One-shot distillation in a general resource theory

One-Shot distillation in a General Resource Theory

arxiv:1906.04959

Madhav Krishnan V

UTS

Collaborators : Min-Hsiu Hsieh (UTS)

Eric Chitambar (U. Illinois)

R.T.
Entanglement		LOCC, SEP, PPT,...
Coherence		MIO, IO, DIO,...

Converse

The one-shot distillable resource can be bounded as

R(\rho, \epsilon) \leq \max\limits_{\overline\rho \in b(\rho, 2\sqrt\epsilon)}G_{min}(\overline\rho) \equiv G^{2\sqrt{\epsilon}}_{min}(\rho)

R(\rho, \epsilon) \leq \max\limits_{\overline\rho \in b(\rho, 2\sqrt\epsilon)}G_{min}(\overline\rho) \equiv G^{2\sqrt{\epsilon}}_{min}(\rho)

\Phi_M \gamma \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F

\Phi_M \gamma \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F

Entanglement
Coherence
Purity

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{\scriptstyle b(\rho, 2\sqrt\epsilon) }

{\scriptstyle b(\rho, 2\sqrt\epsilon) }

{\scriptstyle b(\rho, \epsilon) = \{ \overline\rho \hspace{0.1cm}: \hspace{0.1cm} F(\rho, \overline\rho ) \geq 1 - \epsilon \} }

{\scriptstyle b(\rho, \epsilon) = \{ \overline\rho \hspace{0.1cm}: \hspace{0.1cm} F(\rho, \overline\rho ) \geq 1 - \epsilon \} }

{\scriptstyle b(\rho, \epsilon) = \{ \overline\rho \hspace{0.1cm}: \hspace{0.1cm} F(\rho, \overline\rho ) \geq 1 - \epsilon \} }

{\scriptstyle b(\rho, \epsilon) = \{ \overline\rho \hspace{0.1cm}: \hspace{0.1cm} F(\rho, \overline\rho ) \geq 1 - \epsilon \} }

Proof Sketch

\log M \leq \min\limits_{\gamma \in \mathcal{I}} \left\{ -\log_2\text{Tr}(\overline{\rho}\gamma) \right\}

\log M \leq \min\limits_{\gamma \in \mathcal{I}} \left\{ -\log_2\text{Tr}(\overline{\rho}\gamma) \right\}

\overline{\rho} \text{ depends on } \Lambda, \Phi_M, \rho

\overline{\rho} \text{ depends on } \Lambda, \Phi_M, \rho

\overline{\rho}

\overline{\rho}

\cdot \rho

\cdot \rho

\cdot \overline{\rho}

\cdot \overline{\rho}

= G_{min}(\overline \rho)

= G_{min}(\overline \rho)

2\sqrt{\epsilon}

2\sqrt{\epsilon}

\log_2 M \leq \max\limits_{\overline{\rho} \in b(\rho, 2\sqrt\epsilon)} G_{min}(\overline{\rho})

\log_2 M \leq \max\limits_{\overline{\rho} \in b(\rho, 2\sqrt\epsilon)} G_{min}(\overline{\rho})

Assume log M is the optimal rate

\exists \Lambda \in \mathcal O \text{ } s.t. \text{ } F^2(\Lambda(\rho), \Phi_M ) \geq 1 - \epsilon

\exists \Lambda \in \mathcal O \text{ } s.t. \text{ } F^2(\Lambda(\rho), \Phi_M ) \geq 1 - \epsilon

\Phi_M \Lambda(\gamma) \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F

\Phi_M \Lambda(\gamma) \Phi_M \leq \frac{1}{M} \Phi_M \hspace{0.5cm} \forall \gamma \in \mathcal F

G_{min}(\psi) = S_{min}(\Delta(\psi)) := -\log(\lambda_{max}(\Delta(\psi)))

G_{min}(\psi) = S_{min}(\Delta(\psi)) := -\log(\lambda_{max}(\Delta(\psi)))

Coherence

Entanglement

G_{min}(\psi^{AB}) = S_{min}(\rho^B_{\psi})

G_{min}(\psi^{AB}) = S_{min}(\rho^B_{\psi})

Pure state improvement

\log_2 M \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2 \epsilon)} G_{min}(\overline{\psi})

\log_2 M \leq \max\limits_{\overline{\psi} \in b_*(\psi, 2 \epsilon)} G_{min}(\overline{\psi})

Pure state transformation

\psi \rightarrow \phi

\psi \rightarrow \phi

F^2(\Lambda(\psi), \phi) \geq 1-\epsilon

F^2(\Lambda(\psi), \phi) \geq 1-\epsilon

\Lambda(\omega) = \text{Tr}(\overline\psi\omega)\phi + (1 - \text{Tr}(\overline\psi\omega))\pi_{\phi}

\Lambda(\omega) = \text{Tr}(\overline\psi\omega)\phi + (1 - \text{Tr}(\overline\psi\omega))\pi_{\phi}

\overline \psi \text{ optimizes } G_{min}^{\epsilon}(\psi)

\overline \psi \text{ optimizes } G_{min}^{\epsilon}(\psi)

\Lambda(\omega) = \text{Tr}(\overline\psi\omega)\phi + (1 - \text{Tr}(\overline\psi\omega))\pi_{\phi}

\Lambda(\omega) = \text{Tr}(\overline\psi\omega)\phi + (1 - \text{Tr}(\overline\psi\omega))\pi_{\phi}

\text{If } \mathcal F \text{ is convex, } \Lambda \in \mathcal O, \text{ iff}

\text{If } \mathcal F \text{ is convex, } \Lambda \in \mathcal O, \text{ iff}

\pi_{\phi} \text{ optimizes } \mathcal R_f(\phi)

\pi_{\phi} \text{ optimizes } \mathcal R_f(\phi)

(1 - \text{Tr}(\overline\psi\gamma)) \geq \frac{ \mathcal R_f(\phi)}{1 + \mathcal R_f(\phi) } \hspace{0.1cm}\forall \gamma \in \mathcal F

(1 - \text{Tr}(\overline\psi\gamma)) \geq \frac{ \mathcal R_f(\phi)}{1 + \mathcal R_f(\phi) } \hspace{0.1cm}\forall \gamma \in \mathcal F

\iff G_{min}^{\epsilon}(\psi) \geq \log(1 + \mathcal R_f(\phi))

\iff G_{min}^{\epsilon}(\psi) \geq \log(1 + \mathcal R_f(\phi))

One-Shot distillation in a General Resource Theory arxiv:1906.04959 Madhav Krishnan V UTS Collaborators : Min-Hsiu Hsieh (UTS) Eric Chitambar (U. Illinois)