪ (0, 1). Taking the second-order partial derivative of N(x, α, β) with respect to x, we obtain
$\begin{eqnarray*}\begin{array}{rcl}\frac{{\partial }^{2}N(x,\alpha ,\beta )}{\partial {x}^{2}} & = & \frac{{\partial }^{2}g[w(x,\alpha ),\alpha ,\beta ]}{\partial {[w(x,\alpha )]}^{2}}{\left[\frac{\partial w(x,\alpha )}{\partial x}\right]}^{2}\\ & & +\frac{{\partial }^{2}w(x,\alpha )}{\partial {x}^{2}}\frac{\partial g[w(x,\alpha ),\alpha ,\beta ]}{\partial w(x,\alpha )}.\end{array}\end{eqnarray*}$
For α ∈ (1, 2), β ∈ ( − ∞, 0) ∪ (0, 1) and $x\in (0,\frac{\pi }{16})$, it follows from lemma 3 and lemma 7 that
$\begin{eqnarray*}\begin{array}{r}\begin{array}{l}\frac{{\partial }^{2}g[w(x,\alpha ),\alpha ,\beta ]}{\partial {[w(x,\alpha )]}^{2}}\gt 0,{\left[\frac{\partial w(x,\alpha )}{\partial x}\right]}^{2}\gt 0,\\ \frac{{\partial }^{2}w(x,\alpha )}{\partial {x}^{2}}\lt 0,\frac{\partial g[w(x,\alpha ),\alpha ,\beta ]}{\partial w(x,\alpha )}\lt 0.\end{array}\end{array}\end{eqnarray*}$
Then we have $\begin{eqnarray*}\begin{array}{r}\frac{{\partial }^{2}N(x,\alpha ,\beta )}{\partial {x}^{2}}\gt 0,\end{array}\end{eqnarray*}$
and therefore $\begin{eqnarray*}\begin{array}{r}\frac{\partial K(x,\alpha ,\beta )}{\partial x}=\frac{\partial N(x+\frac{\pi }{32},\alpha ,\beta )}{\partial x}-\frac{\partial N(x,\alpha ,\beta )}{\partial x}\gt 0,\end{array}\end{eqnarray*}$
which implies that K(x, α, β) is strictly monotonically increasing with respect to $x\in (0,\frac{\pi }{16}]$. Hence we have $\begin{eqnarray*}\begin{array}{r}K\left(\frac{\pi }{64},\alpha ,\beta \right)\gt K(0,\alpha ,\beta )=0,\end{array}\end{eqnarray*}$
which completes the proof. □