Mutual Mana: Converting Local Magic into Correlations via Discrete Beamsplitters

TL;DR

The paper introduces mutual mana as a dedicated measure of magic correlations in bipartite quantum systems, drawing an analogy to quantum mutual information. It shows that generalized discrete beamsplitters, which are Clifford operations, can redistribute magic offered by a local magic state into inter-system correlations, achieving complete conversion of local mana into mutual mana while preserving total mana. The authors derive analytic expressions and bounds for mutual mana for representative qutrit states, demonstrate threshold behavior for certain inputs, and compare mutual mana with other correlation quantifiers, highlighting their complementary operational meanings. This work provides a concrete, implementable mechanism to engineer and quantify magic correlations in quantum devices, with potential implications for fault-tolerant quantum computation and resource redistribution in quantum networks.

Abstract

Magic (non-stabilizerness) is a key resource for achieving universal fault-tolerant quantum computation beyond classical computation. While previous studies have primarily focused on magic in single systems, its interactions and distribution in multipartite settings remain largely unexplored. In this work, we introduce mutual mana as a measure of magic correlations defined in close analogy with quantum mutual information. Our definition builds upon mana, which is the established quantifier of magic based on discrete Wigner function negativity. We characterize magic correlations generated by discrete beamsplitters, whose Gaussian counterparts are fundamental components in quantum optics and quantum technologies. We show that coupling a magic state with a stabilizer vacuum state via a discrete beamsplitter will induce a full conversion of local magic into mutual mana, thereby establishing a mechanism for redistributing magic resources as magic correlations. We reveal the fundamental properties of mutual mana and derive its explicit expressions for several prototypical qutrit states subject to a discrete beamsplitter. We make a comparative study of mutual mana with several established quantifiers of correlations generated by the qutrit beamsplitter, including quantum mutual information, mutual $L^1$-norm magic, and mutual stabilizer 2-Rényi entropy.

Figures (4)

• Figure 1: Landscape of mutual mana $\mathcal{M}_{\rm mana}(\rho_{\Phi_\lambda,p}^{\rm out})$ for the qutrit state $\rho_{\Phi_\lambda,p}^{\rm out} = \mathrm{CSUM}_3(\rho_{\Phi_\lambda,p}\otimes|0\rangle\langle0|) \mathrm{CSUM}_3^{\dagger}$ as a function of $p$ and $\lambda$, where $\rho_{\Phi_\lambda,p}=p|\Phi_\lambda\rangle\langle\Phi_\lambda|+ (1-p) {\bf 1}/3$ and $|\Phi_{\lambda}\rangle = \lambda |0\rangle + \lambda |1\rangle + \sqrt{1-2\lambda^{2}}\,|2\rangle$.
• Figure 2: Landscape of mutual mana $\mathcal{M}_{\rm mana}(\rho_{\psi_{\theta,p}}^{\rm out})$ for the qutrit state $\rho_{\psi_{\theta,p}}^{\rm out} = \mathrm{CSUM}_3(\rho_{\psi_{\theta,p}}\otimes|0\rangle\langle0|) \mathrm{CSUM}_3^{\dagger}$ as a function of $p$ and $\theta$, where $\rho_{\psi_{\theta,p}}=p|\psi_\theta\rangle\langle\psi_\theta|+(1-p){\bf1}/3$ and $|\psi_\theta\rangle = \cos\theta\,|0\rangle + \sin\theta\,|1\rangle$.
• Figure 3: Comparison of quantum mutual information $I$ (black solid line), mutual $L^1$-norm magic $\mathcal{M}_{L^1}$ (green dash-dotted line), mutual stabilizer 2-Rényi entropy $\mathcal{M}_{\mathrm{SRE}_2}$ (blue dotted line), and mutual mana $\mathcal{M}_{\rm mana}$ (red dashed line) for the output states: (a) $|\Phi_\lambda^{\mathrm{out}}\rangle =\mathrm{CSUM}_3(|\Phi_{\lambda}\rangle\otimes|0\rangle)= \lambda|00\rangle + \lambda|11\rangle + \sqrt{1-2\lambda^2}|22\rangle$ versus the parameter $\lambda\in[0,1/\sqrt{2}]$ (b) $|\psi_\theta^{\mathrm{out}}\rangle = \mathrm{CSUM}_3(|\psi_{\theta}\rangle\otimes|0\rangle)=\cos\theta\,|00\rangle + \sin\theta\,|11\rangle$ versus the parameter $\theta\in[0,\pi/2]$.
• Figure 4: Comparison of quantum mutual information $I(\rho_{\phi,p}^{\rm out})$ (black solid line), mutual $L^1$-norm magic $\mathcal{M}_{L^1}(\rho_{\phi,p}^{\rm out})$ (green dash-dotted line), mutual stabilizer 2-Rényi entropy $\mathcal{M}_{\mathrm{SRE}_2}(\rho_{\phi,p}^{\rm out})$ (blue dotted line), and mutual mana $\mathcal{M}_{\rm mana}(\rho_{\phi,p}^{\mathrm{out}})$ (red dashed line) versus the noise parameter $p\in[0,1]$ for the output states: (a) $\rho_{S,p}^{\mathrm{out}}$, (b) $\rho_{N,p}^{\mathrm{out}}$, (c) $\rho_{T,p}^{\mathrm{out}}$, (d) $\rho_{H,p}^{\mathrm{out}}$. All output states $\rho_{\phi,p}^{\rm out}=\mathrm{CSUM}_3(\rho_{\phi,p}\otimes|0\rangle\langle 0|)\mathrm{CSUM}^{\dag}_3$ are generated by passing the noisy input state $\rho_{\phi,p}=p|\phi\rangle\langle\phi|+ (1-p) {\bf 1}/3$ through the qutrit beamsplitter $\mathrm{CSUM}_3$.

Theorems & Definitions (7)

• Proposition 1
• Proposition 2
• Proposition 3
• Definition 1
• Proposition 4
• Theorem 1
• Proposition 5