Quantumness of Pure-State Ensembles via Coherence of Gram Matrix Based on Generalized α-z-Relative Rényi Entropy

Abstract

The Gram matrix of a set of quantum pure states plays key roles in quantum information theory. It has been highlighted that the Gram matrix of a pure-state ensemble can be viewed as a quantum state, and the quantumness of a pure-state ensemble can thus be quantified by the coherence of the Gram matrix [Europhys. Lett. 134, 30003]. Instead of the l₁-norm of coherence and the relative entropy of coherence, we utilize the generalized α-z-relative Rényi entropy of coherence of the Gram matrix to quantify the quantumness of a pure-state ensemble and explore its properties. We show the usefulness of this quantifier by calculating the quantumness of six important pure-state ensembles. Furthermore, we compare our quantumness with other existing ones and show their features as well as orderings.

Appendix: Proof of the subadditivity of the quantumness Q α,z(⋅)

According to (6), we have

$$\begin{array}{@{}rcl@{}} Q_{\alpha,z}^{\prime}(\mathcal{E}\otimes\mathcal{F})&=&\underset{\sigma_{1}\in\mathcal{I}_{1},\sigma_{2}\in\mathcal{I}_{2}}{\min}\frac{f^{\frac{1}{\alpha}}_{\alpha,z}(G_{\mathcal{E}\otimes \mathcal{F}},\sigma_{1}\otimes \sigma_{2})-1}{nm(\alpha-1)},\\ Q_{\alpha,z}^{\prime}(\mathcal{E})&=&\underset{\sigma_{1}\in\mathcal{I}_{1}}{\min}\frac{f^{\frac{1}{\alpha}}_{\alpha,z}(G_{\mathcal{E}}, \sigma_{1})-1}{n(\alpha-1)},\\ Q_{\alpha,z}^{\prime}(\mathcal{F})&=&\underset{\sigma_{2}\in\mathcal{I}_{2}}{\min}\frac{f^{\frac{1}{\alpha}}_{\alpha,z}(G_{\mathcal{F}}, \sigma_{2})-1}{m(\alpha-1)}, \end{array}$$

where $\mathcal {I}_{1}$ and $\mathcal {I}_{2}$ denotes the set of incoherent states on the m-dimensional and n-dimensional Hilbert spaces, respectively. By the tensor multiplicability of the Gram matrix, i.e., $G_{\mathcal {E}\otimes \mathcal {F}}=G_{\mathcal {E}}\otimes G_{\mathcal {F}}$, we have

$$\begin{array}{@{}rcl@{}} &&f^{\frac{1}{\alpha}}_{\alpha,z}(G_{\mathcal{E}\otimes \mathcal{F}},\sigma_{1}\otimes \sigma_{2})\\ &=&\{\text{Tr}[(\sigma_{1}\otimes\sigma_{2})^{\frac{1-\alpha}{2z}}{G_{\mathcal{E}\otimes \mathcal{F}}}^{\frac{\alpha}{z}}(\sigma_{1}\otimes\sigma_{2})^{\frac{1-\alpha}{2z}}]^{z}\}^{\frac{1}{\alpha}}\\ &=&[\text{Tr}({\sigma_{1}}^{\frac{1-\alpha}{2z}}{G_{\mathcal{E}}}^{\frac{\alpha}{z}}{\sigma_{1}}^{\frac{1-\alpha}{2z}})^{z}]^{\frac{1}{\alpha}}\cdot [\text{Tr}({\sigma_{2}}^{\frac{1-\alpha}{2z}}{G_{\mathcal{F}}}^{\frac{\alpha}{z}}{\sigma_{2}}^{\frac{1-\alpha}{2z}})^{z}]^{\frac{1}{\alpha}}\\ &=&f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{E}},\sigma_{1})\cdot f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{F}},\sigma_{2}). \end{array}$$

So in order to prove the subadditivity, we only need to prove that

$$\begin{array}{@{}rcl@{}} &&\underset{\sigma_{1}\in\mathcal{I}_{1},\sigma_{2}\in\mathcal{I}_{2}}{\min}\frac{f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{E}},\sigma_{1})\cdot f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{F}},\sigma_{2})-1}{nm(\alpha-1)} \\ &&\le\underset{\sigma_{1}\in\mathcal{I}_{1}}{\min}\frac{f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{E}},\sigma_{1})-1}{n(\alpha-1)} +\underset{\sigma_{2}\in\mathcal{I}_{2}}{\min}\frac{f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{F}},\sigma_{2})-1}{m(\alpha-1)}. \end{array}$$

(15)

• Case (i): 0 < α < 1,z > 0. Since the matrix ${\sigma }^{\frac {1-\alpha }{2z}}{\rho }^{\frac {\alpha }{z}} {\sigma }^{\frac {1-\alpha }{2z}}$ has real, non-negative eigenvalues, we obtain $f_{\alpha ,z}^{\frac {1}{\alpha }}(G_{\mathcal {E}},\sigma _{1})\geq 0$ and $f_{\alpha ,z}^{\frac {1}{\alpha }}(G_{\mathcal {F}},\sigma _{2})\geq 0$. Noting that $f_{\alpha ,z}^{\frac {1}{\alpha }}(\rho ,\sigma )\leq 1$ when 0 < α < 1, we have $0\leq f_{\alpha ,z}^{\frac {1}{\alpha }}(G_{\mathcal {E}},\sigma _{1})\leq 1$ and $0\leq f_{\alpha ,z}^{\frac {1}{\alpha }}(G_{\mathcal {F}},\sigma _{2})\leq 1$, which implies that

  $$(f_{\alpha,z}^{\frac{1}{\alpha}} (G_{\mathcal{E}},\sigma_{1})-n)(f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{F}},\sigma_{2})-m)\geq(1-n)(1-m)$$

  (16)

  for each $\sigma _{1}\in \mathcal {I}_{1}$ and $\sigma _{2}\in \mathcal {I}_{2}$. Hence, (15) holds.

• Case (ii): 1 < α ≤ 2,z > 0. Since the completely mixed state σ_∗ = I/d is a diagonal matrix, which is an incoherent state, we have $\underset {\sigma \in \mathcal {I}}{\min \limits }f_{\alpha ,z}^{\frac {1}{\alpha }}(\rho ,\sigma )\leq f_{\alpha ,z}^{\frac {1}{\alpha }}(\rho ,\sigma _{*})= (d^{\alpha -1}{\text {Tr}(\rho ^{\alpha })})^{\frac {1}{\alpha }}\leq d$. Noting that $f_{\alpha ,z}^{\frac {1}{\alpha }}(\rho ,\sigma )\geq 1$ when α > 1, we have $1\leq \underset {\sigma _{1}\in \mathcal {I}_{1}}{\min \limits }f_{\alpha ,z}^{\frac {1}{\alpha }}(G_{\mathcal {E}},\sigma _{1})\le n$ and $1\leq \underset {\sigma _{2}\in \mathcal {I}_{2}}{\min \limits }f_{\alpha ,z}^{\frac {1}{\alpha }}(G_{\mathcal {F}},\sigma _{2})\le m$, which implies that

  $$\begin{array}{@{}rcl@{}} &\underset{\sigma_{1}\in\mathcal{I}_{1}}{\min}f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{E}},\sigma_{1})\cdot \underset{\sigma_{2}\in\mathcal{I}_{2}}{\min}f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{F}},\sigma_{2})-1 \\ &\leq m\left( \underset{\sigma_{1}\in\mathcal{I}_{1}}{\min}f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{E}},\sigma_{1})-1\right) +n\left( \underset{\sigma_{2}\in\mathcal{I}_{2}}{\min}f_{\alpha,z}^{\frac{1}{\alpha}}(G_{\mathcal{F}},\sigma_{2})-1\right), \end{array}$$

  and thus (15) holds.

In either case, we have proved (15), and so (7) is established. This completes the proof.